Subgroups — D₄×#135×C₆₇ #30740299

Order 17152.

2822 conjugacy classes of subgroups.

The subgroup lattice has 858 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₆₇	C₂	1	2	yes	maximal, complemented
2	D₄×#41×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#3287	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#3285	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3284	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#3282	C₂	1	4	yes	maximal, complemented
9	D₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3280	C₂	1	2	yes	maximal, complemented
11	D₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#3050	C₂	1	2	yes	maximal, complemented
14	C₆₇×#3035	C₂	1	2	yes	maximal, complemented
15	C₆₇×#3002	C₂	1	4	yes	maximal, complemented
16	C₄×#135×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×#41×C₆₇	C₂²	1	4	yes	complemented
19	C₂×#135×C₆₇	—	2	16	no	—
20	C₂×#135×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
22	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
23	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
24	C₂×#135×C₆₇	C₂²	1	4	yes	complemented
25	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
26	C₂×#135×C₆₇	C₂²	1	2	yes	complemented
27	C₂²×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
28	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
29	C₂²×D₈×C₆₇	C₂²	1	2	yes	complemented
30	C₂³×C₈×C₆₇	C₂²	1	2	yes	complemented
31	#550×C₆₇	C₂²	1	2	yes	complemented
32	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
33	#548×C₆₇	C₂²	1	4	yes	complemented
34	#547×C₆₇	—	2	2	no	—
35	#547×C₆₇	C₂²	1	2	yes	complemented
36	#546×C₆₇	—	2	4	no	—
37	#546×C₆₇	C₂²	1	4	yes	complemented
38	D₄²×C₆₇	—	2	2	no	—
39	D₄²×C₆₇	C₂²	1	2	yes	—
40	#532×C₆₇	—	2	2	no	—
41	#532×C₆₇	C₂²	1	2	yes	complemented
42	#525×C₆₇	—	2	4	no	—
43	#525×C₆₇	C₂²	1	4	yes	complemented
44	C₄×#41×C₆₇	C₂²	1	2	yes	complemented
45	C₂×C₄×D₄×C₆₇	—	2	2	no	—
46	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
47	#495×C₆₇	C₂²	1	2	yes	complemented
48	#495×C₆₇	C₂²	1	2	yes	complemented
49	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
50	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
51	#492×C₆₇	C₂²	1	2	yes	complemented
52	#467×C₆₇	C₂²	1	2	yes	complemented
53	#467×C₆₇	C₂²	1	2	yes	complemented
54	#466×C₆₇	C₂²	1	2	yes	complemented
55	#466×C₆₇	C₂²	1	2	yes	complemented
56	#465×C₆₇	C₂²	1	4	yes	complemented
57	#465×C₆₇	C₂²	1	4	yes	complemented
58	#464×C₆₇	C₂²	1	4	yes	complemented
59	#463×C₆₇	C₂²	1	4	yes	complemented
60	#462×C₆₇	C₂²	1	2	yes	complemented
61	#461×C₆₇	C₂²	1	2	yes	complemented
62	#460×C₆₇	C₂²	1	2	yes	complemented
63	#459×C₆₇	C₂²	1	2	yes	complemented
64	#452×C₆₇	C₂²	1	4	yes	complemented
65	#452×C₆₇	C₂²	1	4	yes	complemented
66	#451×C₆₇	C₂²	1	4	yes	complemented
67	#450×C₆₇	C₂²	1	4	yes	complemented
68	#449×C₆₇	C₂²	1	4	yes	complemented
69	#449×C₆₇	C₂²	1	4	yes	complemented
70	#448×C₆₇	C₂²	1	4	yes	complemented
71	#447×C₆₇	C₂²	1	4	yes	complemented
72	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
73	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
74	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
75	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
76	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
77	C₄×D₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
78	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
79	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
80	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
81	C₈×D₄×C₆₇	C₂²	1	2	yes	complemented
82	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
83	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic, complemented
84	#427×C₆₇	C₂²	1	1	yes	characteristic, complemented
85	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
86	#416×C₆₇	C₂²	1	4	yes	complemented
87	#416×C₆₇	C₂²	1	2	yes	complemented
88	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
89	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
90	C₂×#41×C₆₇	—	2	4	no	—
91	C₂×#41×C₆₇	—	2	4	no	—
92	C₂×#41×C₆₇	—	2	16	no	—
93	C₂×#41×C₆₇	—	4	8	no	—
94	C₂×#41×C₆₇	C₂³	1	4	yes	—
95	C₂×#41×C₆₇	C₂³	1	4	yes	—
96	C₂×#41×C₆₇	C₂³	1	4	yes	—
97	C₂×#41×C₆₇	C₂³	1	2	yes	—
98	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
99	C₂²×D₄×C₆₇	—	2	4	no	—
100	C₂²×D₄×C₆₇	—	2	2	no	—
101	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
102	C₂²×D₄×C₆₇	C₂³	1	4	yes	—
103	C₂³×C₄×C₆₇	—	2	4	no	—
104	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
105	#135×C₆₇	—	2	16	no	—
106	#135×C₆₇	—	2	8	no	—
107	#135×C₆₇	—	2	16	no	—
108	#135×C₆₇	—	2	8	no	—
109	#135×C₆₇	—	2	8	no	—
110	#135×C₆₇	—	4	4	no	—
111	#135×C₆₇	D₄	1	8	yes	direct product factor
112	C₂×Q₁₆×C₆₇	—	2	8	no	—
113	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
114	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
115	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
116	C₂×Q₁₆×C₆₇	C₂³	1	1	yes	characteristic
117	C₂×SD₁₆×C₆₇	—	2	16	no	—
118	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
119	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
120	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
121	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
122	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
123	C₂×D₈×C₆₇	—	2	8	no	—
124	C₂×D₈×C₆₇	C₂³	1	4	yes	—
125	C₂×D₈×C₆₇	C₂³	1	2	yes	—
126	C₂×D₈×C₆₇	C₂³	1	2	yes	—
127	C₂×D₈×C₆₇	C₂³	1	1	yes	characteristic
128	C₂²×C₈×C₆₇	—	2	8	no	—
129	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
130	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
131	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
132	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
133	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
134	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
135	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
136	C₄⋊Q₈×C₆₇	—	2	2	no	—
137	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
138	#127×C₆₇	—	2	2	no	—
139	#127×C₆₇	C₂³	1	2	yes	—
140	#124×C₆₇	—	2	4	no	—
141	#124×C₆₇	C₂³	1	4	yes	—
142	#123×C₆₇	—	2	4	no	—
143	#123×C₆₇	—	2	4	no	—
144	#123×C₆₇	—	2	4	no	—
145	C₂²⋊Q₈×C₆₇	—	2	4	no	—
146	C₂²⋊Q₈×C₆₇	—	2	4	no	—
147	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
148	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
149	#121×C₆₇	—	2	4	no	—
150	#121×C₆₇	—	2	2	no	—
151	#121×C₆₇	—	2	2	no	—
152	#121×C₆₇	—	2	2	no	—
153	#121×C₆₇	—	2	2	no	—
154	#121×C₆₇	—	2	4	no	—
155	#121×C₆₇	—	2	4	no	—
156	#121×C₆₇	C₂³	1	4	yes	—
157	#121×C₆₇	C₂³	1	4	yes	—
158	#120×C₆₇	—	2	4	no	—
159	#120×C₆₇	—	2	4	no	—
160	#120×C₆₇	—	2	4	no	—
161	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
162	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
163	C₄×D₄×C₆₇	—	2	2	no	—
164	C₄×D₄×C₆₇	—	2	4	no	—
165	C₄×D₄×C₆₇	—	2	2	no	—
166	C₄×D₄×C₆₇	—	2	2	no	—
167	C₄×D₄×C₆₇	—	2	4	no	—
168	C₄×D₄×C₆₇	—	2	2	no	—
169	C₄×D₄×C₆₇	—	2	4	no	—
170	C₄×D₄×C₆₇	—	2	2	no	—
171	C₄×D₄×C₆₇	—	2	2	no	—
172	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
173	C₄×D₄×C₆₇	D₄	1	2	yes	—
174	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
175	C₄×D₄×C₆₇	C₂³	1	2	yes	—
176	C₄×D₄×C₆₇	C₂³	1	2	yes	—
177	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₄×D₄×C₆₇	C₂³	1	2	yes	—
179	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
180	#117×C₆₇	—	2	2	no	—
181	#117×C₆₇	C₂³	1	2	yes	—
182	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
183	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
185	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
186	C₂×C₄²×C₆₇	—	2	2	no	—
187	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
188	#107×C₆₇	C₂³	1	1	yes	characteristic
189	#107×C₆₇	C₂³	1	1	yes	characteristic
190	#106×C₆₇	C₂³	1	2	yes	—
191	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
192	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
193	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
194	#103×C₆₇	C₂³	1	2	yes	—
195	#103×C₆₇	C₂³	1	4	yes	—
196	#103×C₆₇	C₂³	1	2	yes	—
197	#103×C₆₇	C₂³	1	4	yes	—
198	#102×C₆₇	C₂³	1	2	yes	—
199	#102×C₆₇	C₂³	1	4	yes	—
200	#102×C₆₇	C₂³	1	2	yes	—
201	#102×C₆₇	C₂³	1	4	yes	—
202	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
203	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
204	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
205	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
206	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
207	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
208	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
209	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
210	C₂⁴×C₆₇	—	2	4	no	—
211	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
212	#41×C₆₇	—	2	4	no	—
213	#41×C₆₇	—	2	8	no	—
214	#41×C₆₇	—	2	8	no	—
215	#41×C₆₇	—	2	8	no	—
216	#41×C₆₇	—	2	8	no	—
217	#41×C₆₇	—	4	16	no	—
218	#41×C₆₇	—	4	4	no	—
219	#41×C₆₇	—	4	4	no	—
220	#41×C₆₇	—	4	16	no	—
221	#41×C₆₇	C₂×D₄	1	8	yes	complemented
222	C₂×Q₈×C₆₇	—	2	4	no	—
223	C₂×Q₈×C₆₇	—	2	8	no	—
224	C₂×Q₈×C₆₇	—	4	4	no	—
225	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
226	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
227	C₂×D₄×C₆₇	—	2	2	no	—
228	C₂×D₄×C₆₇	—	2	4	no	—
229	C₂×D₄×C₆₇	—	2	8	no	—
230	C₂×D₄×C₆₇	—	2	4	no	—
231	C₂×D₄×C₆₇	—	2	4	no	—
232	C₂×D₄×C₆₇	—	2	4	no	—
233	C₂×D₄×C₆₇	—	2	4	no	—
234	C₂×D₄×C₆₇	—	2	4	no	—
235	C₂×D₄×C₆₇	—	2	2	no	—
236	C₂×D₄×C₆₇	—	2	4	no	—
237	C₂×D₄×C₆₇	—	2	2	no	—
238	C₂×D₄×C₆₇	—	4	4	no	—
239	C₂×D₄×C₆₇	—	4	16	no	—
240	C₂×D₄×C₆₇	—	4	8	no	—
241	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
242	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
243	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
244	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
245	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
246	C₂²×C₄×C₆₇	—	2	4	no	—
247	C₂²×C₄×C₆₇	—	2	2	no	—
248	C₂²×C₄×C₆₇	—	2	4	no	—
249	C₂²×C₄×C₆₇	—	2	4	no	—
250	C₂²×C₄×C₆₇	—	2	2	no	—
251	C₂²×C₄×C₆₇	—	2	4	no	—
252	C₂²×C₄×C₆₇	—	2	4	no	—
253	C₂²×C₄×C₆₇	—	2	4	no	—
254	C₂²×C₄×C₆₇	—	2	2	no	—
255	C₂²×C₄×C₆₇	—	4	16	no	—
256	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
257	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
258	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
259	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
260	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
261	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
262	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
263	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
264	Q₁₆×C₆₇	—	2	8	no	—
265	Q₁₆×C₆₇	—	2	4	no	—
266	Q₁₆×C₆₇	—	4	2	no	—
267	Q₁₆×C₆₇	C₂×D₄	1	4	yes	complemented
268	SD₁₆×C₆₇	—	2	8	no	—
269	SD₁₆×C₆₇	—	2	8	no	—
270	SD₁₆×C₆₇	—	2	8	no	—
271	SD₁₆×C₆₇	—	4	4	no	—
272	SD₁₆×C₆₇	C₂×D₄	1	8	yes	complemented
273	D₈×C₆₇	—	2	8	no	—
274	D₈×C₆₇	—	2	4	no	—
275	D₈×C₆₇	—	4	2	no	—
276	D₈×C₆₇	C₂×D₄	1	4	yes	complemented
277	C₂×C₈×C₆₇	—	2	4	no	—
278	C₂×C₈×C₆₇	—	2	2	no	—
279	C₂×C₈×C₆₇	—	2	4	no	—
280	C₂×C₈×C₆₇	—	2	4	no	—
281	C₂×C₈×C₆₇	—	2	4	no	—
282	C₂×C₈×C₆₇	—	2	4	no	—
283	C₂×C₈×C₆₇	C₂×D₄	1	4	yes	complemented
284	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
285	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
286	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
287	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
288	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
289	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
290	C₄⋊C₄×C₆₇	—	2	2	no	—
291	C₄⋊C₄×C₆₇	—	2	4	no	—
292	C₄⋊C₄×C₆₇	—	2	2	no	—
293	C₄⋊C₄×C₆₇	—	2	2	no	—
294	C₄⋊C₄×C₆₇	—	2	2	no	—
295	C₄⋊C₄×C₆₇	—	2	4	no	—
296	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
297	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
298	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
299	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
300	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
301	C₂²⋊C₄×C₆₇	—	2	4	no	—
302	C₂²⋊C₄×C₆₇	—	2	2	no	—
303	C₂²⋊C₄×C₆₇	—	2	4	no	—
304	C₂²⋊C₄×C₆₇	—	2	4	no	—
305	C₂²⋊C₄×C₆₇	—	2	4	no	—
306	C₂²⋊C₄×C₆₇	—	2	2	no	—
307	C₂²⋊C₄×C₆₇	—	2	2	no	—
308	C₂²⋊C₄×C₆₇	—	2	4	no	—
309	C₂²⋊C₄×C₆₇	—	2	4	no	—
310	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
311	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
312	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
313	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
314	C₄²×C₆₇	—	2	2	no	—
315	C₄²×C₆₇	—	2	2	no	—
316	C₄²×C₆₇	—	2	2	no	—
317	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
318	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
319	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
320	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
321	C₂³×C₆₇	—	2	4	no	—
322	C₂³×C₆₇	—	2	4	no	—
323	C₂³×C₆₇	—	2	2	no	—
324	C₂³×C₆₇	—	2	4	no	—
325	C₂³×C₆₇	—	4	2	no	—
326	C₂³×C₆₇	—	4	8	no	—
327	C₂³×C₆₇	—	4	8	no	—
328	C₂³×C₆₇	—	8	4	no	—
329	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
330	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
331	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
332	Q₈×C₆₇	—	2	2	no	—
333	Q₈×C₆₇	—	2	4	no	—
334	Q₈×C₆₇	—	2	4	no	—
335	Q₈×C₆₇	—	4	8	no	—
336	Q₈×C₆₇	C₂²×D₄	1	4	yes	—
337	D₄×C₆₇	—	2	8	no	—
338	D₄×C₆₇	—	2	4	no	—
339	D₄×C₆₇	—	2	2	no	—
340	D₄×C₆₇	—	2	4	no	—
341	D₄×C₆₇	—	2	2	no	—
342	D₄×C₆₇	—	2	4	no	—
343	D₄×C₆₇	—	4	8	no	—
344	D₄×C₆₇	—	4	8	no	—
345	D₄×C₆₇	—	4	8	no	—
346	D₄×C₆₇	—	4	8	no	—
347	D₄×C₆₇	—	4	8	no	—
348	D₄×C₆₇	—	4	4	no	—
349	D₄×C₆₇	—	8	2	no	—
350	D₄×C₆₇	—	8	4	no	—
351	D₄×C₆₇	#135	1	4	yes	direct product factor
352	D₄×C₆₇	C₂²×D₄	1	4	yes	—
353	C₂×C₄×C₆₇	—	2	4	no	—
354	C₂×C₄×C₆₇	—	2	2	no	—
355	C₂×C₄×C₆₇	—	2	4	no	—
356	C₂×C₄×C₆₇	—	2	4	no	—
357	C₂×C₄×C₆₇	—	2	8	no	—
358	C₂×C₄×C₆₇	—	2	2	no	—
359	C₂×C₄×C₆₇	—	2	2	no	—
360	C₂×C₄×C₆₇	—	2	2	no	—
361	C₂×C₄×C₆₇	—	2	2	no	—
362	C₂×C₄×C₆₇	—	2	2	no	—
363	C₂×C₄×C₆₇	—	2	2	no	—
364	C₂×C₄×C₆₇	—	2	2	no	—
365	C₂×C₄×C₆₇	—	2	4	no	—
366	C₂×C₄×C₆₇	—	4	1	no	—
367	C₂×C₄×C₆₇	—	4	8	no	—
368	C₂×C₄×C₆₇	—	4	8	no	—
369	C₂×C₄×C₆₇	—	4	8	no	—
370	C₂×C₄×C₆₇	—	4	8	no	—
371	C₂×C₄×C₆₇	—	4	4	no	—
372	C₂×C₄×C₆₇	—	8	2	no	—
373	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
374	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
375	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
376	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
377	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
378	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
379	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
380	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
381	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
382	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
383	C₈×C₆₇	—	2	2	no	—
384	C₈×C₆₇	—	2	2	no	—
385	C₈×C₆₇	—	2	1	no	—
386	C₈×C₆₇	—	2	1	no	—
387	C₈×C₆₇	C₂²×D₄	1	2	yes	—
388	C₈×C₆₇	C₂²×D₄	1	2	yes	—
389	C₂²×C₆₇	—	2	1	no	—
390	C₂²×C₆₇	—	2	2	no	—
391	C₂²×C₆₇	—	2	2	no	—
392	C₂²×C₆₇	—	2	1	no	—
393	C₂²×C₆₇	—	2	2	no	—
394	C₂²×C₆₇	—	2	2	no	—
395	C₂²×C₆₇	—	2	4	no	—
396	C₂²×C₆₇	—	2	2	no	—
397	C₂²×C₆₇	—	4	8	no	—
398	C₂²×C₆₇	—	4	2	no	—
399	C₂²×C₆₇	—	4	2	no	—
400	C₂²×C₆₇	—	4	4	no	—
401	C₂²×C₆₇	—	4	4	no	—
402	C₂²×C₆₇	—	4	4	no	—
403	C₂²×C₆₇	—	8	16	no	—
404	C₂²×C₆₇	D₄²	1	2	yes	—
405	C₂²×C₆₇	C₂×#135	1	4	yes	complemented
406	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
407	C₄×C₆₇	—	2	1	no	—
408	C₄×C₆₇	—	2	1	no	—
409	C₄×C₆₇	—	2	2	no	—
410	C₄×C₆₇	—	2	2	no	—
411	C₄×C₆₇	—	2	4	no	—
412	C₄×C₆₇	—	2	1	no	—
413	C₄×C₆₇	—	4	2	no	—
414	C₄×C₆₇	—	4	4	no	—
415	C₄×C₆₇	—	4	2	no	—
416	C₄×C₆₇	D₄²	1	2	yes	—
417	C₄×C₆₇	C₂×#135	1	2	yes	complemented
418	C₄×C₆₇	C₂³×D₄	1	1	yes	characteristic
419	C₄×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
420	D₄×#135	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
421	C₂×C₆₇	—	2	2	no	—
422	C₂×C₆₇	—	2	4	no	—
423	C₂×C₆₇	—	4	2	no	—
424	C₂×C₆₇	—	4	4	no	—
425	C₂×C₆₇	—	8	4	no	—
426	C₂×C₆₇	C₂×D₄²	1	1	yes	characteristic
427	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
428	C₂×C₆₇	#3584	1	1	yes	characteristic
429	C₂²×#135	C₂×C₆₇	1	2	yes	complemented
430	D₄×#41	C₂×C₆₇	1	2	yes	complemented
431	#3287	C₂×C₆₇	1	1	yes	characteristic, complemented
432	D₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
433	#3285	C₂×C₆₇	1	2	yes	complemented
434	#3284	C₂×C₆₇	1	2	yes	complemented
435	#3283	C₂×C₆₇	1	1	yes	characteristic, complemented
436	#3282	C₂×C₆₇	1	4	yes	complemented
437	D₄×SD₁₆	C₂×C₆₇	1	2	yes	complemented
438	#3280	C₂×C₆₇	1	2	yes	complemented
439	D₄×D₈	C₂×C₆₇	1	1	yes	characteristic, complemented
440	#3147	C₂×C₆₇	1	1	yes	characteristic, complemented
441	#3050	C₂×C₆₇	1	2	yes	complemented
442	#3035	C₂×C₆₇	1	2	yes	complemented
443	#3002	C₂×C₆₇	1	4	yes	complemented
444	C₄×#135	C₂×C₆₇	1	1	yes	characteristic, complemented
445	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic, complemented
446	C₆₇	D₄×#135	1	1	yes	characteristic, direct product factor, Sylow
447	C₂²×#41	C₂²×C₆₇	1	4	yes	complemented
448	C₂×#135	—	2	16	no	—
449	C₂×#135	C₂²×C₆₇	1	1	yes	characteristic
450	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
451	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
452	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
453	C₂×#135	C₂²×C₆₇	1	4	yes	complemented
454	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
455	C₂×#135	C₂²×C₆₇	1	2	yes	complemented
456	C₂²×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
457	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
458	C₂²×D₈	C₂²×C₆₇	1	2	yes	complemented
459	C₂³×C₈	C₂²×C₆₇	1	2	yes	complemented
460	#550	C₂²×C₆₇	1	2	yes	complemented
461	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
462	#548	C₂²×C₆₇	1	4	yes	complemented
463	#547	—	2	2	no	—
464	#547	C₂²×C₆₇	1	2	yes	complemented
465	#546	—	2	4	no	—
466	#546	C₂²×C₆₇	1	4	yes	complemented
467	D₄²	—	2	2	no	—
468	D₄²	C₂²×C₆₇	1	2	yes	—
469	#532	—	2	2	no	—
470	#532	C₂²×C₆₇	1	2	yes	complemented
471	#525	—	2	4	no	—
472	#525	C₂²×C₆₇	1	4	yes	complemented
473	C₄×#41	C₂²×C₆₇	1	2	yes	complemented
474	C₂×C₄×D₄	—	2	2	no	—
475	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
476	#495	C₂²×C₆₇	1	2	yes	complemented
477	#495	C₂²×C₆₇	1	2	yes	complemented
478	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
479	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
480	#492	C₂²×C₆₇	1	2	yes	complemented
481	#467	C₂²×C₆₇	1	2	yes	complemented
482	#467	C₂²×C₆₇	1	2	yes	complemented
483	#466	C₂²×C₆₇	1	2	yes	complemented
484	#466	C₂²×C₆₇	1	2	yes	complemented
485	#465	C₂²×C₆₇	1	4	yes	complemented
486	#465	C₂²×C₆₇	1	4	yes	complemented
487	#464	C₂²×C₆₇	1	4	yes	complemented
488	#463	C₂²×C₆₇	1	4	yes	complemented
489	#462	C₂²×C₆₇	1	2	yes	complemented
490	#461	C₂²×C₆₇	1	2	yes	complemented
491	#460	C₂²×C₆₇	1	2	yes	complemented
492	#459	C₂²×C₆₇	1	2	yes	complemented
493	#452	C₂²×C₆₇	1	4	yes	complemented
494	#452	C₂²×C₆₇	1	4	yes	complemented
495	#451	C₂²×C₆₇	1	4	yes	complemented
496	#450	C₂²×C₆₇	1	4	yes	complemented
497	#449	C₂²×C₆₇	1	4	yes	complemented
498	#449	C₂²×C₆₇	1	4	yes	complemented
499	#448	C₂²×C₆₇	1	4	yes	complemented
500	#447	C₂²×C₆₇	1	4	yes	complemented
501	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
502	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
503	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
504	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
505	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
506	C₄×D₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
507	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
508	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
509	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
510	C₈×D₄	C₂²×C₆₇	1	2	yes	complemented
511	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
512	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic, complemented
513	#427	C₂²×C₆₇	1	1	yes	characteristic, complemented
514	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
515	#416	C₂²×C₆₇	1	2	yes	complemented
516	#416	C₂²×C₆₇	1	4	yes	complemented
517	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
518	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
519	C₂×#41	—	2	4	no	—
520	C₂×#41	—	2	4	no	—
521	C₂×#41	—	2	16	no	—
522	C₂×#41	—	4	8	no	—
523	C₂×#41	C₂³×C₆₇	1	4	yes	—
524	C₂×#41	C₂³×C₆₇	1	4	yes	—
525	C₂×#41	C₂³×C₆₇	1	4	yes	—
526	C₂×#41	C₂³×C₆₇	1	2	yes	—
527	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
528	C₂²×D₄	—	2	4	no	—
529	C₂²×D₄	—	2	2	no	—
530	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
531	C₂²×D₄	C₂³×C₆₇	1	4	yes	—
532	C₂³×C₄	—	2	4	no	—
533	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
534	#135	—	2	16	no	—
535	#135	—	2	8	no	—
536	#135	—	2	16	no	—
537	#135	—	2	8	no	—
538	#135	—	2	8	no	—
539	#135	—	4	4	no	—
540	#135	D₄×C₆₇	1	8	yes	direct product factor
541	C₂×Q₁₆	—	2	8	no	—
542	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
543	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
544	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
545	C₂×Q₁₆	C₂³×C₆₇	1	1	yes	characteristic
546	C₂×SD₁₆	—	2	16	no	—
547	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
548	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
549	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
550	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
551	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
552	C₂×D₈	—	2	8	no	—
553	C₂×D₈	C₂³×C₆₇	1	4	yes	—
554	C₂×D₈	C₂³×C₆₇	1	2	yes	—
555	C₂×D₈	C₂³×C₆₇	1	2	yes	—
556	C₂×D₈	C₂³×C₆₇	1	1	yes	characteristic
557	C₂²×C₈	—	2	8	no	—
558	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
559	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
560	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
561	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
562	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
563	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
564	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
565	C₄⋊Q₈	—	2	2	no	—
566	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
567	#127	—	2	2	no	—
568	#127	C₂³×C₆₇	1	2	yes	—
569	#124	—	2	4	no	—
570	#124	C₂³×C₆₇	1	4	yes	—
571	#123	—	2	4	no	—
572	#123	—	2	4	no	—
573	#123	—	2	4	no	—
574	C₂²⋊Q₈	—	2	4	no	—
575	C₂²⋊Q₈	—	2	4	no	—
576	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
577	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
578	#121	—	2	4	no	—
579	#121	—	2	2	no	—
580	#121	—	2	2	no	—
581	#121	—	2	2	no	—
582	#121	—	2	2	no	—
583	#121	—	2	4	no	—
584	#121	—	2	4	no	—
585	#121	C₂³×C₆₇	1	4	yes	—
586	#121	C₂³×C₆₇	1	4	yes	—
587	#120	—	2	4	no	—
588	#120	—	2	4	no	—
589	#120	—	2	4	no	—
590	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
591	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
592	C₄×D₄	—	2	2	no	—
593	C₄×D₄	—	2	4	no	—
594	C₄×D₄	—	2	2	no	—
595	C₄×D₄	—	2	2	no	—
596	C₄×D₄	—	2	4	no	—
597	C₄×D₄	—	2	2	no	—
598	C₄×D₄	—	2	4	no	—
599	C₄×D₄	—	2	2	no	—
600	C₄×D₄	—	2	2	no	—
601	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
602	C₄×D₄	D₄×C₆₇	1	2	yes	—
603	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
604	C₄×D₄	C₂³×C₆₇	1	2	yes	—
605	C₄×D₄	C₂³×C₆₇	1	2	yes	—
606	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
607	C₄×D₄	C₂³×C₆₇	1	2	yes	—
608	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
609	#117	—	2	2	no	—
610	#117	C₂³×C₆₇	1	2	yes	—
611	C₂×C₄⋊C₄	—	2	2	no	—
612	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
613	C₂×C₂²⋊C₄	—	2	4	no	—
614	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
615	C₂×C₄²	—	2	2	no	—
616	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
617	#107	C₂³×C₆₇	1	1	yes	characteristic
618	#107	C₂³×C₆₇	1	1	yes	characteristic
619	#106	C₂³×C₆₇	1	2	yes	—
620	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
621	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
622	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
623	#103	C₂³×C₆₇	1	2	yes	—
624	#103	C₂³×C₆₇	1	2	yes	—
625	#103	C₂³×C₆₇	1	4	yes	—
626	#103	C₂³×C₆₇	1	4	yes	—
627	#102	C₂³×C₆₇	1	2	yes	—
628	#102	C₂³×C₆₇	1	2	yes	—
629	#102	C₂³×C₆₇	1	4	yes	—
630	#102	C₂³×C₆₇	1	4	yes	—
631	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
632	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
633	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
634	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
635	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
636	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
637	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
638	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
639	C₂⁴	—	2	4	no	—
640	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
641	#41	—	2	4	no	—
642	#41	—	2	8	no	—
643	#41	—	2	8	no	—
644	#41	—	2	8	no	—
645	#41	—	2	8	no	—
646	#41	—	4	16	no	—
647	#41	—	4	4	no	—
648	#41	—	4	4	no	—
649	#41	—	4	16	no	—
650	#41	C₂×D₄×C₆₇	1	8	yes	complemented
651	C₂×Q₈	—	2	4	no	—
652	C₂×Q₈	—	2	8	no	—
653	C₂×Q₈	—	4	4	no	—
654	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
655	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
656	C₂×D₄	—	2	2	no	—
657	C₂×D₄	—	2	4	no	—
658	C₂×D₄	—	2	8	no	—
659	C₂×D₄	—	2	4	no	—
660	C₂×D₄	—	2	4	no	—
661	C₂×D₄	—	2	4	no	—
662	C₂×D₄	—	2	4	no	—
663	C₂×D₄	—	2	4	no	—
664	C₂×D₄	—	2	2	no	—
665	C₂×D₄	—	2	4	no	—
666	C₂×D₄	—	2	2	no	—
667	C₂×D₄	—	4	4	no	—
668	C₂×D₄	—	4	16	no	—
669	C₂×D₄	—	4	8	no	—
670	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
671	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
672	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
673	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
674	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
675	C₂²×C₄	—	2	2	no	—
676	C₂²×C₄	—	2	4	no	—
677	C₂²×C₄	—	2	4	no	—
678	C₂²×C₄	—	2	4	no	—
679	C₂²×C₄	—	2	2	no	—
680	C₂²×C₄	—	2	4	no	—
681	C₂²×C₄	—	2	4	no	—
682	C₂²×C₄	—	2	4	no	—
683	C₂²×C₄	—	2	2	no	—
684	C₂²×C₄	—	4	16	no	—
685	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
686	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
687	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
688	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
689	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
690	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
691	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
692	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
693	Q₁₆	—	2	8	no	—
694	Q₁₆	—	2	4	no	—
695	Q₁₆	—	4	2	no	—
696	Q₁₆	C₂×D₄×C₆₇	1	4	yes	complemented
697	SD₁₆	—	2	8	no	—
698	SD₁₆	—	2	8	no	—
699	SD₁₆	—	2	8	no	—
700	SD₁₆	—	4	4	no	—
701	SD₁₆	C₂×D₄×C₆₇	1	8	yes	complemented
702	D₈	—	2	8	no	—
703	D₈	—	2	4	no	—
704	D₈	—	4	2	no	—
705	D₈	C₂×D₄×C₆₇	1	4	yes	complemented
706	C₂×C₈	—	2	4	no	—
707	C₂×C₈	—	2	2	no	—
708	C₂×C₈	—	2	4	no	—
709	C₂×C₈	—	2	4	no	—
710	C₂×C₈	—	2	4	no	—
711	C₂×C₈	—	2	4	no	—
712	C₂×C₈	C₂×D₄×C₆₇	1	4	yes	complemented
713	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
714	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
715	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
716	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
717	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
718	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
719	C₄⋊C₄	—	2	2	no	—
720	C₄⋊C₄	—	2	4	no	—
721	C₄⋊C₄	—	2	2	no	—
722	C₄⋊C₄	—	2	2	no	—
723	C₄⋊C₄	—	2	2	no	—
724	C₄⋊C₄	—	2	4	no	—
725	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
726	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
727	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
728	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
729	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
730	C₂²⋊C₄	—	2	4	no	—
731	C₂²⋊C₄	—	2	2	no	—
732	C₂²⋊C₄	—	2	4	no	—
733	C₂²⋊C₄	—	2	4	no	—
734	C₂²⋊C₄	—	2	4	no	—
735	C₂²⋊C₄	—	2	2	no	—
736	C₂²⋊C₄	—	2	2	no	—
737	C₂²⋊C₄	—	2	4	no	—
738	C₂²⋊C₄	—	2	4	no	—
739	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
740	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
741	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
742	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
743	C₄²	—	2	2	no	—
744	C₄²	—	2	2	no	—
745	C₄²	—	2	2	no	—
746	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
747	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
748	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
749	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
750	C₂³	—	2	4	no	—
751	C₂³	—	2	4	no	—
752	C₂³	—	2	2	no	—
753	C₂³	—	2	4	no	—
754	C₂³	—	4	2	no	—
755	C₂³	—	4	8	no	—
756	C₂³	—	4	8	no	—
757	C₂³	—	8	4	no	—
758	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
759	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
760	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
761	Q₈	—	2	2	no	—
762	Q₈	—	2	4	no	—
763	Q₈	—	2	4	no	—
764	Q₈	—	4	8	no	—
765	Q₈	C₂²×D₄×C₆₇	1	4	yes	—
766	D₄	—	2	8	no	—
767	D₄	—	2	4	no	—
768	D₄	—	2	2	no	—
769	D₄	—	2	4	no	—
770	D₄	—	2	2	no	—
771	D₄	—	2	4	no	—
772	D₄	—	4	8	no	—
773	D₄	—	4	8	no	—
774	D₄	—	4	8	no	—
775	D₄	—	4	8	no	—
776	D₄	—	4	8	no	—
777	D₄	—	4	4	no	—
778	D₄	—	8	2	no	—
779	D₄	—	8	4	no	—
780	D₄	#135×C₆₇	1	4	yes	direct product factor
781	D₄	C₂²×D₄×C₆₇	1	4	yes	—
782	C₂×C₄	—	2	4	no	—
783	C₂×C₄	—	2	2	no	—
784	C₂×C₄	—	2	4	no	—
785	C₂×C₄	—	2	4	no	—
786	C₂×C₄	—	2	8	no	—
787	C₂×C₄	—	2	2	no	—
788	C₂×C₄	—	2	2	no	—
789	C₂×C₄	—	2	2	no	—
790	C₂×C₄	—	2	2	no	—
791	C₂×C₄	—	2	2	no	—
792	C₂×C₄	—	2	2	no	—
793	C₂×C₄	—	2	2	no	—
794	C₂×C₄	—	2	4	no	—
795	C₂×C₄	—	4	1	no	—
796	C₂×C₄	—	4	8	no	—
797	C₂×C₄	—	4	8	no	—
798	C₂×C₄	—	4	8	no	—
799	C₂×C₄	—	4	8	no	—
800	C₂×C₄	—	4	4	no	—
801	C₂×C₄	—	8	2	no	—
802	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
803	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
804	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
805	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
806	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
807	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
808	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
809	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
810	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
811	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
812	C₈	—	2	2	no	—
813	C₈	—	2	2	no	—
814	C₈	—	2	1	no	—
815	C₈	—	2	1	no	—
816	C₈	C₂²×D₄×C₆₇	1	2	yes	—
817	C₈	C₂²×D₄×C₆₇	1	2	yes	—
818	C₂²	—	2	1	no	—
819	C₂²	—	2	1	no	—
820	C₂²	—	2	2	no	—
821	C₂²	—	2	2	no	—
822	C₂²	—	2	2	no	—
823	C₂²	—	2	2	no	—
824	C₂²	—	2	2	no	—
825	C₂²	—	2	4	no	—
826	C₂²	—	4	8	no	—
827	C₂²	—	4	2	no	—
828	C₂²	—	4	2	no	—
829	C₂²	—	4	4	no	—
830	C₂²	—	4	4	no	—
831	C₂²	—	4	4	no	—
832	C₂²	—	8	16	no	—
833	C₂²	D₄²×C₆₇	1	2	yes	—
834	C₂²	C₂×#135×C₆₇	1	4	yes	complemented
835	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
836	C₄	—	2	1	no	—
837	C₄	—	2	1	no	—
838	C₄	—	2	1	no	—
839	C₄	—	2	2	no	—
840	C₄	—	2	2	no	—
841	C₄	—	2	4	no	—
842	C₄	—	4	2	no	—
843	C₄	—	4	2	no	—
844	C₄	—	4	4	no	—
845	C₄	D₄²×C₆₇	1	2	yes	—
846	C₄	C₂×#135×C₆₇	1	2	yes	complemented
847	C₄	C₂³×D₄×C₆₇	1	1	yes	characteristic
848	C₄	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
849	C₂	—	2	2	no	—
850	C₂	—	2	4	no	—
851	C₂	—	4	2	no	—
852	C₂	—	4	4	no	—
853	C₂	—	8	4	no	—
854	C₂	C₂×D₄²×C₆₇	1	1	yes	characteristic
855	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
856	C₂	C₆₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
socle	C₂²×C₆₇	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#135×C₆₇	C₂	semidirect
2. D₄×#41×C₆₇	C₂	semidirect
3. C₆₇×#3287	C₂	semidirect
4. D₄×Q₁₆×C₆₇	C₂	semidirect
5. C₆₇×#3285	C₂	semidirect
6. C₆₇×#3284	C₂	semidirect
7. C₆₇×#3283	C₂	semidirect
8. C₆₇×#3282	C₂	semidirect
9. D₄×SD₁₆×C₆₇	C₂	semidirect
10. C₆₇×#3280	C₂	semidirect
11. D₄×D₈×C₆₇	C₂	semidirect
12. C₆₇×#3147	C₂	semidirect
13. C₆₇×#3050	C₂	semidirect
14. C₆₇×#3035	C₂	semidirect
15. C₆₇×#3002	C₂	semidirect
16. C₄×#135×C₆₇	C₂	semidirect
17. C₂×C₈×D₄×C₆₇	C₂	semidirect
18. C₂²×#41×C₆₇	C₂²	semidirect
21. C₂×#135×C₆₇	C₂²	semidirect
22. C₂×#135×C₆₇	C₂²	semidirect
23. C₂×#135×C₆₇	C₂²	semidirect
24. C₂×#135×C₆₇	C₂²	semidirect
25. C₂×#135×C₆₇	C₂²	semidirect
26. C₂×#135×C₆₇	C₂²	semidirect
27. C₂²×Q₁₆×C₆₇	C₂²	semidirect
28. C₂²×SD₁₆×C₆₇	C₂²	semidirect
29. C₂²×D₈×C₆₇	C₂²	semidirect
30. C₂³×C₈×C₆₇	C₂²	semidirect
31. #550×C₆₇	C₂²	semidirect
33. #548×C₆₇	C₂²	semidirect
35. #547×C₆₇	C₂²	semidirect
37. #546×C₆₇	C₂²	semidirect
41. #532×C₆₇	C₂²	semidirect
43. #525×C₆₇	C₂²	semidirect
44. C₄×#41×C₆₇	C₂²	semidirect
47. #495×C₆₇	C₂²	semidirect
48. #495×C₆₇	C₂²	semidirect
49. #494×C₆₇	C₂²	semidirect
50. #493×C₆₇	C₂²	semidirect
51. #492×C₆₇	C₂²	semidirect
52. #467×C₆₇	C₂²	semidirect
53. #467×C₆₇	C₂²	semidirect
54. #466×C₆₇	C₂²	semidirect
55. #466×C₆₇	C₂²	semidirect
56. #465×C₆₇	C₂²	semidirect
57. #465×C₆₇	C₂²	semidirect
58. #464×C₆₇	C₂²	semidirect
59. #463×C₆₇	C₂²	semidirect
60. #462×C₆₇	C₂²	semidirect
61. #461×C₆₇	C₂²	semidirect
62. #460×C₆₇	C₂²	semidirect
63. #459×C₆₇	C₂²	semidirect
64. #452×C₆₇	C₂²	semidirect
65. #452×C₆₇	C₂²	semidirect
66. #451×C₆₇	C₂²	semidirect
67. #450×C₆₇	C₂²	semidirect
68. #449×C₆₇	C₂²	semidirect
69. #449×C₆₇	C₂²	semidirect
70. #448×C₆₇	C₂²	semidirect
71. #447×C₆₇	C₂²	semidirect
72. C₄×Q₁₆×C₆₇	C₂²	semidirect
73. C₄×Q₁₆×C₆₇	C₂²	semidirect
74. C₄×SD₁₆×C₆₇	C₂²	semidirect
75. C₄×SD₁₆×C₆₇	C₂²	semidirect
76. C₄×D₈×C₆₇	C₂²	semidirect
77. C₄×D₈×C₆₇	C₂²	semidirect
80. C₈×D₄×C₆₇	C₂²	semidirect
81. C₈×D₄×C₆₇	C₂²	semidirect
82. C₈×D₄×C₆₇	C₂²	semidirect
83. C₈×D₄×C₆₇	C₂²	semidirect
84. #427×C₆₇	C₂²	semidirect
85. C₂×C₄⋊C₈×C₆₇	C₂²	semidirect
86. #416×C₆₇	C₂²	semidirect
87. #416×C₆₇	C₂²	semidirect
88. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
89. C₂×C₄×C₈×C₆₇	C₂²	semidirect
111. #135×C₆₇	D₄	direct (N × H)
221. #41×C₆₇	C₂×D₄	semidirect
267. Q₁₆×C₆₇	C₂×D₄	semidirect
272. SD₁₆×C₆₇	C₂×D₄	semidirect
276. D₈×C₆₇	C₂×D₄	semidirect
283. C₂×C₈×C₆₇	C₂×D₄	semidirect
351. D₄×C₆₇	#135	direct (N × H)
405. C₂²×C₆₇	C₂×#135	semidirect
417. C₄×C₆₇	C₂×#135	semidirect
420. D₄×#135	C₆₇	direct (N × H)
429. C₂²×#135	C₂×C₆₇	semidirect
430. D₄×#41	C₂×C₆₇	semidirect
431. #3287	C₂×C₆₇	semidirect
432. D₄×Q₁₆	C₂×C₆₇	semidirect
433. #3285	C₂×C₆₇	semidirect
434. #3284	C₂×C₆₇	semidirect
435. #3283	C₂×C₆₇	semidirect
436. #3282	C₂×C₆₇	semidirect
437. D₄×SD₁₆	C₂×C₆₇	semidirect
438. #3280	C₂×C₆₇	semidirect
439. D₄×D₈	C₂×C₆₇	semidirect
440. #3147	C₂×C₆₇	semidirect
441. #3050	C₂×C₆₇	semidirect
442. #3035	C₂×C₆₇	semidirect
443. #3002	C₂×C₆₇	semidirect
444. C₄×#135	C₂×C₆₇	semidirect
445. C₂×C₈×D₄	C₂×C₆₇	semidirect
446. C₆₇	D₄×#135	direct (N × H)
447. C₂²×#41	C₂²×C₆₇	semidirect
450. C₂×#135	C₂²×C₆₇	semidirect
451. C₂×#135	C₂²×C₆₇	semidirect
452. C₂×#135	C₂²×C₆₇	semidirect
453. C₂×#135	C₂²×C₆₇	semidirect
454. C₂×#135	C₂²×C₆₇	semidirect
455. C₂×#135	C₂²×C₆₇	semidirect
456. C₂²×Q₁₆	C₂²×C₆₇	semidirect
457. C₂²×SD₁₆	C₂²×C₆₇	semidirect
458. C₂²×D₈	C₂²×C₆₇	semidirect
459. C₂³×C₈	C₂²×C₆₇	semidirect
460. #550	C₂²×C₆₇	semidirect
462. #548	C₂²×C₆₇	semidirect
464. #547	C₂²×C₆₇	semidirect
466. #546	C₂²×C₆₇	semidirect
470. #532	C₂²×C₆₇	semidirect
472. #525	C₂²×C₆₇	semidirect
473. C₄×#41	C₂²×C₆₇	semidirect
476. #495	C₂²×C₆₇	semidirect
477. #495	C₂²×C₆₇	semidirect
478. #494	C₂²×C₆₇	semidirect
479. #493	C₂²×C₆₇	semidirect
480. #492	C₂²×C₆₇	semidirect
481. #467	C₂²×C₆₇	semidirect
482. #467	C₂²×C₆₇	semidirect
483. #466	C₂²×C₆₇	semidirect
484. #466	C₂²×C₆₇	semidirect
485. #465	C₂²×C₆₇	semidirect
486. #465	C₂²×C₆₇	semidirect
487. #464	C₂²×C₆₇	semidirect
488. #463	C₂²×C₆₇	semidirect
489. #462	C₂²×C₆₇	semidirect
490. #461	C₂²×C₆₇	semidirect
491. #460	C₂²×C₆₇	semidirect
492. #459	C₂²×C₆₇	semidirect
493. #452	C₂²×C₆₇	semidirect
494. #452	C₂²×C₆₇	semidirect
495. #451	C₂²×C₆₇	semidirect
496. #450	C₂²×C₆₇	semidirect
497. #449	C₂²×C₆₇	semidirect
498. #449	C₂²×C₆₇	semidirect
499. #448	C₂²×C₆₇	semidirect
500. #447	C₂²×C₆₇	semidirect
501. C₄×Q₁₆	C₂²×C₆₇	semidirect
502. C₄×Q₁₆	C₂²×C₆₇	semidirect
503. C₄×SD₁₆	C₂²×C₆₇	semidirect
504. C₄×SD₁₆	C₂²×C₆₇	semidirect
505. C₄×D₈	C₂²×C₆₇	semidirect
506. C₄×D₈	C₂²×C₆₇	semidirect
509. C₈×D₄	C₂²×C₆₇	semidirect
510. C₈×D₄	C₂²×C₆₇	semidirect
511. C₈×D₄	C₂²×C₆₇	semidirect
512. C₈×D₄	C₂²×C₆₇	semidirect
513. #427	C₂²×C₆₇	semidirect
514. C₂×C₄⋊C₈	C₂²×C₆₇	semidirect
515. #416	C₂²×C₆₇	semidirect
516. #416	C₂²×C₆₇	semidirect
517. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
518. C₂×C₄×C₈	C₂²×C₆₇	semidirect
540. #135	D₄×C₆₇	direct (N × H)
650. #41	C₂×D₄×C₆₇	semidirect
696. Q₁₆	C₂×D₄×C₆₇	semidirect
701. SD₁₆	C₂×D₄×C₆₇	semidirect
705. D₈	C₂×D₄×C₆₇	semidirect
712. C₂×C₈	C₂×D₄×C₆₇	semidirect
780. D₄	#135×C₆₇	direct (N × H)
834. C₂²	C₂×#135×C₆₇	semidirect
846. C₄	C₂×#135×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#135×C₆₇	2
D₄×#41×C₆₇	2
C₆₇×#3287	2
D₄×Q₁₆×C₆₇	2
C₆₇×#3285	2
C₆₇×#3284	2
C₆₇×#3283	2
C₆₇×#3282	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3280	2
D₄×D₈×C₆₇	2
C₆₇×#3147	2
C₆₇×#3050	2
C₆₇×#3035	2
C₆₇×#3002	2
C₄×#135×C₆₇	2
C₂×C₈×D₄×C₆₇	2
D₄×#135	67

Subgroups — D4×#135×C67 #30740299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#135×C₆₇ #30740299