Subgroups — C₆₇×#61602 #30740853

Order 17152.

2084 conjugacy classes of subgroups.

The subgroup lattice has 896 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3520	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3328	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3312	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3298	C₂	1	2	yes	maximal, complemented
6	D₄×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3276	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3270	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3218	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3209	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3195	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3190	C₂	1	2	yes	maximal, complemented
13	C₆₇×#3113	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#3108	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#3004	C₂	1	2	yes	maximal, complemented
16	C₂×#451×C₆₇	C₂	1	2	yes	maximal, complemented
17	C₆₇×#2974	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂³×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
21	#556×C₆₇	C₂²	1	1	yes	characteristic
22	#554×C₆₇	—	2	1	no	—
23	#552×C₆₇	C₂²	1	2	yes	—
24	#551×C₆₇	—	2	2	no	—
25	D₄×Q₈×C₆₇	—	2	1	no	—
26	D₄×Q₈×C₆₇	—	2	2	no	—
27	D₄×Q₈×C₆₇	—	2	1	no	—
28	D₄×Q₈×C₆₇	C₂²	1	4	yes	—
29	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	D₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
32	#548×C₆₇	C₂²	1	2	yes	—
33	#547×C₆₇	C₂²	1	1	yes	characteristic
34	#547×C₆₇	C₂²	1	1	yes	characteristic
35	#546×C₆₇	—	2	1	no	—
36	#542×C₆₇	—	2	1	no	—
37	#539×C₆₇	—	2	2	no	—
38	#533×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
41	#529×C₆₇	C₂²	1	1	yes	characteristic
42	#525×C₆₇	C₂²	1	2	yes	—
43	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
44	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
45	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
46	#520×C₆₇	—	2	1	no	—
47	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×D₄×C₆₇	—	2	1	no	—
49	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
50	#497×C₆₇	C₂²	1	2	yes	complemented
51	#494×C₆₇	C₂²	1	2	yes	complemented
52	#491×C₆₇	C₂²	1	2	yes	complemented
53	#487×C₆₇	C₂²	1	2	yes	complemented
54	#485×C₆₇	C₂²	1	4	yes	complemented
55	#483×C₆₇	C₂²	1	1	yes	characteristic
56	#483×C₆₇	C₂²	1	2	yes	—
57	#483×C₆₇	C₂²	1	4	yes	complemented
58	#480×C₆₇	C₂²	1	1	yes	characteristic
59	#479×C₆₇	C₂²	1	2	yes	complemented
60	#475×C₆₇	C₂²	1	2	yes	complemented
61	#470×C₆₇	C₂²	1	4	yes	complemented
62	#470×C₆₇	C₂²	1	4	yes	complemented
63	#467×C₆₇	C₂²	1	4	yes	complemented
64	#465×C₆₇	C₂²	1	4	yes	complemented
65	#464×C₆₇	C₂²	1	2	yes	complemented
66	#462×C₆₇	C₂²	1	4	yes	complemented
67	#462×C₆₇	C₂²	1	2	yes	complemented
68	#462×C₆₇	C₂²	1	2	yes	complemented
69	#461×C₆₇	C₂²	1	2	yes	complemented
70	#452×C₆₇	C₂²	1	4	yes	complemented
71	#451×C₆₇	C₂²	1	1	yes	characteristic
72	#451×C₆₇	C₂²	1	2	yes	—
73	#451×C₆₇	C₂²	1	2	yes	—
74	#451×C₆₇	C₂²	1	2	yes	—
75	#451×C₆₇	C₂²	1	2	yes	—
76	#451×C₆₇	C₂²	1	4	yes	complemented
77	#451×C₆₇	C₂²	1	4	yes	complemented
78	#451×C₆₇	C₂²	1	4	yes	complemented
79	#451×C₆₇	C₂²	1	4	yes	complemented
80	#448×C₆₇	C₂²	1	1	yes	characteristic
81	#448×C₆₇	C₂²	1	2	yes	—
82	#448×C₆₇	C₂²	1	4	yes	complemented
83	#441×C₆₇	C₂²	1	4	yes	complemented
84	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	complemented
85	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	complemented
86	#436×C₆₇	C₂²	1	2	yes	—
87	#435×C₆₇	C₂²	1	4	yes	—
88	C₈×D₄×C₆₇	C₂²	1	2	yes	—
89	#432×C₆₇	C₂²	1	1	yes	characteristic
90	#431×C₆₇	C₂²	1	1	yes	characteristic
91	#419×C₆₇	C₂²	1	4	yes	complemented
92	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
93	#407×C₆₇	C₂²	1	2	yes	—
94	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
95	C₂×#41×C₆₇	—	2	1	no	—
96	C₂×#41×C₆₇	C₂³	1	2	yes	—
97	C₂²×Q₈×C₆₇	—	2	2	no	—
98	C₂²×Q₈×C₆₇	—	2	2	no	—
99	C₂²×Q₈×C₆₇	—	2	2	no	—
100	C₂²×Q₈×C₆₇	—	4	8	no	—
101	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
102	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
103	C₂²×Q₈×C₆₇	C₂³	1	4	yes	—
104	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
105	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
106	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
107	C₂³×C₄×C₆₇	—	2	2	no	—
108	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
109	#137×C₆₇	—	4	8	no	—
110	C₂×Q₁₆×C₆₇	—	4	8	no	—
111	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
112	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
113	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
114	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
115	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
116	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
117	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
118	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
119	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
120	C₄⋊Q₈×C₆₇	—	2	2	no	—
121	C₄⋊Q₈×C₆₇	—	2	4	no	—
122	C₄⋊Q₈×C₆₇	—	2	2	no	—
123	C₄⋊Q₈×C₆₇	—	2	1	no	—
124	C₄⋊Q₈×C₆₇	—	2	1	no	—
125	C₄⋊Q₈×C₆₇	—	2	1	no	—
126	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
127	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
128	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
129	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
130	#126×C₆₇	—	2	2	no	—
131	#126×C₆₇	—	2	2	no	—
132	#126×C₆₇	—	2	1	no	—
133	#125×C₆₇	—	2	2	no	—
134	#125×C₆₇	—	2	1	no	—
135	#125×C₆₇	C₂³	1	1	yes	characteristic
136	#124×C₆₇	—	2	1	no	—
137	#124×C₆₇	—	2	2	no	—
138	#124×C₆₇	C₂³	1	1	yes	characteristic
139	#123×C₆₇	—	2	2	no	—
140	#123×C₆₇	—	2	2	no	—
141	#123×C₆₇	—	2	1	no	—
142	#123×C₆₇	—	2	2	no	—
143	#123×C₆₇	—	2	2	no	—
144	C₂²⋊Q₈×C₆₇	—	2	2	no	—
145	C₂²⋊Q₈×C₆₇	—	2	2	no	—
146	C₂²⋊Q₈×C₆₇	—	2	2	no	—
147	C₂²⋊Q₈×C₆₇	—	2	2	no	—
148	C₂²⋊Q₈×C₆₇	—	2	1	no	—
149	C₂²⋊Q₈×C₆₇	—	2	2	no	—
150	C₂²⋊Q₈×C₆₇	—	2	1	no	—
151	C₂²⋊Q₈×C₆₇	—	2	4	no	—
152	C₂²⋊Q₈×C₆₇	—	2	4	no	—
153	C₂²⋊Q₈×C₆₇	—	2	4	no	—
154	C₂²⋊Q₈×C₆₇	—	2	4	no	—
155	C₂²⋊Q₈×C₆₇	—	2	2	no	—
156	C₂²⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
157	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
158	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
159	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
160	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
161	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
162	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
163	#121×C₆₇	—	2	2	no	—
164	#121×C₆₇	—	2	1	no	—
165	#121×C₆₇	C₂³	1	1	yes	characteristic
166	#121×C₆₇	C₂³	1	2	yes	—
167	#120×C₆₇	—	2	2	no	—
168	C₄×Q₈×C₆₇	—	2	1	no	—
169	C₄×Q₈×C₆₇	—	2	1	no	—
170	C₄×Q₈×C₆₇	—	2	1	no	—
171	C₄×Q₈×C₆₇	—	2	1	no	—
172	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
173	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
174	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
175	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
176	C₄×D₄×C₆₇	—	2	4	no	—
177	C₄×D₄×C₆₇	—	2	2	no	—
178	C₄×D₄×C₆₇	—	2	2	no	—
179	C₄×D₄×C₆₇	—	2	1	no	—
180	C₄×D₄×C₆₇	—	2	2	no	—
181	C₄×D₄×C₆₇	—	2	2	no	—
182	C₄×D₄×C₆₇	—	2	1	no	—
183	C₄×D₄×C₆₇	—	2	2	no	—
184	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
185	C₄×D₄×C₆₇	D₄	1	2	yes	—
186	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
187	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
188	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
189	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
190	C₄×D₄×C₆₇	C₂³	1	2	yes	—
191	#117×C₆₇	—	2	1	no	—
192	#117×C₆₇	—	2	2	no	—
193	#117×C₆₇	—	2	1	no	—
194	#117×C₆₇	C₂³	1	2	yes	—
195	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
196	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
197	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
199	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
201	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
202	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
203	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
204	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
205	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
206	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
207	C₂×C₄²×C₆₇	—	2	1	no	—
208	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
209	#107×C₆₇	C₂³	1	2	yes	—
210	#106×C₆₇	C₂³	1	4	yes	—
211	#106×C₆₇	C₂³	1	2	yes	—
212	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
213	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
214	#103×C₆₇	C₂³	1	2	yes	—
215	#103×C₆₇	C₂³	1	4	yes	—
216	#103×C₆₇	C₂³	1	4	yes	—
217	#103×C₆₇	C₂³	1	4	yes	—
218	#103×C₆₇	C₂³	1	2	yes	—
219	#103×C₆₇	C₂³	1	4	yes	—
220	#103×C₆₇	C₂³	1	4	yes	—
221	#103×C₆₇	C₂³	1	2	yes	—
222	#103×C₆₇	C₂³	1	4	yes	—
223	#102×C₆₇	C₂³	1	2	yes	—
224	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
225	C₂²⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
226	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
227	C₂²⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
228	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
229	#97×C₆₇	C₂³	1	2	yes	—
230	C₄×C₈×C₆₇	C₂³	1	2	yes	—
231	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
232	#41×C₆₇	—	4	4	no	—
233	#41×C₆₇	—	4	4	no	—
234	C₂×Q₈×C₆₇	—	2	4	no	—
235	C₂×Q₈×C₆₇	—	2	1	no	—
236	C₂×Q₈×C₆₇	—	2	1	no	—
237	C₂×Q₈×C₆₇	—	2	4	no	—
238	C₂×Q₈×C₆₇	—	2	4	no	—
239	C₂×Q₈×C₆₇	—	2	2	no	—
240	C₂×Q₈×C₆₇	—	2	2	no	—
241	C₂×Q₈×C₆₇	—	2	8	no	—
242	C₂×Q₈×C₆₇	—	2	8	no	—
243	C₂×Q₈×C₆₇	—	4	4	no	—
244	C₂×Q₈×C₆₇	—	4	4	no	—
245	C₂×Q₈×C₆₇	—	4	4	no	—
246	C₂×Q₈×C₆₇	—	4	8	no	—
247	C₂×Q₈×C₆₇	—	4	4	no	—
248	C₂×Q₈×C₆₇	—	4	8	no	—
249	C₂×Q₈×C₆₇	—	4	8	no	—
250	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
251	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
252	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
253	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
254	C₂×D₄×C₆₇	—	2	4	no	—
255	C₂×D₄×C₆₇	—	2	1	no	—
256	C₂×D₄×C₆₇	—	2	2	no	—
257	C₂×D₄×C₆₇	—	2	2	no	—
258	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
259	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
260	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
261	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
262	C₂²×C₄×C₆₇	—	2	1	no	—
263	C₂²×C₄×C₆₇	—	2	1	no	—
264	C₂²×C₄×C₆₇	—	2	2	no	—
265	C₂²×C₄×C₆₇	—	2	2	no	—
266	C₂²×C₄×C₆₇	—	2	4	no	—
267	C₂²×C₄×C₆₇	—	2	4	no	—
268	C₂²×C₄×C₆₇	—	2	2	no	—
269	C₂²×C₄×C₆₇	—	2	1	no	—
270	C₂²×C₄×C₆₇	—	2	2	no	—
271	C₂²×C₄×C₆₇	—	4	4	no	—
272	C₂²×C₄×C₆₇	—	4	8	no	—
273	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
274	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
275	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
276	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
277	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
278	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
279	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
280	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
281	Q₁₆×C₆₇	—	2	4	no	—
282	Q₁₆×C₆₇	—	2	8	no	—
283	Q₁₆×C₆₇	—	4	4	no	—
284	Q₁₆×C₆₇	—	4	4	no	—
285	Q₁₆×C₆₇	—	4	4	no	—
286	SD₁₆×C₆₇	—	2	4	no	—
287	SD₁₆×C₆₇	—	4	4	no	—
288	M₁₆×C₆₇	—	4	4	no	—
289	C₂×C₈×C₆₇	—	4	4	no	—
290	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
291	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
292	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
293	C₄⋊C₄×C₆₇	—	2	1	no	—
294	C₄⋊C₄×C₆₇	—	2	2	no	—
295	C₄⋊C₄×C₆₇	—	2	2	no	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	1	no	—
298	C₄⋊C₄×C₆₇	—	2	2	no	—
299	C₄⋊C₄×C₆₇	—	2	2	no	—
300	C₄⋊C₄×C₆₇	—	2	2	no	—
301	C₄⋊C₄×C₆₇	—	2	2	no	—
302	C₄⋊C₄×C₆₇	—	2	2	no	—
303	C₄⋊C₄×C₆₇	—	2	2	no	—
304	C₄⋊C₄×C₆₇	—	2	4	no	—
305	C₄⋊C₄×C₆₇	—	2	4	no	—
306	C₄⋊C₄×C₆₇	—	2	1	no	—
307	C₄⋊C₄×C₆₇	—	2	2	no	—
308	C₄⋊C₄×C₆₇	—	2	1	no	—
309	C₄⋊C₄×C₆₇	—	2	1	no	—
310	C₄⋊C₄×C₆₇	—	2	1	no	—
311	C₄⋊C₄×C₆₇	—	2	2	no	—
312	C₄⋊C₄×C₆₇	—	2	1	no	—
313	C₄⋊C₄×C₆₇	—	2	2	no	—
314	C₄⋊C₄×C₆₇	—	2	1	no	—
315	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
316	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
317	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
318	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
319	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
320	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
321	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
322	C₂²⋊C₄×C₆₇	—	2	1	no	—
323	C₂²⋊C₄×C₆₇	—	2	2	no	—
324	C₂²⋊C₄×C₆₇	—	2	4	no	—
325	C₂²⋊C₄×C₆₇	—	2	4	no	—
326	C₂²⋊C₄×C₆₇	—	2	2	no	—
327	C₂²⋊C₄×C₆₇	—	2	2	no	—
328	C₂²⋊C₄×C₆₇	—	2	2	no	—
329	C₂²⋊C₄×C₆₇	—	2	2	no	—
330	C₂²⋊C₄×C₆₇	—	2	1	no	—
331	C₂²⋊C₄×C₆₇	—	2	1	no	—
332	C₂²⋊C₄×C₆₇	—	2	1	no	—
333	C₂²⋊C₄×C₆₇	—	2	2	no	—
334	C₂²⋊C₄×C₆₇	—	2	1	no	—
335	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
336	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
337	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
338	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
339	C₄²×C₆₇	—	2	2	no	—
340	C₄²×C₆₇	—	2	2	no	—
341	C₄²×C₆₇	—	2	2	no	—
342	C₄²×C₆₇	—	2	2	no	—
343	C₄²×C₆₇	—	2	1	no	—
344	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
345	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
346	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
347	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
348	C₂³×C₆₇	—	2	4	no	—
349	C₂³×C₆₇	—	2	4	no	—
350	C₂³×C₆₇	—	2	1	no	—
351	C₂³×C₆₇	—	4	2	no	—
352	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
353	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
354	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
355	Q₈×C₆₇	—	2	4	no	—
356	Q₈×C₆₇	—	2	2	no	—
357	Q₈×C₆₇	—	2	4	no	—
358	Q₈×C₆₇	—	2	2	no	—
359	Q₈×C₆₇	—	2	4	no	—
360	Q₈×C₆₇	—	2	4	no	—
361	Q₈×C₆₇	—	4	8	no	—
362	Q₈×C₆₇	—	4	8	no	—
363	Q₈×C₆₇	—	4	4	no	—
364	Q₈×C₆₇	—	4	2	no	—
365	Q₈×C₆₇	—	8	2	no	—
366	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
367	D₄×C₆₇	—	2	4	no	—
368	D₄×C₆₇	—	2	2	no	—
369	D₄×C₆₇	—	2	2	no	—
370	D₄×C₆₇	—	4	4	no	—
371	D₄×C₆₇	—	4	2	no	—
372	D₄×C₆₇	—	8	2	no	—
373	D₄×C₆₇	#137	1	4	yes	complemented
374	C₂×C₄×C₆₇	—	2	4	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	2	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	4	no	—
379	C₂×C₄×C₆₇	—	2	4	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	2	1	no	—
382	C₂×C₄×C₆₇	—	2	2	no	—
383	C₂×C₄×C₆₇	—	2	2	no	—
384	C₂×C₄×C₆₇	—	2	1	no	—
385	C₂×C₄×C₆₇	—	2	2	no	—
386	C₂×C₄×C₆₇	—	2	1	no	—
387	C₂×C₄×C₆₇	—	2	4	no	—
388	C₂×C₄×C₆₇	—	4	2	no	—
389	C₂×C₄×C₆₇	—	4	2	no	—
390	C₂×C₄×C₆₇	—	4	2	no	—
391	C₂×C₄×C₆₇	—	4	8	no	—
392	C₂×C₄×C₆₇	—	4	8	no	—
393	C₂×C₄×C₆₇	—	4	4	no	—
394	C₂×C₄×C₆₇	—	4	4	no	—
395	C₂×C₄×C₆₇	—	4	2	no	—
396	C₂×C₄×C₆₇	—	4	2	no	—
397	C₂×C₄×C₆₇	—	8	2	no	—
398	C₂×C₄×C₆₇	—	8	2	no	—
399	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
400	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
401	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
402	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
403	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
404	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
405	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
406	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
407	C₈×C₆₇	—	2	2	no	—
408	C₈×C₆₇	—	2	4	no	—
409	C₈×C₆₇	—	2	2	no	—
410	C₂²×C₆₇	—	2	1	no	—
411	C₂²×C₆₇	—	2	2	no	—
412	C₂²×C₆₇	—	2	1	no	—
413	C₂²×C₆₇	—	2	2	no	—
414	C₂²×C₆₇	—	2	2	no	—
415	C₂²×C₆₇	—	2	2	no	—
416	C₂²×C₆₇	—	2	2	no	—
417	C₂²×C₆₇	—	4	8	no	—
418	C₂²×C₆₇	—	4	1	no	—
419	C₂²×C₆₇	—	4	1	no	—
420	C₂²×C₆₇	—	4	1	no	—
421	C₂²×C₆₇	C₂×#137	1	4	yes	—
422	C₂²×C₆₇	#578	1	2	yes	—
423	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
424	C₄×C₆₇	—	2	4	no	—
425	C₄×C₆₇	—	2	1	no	—
426	C₄×C₆₇	—	2	2	no	—
427	C₄×C₆₇	—	2	2	no	—
428	C₄×C₆₇	—	2	1	no	—
429	C₄×C₆₇	—	2	1	no	—
430	C₄×C₆₇	—	2	1	no	—
431	C₄×C₆₇	—	4	4	no	—
432	C₄×C₆₇	—	4	2	no	—
433	C₄×C₆₇	—	4	1	no	—
434	C₄×C₆₇	—	4	1	no	—
435	C₄×C₆₇	—	4	2	no	—
436	C₄×C₆₇	—	4	2	no	—
437	C₄×C₆₇	—	4	1	no	—
438	C₄×C₆₇	C₂×#137	1	2	yes	—
439	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
440	#61602	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
441	C₂×C₆₇	—	2	2	no	—
442	C₂×C₆₇	—	2	4	no	—
443	C₂×C₆₇	—	4	2	no	—
444	C₂×C₆₇	—	8	1	no	—
445	C₂×C₆₇	#3484	1	1	yes	characteristic
446	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
447	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
448	#3520	C₂×C₆₇	1	1	yes	characteristic
449	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
450	#3328	C₂×C₆₇	1	2	yes	complemented
451	#3312	C₂×C₆₇	1	4	yes	complemented
452	#3298	C₂×C₆₇	1	2	yes	complemented
453	D₄×Q₁₆	C₂×C₆₇	1	2	yes	complemented
454	#3276	C₂×C₆₇	1	2	yes	complemented
455	#3270	C₂×C₆₇	1	4	yes	complemented
456	#3218	C₂×C₆₇	1	1	yes	characteristic, complemented
457	#3209	C₂×C₆₇	1	1	yes	characteristic, complemented
458	#3195	C₂×C₆₇	1	2	yes	complemented
459	#3190	C₂×C₆₇	1	2	yes	complemented
460	#3113	C₂×C₆₇	1	1	yes	characteristic, complemented
461	#3108	C₂×C₆₇	1	1	yes	characteristic, complemented
462	#3004	C₂×C₆₇	1	2	yes	complemented
463	C₂×#451	C₂×C₆₇	1	2	yes	complemented
464	#2974	C₂×C₆₇	1	1	yes	characteristic, complemented
465	C₆₇	#61602	1	1	yes	characteristic, direct product factor, Sylow
466	C₂³×Q₈	C₂²×C₆₇	1	2	yes	—
467	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
468	C₂²×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
469	#556	C₂²×C₆₇	1	1	yes	characteristic
470	#554	—	2	1	no	—
471	#552	C₂²×C₆₇	1	2	yes	—
472	#551	—	2	2	no	—
473	D₄×Q₈	—	2	1	no	—
474	D₄×Q₈	—	2	2	no	—
475	D₄×Q₈	—	2	1	no	—
476	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
477	D₄×Q₈	C₂²×C₆₇	1	4	yes	—
478	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
479	D₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
480	#548	C₂²×C₆₇	1	2	yes	—
481	#547	C₂²×C₆₇	1	1	yes	characteristic
482	#547	C₂²×C₆₇	1	1	yes	characteristic
483	#546	—	2	1	no	—
484	#542	—	2	1	no	—
485	#539	—	2	2	no	—
486	#533	C₂²×C₆₇	1	1	yes	characteristic
487	C₂×C₄⋊Q₈	—	2	1	no	—
488	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
489	#529	C₂²×C₆₇	1	1	yes	characteristic
490	#525	C₂²×C₆₇	1	2	yes	—
491	C₂×C₂²⋊Q₈	—	2	2	no	—
492	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
493	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
494	#520	—	2	1	no	—
495	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
496	C₂×C₄×D₄	—	2	1	no	—
497	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
498	#497	C₂²×C₆₇	1	2	yes	complemented
499	#494	C₂²×C₆₇	1	2	yes	complemented
500	#491	C₂²×C₆₇	1	2	yes	complemented
501	#487	C₂²×C₆₇	1	2	yes	complemented
502	#485	C₂²×C₆₇	1	4	yes	complemented
503	#483	C₂²×C₆₇	1	1	yes	characteristic
504	#483	C₂²×C₆₇	1	2	yes	—
505	#483	C₂²×C₆₇	1	4	yes	complemented
506	#480	C₂²×C₆₇	1	1	yes	characteristic
507	#479	C₂²×C₆₇	1	2	yes	complemented
508	#475	C₂²×C₆₇	1	2	yes	complemented
509	#470	C₂²×C₆₇	1	4	yes	complemented
510	#470	C₂²×C₆₇	1	4	yes	complemented
511	#467	C₂²×C₆₇	1	4	yes	complemented
512	#465	C₂²×C₆₇	1	4	yes	complemented
513	#464	C₂²×C₆₇	1	2	yes	complemented
514	#462	C₂²×C₆₇	1	2	yes	complemented
515	#462	C₂²×C₆₇	1	2	yes	complemented
516	#462	C₂²×C₆₇	1	4	yes	complemented
517	#461	C₂²×C₆₇	1	2	yes	complemented
518	#452	C₂²×C₆₇	1	4	yes	complemented
519	#451	C₂²×C₆₇	1	1	yes	characteristic
520	#451	C₂²×C₆₇	1	2	yes	—
521	#451	C₂²×C₆₇	1	2	yes	—
522	#451	C₂²×C₆₇	1	2	yes	—
523	#451	C₂²×C₆₇	1	2	yes	—
524	#451	C₂²×C₆₇	1	4	yes	complemented
525	#451	C₂²×C₆₇	1	4	yes	complemented
526	#451	C₂²×C₆₇	1	4	yes	complemented
527	#451	C₂²×C₆₇	1	4	yes	complemented
528	#448	C₂²×C₆₇	1	1	yes	characteristic
529	#448	C₂²×C₆₇	1	2	yes	—
530	#448	C₂²×C₆₇	1	4	yes	complemented
531	#441	C₂²×C₆₇	1	4	yes	complemented
532	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
533	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
534	#436	C₂²×C₆₇	1	2	yes	—
535	#435	C₂²×C₆₇	1	4	yes	—
536	C₈×D₄	C₂²×C₆₇	1	2	yes	—
537	#432	C₂²×C₆₇	1	1	yes	characteristic
538	#431	C₂²×C₆₇	1	1	yes	characteristic
539	#419	C₂²×C₆₇	1	4	yes	complemented
540	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
541	#407	C₂²×C₆₇	1	2	yes	—
542	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
543	C₂×#41	—	2	1	no	—
544	C₂×#41	C₂³×C₆₇	1	2	yes	—
545	C₂²×Q₈	—	2	2	no	—
546	C₂²×Q₈	—	2	2	no	—
547	C₂²×Q₈	—	2	2	no	—
548	C₂²×Q₈	—	4	8	no	—
549	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
550	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
551	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
552	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
553	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
554	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
555	C₂³×C₄	—	2	2	no	—
556	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
557	#137	—	4	8	no	—
558	C₂×Q₁₆	—	4	8	no	—
559	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
560	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
561	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
562	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
563	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
564	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
565	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
566	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
567	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
568	C₄⋊Q₈	—	2	2	no	—
569	C₄⋊Q₈	—	2	4	no	—
570	C₄⋊Q₈	—	2	2	no	—
571	C₄⋊Q₈	—	2	1	no	—
572	C₄⋊Q₈	—	2	1	no	—
573	C₄⋊Q₈	—	2	1	no	—
574	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
575	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
576	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
577	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
578	#126	—	2	2	no	—
579	#126	—	2	2	no	—
580	#126	—	2	1	no	—
581	#125	—	2	2	no	—
582	#125	—	2	1	no	—
583	#125	C₂³×C₆₇	1	1	yes	characteristic
584	#124	—	2	1	no	—
585	#124	—	2	2	no	—
586	#124	C₂³×C₆₇	1	1	yes	characteristic
587	#123	—	2	2	no	—
588	#123	—	2	2	no	—
589	#123	—	2	1	no	—
590	#123	—	2	2	no	—
591	#123	—	2	2	no	—
592	C₂²⋊Q₈	—	2	2	no	—
593	C₂²⋊Q₈	—	2	2	no	—
594	C₂²⋊Q₈	—	2	2	no	—
595	C₂²⋊Q₈	—	2	2	no	—
596	C₂²⋊Q₈	—	2	1	no	—
597	C₂²⋊Q₈	—	2	2	no	—
598	C₂²⋊Q₈	—	2	1	no	—
599	C₂²⋊Q₈	—	2	4	no	—
600	C₂²⋊Q₈	—	2	4	no	—
601	C₂²⋊Q₈	—	2	4	no	—
602	C₂²⋊Q₈	—	2	4	no	—
603	C₂²⋊Q₈	—	2	2	no	—
604	C₂²⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
605	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
606	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
607	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
608	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
609	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
610	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
611	#121	—	2	2	no	—
612	#121	—	2	1	no	—
613	#121	C₂³×C₆₇	1	1	yes	characteristic
614	#121	C₂³×C₆₇	1	2	yes	—
615	#120	—	2	2	no	—
616	C₄×Q₈	—	2	1	no	—
617	C₄×Q₈	—	2	1	no	—
618	C₄×Q₈	—	2	1	no	—
619	C₄×Q₈	—	2	1	no	—
620	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
621	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
622	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
623	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
624	C₄×D₄	—	2	4	no	—
625	C₄×D₄	—	2	2	no	—
626	C₄×D₄	—	2	2	no	—
627	C₄×D₄	—	2	1	no	—
628	C₄×D₄	—	2	2	no	—
629	C₄×D₄	—	2	2	no	—
630	C₄×D₄	—	2	1	no	—
631	C₄×D₄	—	2	2	no	—
632	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
633	C₄×D₄	D₄×C₆₇	1	2	yes	—
634	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
635	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
636	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
637	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
638	C₄×D₄	C₂³×C₆₇	1	2	yes	—
639	#117	—	2	1	no	—
640	#117	—	2	2	no	—
641	#117	—	2	1	no	—
642	#117	C₂³×C₆₇	1	2	yes	—
643	C₂×C₄⋊C₄	—	2	1	no	—
644	C₂×C₄⋊C₄	—	2	1	no	—
645	C₂×C₄⋊C₄	—	2	2	no	—
646	C₂×C₄⋊C₄	—	2	1	no	—
647	C₂×C₄⋊C₄	—	2	2	no	—
648	C₂×C₄⋊C₄	—	2	1	no	—
649	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
650	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
651	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
652	C₂×C₂²⋊C₄	—	2	2	no	—
653	C₂×C₂²⋊C₄	—	2	2	no	—
654	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
655	C₂×C₄²	—	2	1	no	—
656	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
657	#107	C₂³×C₆₇	1	2	yes	—
658	#106	C₂³×C₆₇	1	2	yes	—
659	#106	C₂³×C₆₇	1	4	yes	—
660	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
661	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
662	#103	C₂³×C₆₇	1	2	yes	—
663	#103	C₂³×C₆₇	1	4	yes	—
664	#103	C₂³×C₆₇	1	4	yes	—
665	#103	C₂³×C₆₇	1	4	yes	—
666	#103	C₂³×C₆₇	1	2	yes	—
667	#103	C₂³×C₆₇	1	4	yes	—
668	#103	C₂³×C₆₇	1	4	yes	—
669	#103	C₂³×C₆₇	1	2	yes	—
670	#103	C₂³×C₆₇	1	4	yes	—
671	#102	C₂³×C₆₇	1	2	yes	—
672	C₂²⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
673	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
674	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
675	C₂²⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
676	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
677	#97	C₂³×C₆₇	1	2	yes	—
678	C₄×C₈	C₂³×C₆₇	1	2	yes	—
679	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
680	#41	—	4	4	no	—
681	#41	—	4	4	no	—
682	C₂×Q₈	—	2	4	no	—
683	C₂×Q₈	—	2	1	no	—
684	C₂×Q₈	—	2	1	no	—
685	C₂×Q₈	—	2	4	no	—
686	C₂×Q₈	—	2	4	no	—
687	C₂×Q₈	—	2	2	no	—
688	C₂×Q₈	—	2	2	no	—
689	C₂×Q₈	—	2	8	no	—
690	C₂×Q₈	—	2	8	no	—
691	C₂×Q₈	—	4	4	no	—
692	C₂×Q₈	—	4	4	no	—
693	C₂×Q₈	—	4	4	no	—
694	C₂×Q₈	—	4	8	no	—
695	C₂×Q₈	—	4	4	no	—
696	C₂×Q₈	—	4	8	no	—
697	C₂×Q₈	—	4	8	no	—
698	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
699	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
700	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
701	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
702	C₂×D₄	—	2	4	no	—
703	C₂×D₄	—	2	1	no	—
704	C₂×D₄	—	2	2	no	—
705	C₂×D₄	—	2	2	no	—
706	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
707	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
708	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
709	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
710	C₂²×C₄	—	2	1	no	—
711	C₂²×C₄	—	2	1	no	—
712	C₂²×C₄	—	2	2	no	—
713	C₂²×C₄	—	2	2	no	—
714	C₂²×C₄	—	2	4	no	—
715	C₂²×C₄	—	2	4	no	—
716	C₂²×C₄	—	2	2	no	—
717	C₂²×C₄	—	2	1	no	—
718	C₂²×C₄	—	2	2	no	—
719	C₂²×C₄	—	4	4	no	—
720	C₂²×C₄	—	4	8	no	—
721	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
722	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
723	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
724	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
725	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
726	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
727	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
728	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
729	Q₁₆	—	2	4	no	—
730	Q₁₆	—	2	8	no	—
731	Q₁₆	—	4	4	no	—
732	Q₁₆	—	4	4	no	—
733	Q₁₆	—	4	4	no	—
734	SD₁₆	—	2	4	no	—
735	SD₁₆	—	4	4	no	—
736	M₁₆	—	4	4	no	—
737	C₂×C₈	—	4	4	no	—
738	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
739	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
740	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
741	C₄⋊C₄	—	2	1	no	—
742	C₄⋊C₄	—	2	2	no	—
743	C₄⋊C₄	—	2	2	no	—
744	C₄⋊C₄	—	2	2	no	—
745	C₄⋊C₄	—	2	1	no	—
746	C₄⋊C₄	—	2	2	no	—
747	C₄⋊C₄	—	2	2	no	—
748	C₄⋊C₄	—	2	2	no	—
749	C₄⋊C₄	—	2	2	no	—
750	C₄⋊C₄	—	2	2	no	—
751	C₄⋊C₄	—	2	2	no	—
752	C₄⋊C₄	—	2	4	no	—
753	C₄⋊C₄	—	2	4	no	—
754	C₄⋊C₄	—	2	1	no	—
755	C₄⋊C₄	—	2	2	no	—
756	C₄⋊C₄	—	2	1	no	—
757	C₄⋊C₄	—	2	1	no	—
758	C₄⋊C₄	—	2	1	no	—
759	C₄⋊C₄	—	2	2	no	—
760	C₄⋊C₄	—	2	1	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	1	no	—
763	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
764	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
765	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
766	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
767	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
768	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
769	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
770	C₂²⋊C₄	—	2	1	no	—
771	C₂²⋊C₄	—	2	2	no	—
772	C₂²⋊C₄	—	2	4	no	—
773	C₂²⋊C₄	—	2	4	no	—
774	C₂²⋊C₄	—	2	2	no	—
775	C₂²⋊C₄	—	2	2	no	—
776	C₂²⋊C₄	—	2	2	no	—
777	C₂²⋊C₄	—	2	2	no	—
778	C₂²⋊C₄	—	2	1	no	—
779	C₂²⋊C₄	—	2	1	no	—
780	C₂²⋊C₄	—	2	1	no	—
781	C₂²⋊C₄	—	2	2	no	—
782	C₂²⋊C₄	—	2	1	no	—
783	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
784	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
785	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
786	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
787	C₄²	—	2	2	no	—
788	C₄²	—	2	2	no	—
789	C₄²	—	2	2	no	—
790	C₄²	—	2	2	no	—
791	C₄²	—	2	1	no	—
792	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
793	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
794	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
795	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
796	C₂³	—	2	4	no	—
797	C₂³	—	2	4	no	—
798	C₂³	—	2	1	no	—
799	C₂³	—	4	2	no	—
800	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
801	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
802	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
803	Q₈	—	2	4	no	—
804	Q₈	—	2	2	no	—
805	Q₈	—	2	4	no	—
806	Q₈	—	2	2	no	—
807	Q₈	—	2	4	no	—
808	Q₈	—	2	4	no	—
809	Q₈	—	4	8	no	—
810	Q₈	—	4	8	no	—
811	Q₈	—	4	4	no	—
812	Q₈	—	4	2	no	—
813	Q₈	—	8	2	no	—
814	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
815	D₄	—	2	4	no	—
816	D₄	—	2	2	no	—
817	D₄	—	2	2	no	—
818	D₄	—	4	4	no	—
819	D₄	—	4	2	no	—
820	D₄	—	8	2	no	—
821	D₄	#137×C₆₇	1	4	yes	complemented
822	C₂×C₄	—	2	4	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	2	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	4	no	—
827	C₂×C₄	—	2	4	no	—
828	C₂×C₄	—	2	2	no	—
829	C₂×C₄	—	2	1	no	—
830	C₂×C₄	—	2	2	no	—
831	C₂×C₄	—	2	2	no	—
832	C₂×C₄	—	2	1	no	—
833	C₂×C₄	—	2	2	no	—
834	C₂×C₄	—	2	1	no	—
835	C₂×C₄	—	2	4	no	—
836	C₂×C₄	—	4	2	no	—
837	C₂×C₄	—	4	2	no	—
838	C₂×C₄	—	4	2	no	—
839	C₂×C₄	—	4	8	no	—
840	C₂×C₄	—	4	8	no	—
841	C₂×C₄	—	4	4	no	—
842	C₂×C₄	—	4	4	no	—
843	C₂×C₄	—	4	2	no	—
844	C₂×C₄	—	4	2	no	—
845	C₂×C₄	—	8	2	no	—
846	C₂×C₄	—	8	2	no	—
847	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
848	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
849	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
850	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
851	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
852	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
853	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
854	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
855	C₈	—	2	2	no	—
856	C₈	—	2	4	no	—
857	C₈	—	2	2	no	—
858	C₂²	—	2	1	no	—
859	C₂²	—	2	2	no	—
860	C₂²	—	2	1	no	—
861	C₂²	—	2	2	no	—
862	C₂²	—	2	2	no	—
863	C₂²	—	2	2	no	—
864	C₂²	—	2	2	no	—
865	C₂²	—	4	8	no	—
866	C₂²	—	4	1	no	—
867	C₂²	—	4	1	no	—
868	C₂²	—	4	1	no	—
869	C₂²	C₂×#137×C₆₇	1	4	yes	—
870	C₂²	#578×C₆₇	1	2	yes	—
871	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
872	C₄	—	2	1	no	—
873	C₄	—	2	1	no	—
874	C₄	—	2	1	no	—
875	C₄	—	2	4	no	—
876	C₄	—	2	1	no	—
877	C₄	—	2	2	no	—
878	C₄	—	2	2	no	—
879	C₄	—	4	1	no	—
880	C₄	—	4	2	no	—
881	C₄	—	4	2	no	—
882	C₄	—	4	1	no	—
883	C₄	—	4	4	no	—
884	C₄	—	4	2	no	—
885	C₄	—	4	1	no	—
886	C₄	C₂×#137×C₆₇	1	2	yes	—
887	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
888	C₂	—	2	2	no	—
889	C₂	—	2	4	no	—
890	C₂	—	4	2	no	—
891	C₂	—	8	1	no	—
892	C₂	C₆₇×#3484	1	1	yes	characteristic
893	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
894	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3520	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×D₄×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#3328	C₂	semidirect
4. C₆₇×#3312	C₂	semidirect
5. C₆₇×#3298	C₂	semidirect
6. D₄×Q₁₆×C₆₇	C₂	semidirect
7. C₆₇×#3276	C₂	semidirect
8. C₆₇×#3270	C₂	semidirect
9. C₆₇×#3218	C₂	semidirect
10. C₆₇×#3209	C₂	semidirect
11. C₆₇×#3195	C₂	semidirect
12. C₆₇×#3190	C₂	semidirect
13. C₆₇×#3113	C₂	semidirect
14. C₆₇×#3108	C₂	semidirect
15. C₆₇×#3004	C₂	semidirect
16. C₂×#451×C₆₇	C₂	semidirect
17. C₆₇×#2974	C₂	semidirect
19. C₂×#137×C₆₇	C₂²	semidirect
20. C₂²×Q₁₆×C₆₇	C₂²	semidirect
50. #497×C₆₇	C₂²	semidirect
51. #494×C₆₇	C₂²	semidirect
52. #491×C₆₇	C₂²	semidirect
53. #487×C₆₇	C₂²	semidirect
54. #485×C₆₇	C₂²	semidirect
57. #483×C₆₇	C₂²	semidirect
59. #479×C₆₇	C₂²	semidirect
60. #475×C₆₇	C₂²	semidirect
61. #470×C₆₇	C₂²	semidirect
62. #470×C₆₇	C₂²	semidirect
63. #467×C₆₇	C₂²	semidirect
64. #465×C₆₇	C₂²	semidirect
65. #464×C₆₇	C₂²	semidirect
66. #462×C₆₇	C₂²	semidirect
67. #462×C₆₇	C₂²	semidirect
68. #462×C₆₇	C₂²	semidirect
69. #461×C₆₇	C₂²	semidirect
70. #452×C₆₇	C₂²	semidirect
76. #451×C₆₇	C₂²	semidirect
77. #451×C₆₇	C₂²	semidirect
78. #451×C₆₇	C₂²	semidirect
79. #451×C₆₇	C₂²	semidirect
82. #448×C₆₇	C₂²	semidirect
83. #441×C₆₇	C₂²	semidirect
84. C₄×Q₁₆×C₆₇	C₂²	semidirect
85. C₄×SD₁₆×C₆₇	C₂²	semidirect
91. #419×C₆₇	C₂²	semidirect
92. C₂×#103×C₆₇	C₂²	semidirect
373. D₄×C₆₇	#137	semidirect
440. #61602	C₆₇	direct (N × H)
449. C₂×D₄×Q₈	C₂×C₆₇	semidirect
450. #3328	C₂×C₆₇	semidirect
451. #3312	C₂×C₆₇	semidirect
452. #3298	C₂×C₆₇	semidirect
453. D₄×Q₁₆	C₂×C₆₇	semidirect
454. #3276	C₂×C₆₇	semidirect
455. #3270	C₂×C₆₇	semidirect
456. #3218	C₂×C₆₇	semidirect
457. #3209	C₂×C₆₇	semidirect
458. #3195	C₂×C₆₇	semidirect
459. #3190	C₂×C₆₇	semidirect
460. #3113	C₂×C₆₇	semidirect
461. #3108	C₂×C₆₇	semidirect
462. #3004	C₂×C₆₇	semidirect
463. C₂×#451	C₂×C₆₇	semidirect
464. #2974	C₂×C₆₇	semidirect
465. C₆₇	#61602	direct (N × H)
467. C₂×#137	C₂²×C₆₇	semidirect
468. C₂²×Q₁₆	C₂²×C₆₇	semidirect
498. #497	C₂²×C₆₇	semidirect
499. #494	C₂²×C₆₇	semidirect
500. #491	C₂²×C₆₇	semidirect
501. #487	C₂²×C₆₇	semidirect
502. #485	C₂²×C₆₇	semidirect
505. #483	C₂²×C₆₇	semidirect
507. #479	C₂²×C₆₇	semidirect
508. #475	C₂²×C₆₇	semidirect
509. #470	C₂²×C₆₇	semidirect
510. #470	C₂²×C₆₇	semidirect
511. #467	C₂²×C₆₇	semidirect
512. #465	C₂²×C₆₇	semidirect
513. #464	C₂²×C₆₇	semidirect
514. #462	C₂²×C₆₇	semidirect
515. #462	C₂²×C₆₇	semidirect
516. #462	C₂²×C₆₇	semidirect
517. #461	C₂²×C₆₇	semidirect
518. #452	C₂²×C₆₇	semidirect
524. #451	C₂²×C₆₇	semidirect
525. #451	C₂²×C₆₇	semidirect
526. #451	C₂²×C₆₇	semidirect
527. #451	C₂²×C₆₇	semidirect
530. #448	C₂²×C₆₇	semidirect
531. #441	C₂²×C₆₇	semidirect
532. C₄×Q₁₆	C₂²×C₆₇	semidirect
533. C₄×SD₁₆	C₂²×C₆₇	semidirect
539. #419	C₂²×C₆₇	semidirect
540. C₂×#103	C₂²×C₆₇	semidirect
821. D₄	#137×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3520	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3328	2
C₆₇×#3312	2
C₆₇×#3298	2
D₄×Q₁₆×C₆₇	2
C₆₇×#3276	2
C₆₇×#3270	2
C₆₇×#3218	2
C₆₇×#3209	2
C₆₇×#3195	2
C₆₇×#3190	2
C₆₇×#3113	2
C₆₇×#3108	2
C₆₇×#3004	2
C₂×#451×C₆₇	2
C₆₇×#2974	2
#61602	67

Subgroups — C67×#61602 #30740853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61602 #30740853