Subgroups — C₆₇×#61593 #30740844

Order 17152.

2148 conjugacy classes of subgroups.

The subgroup lattice has 924 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3519	C₂	1	1	yes	maximal, characteristic
2	C₂×D₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#3329	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3320	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3299	C₂	1	2	yes	maximal, complemented
6	D₄×SD₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3278	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3266	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3218	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3206	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3193	C₂	1	2	yes	maximal, complemented
12	C₆₇×#3190	C₂	1	2	yes	maximal, complemented
13	C₆₇×#3112	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×#448×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	complemented
19	C₂×#137×C₆₇	C₂²	1	4	yes	complemented
20	C₂²×SD₁₆×C₆₇	C₂²	1	4	yes	complemented
21	#555×C₆₇	C₂²	1	1	yes	characteristic
22	#554×C₆₇	C₂²	1	1	yes	characteristic
23	#553×C₆₇	—	2	1	no	—
24	#553×C₆₇	C₂²	1	2	yes	—
25	#550×C₆₇	C₂²	1	1	yes	characteristic
26	D₄×Q₈×C₆₇	—	2	1	no	—
27	D₄×Q₈×C₆₇	—	2	2	no	—
28	D₄×Q₈×C₆₇	—	2	1	no	—
29	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
30	D₄×Q₈×C₆₇	C₂²	1	4	yes	complemented
31	D₄×Q₈×C₆₇	C₂²	1	2	yes	complemented
32	#546×C₆₇	—	2	2	no	—
33	#546×C₆₇	C₂²	1	2	yes	—
34	D₄²×C₆₇	—	2	1	no	—
35	D₄²×C₆₇	C₂²	1	1	yes	characteristic
36	#540×C₆₇	—	2	1	no	—
37	#538×C₆₇	—	2	2	no	—
38	#532×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
40	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
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	complemented
45	C₂×#121×C₆₇	C₂²	1	2	yes	—
46	#518×C₆₇	—	2	1	no	—
47	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
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	#492×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	#484×C₆₇	C₂²	1	4	yes	complemented
56	#483×C₆₇	C₂²	1	1	yes	characteristic, complemented
57	#480×C₆₇	C₂²	1	1	yes	characteristic
58	#480×C₆₇	C₂²	1	2	yes	complemented
59	#479×C₆₇	C₂²	1	2	yes	complemented
60	#477×C₆₇	C₂²	1	2	yes	complemented
61	#468×C₆₇	C₂²	1	4	yes	complemented
62	#468×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	2	yes	complemented
67	#461×C₆₇	C₂²	1	4	yes	complemented
68	#461×C₆₇	C₂²	1	2	yes	complemented
69	#460×C₆₇	C₂²	1	2	yes	complemented
70	#452×C₆₇	C₂²	1	4	yes	complemented
71	#451×C₆₇	C₂²	1	4	yes	complemented
72	#451×C₆₇	C₂²	1	1	yes	characteristic, complemented
73	#451×C₆₇	C₂²	1	2	yes	complemented
74	#450×C₆₇	C₂²	1	4	yes	complemented
75	#448×C₆₇	C₂²	1	1	yes	characteristic
76	#448×C₆₇	C₂²	1	2	yes	—
77	#448×C₆₇	C₂²	1	4	yes	complemented
78	#448×C₆₇	C₂²	1	4	yes	complemented
79	#448×C₆₇	C₂²	1	4	yes	complemented
80	#448×C₆₇	C₂²	1	2	yes	complemented
81	#448×C₆₇	C₂²	1	2	yes	complemented
82	#448×C₆₇	C₂²	1	2	yes	complemented
83	#440×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	complemented
87	#435×C₆₇	C₂²	1	4	yes	complemented
88	C₈×D₄×C₆₇	C₂²	1	2	yes	—
89	#432×C₆₇	C₂²	1	1	yes	characteristic, complemented
90	#431×C₆₇	C₂²	1	1	yes	characteristic, complemented
91	#419×C₆₇	C₂²	1	4	yes	complemented
92	C₂×#103×C₆₇	C₂²	1	4	yes	complemented
93	#407×C₆₇	C₂²	1	2	yes	complemented
94	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	complemented
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×D₄×C₆₇	—	2	2	no	—
106	C₂²×D₄×C₆₇	—	2	1	no	—
107	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
108	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
109	C₂³×C₄×C₆₇	—	2	2	no	—
110	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
111	#137×C₆₇	—	4	8	no	—
112	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
113	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
114	C₂×SD₁₆×C₆₇	—	4	8	no	—
115	C₂×SD₁₆×C₆₇	C₂³	1	2	yes	—
116	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
117	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
118	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
119	C₂×SD₁₆×C₆₇	C₂³	1	4	yes	—
120	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
121	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
122	C₄⋊Q₈×C₆₇	—	2	2	no	—
123	C₄⋊Q₈×C₆₇	—	2	2	no	—
124	C₄⋊Q₈×C₆₇	—	2	4	no	—
125	C₄⋊Q₈×C₆₇	—	2	1	no	—
126	C₄⋊Q₈×C₆₇	—	2	1	no	—
127	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
128	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
129	C₄⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
130	#127×C₆₇	—	2	1	no	—
131	#127×C₆₇	C₂³	1	1	yes	characteristic
132	#126×C₆₇	—	2	2	no	—
133	#126×C₆₇	—	2	1	no	—
134	#125×C₆₇	—	2	1	no	—
135	#125×C₆₇	C₂³	1	1	yes	characteristic
136	#124×C₆₇	—	2	2	no	—
137	#124×C₆₇	—	2	1	no	—
138	#124×C₆₇	C₂³	1	1	yes	characteristic
139	#123×C₆₇	—	2	2	no	—
140	#123×C₆₇	—	2	1	no	—
141	#123×C₆₇	—	2	2	no	—
142	#123×C₆₇	—	2	2	no	—
143	C₂²⋊Q₈×C₆₇	—	2	2	no	—
144	C₂²⋊Q₈×C₆₇	—	2	2	no	—
145	C₂²⋊Q₈×C₆₇	—	2	4	no	—
146	C₂²⋊Q₈×C₆₇	—	2	4	no	—
147	C₂²⋊Q₈×C₆₇	—	2	4	no	—
148	C₂²⋊Q₈×C₆₇	—	2	4	no	—
149	C₂²⋊Q₈×C₆₇	—	2	2	no	—
150	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
151	C₂²⋊Q₈×C₆₇	C₂³	1	1	yes	characteristic
152	C₂²⋊Q₈×C₆₇	C₂³	1	2	yes	—
153	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
154	#121×C₆₇	—	2	2	no	—
155	#121×C₆₇	—	2	1	no	—
156	#121×C₆₇	—	2	2	no	—
157	#121×C₆₇	—	2	2	no	—
158	#121×C₆₇	—	2	1	no	—
159	#121×C₆₇	—	2	1	no	—
160	#121×C₆₇	—	2	2	no	—
161	#121×C₆₇	—	2	2	no	—
162	#121×C₆₇	C₂³	1	1	yes	characteristic
163	#121×C₆₇	C₂³	1	2	yes	—
164	#121×C₆₇	C₂³	1	2	yes	—
165	#121×C₆₇	C₂³	1	2	yes	—
166	#121×C₆₇	C₂³	1	2	yes	—
167	#120×C₆₇	—	2	2	no	—
168	#120×C₆₇	—	2	2	no	—
169	#120×C₆₇	—	2	2	no	—
170	C₄×Q₈×C₆₇	—	2	1	no	—
171	C₄×Q₈×C₆₇	—	2	1	no	—
172	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
173	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
174	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
175	C₄×D₄×C₆₇	—	2	2	no	—
176	C₄×D₄×C₆₇	—	2	4	no	—
177	C₄×D₄×C₆₇	—	2	2	no	—
178	C₄×D₄×C₆₇	—	2	1	no	—
179	C₄×D₄×C₆₇	—	2	2	no	—
180	C₄×D₄×C₆₇	—	2	2	no	—
181	C₄×D₄×C₆₇	—	2	1	no	—
182	C₄×D₄×C₆₇	—	2	1	no	—
183	C₄×D₄×C₆₇	—	2	2	no	—
184	C₄×D₄×C₆₇	—	2	1	no	—
185	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
186	C₄×D₄×C₆₇	D₄	1	2	yes	—
187	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
188	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
189	C₄×D₄×C₆₇	C₂³	1	2	yes	—
190	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
191	C₄×D₄×C₆₇	C₂³	1	2	yes	—
192	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
193	#117×C₆₇	—	2	1	no	—
194	#117×C₆₇	—	2	1	no	—
195	#117×C₆₇	C₂³	1	2	yes	—
196	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
197	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
198	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
199	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
200	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
201	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
202	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
203	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
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	4	yes	—
210	#107×C₆₇	C₂³	1	2	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	2	yes	—
218	#103×C₆₇	C₂³	1	4	yes	—
219	#103×C₆₇	C₂³	1	4	yes	—
220	#103×C₆₇	C₂³	1	4	yes	—
221	#103×C₆₇	C₂³	1	2	yes	—
222	#102×C₆₇	C₂³	1	2	yes	—
223	#102×C₆₇	C₂³	1	4	yes	—
224	C₂²⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
225	C₂²⋊C₈×C₆₇	C₂³	1	2	yes	—
226	C₂²⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
227	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
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	—	2	2	no	—
232	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
233	#41×C₆₇	—	4	4	no	—
234	#41×C₆₇	—	8	2	no	—
235	C₂×Q₈×C₆₇	—	2	4	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	1	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	8	no	—
248	C₂×Q₈×C₆₇	—	4	8	no	—
249	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
250	C₂×Q₈×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₂×Q₈×C₆₇	C₂⁴	1	2	yes	—
252	C₂×D₄×C₆₇	—	2	4	no	—
253	C₂×D₄×C₆₇	—	2	2	no	—
254	C₂×D₄×C₆₇	—	2	2	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	—	2	2	no	—
259	C₂×D₄×C₆₇	—	2	1	no	—
260	C₂×D₄×C₆₇	—	2	1	no	—
261	C₂×D₄×C₆₇	—	2	2	no	—
262	C₂×D₄×C₆₇	—	4	4	no	—
263	C₂×D₄×C₆₇	—	4	4	no	—
264	C₂×D₄×C₆₇	—	8	4	no	—
265	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
266	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
267	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
268	C₂×D₄×C₆₇	C₂⁴	1	1	yes	characteristic
269	C₂×D₄×C₆₇	C₂⁴	1	2	yes	—
270	C₂²×C₄×C₆₇	—	2	1	no	—
271	C₂²×C₄×C₆₇	—	2	1	no	—
272	C₂²×C₄×C₆₇	—	2	2	no	—
273	C₂²×C₄×C₆₇	—	2	4	no	—
274	C₂²×C₄×C₆₇	—	2	4	no	—
275	C₂²×C₄×C₆₇	—	2	2	no	—
276	C₂²×C₄×C₆₇	—	2	1	no	—
277	C₂²×C₄×C₆₇	—	2	2	no	—
278	C₂²×C₄×C₆₇	—	2	2	no	—
279	C₂²×C₄×C₆₇	—	4	4	no	—
280	C₂²×C₄×C₆₇	—	4	8	no	—
281	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
282	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
283	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
284	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
285	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
286	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
287	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
288	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
289	Q₁₆×C₆₇	—	2	4	no	—
290	Q₁₆×C₆₇	—	4	4	no	—
291	SD₁₆×C₆₇	—	2	4	no	—
292	SD₁₆×C₆₇	—	2	8	no	—
293	SD₁₆×C₆₇	—	4	4	no	—
294	SD₁₆×C₆₇	—	4	4	no	—
295	SD₁₆×C₆₇	—	4	4	no	—
296	M₁₆×C₆₇	—	4	4	no	—
297	C₂×C₈×C₆₇	—	4	4	no	—
298	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
299	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
300	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
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	2	no	—
305	C₄⋊C₄×C₆₇	—	2	2	no	—
306	C₄⋊C₄×C₆₇	—	2	2	no	—
307	C₄⋊C₄×C₆₇	—	2	4	no	—
308	C₄⋊C₄×C₆₇	—	2	4	no	—
309	C₄⋊C₄×C₆₇	—	2	1	no	—
310	C₄⋊C₄×C₆₇	—	2	2	no	—
311	C₄⋊C₄×C₆₇	—	2	1	no	—
312	C₄⋊C₄×C₆₇	—	2	1	no	—
313	C₄⋊C₄×C₆₇	—	2	1	no	—
314	C₄⋊C₄×C₆₇	—	2	1	no	—
315	C₄⋊C₄×C₆₇	—	2	2	no	—
316	C₄⋊C₄×C₆₇	—	2	1	no	—
317	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
318	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
319	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
320	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
321	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
322	C₄⋊C₄×C₆₇	C₂⁴	1	1	yes	characteristic
323	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
324	C₂²⋊C₄×C₆₇	—	2	2	no	—
325	C₂²⋊C₄×C₆₇	—	2	4	no	—
326	C₂²⋊C₄×C₆₇	—	2	4	no	—
327	C₂²⋊C₄×C₆₇	—	2	2	no	—
328	C₂²⋊C₄×C₆₇	—	2	1	no	—
329	C₂²⋊C₄×C₆₇	—	2	1	no	—
330	C₂²⋊C₄×C₆₇	—	2	1	no	—
331	C₂²⋊C₄×C₆₇	—	2	2	no	—
332	C₂²⋊C₄×C₆₇	—	2	2	no	—
333	C₂²⋊C₄×C₆₇	—	2	2	no	—
334	C₂²⋊C₄×C₆₇	—	2	1	no	—
335	C₂²⋊C₄×C₆₇	—	2	2	no	—
336	C₂²⋊C₄×C₆₇	—	2	1	no	—
337	C₂²⋊C₄×C₆₇	—	2	2	no	—
338	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
339	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
340	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
341	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
342	C₄²×C₆₇	—	2	2	no	—
343	C₄²×C₆₇	—	2	2	no	—
344	C₄²×C₆₇	—	2	2	no	—
345	C₄²×C₆₇	—	2	1	no	—
346	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
347	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
348	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
349	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
350	C₂³×C₆₇	—	2	4	no	—
351	C₂³×C₆₇	—	2	4	no	—
352	C₂³×C₆₇	—	2	1	no	—
353	C₂³×C₆₇	—	2	2	no	—
354	C₂³×C₆₇	—	4	2	no	—
355	C₂³×C₆₇	—	4	4	no	—
356	C₂³×C₆₇	—	8	2	no	—
357	C₂³×C₆₇	—	8	2	no	—
358	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
359	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
360	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
361	Q₈×C₆₇	—	2	4	no	—
362	Q₈×C₆₇	—	2	2	no	—
363	Q₈×C₆₇	—	2	4	no	—
364	Q₈×C₆₇	—	2	4	no	—
365	Q₈×C₆₇	—	2	2	no	—
366	Q₈×C₆₇	—	4	8	no	—
367	Q₈×C₆₇	—	4	8	no	—
368	Q₈×C₆₇	—	4	4	no	—
369	Q₈×C₆₇	—	8	1	no	—
370	Q₈×C₆₇	H(5,𝔽₂)	1	4	yes	—
371	D₄×C₆₇	—	2	4	no	—
372	D₄×C₆₇	—	2	2	no	—
373	D₄×C₆₇	—	2	2	no	—
374	D₄×C₆₇	—	2	4	no	—
375	D₄×C₆₇	—	4	4	no	—
376	D₄×C₆₇	—	4	2	no	—
377	D₄×C₆₇	—	4	4	no	—
378	D₄×C₆₇	—	4	2	no	—
379	D₄×C₆₇	—	8	2	no	—
380	D₄×C₆₇	—	8	1	no	—
381	D₄×C₆₇	—	8	2	no	—
382	D₄×C₆₇	—	8	2	no	—
383	D₄×C₆₇	—	8	2	no	—
384	D₄×C₆₇	#137	1	4	yes	complemented
385	C₂×C₄×C₆₇	—	2	4	no	—
386	C₂×C₄×C₆₇	—	2	2	no	—
387	C₂×C₄×C₆₇	—	2	2	no	—
388	C₂×C₄×C₆₇	—	2	2	no	—
389	C₂×C₄×C₆₇	—	2	4	no	—
390	C₂×C₄×C₆₇	—	2	4	no	—
391	C₂×C₄×C₆₇	—	2	2	no	—
392	C₂×C₄×C₆₇	—	2	1	no	—
393	C₂×C₄×C₆₇	—	2	2	no	—
394	C₂×C₄×C₆₇	—	2	1	no	—
395	C₂×C₄×C₆₇	—	2	2	no	—
396	C₂×C₄×C₆₇	—	2	1	no	—
397	C₂×C₄×C₆₇	—	2	4	no	—
398	C₂×C₄×C₆₇	—	4	2	no	—
399	C₂×C₄×C₆₇	—	4	2	no	—
400	C₂×C₄×C₆₇	—	4	2	no	—
401	C₂×C₄×C₆₇	—	4	8	no	—
402	C₂×C₄×C₆₇	—	4	8	no	—
403	C₂×C₄×C₆₇	—	4	4	no	—
404	C₂×C₄×C₆₇	—	4	2	no	—
405	C₂×C₄×C₆₇	—	4	2	no	—
406	C₂×C₄×C₆₇	—	8	2	no	—
407	C₂×C₄×C₆₇	—	8	2	no	—
408	C₂×C₄×C₆₇	—	8	2	no	—
409	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
410	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
411	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
412	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
413	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
414	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
415	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
416	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
417	C₈×C₆₇	—	2	2	no	—
418	C₈×C₆₇	—	2	4	no	—
419	C₈×C₆₇	—	2	2	no	—
420	C₂²×C₆₇	—	2	1	no	—
421	C₂²×C₆₇	—	2	2	no	—
422	C₂²×C₆₇	—	2	1	no	—
423	C₂²×C₆₇	—	2	2	no	—
424	C₂²×C₆₇	—	2	2	no	—
425	C₂²×C₆₇	—	2	2	no	—
426	C₂²×C₆₇	—	2	2	no	—
427	C₂²×C₆₇	—	4	8	no	—
428	C₂²×C₆₇	—	4	2	no	—
429	C₂²×C₆₇	—	4	1	no	—
430	C₂²×C₆₇	—	4	1	no	—
431	C₂²×C₆₇	—	4	2	no	—
432	C₂²×C₆₇	—	4	2	no	—
433	C₂²×C₆₇	—	4	1	no	—
434	C₂²×C₆₇	—	16	4	no	—
435	C₂²×C₆₇	C₂×#137	1	4	yes	—
436	C₂²×C₆₇	#577	1	2	yes	—
437	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
438	C₄×C₆₇	—	2	4	no	—
439	C₄×C₆₇	—	2	1	no	—
440	C₄×C₆₇	—	2	2	no	—
441	C₄×C₆₇	—	2	2	no	—
442	C₄×C₆₇	—	2	1	no	—
443	C₄×C₆₇	—	2	1	no	—
444	C₄×C₆₇	—	2	1	no	—
445	C₄×C₆₇	—	4	4	no	—
446	C₄×C₆₇	—	4	1	no	—
447	C₄×C₆₇	—	4	1	no	—
448	C₄×C₆₇	—	4	2	no	—
449	C₄×C₆₇	—	4	1	no	—
450	C₄×C₆₇	—	4	2	no	—
451	C₄×C₆₇	C₂×#137	1	2	yes	—
452	C₄×C₆₇	C₂×H(5,𝔽₂)	1	2	yes	—
453	#61593	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
454	C₂×C₆₇	—	2	2	no	—
455	C₂×C₆₇	—	2	4	no	—
456	C₂×C₆₇	—	4	2	no	—
457	C₂×C₆₇	—	8	2	no	—
458	C₂×C₆₇	—	8	1	no	—
459	C₂×C₆₇	#3484	1	1	yes	characteristic
460	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
461	C₂×C₆₇	C₂×#577	1	1	yes	characteristic
462	#3519	C₂×C₆₇	1	1	yes	characteristic
463	C₂×D₄×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
464	#3329	C₂×C₆₇	1	2	yes	complemented
465	#3320	C₂×C₆₇	1	4	yes	complemented
466	#3299	C₂×C₆₇	1	2	yes	complemented
467	D₄×SD₁₆	C₂×C₆₇	1	2	yes	complemented
468	#3278	C₂×C₆₇	1	2	yes	complemented
469	#3266	C₂×C₆₇	1	4	yes	complemented
470	#3218	C₂×C₆₇	1	1	yes	characteristic, complemented
471	#3206	C₂×C₆₇	1	1	yes	characteristic, complemented
472	#3193	C₂×C₆₇	1	2	yes	complemented
473	#3190	C₂×C₆₇	1	2	yes	complemented
474	#3112	C₂×C₆₇	1	1	yes	characteristic, complemented
475	#3108	C₂×C₆₇	1	1	yes	characteristic, complemented
476	#3004	C₂×C₆₇	1	2	yes	complemented
477	C₂×#448	C₂×C₆₇	1	2	yes	complemented
478	#2974	C₂×C₆₇	1	1	yes	characteristic, complemented
479	C₆₇	#61593	1	1	yes	characteristic, direct product factor, Sylow
480	C₂³×Q₈	C₂²×C₆₇	1	2	yes	complemented
481	C₂×#137	C₂²×C₆₇	1	4	yes	complemented
482	C₂²×SD₁₆	C₂²×C₆₇	1	4	yes	complemented
483	#555	C₂²×C₆₇	1	1	yes	characteristic
484	#554	C₂²×C₆₇	1	1	yes	characteristic
485	#553	—	2	1	no	—
486	#553	C₂²×C₆₇	1	2	yes	—
487	#550	C₂²×C₆₇	1	1	yes	characteristic
488	D₄×Q₈	—	2	1	no	—
489	D₄×Q₈	—	2	2	no	—
490	D₄×Q₈	—	2	1	no	—
491	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
492	D₄×Q₈	C₂²×C₆₇	1	2	yes	complemented
493	D₄×Q₈	C₂²×C₆₇	1	4	yes	complemented
494	#546	—	2	2	no	—
495	#546	C₂²×C₆₇	1	2	yes	—
496	D₄²	—	2	1	no	—
497	D₄²	C₂²×C₆₇	1	1	yes	characteristic
498	#540	—	2	1	no	—
499	#538	—	2	2	no	—
500	#532	C₂²×C₆₇	1	1	yes	characteristic
501	C₂×C₄⋊Q₈	—	2	1	no	—
502	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
503	#529	C₂²×C₆₇	1	1	yes	characteristic
504	#525	C₂²×C₆₇	1	2	yes	—
505	C₂×C₂²⋊Q₈	—	2	2	no	—
506	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	complemented
507	C₂×#121	C₂²×C₆₇	1	2	yes	—
508	#518	—	2	1	no	—
509	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
510	C₂×C₄×D₄	—	2	1	no	—
511	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
512	#497	C₂²×C₆₇	1	2	yes	complemented
513	#492	C₂²×C₆₇	1	2	yes	complemented
514	#491	C₂²×C₆₇	1	2	yes	complemented
515	#487	C₂²×C₆₇	1	2	yes	complemented
516	#485	C₂²×C₆₇	1	4	yes	complemented
517	#484	C₂²×C₆₇	1	4	yes	complemented
518	#483	C₂²×C₆₇	1	1	yes	characteristic, complemented
519	#480	C₂²×C₆₇	1	1	yes	characteristic
520	#480	C₂²×C₆₇	1	2	yes	complemented
521	#479	C₂²×C₆₇	1	2	yes	complemented
522	#477	C₂²×C₆₇	1	2	yes	complemented
523	#468	C₂²×C₆₇	1	4	yes	complemented
524	#468	C₂²×C₆₇	1	4	yes	complemented
525	#467	C₂²×C₆₇	1	4	yes	complemented
526	#465	C₂²×C₆₇	1	4	yes	complemented
527	#464	C₂²×C₆₇	1	2	yes	complemented
528	#462	C₂²×C₆₇	1	2	yes	complemented
529	#461	C₂²×C₆₇	1	2	yes	complemented
530	#461	C₂²×C₆₇	1	4	yes	complemented
531	#460	C₂²×C₆₇	1	2	yes	complemented
532	#452	C₂²×C₆₇	1	4	yes	complemented
533	#451	C₂²×C₆₇	1	1	yes	characteristic, complemented
534	#451	C₂²×C₆₇	1	2	yes	complemented
535	#451	C₂²×C₆₇	1	4	yes	complemented
536	#450	C₂²×C₆₇	1	4	yes	complemented
537	#448	C₂²×C₆₇	1	1	yes	characteristic
538	#448	C₂²×C₆₇	1	2	yes	—
539	#448	C₂²×C₆₇	1	4	yes	complemented
540	#448	C₂²×C₆₇	1	2	yes	complemented
541	#448	C₂²×C₆₇	1	2	yes	complemented
542	#448	C₂²×C₆₇	1	2	yes	complemented
543	#448	C₂²×C₆₇	1	4	yes	complemented
544	#448	C₂²×C₆₇	1	4	yes	complemented
545	#440	C₂²×C₆₇	1	4	yes	complemented
546	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	complemented
547	C₄×SD₁₆	C₂²×C₆₇	1	2	yes	complemented
548	#436	C₂²×C₆₇	1	2	yes	complemented
549	#435	C₂²×C₆₇	1	4	yes	complemented
550	C₈×D₄	C₂²×C₆₇	1	2	yes	—
551	#432	C₂²×C₆₇	1	1	yes	characteristic, complemented
552	#431	C₂²×C₆₇	1	1	yes	characteristic, complemented
553	#419	C₂²×C₆₇	1	4	yes	complemented
554	C₂×#103	C₂²×C₆₇	1	4	yes	complemented
555	#407	C₂²×C₆₇	1	2	yes	complemented
556	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	complemented
557	C₂×#41	—	2	1	no	—
558	C₂×#41	C₂³×C₆₇	1	2	yes	—
559	C₂²×Q₈	—	2	2	no	—
560	C₂²×Q₈	—	2	2	no	—
561	C₂²×Q₈	—	2	2	no	—
562	C₂²×Q₈	—	4	8	no	—
563	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
564	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
565	C₂²×Q₈	C₂³×C₆₇	1	4	yes	—
566	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
567	C₂²×D₄	—	2	2	no	—
568	C₂²×D₄	—	2	1	no	—
569	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
570	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
571	C₂³×C₄	—	2	2	no	—
572	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
573	#137	—	4	8	no	—
574	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
575	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
576	C₂×SD₁₆	—	4	8	no	—
577	C₂×SD₁₆	C₂³×C₆₇	1	2	yes	—
578	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
579	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
580	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
581	C₂×SD₁₆	C₂³×C₆₇	1	4	yes	—
582	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
583	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
584	C₄⋊Q₈	—	2	2	no	—
585	C₄⋊Q₈	—	2	2	no	—
586	C₄⋊Q₈	—	2	4	no	—
587	C₄⋊Q₈	—	2	1	no	—
588	C₄⋊Q₈	—	2	1	no	—
589	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
590	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
591	C₄⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
592	#127	—	2	1	no	—
593	#127	C₂³×C₆₇	1	1	yes	characteristic
594	#126	—	2	2	no	—
595	#126	—	2	1	no	—
596	#125	—	2	1	no	—
597	#125	C₂³×C₆₇	1	1	yes	characteristic
598	#124	—	2	2	no	—
599	#124	—	2	1	no	—
600	#124	C₂³×C₆₇	1	1	yes	characteristic
601	#123	—	2	2	no	—
602	#123	—	2	1	no	—
603	#123	—	2	2	no	—
604	#123	—	2	2	no	—
605	C₂²⋊Q₈	—	2	2	no	—
606	C₂²⋊Q₈	—	2	2	no	—
607	C₂²⋊Q₈	—	2	4	no	—
608	C₂²⋊Q₈	—	2	4	no	—
609	C₂²⋊Q₈	—	2	4	no	—
610	C₂²⋊Q₈	—	2	4	no	—
611	C₂²⋊Q₈	—	2	2	no	—
612	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
613	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
614	C₂²⋊Q₈	C₂³×C₆₇	1	1	yes	characteristic
615	C₂²⋊Q₈	C₂³×C₆₇	1	2	yes	—
616	#121	—	2	2	no	—
617	#121	—	2	1	no	—
618	#121	—	2	2	no	—
619	#121	—	2	2	no	—
620	#121	—	2	1	no	—
621	#121	—	2	1	no	—
622	#121	—	2	2	no	—
623	#121	—	2	2	no	—
624	#121	C₂³×C₆₇	1	1	yes	characteristic
625	#121	C₂³×C₆₇	1	2	yes	—
626	#121	C₂³×C₆₇	1	2	yes	—
627	#121	C₂³×C₆₇	1	2	yes	—
628	#121	C₂³×C₆₇	1	2	yes	—
629	#120	—	2	2	no	—
630	#120	—	2	2	no	—
631	#120	—	2	2	no	—
632	C₄×Q₈	—	2	1	no	—
633	C₄×Q₈	—	2	1	no	—
634	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
635	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
636	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
637	C₄×D₄	—	2	2	no	—
638	C₄×D₄	—	2	4	no	—
639	C₄×D₄	—	2	2	no	—
640	C₄×D₄	—	2	1	no	—
641	C₄×D₄	—	2	2	no	—
642	C₄×D₄	—	2	2	no	—
643	C₄×D₄	—	2	1	no	—
644	C₄×D₄	—	2	1	no	—
645	C₄×D₄	—	2	2	no	—
646	C₄×D₄	—	2	1	no	—
647	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
648	C₄×D₄	D₄×C₆₇	1	2	yes	—
649	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
650	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
651	C₄×D₄	C₂³×C₆₇	1	2	yes	—
652	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
653	C₄×D₄	C₂³×C₆₇	1	2	yes	—
654	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
655	#117	—	2	1	no	—
656	#117	—	2	1	no	—
657	#117	C₂³×C₆₇	1	2	yes	—
658	C₂×C₄⋊C₄	—	2	1	no	—
659	C₂×C₄⋊C₄	—	2	1	no	—
660	C₂×C₄⋊C₄	—	2	1	no	—
661	C₂×C₄⋊C₄	—	2	2	no	—
662	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
663	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
664	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
665	C₂×C₂²⋊C₄	—	2	2	no	—
666	C₂×C₂²⋊C₄	—	2	2	no	—
667	C₂×C₂²⋊C₄	—	2	2	no	—
668	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
669	C₂×C₄²	—	2	1	no	—
670	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
671	#107	C₂³×C₆₇	1	2	yes	—
672	#107	C₂³×C₆₇	1	4	yes	—
673	#106	C₂³×C₆₇	1	2	yes	—
674	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
675	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
676	#103	C₂³×C₆₇	1	2	yes	—
677	#103	C₂³×C₆₇	1	4	yes	—
678	#103	C₂³×C₆₇	1	4	yes	—
679	#103	C₂³×C₆₇	1	4	yes	—
680	#103	C₂³×C₆₇	1	2	yes	—
681	#103	C₂³×C₆₇	1	4	yes	—
682	#103	C₂³×C₆₇	1	4	yes	—
683	#103	C₂³×C₆₇	1	2	yes	—
684	#102	C₂³×C₆₇	1	2	yes	—
685	#102	C₂³×C₆₇	1	4	yes	—
686	C₂²⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
687	C₂²⋊C₈	C₂³×C₆₇	1	2	yes	—
688	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
689	C₂²⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
690	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
691	#97	C₂³×C₆₇	1	2	yes	—
692	C₄×C₈	C₂³×C₆₇	1	2	yes	—
693	C₂⁴	—	2	2	no	—
694	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
695	#41	—	4	4	no	—
696	#41	—	8	2	no	—
697	C₂×Q₈	—	2	4	no	—
698	C₂×Q₈	—	2	1	no	—
699	C₂×Q₈	—	2	4	no	—
700	C₂×Q₈	—	2	4	no	—
701	C₂×Q₈	—	2	2	no	—
702	C₂×Q₈	—	2	1	no	—
703	C₂×Q₈	—	2	8	no	—
704	C₂×Q₈	—	2	8	no	—
705	C₂×Q₈	—	4	4	no	—
706	C₂×Q₈	—	4	4	no	—
707	C₂×Q₈	—	4	4	no	—
708	C₂×Q₈	—	4	8	no	—
709	C₂×Q₈	—	4	8	no	—
710	C₂×Q₈	—	4	8	no	—
711	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
712	C₂×Q₈	C₂⁴×C₆₇	1	1	yes	characteristic
713	C₂×Q₈	C₂⁴×C₆₇	1	2	yes	—
714	C₂×D₄	—	2	4	no	—
715	C₂×D₄	—	2	2	no	—
716	C₂×D₄	—	2	2	no	—
717	C₂×D₄	—	2	1	no	—
718	C₂×D₄	—	2	2	no	—
719	C₂×D₄	—	2	2	no	—
720	C₂×D₄	—	2	2	no	—
721	C₂×D₄	—	2	1	no	—
722	C₂×D₄	—	2	1	no	—
723	C₂×D₄	—	2	2	no	—
724	C₂×D₄	—	4	4	no	—
725	C₂×D₄	—	4	4	no	—
726	C₂×D₄	—	8	4	no	—
727	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
728	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
729	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
730	C₂×D₄	C₂⁴×C₆₇	1	1	yes	characteristic
731	C₂×D₄	C₂⁴×C₆₇	1	2	yes	—
732	C₂²×C₄	—	2	1	no	—
733	C₂²×C₄	—	2	1	no	—
734	C₂²×C₄	—	2	2	no	—
735	C₂²×C₄	—	2	4	no	—
736	C₂²×C₄	—	2	4	no	—
737	C₂²×C₄	—	2	2	no	—
738	C₂²×C₄	—	2	1	no	—
739	C₂²×C₄	—	2	2	no	—
740	C₂²×C₄	—	2	2	no	—
741	C₂²×C₄	—	4	4	no	—
742	C₂²×C₄	—	4	8	no	—
743	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
744	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
745	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
746	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
747	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
748	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
749	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
750	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
751	Q₁₆	—	2	4	no	—
752	Q₁₆	—	4	4	no	—
753	SD₁₆	—	2	4	no	—
754	SD₁₆	—	2	8	no	—
755	SD₁₆	—	4	4	no	—
756	SD₁₆	—	4	4	no	—
757	SD₁₆	—	4	4	no	—
758	M₁₆	—	4	4	no	—
759	C₂×C₈	—	4	4	no	—
760	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
761	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
762	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	2	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	—	2	2	no	—
767	C₄⋊C₄	—	2	2	no	—
768	C₄⋊C₄	—	2	2	no	—
769	C₄⋊C₄	—	2	4	no	—
770	C₄⋊C₄	—	2	4	no	—
771	C₄⋊C₄	—	2	1	no	—
772	C₄⋊C₄	—	2	2	no	—
773	C₄⋊C₄	—	2	1	no	—
774	C₄⋊C₄	—	2	1	no	—
775	C₄⋊C₄	—	2	1	no	—
776	C₄⋊C₄	—	2	1	no	—
777	C₄⋊C₄	—	2	2	no	—
778	C₄⋊C₄	—	2	1	no	—
779	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
780	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
781	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
782	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
783	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
784	C₄⋊C₄	C₂⁴×C₆₇	1	1	yes	characteristic
785	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
786	C₂²⋊C₄	—	2	2	no	—
787	C₂²⋊C₄	—	2	4	no	—
788	C₂²⋊C₄	—	2	4	no	—
789	C₂²⋊C₄	—	2	2	no	—
790	C₂²⋊C₄	—	2	1	no	—
791	C₂²⋊C₄	—	2	1	no	—
792	C₂²⋊C₄	—	2	1	no	—
793	C₂²⋊C₄	—	2	2	no	—
794	C₂²⋊C₄	—	2	2	no	—
795	C₂²⋊C₄	—	2	2	no	—
796	C₂²⋊C₄	—	2	1	no	—
797	C₂²⋊C₄	—	2	2	no	—
798	C₂²⋊C₄	—	2	1	no	—
799	C₂²⋊C₄	—	2	2	no	—
800	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
801	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
802	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
803	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
804	C₄²	—	2	2	no	—
805	C₄²	—	2	2	no	—
806	C₄²	—	2	2	no	—
807	C₄²	—	2	1	no	—
808	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
809	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
810	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
811	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
812	C₂³	—	2	4	no	—
813	C₂³	—	2	4	no	—
814	C₂³	—	2	1	no	—
815	C₂³	—	2	2	no	—
816	C₂³	—	4	2	no	—
817	C₂³	—	4	4	no	—
818	C₂³	—	8	2	no	—
819	C₂³	—	8	2	no	—
820	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
821	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
822	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
823	Q₈	—	2	4	no	—
824	Q₈	—	2	2	no	—
825	Q₈	—	2	4	no	—
826	Q₈	—	2	4	no	—
827	Q₈	—	2	2	no	—
828	Q₈	—	4	8	no	—
829	Q₈	—	4	8	no	—
830	Q₈	—	4	4	no	—
831	Q₈	—	8	1	no	—
832	Q₈	H(5,𝔽₂)×C₆₇	1	4	yes	—
833	D₄	—	2	4	no	—
834	D₄	—	2	2	no	—
835	D₄	—	2	2	no	—
836	D₄	—	2	4	no	—
837	D₄	—	4	4	no	—
838	D₄	—	4	2	no	—
839	D₄	—	4	4	no	—
840	D₄	—	4	2	no	—
841	D₄	—	8	2	no	—
842	D₄	—	8	1	no	—
843	D₄	—	8	2	no	—
844	D₄	—	8	2	no	—
845	D₄	—	8	2	no	—
846	D₄	#137×C₆₇	1	4	yes	complemented
847	C₂×C₄	—	2	4	no	—
848	C₂×C₄	—	2	2	no	—
849	C₂×C₄	—	2	2	no	—
850	C₂×C₄	—	2	2	no	—
851	C₂×C₄	—	2	4	no	—
852	C₂×C₄	—	2	4	no	—
853	C₂×C₄	—	2	2	no	—
854	C₂×C₄	—	2	1	no	—
855	C₂×C₄	—	2	2	no	—
856	C₂×C₄	—	2	1	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	1	no	—
859	C₂×C₄	—	2	4	no	—
860	C₂×C₄	—	4	2	no	—
861	C₂×C₄	—	4	2	no	—
862	C₂×C₄	—	4	2	no	—
863	C₂×C₄	—	4	8	no	—
864	C₂×C₄	—	4	8	no	—
865	C₂×C₄	—	4	4	no	—
866	C₂×C₄	—	4	2	no	—
867	C₂×C₄	—	4	2	no	—
868	C₂×C₄	—	8	2	no	—
869	C₂×C₄	—	8	2	no	—
870	C₂×C₄	—	8	2	no	—
871	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
872	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
873	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
874	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
875	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
876	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
877	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
878	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
879	C₈	—	2	2	no	—
880	C₈	—	2	4	no	—
881	C₈	—	2	2	no	—
882	C₂²	—	2	1	no	—
883	C₂²	—	2	2	no	—
884	C₂²	—	2	1	no	—
885	C₂²	—	2	2	no	—
886	C₂²	—	2	2	no	—
887	C₂²	—	2	2	no	—
888	C₂²	—	2	2	no	—
889	C₂²	—	4	8	no	—
890	C₂²	—	4	1	no	—
891	C₂²	—	4	2	no	—
892	C₂²	—	4	2	no	—
893	C₂²	—	4	1	no	—
894	C₂²	—	4	2	no	—
895	C₂²	—	4	1	no	—
896	C₂²	—	16	4	no	—
897	C₂²	C₂×#137×C₆₇	1	4	yes	—
898	C₂²	#577×C₆₇	1	2	yes	—
899	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
900	C₄	—	2	1	no	—
901	C₄	—	2	1	no	—
902	C₄	—	2	1	no	—
903	C₄	—	2	4	no	—
904	C₄	—	2	1	no	—
905	C₄	—	2	2	no	—
906	C₄	—	2	2	no	—
907	C₄	—	4	1	no	—
908	C₄	—	4	2	no	—
909	C₄	—	4	1	no	—
910	C₄	—	4	2	no	—
911	C₄	—	4	4	no	—
912	C₄	—	4	1	no	—
913	C₄	C₂×#137×C₆₇	1	2	yes	—
914	C₄	C₂×H(5,𝔽₂)×C₆₇	1	2	yes	—
915	C₂	—	2	2	no	—
916	C₂	—	2	4	no	—
917	C₂	—	4	2	no	—
918	C₂	—	8	1	no	—
919	C₂	—	8	2	no	—
920	C₂	C₆₇×#3484	1	1	yes	characteristic
921	C₂	C₂²×#137×C₆₇	1	1	yes	characteristic
922	C₂	C₂×#577×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3519	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×#3329	C₂	semidirect
4. C₆₇×#3320	C₂	semidirect
5. C₆₇×#3299	C₂	semidirect
6. D₄×SD₁₆×C₆₇	C₂	semidirect
7. C₆₇×#3278	C₂	semidirect
8. C₆₇×#3266	C₂	semidirect
9. C₆₇×#3218	C₂	semidirect
10. C₆₇×#3206	C₂	semidirect
11. C₆₇×#3193	C₂	semidirect
12. C₆₇×#3190	C₂	semidirect
13. C₆₇×#3112	C₂	semidirect
14. C₆₇×#3108	C₂	semidirect
15. C₆₇×#3004	C₂	semidirect
16. C₂×#448×C₆₇	C₂	semidirect
17. C₆₇×#2974	C₂	semidirect
18. C₂³×Q₈×C₆₇	C₂²	semidirect
19. C₂×#137×C₆₇	C₂²	semidirect
20. C₂²×SD₁₆×C₆₇	C₂²	semidirect
30. D₄×Q₈×C₆₇	C₂²	semidirect
31. D₄×Q₈×C₆₇	C₂²	semidirect
40. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
44. C₂×C₂²⋊Q₈×C₆₇	C₂²	semidirect
47. C₂×C₄×Q₈×C₆₇	C₂²	semidirect
50. #497×C₆₇	C₂²	semidirect
51. #492×C₆₇	C₂²	semidirect
52. #491×C₆₇	C₂²	semidirect
53. #487×C₆₇	C₂²	semidirect
54. #485×C₆₇	C₂²	semidirect
55. #484×C₆₇	C₂²	semidirect
56. #483×C₆₇	C₂²	semidirect
58. #480×C₆₇	C₂²	semidirect
59. #479×C₆₇	C₂²	semidirect
60. #477×C₆₇	C₂²	semidirect
61. #468×C₆₇	C₂²	semidirect
62. #468×C₆₇	C₂²	semidirect
63. #467×C₆₇	C₂²	semidirect
64. #465×C₆₇	C₂²	semidirect
65. #464×C₆₇	C₂²	semidirect
66. #462×C₆₇	C₂²	semidirect
67. #461×C₆₇	C₂²	semidirect
68. #461×C₆₇	C₂²	semidirect
69. #460×C₆₇	C₂²	semidirect
70. #452×C₆₇	C₂²	semidirect
71. #451×C₆₇	C₂²	semidirect
72. #451×C₆₇	C₂²	semidirect
73. #451×C₆₇	C₂²	semidirect
74. #450×C₆₇	C₂²	semidirect
77. #448×C₆₇	C₂²	semidirect
78. #448×C₆₇	C₂²	semidirect
79. #448×C₆₇	C₂²	semidirect
80. #448×C₆₇	C₂²	semidirect
81. #448×C₆₇	C₂²	semidirect
82. #448×C₆₇	C₂²	semidirect
83. #440×C₆₇	C₂²	semidirect
84. C₄×Q₁₆×C₆₇	C₂²	semidirect
85. C₄×SD₁₆×C₆₇	C₂²	semidirect
86. #436×C₆₇	C₂²	semidirect
87. #435×C₆₇	C₂²	semidirect
89. #432×C₆₇	C₂²	semidirect
90. #431×C₆₇	C₂²	semidirect
91. #419×C₆₇	C₂²	semidirect
92. C₂×#103×C₆₇	C₂²	semidirect
93. #407×C₆₇	C₂²	semidirect
94. C₂×C₂²⋊C₈×C₆₇	C₂²	semidirect
384. D₄×C₆₇	#137	semidirect
453. #61593	C₆₇	direct (N × H)
463. C₂×D₄×Q₈	C₂×C₆₇	semidirect
464. #3329	C₂×C₆₇	semidirect
465. #3320	C₂×C₆₇	semidirect
466. #3299	C₂×C₆₇	semidirect
467. D₄×SD₁₆	C₂×C₆₇	semidirect
468. #3278	C₂×C₆₇	semidirect
469. #3266	C₂×C₆₇	semidirect
470. #3218	C₂×C₆₇	semidirect
471. #3206	C₂×C₆₇	semidirect
472. #3193	C₂×C₆₇	semidirect
473. #3190	C₂×C₆₇	semidirect
474. #3112	C₂×C₆₇	semidirect
475. #3108	C₂×C₆₇	semidirect
476. #3004	C₂×C₆₇	semidirect
477. C₂×#448	C₂×C₆₇	semidirect
478. #2974	C₂×C₆₇	semidirect
479. C₆₇	#61593	direct (N × H)
480. C₂³×Q₈	C₂²×C₆₇	semidirect
481. C₂×#137	C₂²×C₆₇	semidirect
482. C₂²×SD₁₆	C₂²×C₆₇	semidirect
492. D₄×Q₈	C₂²×C₆₇	semidirect
493. D₄×Q₈	C₂²×C₆₇	semidirect
502. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
506. C₂×C₂²⋊Q₈	C₂²×C₆₇	semidirect
509. C₂×C₄×Q₈	C₂²×C₆₇	semidirect
512. #497	C₂²×C₆₇	semidirect
513. #492	C₂²×C₆₇	semidirect
514. #491	C₂²×C₆₇	semidirect
515. #487	C₂²×C₆₇	semidirect
516. #485	C₂²×C₆₇	semidirect
517. #484	C₂²×C₆₇	semidirect
518. #483	C₂²×C₆₇	semidirect
520. #480	C₂²×C₆₇	semidirect
521. #479	C₂²×C₆₇	semidirect
522. #477	C₂²×C₆₇	semidirect
523. #468	C₂²×C₆₇	semidirect
524. #468	C₂²×C₆₇	semidirect
525. #467	C₂²×C₆₇	semidirect
526. #465	C₂²×C₆₇	semidirect
527. #464	C₂²×C₆₇	semidirect
528. #462	C₂²×C₆₇	semidirect
529. #461	C₂²×C₆₇	semidirect
530. #461	C₂²×C₆₇	semidirect
531. #460	C₂²×C₆₇	semidirect
532. #452	C₂²×C₆₇	semidirect
533. #451	C₂²×C₆₇	semidirect
534. #451	C₂²×C₆₇	semidirect
535. #451	C₂²×C₆₇	semidirect
536. #450	C₂²×C₆₇	semidirect
539. #448	C₂²×C₆₇	semidirect
540. #448	C₂²×C₆₇	semidirect
541. #448	C₂²×C₆₇	semidirect
542. #448	C₂²×C₆₇	semidirect
543. #448	C₂²×C₆₇	semidirect
544. #448	C₂²×C₆₇	semidirect
545. #440	C₂²×C₆₇	semidirect
546. C₄×Q₁₆	C₂²×C₆₇	semidirect
547. C₄×SD₁₆	C₂²×C₆₇	semidirect
548. #436	C₂²×C₆₇	semidirect
549. #435	C₂²×C₆₇	semidirect
551. #432	C₂²×C₆₇	semidirect
552. #431	C₂²×C₆₇	semidirect
553. #419	C₂²×C₆₇	semidirect
554. C₂×#103	C₂²×C₆₇	semidirect
555. #407	C₂²×C₆₇	semidirect
556. C₂×C₂²⋊C₈	C₂²×C₆₇	semidirect
846. 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×#3519	2
C₂×D₄×Q₈×C₆₇	2
C₆₇×#3329	2
C₆₇×#3320	2
C₆₇×#3299	2
D₄×SD₁₆×C₆₇	2
C₆₇×#3278	2
C₆₇×#3266	2
C₆₇×#3218	2
C₆₇×#3206	2
C₆₇×#3193	2
C₆₇×#3190	2
C₆₇×#3112	2
C₆₇×#3108	2
C₆₇×#3004	2
C₂×#448×C₆₇	2
C₆₇×#2974	2
#61593	67

Subgroups — C67×#61593 #30740844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61593 #30740844