Subgroups — C₆₇×#61269 #30740520

Order 17152.

1750 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3495	C₂	1	2	yes	maximal
2	C₆₇×#3356	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3352	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3349	C₂	1	4	yes	maximal, complemented
5	C₆₇×#3344	C₂	1	4	yes	maximal, complemented
6	C₆₇×#3163	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#3097	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3095	C₂	1	2	yes	maximal, complemented
9	C₆₇×#3082	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#470×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₇×#2950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2930	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2912	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×#2911	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	#556×C₆₇	C₂²	1	4	yes	—
20	#554×C₆₇	C₂²	1	4	yes	—
21	#552×C₆₇	—	2	4	no	—
22	#552×C₆₇	C₂²	1	4	yes	—
23	#552×C₆₇	C₂²	1	4	yes	—
24	#551×C₆₇	—	2	4	no	—
25	#544×C₆₇	C₂²	1	2	yes	—
26	#542×C₆₇	—	2	2	no	—
27	#541×C₆₇	—	2	2	no	—
28	#536×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	—	2	2	no	—
30	#529×C₆₇	—	2	2	no	—
31	#525×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
33	#520×C₆₇	C₂²	1	2	yes	—
34	#519×C₆₇	C₂²	1	2	yes	—
35	C₂×#117×C₆₇	C₂²	1	2	yes	—
36	#501×C₆₇	C₂²	1	4	yes	complemented
37	#499×C₆₇	C₂²	1	2	yes	complemented
38	#498×C₆₇	C₂²	1	2	yes	complemented
39	#485×C₆₇	C₂²	1	4	yes	—
40	#485×C₆₇	C₂²	1	4	yes	—
41	#485×C₆₇	C₂²	1	4	yes	complemented
42	#485×C₆₇	C₂²	1	4	yes	complemented
43	#484×C₆₇	C₂²	1	4	yes	—
44	#484×C₆₇	C₂²	1	4	yes	complemented
45	#483×C₆₇	C₂²	1	4	yes	—
46	#483×C₆₇	C₂²	1	4	yes	complemented
47	#479×C₆₇	C₂²	1	4	yes	complemented
48	#479×C₆₇	C₂²	1	4	yes	complemented
49	#477×C₆₇	C₂²	1	4	yes	complemented
50	#475×C₆₇	C₂²	1	4	yes	complemented
51	#470×C₆₇	C₂²	1	4	yes	—
52	#470×C₆₇	C₂²	1	4	yes	—
53	#470×C₆₇	C₂²	1	2	yes	complemented
54	#470×C₆₇	C₂²	1	2	yes	complemented
55	#467×C₆₇	C₂²	1	4	yes	complemented
56	#441×C₆₇	C₂²	1	2	yes	complemented
57	#441×C₆₇	C₂²	1	2	yes	complemented
58	C₄×Q₁₆×C₆₇	C₂²	1	4	yes	complemented
59	#436×C₆₇	C₂²	1	4	yes	—
60	#435×C₆₇	C₂²	1	4	yes	—
61	#428×C₆₇	C₂²	1	2	yes	—
62	#428×C₆₇	C₂²	1	2	yes	—
63	#428×C₆₇	C₂²	1	2	yes	—
64	#428×C₆₇	C₂²	1	2	yes	—
65	#428×C₆₇	C₂²	1	2	yes	—
66	#428×C₆₇	C₂²	1	2	yes	—
67	#428×C₆₇	C₂²	1	4	yes	complemented
68	#427×C₆₇	C₂²	1	1	yes	characteristic
69	#427×C₆₇	C₂²	1	2	yes	—
70	#427×C₆₇	C₂²	1	1	yes	characteristic
71	#427×C₆₇	C₂²	1	2	yes	complemented
72	C₂×#106×C₆₇	C₂²	1	1	yes	characteristic
73	C₂×#106×C₆₇	C₂²	1	2	yes	complemented
74	#424×C₆₇	C₂²	1	1	yes	characteristic
75	#419×C₆₇	C₂²	1	2	yes	—
76	#419×C₆₇	C₂²	1	2	yes	—
77	C₂×#103×C₆₇	C₂²	1	2	yes	—
78	#408×C₆₇	C₂²	1	1	yes	characteristic
79	#407×C₆₇	C₂²	1	1	yes	characteristic
80	#405×C₆₇	C₂²	1	1	yes	characteristic
81	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
82	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
83	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
84	C₂×Q₁₆×C₆₇	—	2	4	no	—
85	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
87	#131×C₆₇	—	2	4	no	—
88	C₂×M₁₆×C₆₇	—	2	4	no	—
89	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
90	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
91	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
92	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
93	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
94	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
95	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
97	C₄⋊Q₈×C₆₇	—	2	2	no	—
98	C₄⋊Q₈×C₆₇	—	2	2	no	—
99	C₄⋊Q₈×C₆₇	C₂³	1	4	yes	—
100	#126×C₆₇	—	2	4	no	—
101	#126×C₆₇	—	2	4	no	—
102	#126×C₆₇	—	2	4	no	—
103	#126×C₆₇	—	2	4	no	—
104	#125×C₆₇	—	2	2	no	—
105	#125×C₆₇	—	2	4	no	—
106	#125×C₆₇	—	2	4	no	—
107	#125×C₆₇	—	2	4	no	—
108	#125×C₆₇	C₂³	1	4	yes	—
109	#124×C₆₇	—	2	2	no	—
110	#124×C₆₇	—	2	4	no	—
111	#123×C₆₇	—	2	4	no	—
112	#123×C₆₇	—	2	4	no	—
113	#123×C₆₇	—	2	4	no	—
114	#123×C₆₇	D₄	1	2	yes	—
115	C₂²⋊Q₈×C₆₇	—	2	2	no	—
116	C₂²⋊Q₈×C₆₇	—	2	2	no	—
117	C₂²⋊Q₈×C₆₇	—	2	4	no	—
118	C₂²⋊Q₈×C₆₇	—	2	4	no	—
119	C₂²⋊Q₈×C₆₇	—	2	4	no	—
120	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
121	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
122	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
123	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
124	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
125	#121×C₆₇	D₄	1	2	yes	—
126	#120×C₆₇	D₄	1	2	yes	—
127	C₄×Q₈×C₆₇	—	2	2	no	—
128	C₄×Q₈×C₆₇	—	2	2	no	—
129	C₄×Q₈×C₆₇	—	2	2	no	—
130	C₄×Q₈×C₆₇	—	2	4	no	—
131	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
132	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
133	C₄×D₄×C₆₇	—	2	2	no	—
134	C₄×D₄×C₆₇	—	2	4	no	—
135	C₄×D₄×C₆₇	—	2	4	no	—
136	C₄×D₄×C₆₇	C₂³	1	4	yes	—
137	#117×C₆₇	—	2	2	no	—
138	#117×C₆₇	—	2	2	no	—
139	#117×C₆₇	—	2	4	no	—
140	#117×C₆₇	—	2	4	no	—
141	#117×C₆₇	—	2	4	no	—
142	#117×C₆₇	C₂³	1	2	yes	—
143	#117×C₆₇	C₂³	1	2	yes	—
144	#117×C₆₇	C₂³	1	4	yes	—
145	#117×C₆₇	C₂³	1	2	yes	—
146	#117×C₆₇	C₂³	1	2	yes	—
147	#117×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
153	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
154	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
155	C₂×C₄²×C₆₇	—	2	2	no	—
156	#107×C₆₇	C₂³	1	2	yes	—
157	#107×C₆₇	C₂³	1	2	yes	—
158	#107×C₆₇	C₂³	1	4	yes	—
159	#107×C₆₇	C₂³	1	2	yes	—
160	#107×C₆₇	C₂³	1	4	yes	—
161	#107×C₆₇	C₂³	1	2	yes	—
162	#106×C₆₇	C₂³	1	2	yes	—
163	#106×C₆₇	C₂³	1	2	yes	—
164	#106×C₆₇	C₂³	1	1	yes	characteristic
165	#106×C₆₇	C₂³	1	1	yes	characteristic
166	#106×C₆₇	C₂³	1	2	yes	—
167	#106×C₆₇	C₂³	1	4	yes	—
168	#106×C₆₇	C₂³	1	2	yes	—
169	#106×C₆₇	C₂³	1	4	yes	—
170	#106×C₆₇	C₂³	1	1	yes	characteristic
171	#106×C₆₇	C₂³	1	1	yes	characteristic
172	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
173	#103×C₆₇	C₂³	1	4	yes	—
174	#103×C₆₇	C₂³	1	4	yes	—
175	#103×C₆₇	C₂³	1	4	yes	—
176	#103×C₆₇	C₂³	1	4	yes	—
177	#103×C₆₇	C₂³	1	4	yes	—
178	#103×C₆₇	C₂³	1	4	yes	—
179	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
180	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
181	#97×C₆₇	C₂³	1	2	yes	—
182	C₄×C₈×C₆₇	C₂³	1	2	yes	—
183	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
184	#41×C₆₇	—	2	2	no	—
185	#41×C₆₇	—	2	2	no	—
186	C₂×Q₈×C₆₇	—	2	2	no	—
187	C₂×Q₈×C₆₇	—	2	4	no	—
188	C₂×Q₈×C₆₇	—	2	4	no	—
189	C₂×Q₈×C₆₇	—	2	4	no	—
190	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
192	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
193	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
194	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₆₇	—	2	2	no	—
196	C₂²×C₄×C₆₇	—	2	2	no	—
197	C₂²×C₄×C₆₇	—	2	2	no	—
198	C₂²×C₄×C₆₇	—	2	2	no	—
199	C₂²×C₄×C₆₇	—	2	2	no	—
200	C₂²×C₄×C₆₇	—	2	2	no	—
201	C₂²×C₄×C₆₇	—	2	4	no	—
202	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
203	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
204	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
205	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
206	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
207	Q₁₆×C₆₇	—	2	4	no	—
208	Q₁₆×C₆₇	—	2	4	no	—
209	M₁₆×C₆₇	—	2	2	no	—
210	M₁₆×C₆₇	—	2	2	no	—
211	M₁₆×C₆₇	—	2	4	no	—
212	M₁₆×C₆₇	—	2	4	no	—
213	M₁₆×C₆₇	—	2	4	no	—
214	C₂×C₈×C₆₇	—	2	2	no	—
215	C₂×C₈×C₆₇	—	2	4	no	—
216	C₂×C₈×C₆₇	#41	1	4	yes	—
217	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
218	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
219	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
220	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
221	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
222	C₄⋊C₄×C₆₇	—	2	4	no	—
223	C₄⋊C₄×C₆₇	—	2	4	no	—
224	C₄⋊C₄×C₆₇	—	2	2	no	—
225	C₄⋊C₄×C₆₇	—	2	2	no	—
226	C₄⋊C₄×C₆₇	—	2	2	no	—
227	C₄⋊C₄×C₆₇	—	2	4	no	—
228	C₄⋊C₄×C₆₇	—	2	4	no	—
229	C₄⋊C₄×C₆₇	—	2	4	no	—
230	C₄⋊C₄×C₆₇	—	2	4	no	—
231	C₄⋊C₄×C₆₇	—	2	4	no	—
232	C₄⋊C₄×C₆₇	—	2	4	no	—
233	C₄⋊C₄×C₆₇	—	2	4	no	—
234	C₄⋊C₄×C₆₇	—	2	4	no	—
235	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
236	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
237	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
238	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
239	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
240	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
241	C₄⋊C₄×C₆₇	C₂⁴	1	2	yes	—
242	C₂²⋊C₄×C₆₇	—	2	2	no	—
243	C₂²⋊C₄×C₆₇	—	2	2	no	—
244	C₂²⋊C₄×C₆₇	—	2	2	no	—
245	C₂²⋊C₄×C₆₇	—	2	2	no	—
246	C₂²⋊C₄×C₆₇	—	2	4	no	—
247	C₂²⋊C₄×C₆₇	—	2	4	no	—
248	C₂²⋊C₄×C₆₇	—	2	4	no	—
249	C₂²⋊C₄×C₆₇	—	2	4	no	—
250	C₂²⋊C₄×C₆₇	—	2	4	no	—
251	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
252	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
253	C₂²⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
254	C₄²×C₆₇	—	2	2	no	—
255	C₄²×C₆₇	—	2	2	no	—
256	C₄²×C₆₇	—	2	2	no	—
257	C₄²×C₆₇	—	2	2	no	—
258	C₄²×C₆₇	—	2	4	no	—
259	C₄²×C₆₇	—	2	4	no	—
260	C₄²×C₆₇	—	2	4	no	—
261	C₄²×C₆₇	C₂⁴	1	2	yes	—
262	C₂³×C₆₇	—	2	2	no	—
263	C₂³×C₆₇	—	4	2	no	—
264	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
265	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
266	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
267	Q₈×C₆₇	—	2	2	no	—
268	Q₈×C₆₇	—	2	4	no	—
269	Q₈×C₆₇	—	2	4	no	—
270	Q₈×C₆₇	—	4	4	no	—
271	Q₈×C₆₇	—	8	4	no	—
272	D₄×C₆₇	—	2	4	no	—
273	D₄×C₆₇	—	2	2	no	—
274	D₄×C₆₇	—	4	4	no	—
275	C₂×C₄×C₆₇	—	2	2	no	—
276	C₂×C₄×C₆₇	—	2	4	no	—
277	C₂×C₄×C₆₇	—	2	2	no	—
278	C₂×C₄×C₆₇	—	2	2	no	—
279	C₂×C₄×C₆₇	—	2	2	no	—
280	C₂×C₄×C₆₇	—	2	2	no	—
281	C₂×C₄×C₆₇	—	2	2	no	—
282	C₂×C₄×C₆₇	—	2	2	no	—
283	C₂×C₄×C₆₇	—	2	2	no	—
284	C₂×C₄×C₆₇	—	2	4	no	—
285	C₂×C₄×C₆₇	—	2	2	no	—
286	C₂×C₄×C₆₇	—	2	4	no	—
287	C₂×C₄×C₆₇	—	4	4	no	—
288	C₂×C₄×C₆₇	—	4	4	no	—
289	C₂×C₄×C₆₇	—	8	4	no	—
290	C₂×C₄×C₆₇	—	8	2	no	—
291	C₂×C₄×C₆₇	—	8	2	no	—
292	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
293	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
297	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
298	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
299	C₈×C₆₇	—	2	2	no	—
300	C₈×C₆₇	—	2	2	no	—
301	C₈×C₆₇	—	2	2	no	—
302	C₈×C₆₇	C₂×#41	1	2	yes	—
303	C₈×C₆₇	C₂×#41	1	2	yes	—
304	C₂²×C₆₇	—	2	2	no	—
305	C₂²×C₆₇	—	2	2	no	—
306	C₂²×C₆₇	—	2	4	no	—
307	C₂²×C₆₇	—	2	2	no	—
308	C₂²×C₆₇	—	2	2	no	—
309	C₂²×C₆₇	—	2	2	no	—
310	C₂²×C₆₇	—	8	2	no	—
311	C₂²×C₆₇	#547	1	2	yes	—
312	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₆₇	—	2	4	no	—
314	C₄×C₆₇	—	2	2	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	4	2	no	—
317	C₄×C₆₇	—	4	2	no	—
318	C₄×C₆₇	—	4	2	no	—
319	C₄×C₆₇	—	4	4	no	—
320	C₄×C₆₇	—	4	2	no	—
321	C₄×C₆₇	—	4	4	no	—
322	C₄×C₆₇	#547	1	2	yes	—
323	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
324	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
325	#61269	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
326	C₂×C₆₇	—	2	2	no	—
327	C₂×C₆₇	—	4	2	no	—
328	C₂×C₆₇	—	4	2	no	—
329	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
330	C₂×C₆₇	#3586	1	2	yes	—
331	#3495	C₂×C₆₇	1	2	yes	—
332	#3356	C₂×C₆₇	1	4	yes	complemented
333	#3352	C₂×C₆₇	1	4	yes	complemented
334	#3349	C₂×C₆₇	1	4	yes	complemented
335	#3344	C₂×C₆₇	1	4	yes	complemented
336	#3163	C₂×C₆₇	1	1	yes	characteristic, complemented
337	#3097	C₂×C₆₇	1	2	yes	complemented
338	#3095	C₂×C₆₇	1	2	yes	complemented
339	#3082	C₂×C₆₇	1	2	yes	complemented
340	#3059	C₂×C₆₇	1	1	yes	characteristic, complemented
341	C₂×#470	C₂×C₆₇	1	1	yes	characteristic, complemented
342	#2950	C₂×C₆₇	1	1	yes	characteristic, complemented
343	#2930	C₂×C₆₇	1	1	yes	characteristic
344	#2912	C₂×C₆₇	1	1	yes	characteristic, complemented
345	#2911	C₂×C₆₇	1	1	yes	characteristic, complemented
346	C₆₇	#61269	1	1	yes	characteristic, direct product factor, Sylow
347	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#131	C₂²×C₆₇	1	1	yes	characteristic
349	C₂²×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
350	#556	C₂²×C₆₇	1	4	yes	—
351	#554	C₂²×C₆₇	1	4	yes	—
352	#552	—	2	4	no	—
353	#552	C₂²×C₆₇	1	4	yes	—
354	#552	C₂²×C₆₇	1	4	yes	—
355	#551	—	2	4	no	—
356	#544	C₂²×C₆₇	1	2	yes	—
357	#542	—	2	2	no	—
358	#541	—	2	2	no	—
359	#536	C₂²×C₆₇	1	2	yes	—
360	#533	—	2	2	no	—
361	#529	—	2	2	no	—
362	#525	C₂²×C₆₇	1	1	yes	characteristic
363	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
364	#520	C₂²×C₆₇	1	2	yes	—
365	#519	C₂²×C₆₇	1	2	yes	—
366	C₂×#117	C₂²×C₆₇	1	2	yes	—
367	#501	C₂²×C₆₇	1	4	yes	complemented
368	#499	C₂²×C₆₇	1	2	yes	complemented
369	#498	C₂²×C₆₇	1	2	yes	complemented
370	#485	C₂²×C₆₇	1	4	yes	—
371	#485	C₂²×C₆₇	1	4	yes	—
372	#485	C₂²×C₆₇	1	4	yes	complemented
373	#485	C₂²×C₆₇	1	4	yes	complemented
374	#484	C₂²×C₆₇	1	4	yes	—
375	#484	C₂²×C₆₇	1	4	yes	complemented
376	#483	C₂²×C₆₇	1	4	yes	—
377	#483	C₂²×C₆₇	1	4	yes	complemented
378	#479	C₂²×C₆₇	1	4	yes	complemented
379	#479	C₂²×C₆₇	1	4	yes	complemented
380	#477	C₂²×C₆₇	1	4	yes	complemented
381	#475	C₂²×C₆₇	1	4	yes	complemented
382	#470	C₂²×C₆₇	1	4	yes	—
383	#470	C₂²×C₆₇	1	4	yes	—
384	#470	C₂²×C₆₇	1	2	yes	complemented
385	#470	C₂²×C₆₇	1	2	yes	complemented
386	#467	C₂²×C₆₇	1	4	yes	complemented
387	#441	C₂²×C₆₇	1	2	yes	complemented
388	#441	C₂²×C₆₇	1	2	yes	complemented
389	C₄×Q₁₆	C₂²×C₆₇	1	4	yes	complemented
390	#436	C₂²×C₆₇	1	4	yes	—
391	#435	C₂²×C₆₇	1	4	yes	—
392	#428	C₂²×C₆₇	1	2	yes	—
393	#428	C₂²×C₆₇	1	2	yes	—
394	#428	C₂²×C₆₇	1	2	yes	—
395	#428	C₂²×C₆₇	1	2	yes	—
396	#428	C₂²×C₆₇	1	2	yes	—
397	#428	C₂²×C₆₇	1	2	yes	—
398	#428	C₂²×C₆₇	1	4	yes	complemented
399	#427	C₂²×C₆₇	1	2	yes	—
400	#427	C₂²×C₆₇	1	1	yes	characteristic
401	#427	C₂²×C₆₇	1	1	yes	characteristic
402	#427	C₂²×C₆₇	1	2	yes	complemented
403	C₂×#106	C₂²×C₆₇	1	1	yes	characteristic
404	C₂×#106	C₂²×C₆₇	1	2	yes	complemented
405	#424	C₂²×C₆₇	1	1	yes	characteristic
406	#419	C₂²×C₆₇	1	2	yes	—
407	#419	C₂²×C₆₇	1	2	yes	—
408	C₂×#103	C₂²×C₆₇	1	2	yes	—
409	#408	C₂²×C₆₇	1	1	yes	characteristic
410	#407	C₂²×C₆₇	1	1	yes	characteristic
411	#405	C₂²×C₆₇	1	1	yes	characteristic
412	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
413	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
414	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
415	C₂×Q₁₆	—	2	4	no	—
416	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
417	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
418	#131	—	2	4	no	—
419	C₂×M₁₆	—	2	4	no	—
420	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
421	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
422	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
423	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
424	C₂×M₁₆	C₂³×C₆₇	1	1	yes	characteristic
425	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
426	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
427	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	C₂³×C₆₇	1	4	yes	—
431	#126	—	2	4	no	—
432	#126	—	2	4	no	—
433	#126	—	2	4	no	—
434	#126	—	2	4	no	—
435	#125	—	2	2	no	—
436	#125	—	2	4	no	—
437	#125	—	2	4	no	—
438	#125	—	2	4	no	—
439	#125	C₂³×C₆₇	1	4	yes	—
440	#124	—	2	2	no	—
441	#124	—	2	4	no	—
442	#123	—	2	4	no	—
443	#123	—	2	4	no	—
444	#123	—	2	4	no	—
445	#123	D₄×C₆₇	1	2	yes	—
446	C₂²⋊Q₈	—	2	2	no	—
447	C₂²⋊Q₈	—	2	2	no	—
448	C₂²⋊Q₈	—	2	4	no	—
449	C₂²⋊Q₈	—	2	4	no	—
450	C₂²⋊Q₈	—	2	4	no	—
451	C₂²⋊Q₈	D₄×C₆₇	1	2	yes	—
452	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
453	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
454	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
455	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
456	#121	D₄×C₆₇	1	2	yes	—
457	#120	D₄×C₆₇	1	2	yes	—
458	C₄×Q₈	—	2	2	no	—
459	C₄×Q₈	—	2	2	no	—
460	C₄×Q₈	—	2	2	no	—
461	C₄×Q₈	—	2	4	no	—
462	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
463	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
464	C₄×D₄	—	2	2	no	—
465	C₄×D₄	—	2	4	no	—
466	C₄×D₄	—	2	4	no	—
467	C₄×D₄	C₂³×C₆₇	1	4	yes	—
468	#117	—	2	2	no	—
469	#117	—	2	2	no	—
470	#117	—	2	4	no	—
471	#117	—	2	4	no	—
472	#117	—	2	4	no	—
473	#117	C₂³×C₆₇	1	2	yes	—
474	#117	C₂³×C₆₇	1	2	yes	—
475	#117	C₂³×C₆₇	1	2	yes	—
476	#117	C₂³×C₆₇	1	2	yes	—
477	#117	C₂³×C₆₇	1	4	yes	—
478	#117	C₂³×C₆₇	1	1	yes	characteristic
479	C₂×C₄⋊C₄	—	2	2	no	—
480	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
481	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
482	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
483	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
484	C₂×C₂²⋊C₄	—	2	2	no	—
485	C₂×C₂²⋊C₄	—	2	2	no	—
486	C₂×C₄²	—	2	2	no	—
487	#107	C₂³×C₆₇	1	2	yes	—
488	#107	C₂³×C₆₇	1	2	yes	—
489	#107	C₂³×C₆₇	1	2	yes	—
490	#107	C₂³×C₆₇	1	2	yes	—
491	#107	C₂³×C₆₇	1	4	yes	—
492	#107	C₂³×C₆₇	1	4	yes	—
493	#106	C₂³×C₆₇	1	2	yes	—
494	#106	C₂³×C₆₇	1	2	yes	—
495	#106	C₂³×C₆₇	1	1	yes	characteristic
496	#106	C₂³×C₆₇	1	4	yes	—
497	#106	C₂³×C₆₇	1	1	yes	characteristic
498	#106	C₂³×C₆₇	1	1	yes	characteristic
499	#106	C₂³×C₆₇	1	1	yes	characteristic
500	#106	C₂³×C₆₇	1	2	yes	—
501	#106	C₂³×C₆₇	1	2	yes	—
502	#106	C₂³×C₆₇	1	4	yes	—
503	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
504	#103	C₂³×C₆₇	1	4	yes	—
505	#103	C₂³×C₆₇	1	4	yes	—
506	#103	C₂³×C₆₇	1	4	yes	—
507	#103	C₂³×C₆₇	1	4	yes	—
508	#103	C₂³×C₆₇	1	4	yes	—
509	#103	C₂³×C₆₇	1	4	yes	—
510	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
511	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
512	#97	C₂³×C₆₇	1	2	yes	—
513	C₄×C₈	C₂³×C₆₇	1	2	yes	—
514	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
515	#41	—	2	2	no	—
516	#41	—	2	2	no	—
517	C₂×Q₈	—	2	2	no	—
518	C₂×Q₈	—	2	4	no	—
519	C₂×Q₈	—	2	4	no	—
520	C₂×Q₈	—	2	4	no	—
521	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
522	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
523	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
524	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
525	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	2	no	—
528	C₂²×C₄	—	2	2	no	—
529	C₂²×C₄	—	2	2	no	—
530	C₂²×C₄	—	2	2	no	—
531	C₂²×C₄	—	2	2	no	—
532	C₂²×C₄	—	2	4	no	—
533	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
538	Q₁₆	—	2	4	no	—
539	Q₁₆	—	2	4	no	—
540	M₁₆	—	2	2	no	—
541	M₁₆	—	2	2	no	—
542	M₁₆	—	2	4	no	—
543	M₁₆	—	2	4	no	—
544	M₁₆	—	2	4	no	—
545	C₂×C₈	—	2	2	no	—
546	C₂×C₈	—	2	4	no	—
547	C₂×C₈	#41×C₆₇	1	4	yes	—
548	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
549	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
550	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
551	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
552	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
553	C₄⋊C₄	—	2	4	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	2	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	—	2	2	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	4	no	—
563	C₄⋊C₄	—	2	4	no	—
564	C₄⋊C₄	—	2	4	no	—
565	C₄⋊C₄	—	2	4	no	—
566	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
567	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
568	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
569	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
570	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
571	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
572	C₄⋊C₄	C₂⁴×C₆₇	1	2	yes	—
573	C₂²⋊C₄	—	2	2	no	—
574	C₂²⋊C₄	—	2	2	no	—
575	C₂²⋊C₄	—	2	2	no	—
576	C₂²⋊C₄	—	2	2	no	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	4	no	—
580	C₂²⋊C₄	—	2	4	no	—
581	C₂²⋊C₄	—	2	4	no	—
582	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
583	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
584	C₂²⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
585	C₄²	—	2	2	no	—
586	C₄²	—	2	2	no	—
587	C₄²	—	2	2	no	—
588	C₄²	—	2	2	no	—
589	C₄²	—	2	4	no	—
590	C₄²	—	2	4	no	—
591	C₄²	—	2	4	no	—
592	C₄²	C₂⁴×C₆₇	1	2	yes	—
593	C₂³	—	2	2	no	—
594	C₂³	—	4	2	no	—
595	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
596	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
597	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
598	Q₈	—	2	2	no	—
599	Q₈	—	2	4	no	—
600	Q₈	—	2	4	no	—
601	Q₈	—	4	4	no	—
602	Q₈	—	8	4	no	—
603	D₄	—	2	4	no	—
604	D₄	—	2	2	no	—
605	D₄	—	4	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	4	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	4	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	4	4	no	—
620	C₂×C₄	—	8	4	no	—
621	C₂×C₄	—	8	2	no	—
622	C₂×C₄	—	8	2	no	—
623	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
624	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
625	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
628	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
629	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
630	C₈	—	2	2	no	—
631	C₈	—	2	2	no	—
632	C₈	—	2	2	no	—
633	C₈	C₂×#41×C₆₇	1	2	yes	—
634	C₈	C₂×#41×C₆₇	1	2	yes	—
635	C₂²	—	2	2	no	—
636	C₂²	—	2	2	no	—
637	C₂²	—	2	4	no	—
638	C₂²	—	2	2	no	—
639	C₂²	—	2	2	no	—
640	C₂²	—	2	2	no	—
641	C₂²	—	8	2	no	—
642	C₂²	#547×C₆₇	1	2	yes	—
643	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
644	C₄	—	2	4	no	—
645	C₄	—	2	2	no	—
646	C₄	—	2	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	—	4	2	no	—
650	C₄	—	4	4	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	4	no	—
653	C₄	#547×C₆₇	1	2	yes	—
654	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
655	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
656	C₂	—	2	2	no	—
657	C₂	—	4	2	no	—
658	C₂	—	4	2	no	—
659	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
660	C₂	C₆₇×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	#525×C₆₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ #525×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3356	C₂	semidirect
3. C₆₇×#3352	C₂	semidirect
4. C₆₇×#3349	C₂	semidirect
5. C₆₇×#3344	C₂	semidirect
6. C₆₇×#3163	C₂	semidirect
7. C₆₇×#3097	C₂	semidirect
8. C₆₇×#3095	C₂	semidirect
9. C₆₇×#3082	C₂	semidirect
10. C₆₇×#3059	C₂	semidirect
11. C₂×#470×C₆₇	C₂	semidirect
12. C₆₇×#2950	C₂	semidirect
14. C₆₇×#2912	C₂	semidirect
15. C₆₇×#2911	C₂	semidirect
36. #501×C₆₇	C₂²	semidirect
37. #499×C₆₇	C₂²	semidirect
38. #498×C₆₇	C₂²	semidirect
41. #485×C₆₇	C₂²	semidirect
42. #485×C₆₇	C₂²	semidirect
44. #484×C₆₇	C₂²	semidirect
46. #483×C₆₇	C₂²	semidirect
47. #479×C₆₇	C₂²	semidirect
48. #479×C₆₇	C₂²	semidirect
49. #477×C₆₇	C₂²	semidirect
50. #475×C₆₇	C₂²	semidirect
53. #470×C₆₇	C₂²	semidirect
54. #470×C₆₇	C₂²	semidirect
55. #467×C₆₇	C₂²	semidirect
56. #441×C₆₇	C₂²	semidirect
57. #441×C₆₇	C₂²	semidirect
58. C₄×Q₁₆×C₆₇	C₂²	semidirect
67. #428×C₆₇	C₂²	semidirect
71. #427×C₆₇	C₂²	semidirect
73. C₂×#106×C₆₇	C₂²	semidirect
325. #61269	C₆₇	direct (N × H)
332. #3356	C₂×C₆₇	semidirect
333. #3352	C₂×C₆₇	semidirect
334. #3349	C₂×C₆₇	semidirect
335. #3344	C₂×C₆₇	semidirect
336. #3163	C₂×C₆₇	semidirect
337. #3097	C₂×C₆₇	semidirect
338. #3095	C₂×C₆₇	semidirect
339. #3082	C₂×C₆₇	semidirect
340. #3059	C₂×C₆₇	semidirect
341. C₂×#470	C₂×C₆₇	semidirect
342. #2950	C₂×C₆₇	semidirect
344. #2912	C₂×C₆₇	semidirect
345. #2911	C₂×C₆₇	semidirect
346. C₆₇	#61269	direct (N × H)
367. #501	C₂²×C₆₇	semidirect
368. #499	C₂²×C₆₇	semidirect
369. #498	C₂²×C₆₇	semidirect
372. #485	C₂²×C₆₇	semidirect
373. #485	C₂²×C₆₇	semidirect
375. #484	C₂²×C₆₇	semidirect
377. #483	C₂²×C₆₇	semidirect
378. #479	C₂²×C₆₇	semidirect
379. #479	C₂²×C₆₇	semidirect
380. #477	C₂²×C₆₇	semidirect
381. #475	C₂²×C₆₇	semidirect
384. #470	C₂²×C₆₇	semidirect
385. #470	C₂²×C₆₇	semidirect
386. #467	C₂²×C₆₇	semidirect
387. #441	C₂²×C₆₇	semidirect
388. #441	C₂²×C₆₇	semidirect
389. C₄×Q₁₆	C₂²×C₆₇	semidirect
398. #428	C₂²×C₆₇	semidirect
402. #427	C₂²×C₆₇	semidirect
404. C₂×#106	C₂²×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3495	2
C₆₇×#3356	2
C₆₇×#3352	2
C₆₇×#3349	2
C₆₇×#3344	2
C₆₇×#3163	2
C₆₇×#3097	2
C₆₇×#3095	2
C₆₇×#3082	2
C₆₇×#3059	2
C₂×#470×C₆₇	2
C₆₇×#2950	2
C₆₇×#2930	2
C₆₇×#2912	2
C₆₇×#2911	2
#61269	67

Subgroups — C67×#61269 #30740520

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61269 #30740520