Subgroups — C₆₇×#61012 #30740263

Order 17152.

1776 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3509	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#3508	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#3260	C₂	1	4	yes	maximal, complemented
4	C₆₇×#3231	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3208	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3206	C₂	1	2	yes	maximal, complemented
8	C₆₇×#3200	C₂	1	4	yes	maximal, complemented
9	C₆₇×#3182	C₂	1	2	yes	maximal, complemented
10	C₆₇×#3167	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3126	C₂	1	4	yes	maximal, complemented
12	C₆₇×#3123	C₂	1	4	yes	maximal, complemented
13	C₆₇×#2922	C₂	1	1	yes	maximal, characteristic, complemented
14	#559×C₆₇	—	2	2	no	—
15	#557×C₆₇	—	2	2	no	—
16	#556×C₆₇	—	2	2	no	—
17	#552×C₆₇	—	2	2	no	—
18	#552×C₆₇	—	2	2	no	—
19	#550×C₆₇	—	2	2	no	—
20	#546×C₆₇	—	2	2	no	—
21	#546×C₆₇	—	2	2	no	—
22	#541×C₆₇	C₂²	1	2	yes	—
23	#541×C₆₇	C₂²	1	2	yes	complemented
24	#540×C₆₇	C₂²	1	2	yes	—
25	#540×C₆₇	C₂²	1	2	yes	—
26	#536×C₆₇	C₂²	1	2	yes	—
27	#535×C₆₇	C₂²	1	2	yes	—
28	#533×C₆₇	C₂²	1	2	yes	—
29	#533×C₆₇	C₂²	1	1	yes	characteristic
30	#533×C₆₇	C₂²	1	2	yes	complemented
31	#533×C₆₇	C₂²	1	1	yes	characteristic, complemented
32	#532×C₆₇	C₂²	1	2	yes	—
33	#532×C₆₇	C₂²	1	2	yes	—
34	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
36	#519×C₆₇	C₂²	1	2	yes	complemented
37	#518×C₆₇	C₂²	1	2	yes	—
38	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
39	#500×C₆₇	C₂²	1	2	yes	complemented
40	#498×C₆₇	C₂²	1	2	yes	complemented
41	#492×C₆₇	C₂²	1	2	yes	—
42	#492×C₆₇	C₂²	1	2	yes	—
43	#487×C₆₇	C₂²	1	2	yes	complemented
44	#487×C₆₇	C₂²	1	2	yes	complemented
45	#486×C₆₇	C₂²	1	2	yes	—
46	#486×C₆₇	C₂²	1	2	yes	—
47	#485×C₆₇	C₂²	1	8	yes	—
48	#482×C₆₇	C₂²	1	8	yes	complemented
49	#479×C₆₇	C₂²	1	8	yes	complemented
50	#478×C₆₇	C₂²	1	8	yes	complemented
51	#468×C₆₇	C₂²	1	8	yes	complemented
52	#465×C₆₇	C₂²	1	8	yes	complemented
53	#461×C₆₇	C₂²	1	8	yes	complemented
54	#460×C₆₇	C₂²	1	8	yes	—
55	#450×C₆₇	C₂²	1	8	yes	—
56	#448×C₆₇	C₂²	1	8	yes	complemented
57	#440×C₆₇	C₂²	1	8	yes	complemented
58	C₄×SD₁₆×C₆₇	C₂²	1	8	yes	complemented
59	#433×C₆₇	C₂²	1	2	yes	—
60	#433×C₆₇	C₂²	1	2	yes	complemented
61	#431×C₆₇	C₂²	1	2	yes	—
62	#431×C₆₇	C₂²	1	2	yes	complemented
63	#424×C₆₇	C₂²	1	2	yes	complemented
64	#408×C₆₇	C₂²	1	2	yes	—
65	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
67	C₂×#41×C₆₇	—	2	2	no	—
68	C₂×#41×C₆₇	—	2	2	no	—
69	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
71	C₂²×D₄×C₆₇	—	2	2	no	—
72	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
75	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
76	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
77	C₄⋊Q₈×C₆₇	—	2	2	no	—
78	C₄⋊Q₈×C₆₇	—	2	2	no	—
79	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
82	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
83	#127×C₆₇	—	2	2	no	—
84	#127×C₆₇	—	2	2	no	—
85	#127×C₆₇	C₂³	1	2	yes	—
86	#127×C₆₇	C₂³	1	2	yes	—
87	#126×C₆₇	—	2	4	no	—
88	#126×C₆₇	—	2	4	no	—
89	#125×C₆₇	—	2	4	no	—
90	#125×C₆₇	—	2	4	no	—
91	#125×C₆₇	—	2	2	no	—
92	#125×C₆₇	—	2	2	no	—
93	#125×C₆₇	C₂³	1	2	yes	—
94	#125×C₆₇	C₂³	1	2	yes	—
95	#124×C₆₇	—	2	2	no	—
96	#124×C₆₇	—	2	2	no	—
97	#124×C₆₇	—	2	2	no	—
98	#124×C₆₇	—	2	2	no	—
99	#123×C₆₇	—	2	2	no	—
100	#123×C₆₇	—	2	2	no	—
101	#123×C₆₇	—	2	4	no	—
102	#123×C₆₇	—	2	4	no	—
103	C₂²⋊Q₈×C₆₇	—	2	2	no	—
104	C₂²⋊Q₈×C₆₇	—	2	2	no	—
105	C₂²⋊Q₈×C₆₇	—	2	2	no	—
106	C₂²⋊Q₈×C₆₇	—	2	2	no	—
107	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
108	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
109	#121×C₆₇	—	2	2	no	—
110	#121×C₆₇	—	2	2	no	—
111	#121×C₆₇	—	2	4	no	—
112	#121×C₆₇	—	2	4	no	—
113	#121×C₆₇	C₂³	1	4	yes	—
114	#121×C₆₇	C₂³	1	4	yes	—
115	#120×C₆₇	—	2	2	no	—
116	#120×C₆₇	—	2	2	no	—
117	C₄×Q₈×C₆₇	—	2	4	no	—
118	C₄×Q₈×C₆₇	—	2	2	no	—
119	C₄×Q₈×C₆₇	—	2	2	no	—
120	C₄×Q₈×C₆₇	—	2	2	no	—
121	C₄×Q₈×C₆₇	—	2	2	no	—
122	C₄×Q₈×C₆₇	D₄	1	2	yes	—
123	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
124	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
125	C₄×D₄×C₆₇	—	2	4	no	—
126	C₄×D₄×C₆₇	—	2	2	no	—
127	C₄×D₄×C₆₇	—	2	2	no	—
128	C₄×D₄×C₆₇	D₄	1	2	yes	—
129	C₄×D₄×C₆₇	D₄	1	2	yes	—
130	C₄×D₄×C₆₇	D₄	1	2	yes	—
131	C₄×D₄×C₆₇	C₂³	1	4	yes	—
132	C₄×D₄×C₆₇	C₂³	1	4	yes	—
133	#117×C₆₇	—	2	2	no	—
134	#117×C₆₇	—	2	2	no	—
135	#117×C₆₇	—	2	4	no	—
136	#117×C₆₇	C₂³	1	2	yes	—
137	#117×C₆₇	C₂³	1	2	yes	—
138	#117×C₆₇	C₂³	1	1	yes	characteristic
139	#117×C₆₇	C₂³	1	1	yes	characteristic
140	#117×C₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
143	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
145	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
146	#107×C₆₇	C₂³	1	8	yes	—
147	#106×C₆₇	C₂³	1	8	yes	—
148	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
149	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
150	#103×C₆₇	C₂³	1	8	yes	—
151	#103×C₆₇	C₂³	1	8	yes	—
152	#102×C₆₇	C₂³	1	8	yes	—
153	#102×C₆₇	C₂³	1	8	yes	—
154	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
155	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
156	#97×C₆₇	C₂³	1	2	yes	—
157	C₄×C₈×C₆₇	C₂³	1	2	yes	—
158	C₄×C₈×C₆₇	C₂³	1	2	yes	—
159	C₄×C₈×C₆₇	C₂³	1	2	yes	—
160	C₂⁴×C₆₇	—	2	2	no	—
161	#41×C₆₇	—	4	2	no	—
162	#41×C₆₇	—	8	4	no	—
163	#41×C₆₇	—	8	4	no	—
164	C₂×Q₈×C₆₇	—	2	2	no	—
165	C₂×Q₈×C₆₇	—	2	2	no	—
166	C₂×Q₈×C₆₇	—	2	2	no	—
167	C₂×Q₈×C₆₇	—	4	4	no	—
168	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
170	C₂×D₄×C₆₇	—	2	4	no	—
171	C₂×D₄×C₆₇	—	2	4	no	—
172	C₂×D₄×C₆₇	—	2	2	no	—
173	C₂×D₄×C₆₇	—	2	2	no	—
174	C₂×D₄×C₆₇	—	2	2	no	—
175	C₂×D₄×C₆₇	—	2	4	no	—
176	C₂×D₄×C₆₇	—	4	4	no	—
177	C₂×D₄×C₆₇	—	8	4	no	—
178	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
179	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
180	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
181	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
182	C₂²×C₄×C₆₇	—	2	2	no	—
183	C₂²×C₄×C₆₇	—	2	4	no	—
184	C₂²×C₄×C₆₇	—	2	4	no	—
185	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
190	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
191	SD₁₆×C₆₇	—	4	8	no	—
192	SD₁₆×C₆₇	—	4	8	no	—
193	M₁₆×C₆₇	—	4	2	no	—
194	C₂×C₈×C₆₇	—	4	2	no	—
195	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
196	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
197	C₄⋊C₄×C₆₇	—	2	2	no	—
198	C₄⋊C₄×C₆₇	—	2	2	no	—
199	C₄⋊C₄×C₆₇	—	2	4	no	—
200	C₄⋊C₄×C₆₇	—	2	2	no	—
201	C₄⋊C₄×C₆₇	—	2	4	no	—
202	C₄⋊C₄×C₆₇	—	2	4	no	—
203	C₄⋊C₄×C₆₇	—	2	2	no	—
204	C₄⋊C₄×C₆₇	—	2	4	no	—
205	C₄⋊C₄×C₆₇	—	2	4	no	—
206	C₄⋊C₄×C₆₇	—	2	4	no	—
207	C₄⋊C₄×C₆₇	—	2	4	no	—
208	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
210	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
211	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
212	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
213	C₂²⋊C₄×C₆₇	—	2	2	no	—
214	C₂²⋊C₄×C₆₇	—	2	2	no	—
215	C₂²⋊C₄×C₆₇	—	2	4	no	—
216	C₂²⋊C₄×C₆₇	—	2	4	no	—
217	C₂²⋊C₄×C₆₇	—	2	4	no	—
218	C₂²⋊C₄×C₆₇	—	2	4	no	—
219	C₂²⋊C₄×C₆₇	—	2	4	no	—
220	C₂²⋊C₄×C₆₇	—	2	4	no	—
221	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
222	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
223	C₂²⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
224	C₄²×C₆₇	—	2	4	no	—
225	C₄²×C₆₇	—	2	4	no	—
226	C₄²×C₆₇	—	2	2	no	—
227	C₄²×C₆₇	—	2	2	no	—
228	C₄²×C₆₇	C₂×D₄	1	2	yes	—
229	C₄²×C₆₇	C₂×D₄	1	2	yes	—
230	C₄²×C₆₇	C₂×D₄	1	2	yes	—
231	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
232	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
233	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
235	C₂³×C₆₇	—	2	4	no	—
236	C₂³×C₆₇	—	4	4	no	—
237	C₂³×C₆₇	—	8	4	no	—
238	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
239	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
240	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
241	Q₈×C₆₇	—	2	8	no	—
242	Q₈×C₆₇	—	4	1	no	—
243	Q₈×C₆₇	—	4	4	no	—
244	Q₈×C₆₇	—	4	4	no	—
245	Q₈×C₆₇	—	4	4	no	—
246	D₄×C₆₇	—	2	8	no	—
247	D₄×C₆₇	—	4	1	no	—
248	D₄×C₆₇	—	4	1	no	—
249	D₄×C₆₇	—	4	1	no	—
250	D₄×C₆₇	—	4	4	no	—
251	D₄×C₆₇	—	4	4	no	—
252	D₄×C₆₇	—	4	4	no	—
253	D₄×C₆₇	—	8	4	no	—
254	D₄×C₆₇	—	8	4	no	—
255	D₄×C₆₇	—	8	4	no	—
256	C₂×C₄×C₆₇	—	2	4	no	—
257	C₂×C₄×C₆₇	—	2	4	no	—
258	C₂×C₄×C₆₇	—	2	4	no	—
259	C₂×C₄×C₆₇	—	2	2	no	—
260	C₂×C₄×C₆₇	—	2	2	no	—
261	C₂×C₄×C₆₇	—	2	2	no	—
262	C₂×C₄×C₆₇	—	2	4	no	—
263	C₂×C₄×C₆₇	—	2	4	no	—
264	C₂×C₄×C₆₇	—	2	4	no	—
265	C₂×C₄×C₆₇	—	4	4	no	—
266	C₂×C₄×C₆₇	—	8	4	no	—
267	C₂×C₄×C₆₇	—	8	4	no	—
268	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
276	C₈×C₆₇	—	2	4	no	—
277	C₈×C₆₇	—	2	4	no	—
278	C₂²×C₆₇	—	2	1	no	—
279	C₂²×C₆₇	—	2	1	no	—
280	C₂²×C₆₇	—	2	1	no	—
281	C₂²×C₆₇	—	2	2	no	—
282	C₂²×C₆₇	—	2	2	no	—
283	C₂²×C₆₇	—	2	2	no	—
284	C₂²×C₆₇	—	4	4	no	—
285	C₂²×C₆₇	—	4	4	no	—
286	C₂²×C₆₇	—	4	4	no	—
287	C₂²×C₆₇	—	16	4	no	—
288	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
289	C₄×C₆₇	—	2	1	no	—
290	C₄×C₆₇	—	2	1	no	—
291	C₄×C₆₇	—	2	1	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	2	no	—
294	C₄×C₆₇	—	2	2	no	—
295	C₄×C₆₇	—	4	4	no	—
296	C₄×C₆₇	—	4	4	no	—
297	C₄×C₆₇	—	4	4	no	—
298	C₄×C₆₇	#577	1	4	yes	—
299	C₄×C₆₇	#585	1	2	yes	—
300	#61012	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
301	C₂×C₆₇	—	4	1	no	—
302	C₂×C₆₇	—	4	1	no	—
303	C₂×C₆₇	—	4	1	no	—
304	C₂×C₆₇	—	8	4	no	—
305	C₂×C₆₇	#3449	1	1	yes	characteristic
306	C₂×C₆₇	C₂×#577	1	2	yes	—
307	#3509	C₂×C₆₇	1	1	yes	characteristic, complemented
308	#3508	C₂×C₆₇	1	1	yes	characteristic
309	#3260	C₂×C₆₇	1	4	yes	complemented
310	#3231	C₂×C₆₇	1	2	yes	complemented
311	#3228	C₂×C₆₇	1	2	yes	complemented
312	#3208	C₂×C₆₇	1	2	yes	complemented
313	#3206	C₂×C₆₇	1	2	yes	complemented
314	#3200	C₂×C₆₇	1	4	yes	complemented
315	#3182	C₂×C₆₇	1	2	yes	complemented
316	#3167	C₂×C₆₇	1	2	yes	complemented
317	#3126	C₂×C₆₇	1	4	yes	complemented
318	#3123	C₂×C₆₇	1	4	yes	complemented
319	#2922	C₂×C₆₇	1	1	yes	characteristic, complemented
320	C₆₇	#61012	1	1	yes	characteristic, direct product factor, Sylow
321	#559	—	2	2	no	—
322	#557	—	2	2	no	—
323	#556	—	2	2	no	—
324	#552	—	2	2	no	—
325	#552	—	2	2	no	—
326	#550	—	2	2	no	—
327	#546	—	2	2	no	—
328	#546	—	2	2	no	—
329	#541	C₂²×C₆₇	1	2	yes	—
330	#541	C₂²×C₆₇	1	2	yes	complemented
331	#540	C₂²×C₆₇	1	2	yes	—
332	#540	C₂²×C₆₇	1	2	yes	—
333	#536	C₂²×C₆₇	1	2	yes	—
334	#535	C₂²×C₆₇	1	2	yes	—
335	#533	C₂²×C₆₇	1	2	yes	—
336	#533	C₂²×C₆₇	1	1	yes	characteristic
337	#533	C₂²×C₆₇	1	2	yes	complemented
338	#533	C₂²×C₆₇	1	1	yes	characteristic, complemented
339	#532	C₂²×C₆₇	1	2	yes	—
340	#532	C₂²×C₆₇	1	2	yes	—
341	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
342	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
343	#519	C₂²×C₆₇	1	2	yes	complemented
344	#518	C₂²×C₆₇	1	2	yes	—
345	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
346	#500	C₂²×C₆₇	1	2	yes	complemented
347	#498	C₂²×C₆₇	1	2	yes	complemented
348	#492	C₂²×C₆₇	1	2	yes	—
349	#492	C₂²×C₆₇	1	2	yes	—
350	#487	C₂²×C₆₇	1	2	yes	complemented
351	#487	C₂²×C₆₇	1	2	yes	complemented
352	#486	C₂²×C₆₇	1	2	yes	—
353	#486	C₂²×C₆₇	1	2	yes	—
354	#485	C₂²×C₆₇	1	8	yes	—
355	#482	C₂²×C₆₇	1	8	yes	complemented
356	#479	C₂²×C₆₇	1	8	yes	complemented
357	#478	C₂²×C₆₇	1	8	yes	complemented
358	#468	C₂²×C₆₇	1	8	yes	complemented
359	#465	C₂²×C₆₇	1	8	yes	complemented
360	#461	C₂²×C₆₇	1	8	yes	complemented
361	#460	C₂²×C₆₇	1	8	yes	—
362	#450	C₂²×C₆₇	1	8	yes	—
363	#448	C₂²×C₆₇	1	8	yes	complemented
364	#440	C₂²×C₆₇	1	8	yes	complemented
365	C₄×SD₁₆	C₂²×C₆₇	1	8	yes	complemented
366	#433	C₂²×C₆₇	1	2	yes	—
367	#433	C₂²×C₆₇	1	2	yes	complemented
368	#431	C₂²×C₆₇	1	2	yes	—
369	#431	C₂²×C₆₇	1	2	yes	complemented
370	#424	C₂²×C₆₇	1	2	yes	complemented
371	#408	C₂²×C₆₇	1	2	yes	—
372	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic, complemented
373	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic, complemented
374	C₂×#41	—	2	2	no	—
375	C₂×#41	—	2	2	no	—
376	C₂×#41	C₂³×C₆₇	1	1	yes	characteristic
377	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
378	C₂²×D₄	—	2	2	no	—
379	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
380	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
381	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
382	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
383	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
384	C₄⋊Q₈	—	2	2	no	—
385	C₄⋊Q₈	—	2	2	no	—
386	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
387	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
388	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
389	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
390	#127	—	2	2	no	—
391	#127	—	2	2	no	—
392	#127	C₂³×C₆₇	1	2	yes	—
393	#127	C₂³×C₆₇	1	2	yes	—
394	#126	—	2	4	no	—
395	#126	—	2	4	no	—
396	#125	—	2	4	no	—
397	#125	—	2	4	no	—
398	#125	—	2	2	no	—
399	#125	—	2	2	no	—
400	#125	C₂³×C₆₇	1	2	yes	—
401	#125	C₂³×C₆₇	1	2	yes	—
402	#124	—	2	2	no	—
403	#124	—	2	2	no	—
404	#124	—	2	2	no	—
405	#124	—	2	2	no	—
406	#123	—	2	2	no	—
407	#123	—	2	2	no	—
408	#123	—	2	4	no	—
409	#123	—	2	4	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	—	2	2	no	—
412	C₂²⋊Q₈	—	2	2	no	—
413	C₂²⋊Q₈	—	2	2	no	—
414	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
415	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
416	#121	—	2	2	no	—
417	#121	—	2	2	no	—
418	#121	—	2	4	no	—
419	#121	—	2	4	no	—
420	#121	C₂³×C₆₇	1	4	yes	—
421	#121	C₂³×C₆₇	1	4	yes	—
422	#120	—	2	2	no	—
423	#120	—	2	2	no	—
424	C₄×Q₈	—	2	4	no	—
425	C₄×Q₈	—	2	2	no	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×Q₈	—	2	2	no	—
429	C₄×Q₈	D₄×C₆₇	1	2	yes	—
430	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
431	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	—	2	2	no	—
434	C₄×D₄	—	2	2	no	—
435	C₄×D₄	D₄×C₆₇	1	2	yes	—
436	C₄×D₄	D₄×C₆₇	1	2	yes	—
437	C₄×D₄	D₄×C₆₇	1	2	yes	—
438	C₄×D₄	C₂³×C₆₇	1	4	yes	—
439	C₄×D₄	C₂³×C₆₇	1	4	yes	—
440	#117	—	2	2	no	—
441	#117	—	2	2	no	—
442	#117	—	2	4	no	—
443	#117	C₂³×C₆₇	1	2	yes	—
444	#117	C₂³×C₆₇	1	2	yes	—
445	#117	C₂³×C₆₇	1	1	yes	characteristic
446	#117	C₂³×C₆₇	1	1	yes	characteristic
447	#117	C₂³×C₆₇	1	1	yes	characteristic
448	C₂×C₄⋊C₄	—	2	2	no	—
449	C₂×C₂²⋊C₄	—	2	4	no	—
450	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
451	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
452	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
453	#107	C₂³×C₆₇	1	8	yes	—
454	#106	C₂³×C₆₇	1	8	yes	—
455	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
456	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
457	#103	C₂³×C₆₇	1	8	yes	—
458	#103	C₂³×C₆₇	1	8	yes	—
459	#102	C₂³×C₆₇	1	8	yes	—
460	#102	C₂³×C₆₇	1	8	yes	—
461	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
462	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
463	#97	C₂³×C₆₇	1	2	yes	—
464	C₄×C₈	C₂³×C₆₇	1	2	yes	—
465	C₄×C₈	C₂³×C₆₇	1	2	yes	—
466	C₄×C₈	C₂³×C₆₇	1	2	yes	—
467	C₂⁴	—	2	2	no	—
468	#41	—	4	2	no	—
469	#41	—	8	4	no	—
470	#41	—	8	4	no	—
471	C₂×Q₈	—	2	2	no	—
472	C₂×Q₈	—	2	2	no	—
473	C₂×Q₈	—	2	2	no	—
474	C₂×Q₈	—	4	4	no	—
475	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
476	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	2	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	2	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	—	8	4	no	—
485	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
486	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
487	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
488	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
493	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
495	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
496	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
498	SD₁₆	—	4	8	no	—
499	SD₁₆	—	4	8	no	—
500	M₁₆	—	4	2	no	—
501	C₂×C₈	—	4	2	no	—
502	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
503	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	2	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	4	no	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	4	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
516	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
517	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
518	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
519	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
520	C₂²⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	2	2	no	—
522	C₂²⋊C₄	—	2	4	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	—	2	4	no	—
525	C₂²⋊C₄	—	2	4	no	—
526	C₂²⋊C₄	—	2	4	no	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
529	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
530	C₂²⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
531	C₄²	—	2	4	no	—
532	C₄²	—	2	4	no	—
533	C₄²	—	2	2	no	—
534	C₄²	—	2	2	no	—
535	C₄²	C₂×D₄×C₆₇	1	2	yes	—
536	C₄²	C₂×D₄×C₆₇	1	2	yes	—
537	C₄²	C₂×D₄×C₆₇	1	2	yes	—
538	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
539	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
540	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
541	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
542	C₂³	—	2	4	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	8	4	no	—
545	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
546	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
547	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
548	Q₈	—	2	8	no	—
549	Q₈	—	4	1	no	—
550	Q₈	—	4	4	no	—
551	Q₈	—	4	4	no	—
552	Q₈	—	4	4	no	—
553	D₄	—	2	8	no	—
554	D₄	—	4	1	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	1	no	—
557	D₄	—	4	4	no	—
558	D₄	—	4	4	no	—
559	D₄	—	4	4	no	—
560	D₄	—	8	4	no	—
561	D₄	—	8	4	no	—
562	D₄	—	8	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	4	4	no	—
573	C₂×C₄	—	8	4	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
576	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
577	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
578	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
582	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	—	2	4	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	2	no	—
587	C₂²	—	2	2	no	—
588	C₂²	—	2	1	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	4	4	no	—
592	C₂²	—	4	4	no	—
593	C₂²	—	4	4	no	—
594	C₂²	—	16	4	no	—
595	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
596	C₄	—	2	2	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	4	no	—
604	C₄	—	4	4	no	—
605	C₄	#577×C₆₇	1	4	yes	—
606	C₄	#585×C₆₇	1	2	yes	—
607	C₂	—	4	1	no	—
608	C₂	—	4	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	8	4	no	—
611	C₂	C₆₇×#3449	1	1	yes	characteristic
612	C₂	C₂×#577×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₆₇×#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×#41×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3509	C₂	semidirect
3. C₆₇×#3260	C₂	semidirect
4. C₆₇×#3231	C₂	semidirect
5. C₆₇×#3228	C₂	semidirect
6. C₆₇×#3208	C₂	semidirect
7. C₆₇×#3206	C₂	semidirect
8. C₆₇×#3200	C₂	semidirect
9. C₆₇×#3182	C₂	semidirect
10. C₆₇×#3167	C₂	semidirect
11. C₆₇×#3126	C₂	semidirect
12. C₆₇×#3123	C₂	semidirect
13. C₆₇×#2922	C₂	semidirect
23. #541×C₆₇	C₂²	semidirect
30. #533×C₆₇	C₂²	semidirect
31. #533×C₆₇	C₂²	semidirect
34. C₂×C₄⋊Q₈×C₆₇	C₂²	semidirect
36. #519×C₆₇	C₂²	semidirect
39. #500×C₆₇	C₂²	semidirect
40. #498×C₆₇	C₂²	semidirect
43. #487×C₆₇	C₂²	semidirect
44. #487×C₆₇	C₂²	semidirect
48. #482×C₆₇	C₂²	semidirect
49. #479×C₆₇	C₂²	semidirect
50. #478×C₆₇	C₂²	semidirect
51. #468×C₆₇	C₂²	semidirect
52. #465×C₆₇	C₂²	semidirect
53. #461×C₆₇	C₂²	semidirect
56. #448×C₆₇	C₂²	semidirect
57. #440×C₆₇	C₂²	semidirect
58. C₄×SD₁₆×C₆₇	C₂²	semidirect
60. #433×C₆₇	C₂²	semidirect
62. #431×C₆₇	C₂²	semidirect
63. #424×C₆₇	C₂²	semidirect
65. C₄×M₁₆×C₆₇	C₂²	semidirect
66. C₂×C₄×C₈×C₆₇	C₂²	semidirect
300. #61012	C₆₇	direct (N × H)
307. #3509	C₂×C₆₇	semidirect
309. #3260	C₂×C₆₇	semidirect
310. #3231	C₂×C₆₇	semidirect
311. #3228	C₂×C₆₇	semidirect
312. #3208	C₂×C₆₇	semidirect
313. #3206	C₂×C₆₇	semidirect
314. #3200	C₂×C₆₇	semidirect
315. #3182	C₂×C₆₇	semidirect
316. #3167	C₂×C₆₇	semidirect
317. #3126	C₂×C₆₇	semidirect
318. #3123	C₂×C₆₇	semidirect
319. #2922	C₂×C₆₇	semidirect
320. C₆₇	#61012	direct (N × H)
330. #541	C₂²×C₆₇	semidirect
337. #533	C₂²×C₆₇	semidirect
338. #533	C₂²×C₆₇	semidirect
341. C₂×C₄⋊Q₈	C₂²×C₆₇	semidirect
343. #519	C₂²×C₆₇	semidirect
346. #500	C₂²×C₆₇	semidirect
347. #498	C₂²×C₆₇	semidirect
350. #487	C₂²×C₆₇	semidirect
351. #487	C₂²×C₆₇	semidirect
355. #482	C₂²×C₆₇	semidirect
356. #479	C₂²×C₆₇	semidirect
357. #478	C₂²×C₆₇	semidirect
358. #468	C₂²×C₆₇	semidirect
359. #465	C₂²×C₆₇	semidirect
360. #461	C₂²×C₆₇	semidirect
363. #448	C₂²×C₆₇	semidirect
364. #440	C₂²×C₆₇	semidirect
365. C₄×SD₁₆	C₂²×C₆₇	semidirect
367. #433	C₂²×C₆₇	semidirect
369. #431	C₂²×C₆₇	semidirect
370. #424	C₂²×C₆₇	semidirect
372. C₄×M₁₆	C₂²×C₆₇	semidirect
373. C₂×C₄×C₈	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×#3509	2
C₆₇×#3508	2
C₆₇×#3260	2
C₆₇×#3231	2
C₆₇×#3228	2
C₆₇×#3208	2
C₆₇×#3206	2
C₆₇×#3200	2
C₆₇×#3182	2
C₆₇×#3167	2
C₆₇×#3126	2
C₆₇×#3123	2
C₆₇×#2922	2
#61012	67

Subgroups — C67×#61012 #30740263

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#61012 #30740263