Subgroups — C₆₇×#60972 #30740223

Order 17152.

1824 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3444	C₂	1	2	yes	maximal, complemented
2	C₆₇×#3256	C₂	1	4	yes	maximal, complemented
3	C₆₇×#3231	C₂	1	2	yes	maximal, complemented
4	C₆₇×#3228	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3210	C₂	1	2	yes	maximal, complemented
6	C₆₇×#3207	C₂	1	2	yes	maximal, complemented
7	C₆₇×#3197	C₂	1	4	yes	maximal, complemented
8	C₆₇×#3185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#3184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#3182	C₂	1	2	yes	maximal, complemented
11	C₆₇×#3111	C₂	1	8	yes	maximal, complemented
12	C₆₇×#2920	C₂	1	1	yes	maximal, characteristic, complemented
13	#533×C₆₇	C₂²	1	2	yes	—
14	#533×C₆₇	C₂²	1	2	yes	—
15	#533×C₆₇	C₂²	1	2	yes	—
16	#532×C₆₇	C₂²	1	2	yes	—
17	#532×C₆₇	C₂²	1	2	yes	—
18	#532×C₆₇	C₂²	1	2	yes	—
19	#529×C₆₇	—	2	2	no	—
20	#529×C₆₇	—	2	2	no	—
21	#529×C₆₇	—	2	2	no	—
22	#529×C₆₇	—	2	2	no	—
23	#529×C₆₇	C₂²	1	4	yes	—
24	#529×C₆₇	C₂²	1	4	yes	—
25	#525×C₆₇	—	2	2	no	—
26	#525×C₆₇	—	2	2	no	—
27	#525×C₆₇	C₂²	1	4	yes	—
28	C₄×#41×C₆₇	—	2	2	no	—
29	C₄×#41×C₆₇	—	2	2	no	—
30	C₄×#41×C₆₇	C₂²	1	4	yes	—
31	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	#500×C₆₇	C₂²	1	2	yes	complemented
33	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
34	#498×C₆₇	C₂²	1	1	yes	characteristic, complemented
35	#494×C₆₇	C₂²	1	1	yes	characteristic, complemented
36	#493×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#492×C₆₇	C₂²	1	2	yes	complemented
38	#487×C₆₇	C₂²	1	2	yes	complemented
39	#487×C₆₇	C₂²	1	2	yes	complemented
40	#486×C₆₇	C₂²	1	2	yes	complemented
41	#486×C₆₇	C₂²	1	2	yes	complemented
42	#485×C₆₇	C₂²	1	8	yes	—
43	#482×C₆₇	C₂²	1	8	yes	—
44	#479×C₆₇	C₂²	1	8	yes	complemented
45	#478×C₆₇	C₂²	1	8	yes	complemented
46	#470×C₆₇	C₂²	1	4	yes	—
47	#469×C₆₇	C₂²	1	4	yes	—
48	#468×C₆₇	C₂²	1	8	yes	—
49	#464×C₆₇	C₂²	1	8	yes	complemented
50	#463×C₆₇	C₂²	1	8	yes	complemented
51	#452×C₆₇	C₂²	1	8	yes	—
52	#449×C₆₇	C₂²	1	8	yes	—
53	#442×C₆₇	C₂²	1	4	yes	complemented
54	#441×C₆₇	C₂²	1	4	yes	complemented
55	#440×C₆₇	C₂²	1	8	yes	complemented
56	#432×C₆₇	C₂²	1	4	yes	—
57	#431×C₆₇	C₂²	1	2	yes	—
58	#431×C₆₇	C₂²	1	2	yes	—
59	#423×C₆₇	C₂²	1	2	yes	—
60	#407×C₆₇	C₂²	1	2	yes	—
61	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
62	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
63	C₂×#41×C₆₇	—	2	2	no	—
64	C₂×#41×C₆₇	—	2	2	no	—
65	C₂×#41×C₆₇	C₂³	1	4	yes	—
66	C₂³×C₄×C₆₇	D₄	1	2	yes	—
67	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
68	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
69	C₂×SD₁₆×C₆₇	C₂³	1	8	yes	—
70	C₂×D₈×C₆₇	C₂³	1	4	yes	—
71	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
72	C₂×M₁₆×C₆₇	C₂³	1	2	yes	—
73	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
74	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
75	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
76	#127×C₆₇	C₂³	1	2	yes	—
77	#126×C₆₇	—	2	4	no	—
78	#126×C₆₇	—	2	4	no	—
79	#125×C₆₇	—	2	2	no	—
80	#125×C₆₇	—	2	2	no	—
81	#125×C₆₇	C₂³	1	4	yes	—
82	#124×C₆₇	—	2	2	no	—
83	#124×C₆₇	—	2	2	no	—
84	#124×C₆₇	C₂³	1	4	yes	—
85	#123×C₆₇	—	2	4	no	—
86	#123×C₆₇	—	2	4	no	—
87	#123×C₆₇	—	2	4	no	—
88	C₂²⋊Q₈×C₆₇	—	2	2	no	—
89	C₂²⋊Q₈×C₆₇	—	2	2	no	—
90	C₂²⋊Q₈×C₆₇	—	2	2	no	—
91	C₂²⋊Q₈×C₆₇	—	2	2	no	—
92	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
93	C₂²⋊Q₈×C₆₇	C₂³	1	4	yes	—
94	#121×C₆₇	—	2	2	no	—
95	#121×C₆₇	—	2	2	no	—
96	#121×C₆₇	—	2	2	no	—
97	#121×C₆₇	—	2	2	no	—
98	#121×C₆₇	C₂³	1	4	yes	—
99	#121×C₆₇	C₂³	1	4	yes	—
100	#120×C₆₇	—	2	4	no	—
101	C₄×Q₈×C₆₇	—	2	2	no	—
102	C₄×Q₈×C₆₇	—	2	2	no	—
103	C₄×Q₈×C₆₇	—	2	2	no	—
104	C₄×Q₈×C₆₇	—	2	2	no	—
105	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
106	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
107	C₄×D₄×C₆₇	—	2	4	no	—
108	C₄×D₄×C₆₇	—	2	4	no	—
109	C₄×D₄×C₆₇	—	2	4	no	—
110	C₄×D₄×C₆₇	—	2	4	no	—
111	C₄×D₄×C₆₇	—	2	4	no	—
112	C₄×D₄×C₆₇	—	2	4	no	—
113	C₄×D₄×C₆₇	—	2	2	no	—
114	C₄×D₄×C₆₇	—	2	2	no	—
115	C₄×D₄×C₆₇	—	2	2	no	—
116	C₄×D₄×C₆₇	—	2	2	no	—
117	C₄×D₄×C₆₇	C₂³	1	4	yes	—
118	C₄×D₄×C₆₇	C₂³	1	4	yes	—
119	#117×C₆₇	—	2	2	no	—
120	#117×C₆₇	—	2	2	no	—
121	#117×C₆₇	—	2	4	no	—
122	#117×C₆₇	—	2	4	no	—
123	#117×C₆₇	—	2	4	no	—
124	#117×C₆₇	C₂³	1	4	yes	—
125	C₂×C₄²×C₆₇	—	2	4	no	—
126	C₂×C₄²×C₆₇	D₄	1	2	yes	—
127	C₂×C₄²×C₆₇	D₄	1	2	yes	—
128	C₂×C₄²×C₆₇	D₄	1	2	yes	—
129	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
133	#107×C₆₇	C₂³	1	8	yes	—
134	#106×C₆₇	C₂³	1	4	yes	—
135	#106×C₆₇	C₂³	1	4	yes	—
136	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
137	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
138	#103×C₆₇	C₂³	1	8	yes	—
139	#103×C₆₇	C₂³	1	8	yes	—
140	#102×C₆₇	C₂³	1	8	yes	—
141	#102×C₆₇	C₂³	1	8	yes	—
142	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
143	C₂²⋊C₈×C₆₇	C₂³	1	4	yes	—
144	#97×C₆₇	C₂³	1	2	yes	—
145	#97×C₆₇	C₂³	1	2	yes	—
146	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₆₇	C₂³	1	2	yes	—
148	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
149	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
150	#41×C₆₇	—	4	8	no	—
151	#41×C₆₇	—	4	8	no	—
152	#41×C₆₇	—	4	8	no	—
153	C₂×Q₈×C₆₇	—	2	2	no	—
154	C₂×Q₈×C₆₇	—	2	2	no	—
155	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
156	C₂×D₄×C₆₇	—	2	4	no	—
157	C₂×D₄×C₆₇	—	2	4	no	—
158	C₂×D₄×C₆₇	—	2	4	no	—
159	C₂×D₄×C₆₇	—	2	2	no	—
160	C₂×D₄×C₆₇	—	2	2	no	—
161	C₂×D₄×C₆₇	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₆₇	—	2	4	no	—
163	C₂²×C₄×C₆₇	—	2	4	no	—
164	C₂²×C₄×C₆₇	—	2	4	no	—
165	C₂²×C₄×C₆₇	—	4	4	no	—
166	C₂²×C₄×C₆₇	—	4	2	no	—
167	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
170	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
171	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
177	Q₁₆×C₆₇	—	4	4	no	—
178	SD₁₆×C₆₇	—	4	8	no	—
179	D₈×C₆₇	—	4	4	no	—
180	M₁₆×C₆₇	—	4	2	no	—
181	M₁₆×C₆₇	—	4	2	no	—
182	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
183	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
184	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
185	C₄⋊C₄×C₆₇	—	2	2	no	—
186	C₄⋊C₄×C₆₇	—	2	2	no	—
187	C₄⋊C₄×C₆₇	—	2	2	no	—
188	C₄⋊C₄×C₆₇	—	2	2	no	—
189	C₄⋊C₄×C₆₇	—	2	4	no	—
190	C₄⋊C₄×C₆₇	—	2	4	no	—
191	C₄⋊C₄×C₆₇	—	2	4	no	—
192	C₄⋊C₄×C₆₇	—	2	4	no	—
193	C₄⋊C₄×C₆₇	—	2	4	no	—
194	C₄⋊C₄×C₆₇	—	2	4	no	—
195	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₆₇	—	2	4	no	—
198	C₂²⋊C₄×C₆₇	—	2	4	no	—
199	C₂²⋊C₄×C₆₇	—	2	4	no	—
200	C₂²⋊C₄×C₆₇	—	2	4	no	—
201	C₂²⋊C₄×C₆₇	—	2	4	no	—
202	C₂²⋊C₄×C₆₇	—	2	4	no	—
203	C₂²⋊C₄×C₆₇	—	2	4	no	—
204	C₂²⋊C₄×C₆₇	—	2	4	no	—
205	C₂²⋊C₄×C₆₇	—	2	4	no	—
206	C₄²×C₆₇	—	2	4	no	—
207	C₄²×C₆₇	—	2	4	no	—
208	C₄²×C₆₇	—	2	4	no	—
209	C₄²×C₆₇	C₂×D₄	1	2	yes	—
210	C₄²×C₆₇	C₂×D₄	1	2	yes	—
211	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
212	C₄²×C₆₇	C₂×D₄	1	2	yes	—
213	C₄²×C₆₇	C₂×D₄	1	2	yes	—
214	C₄²×C₆₇	C₂×D₄	1	2	yes	—
215	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
216	C₄²×C₆₇	C₂⁴	1	2	yes	—
217	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
218	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
219	C₂³×C₆₇	—	2	4	no	—
220	C₂³×C₆₇	—	4	1	no	—
221	C₂³×C₆₇	—	4	2	no	—
222	C₂³×C₆₇	C₂²×D₄	1	2	yes	—
223	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
224	Q₈×C₆₇	—	2	8	no	—
225	Q₈×C₆₇	—	4	4	no	—
226	Q₈×C₆₇	—	4	4	no	—
227	D₄×C₆₇	—	2	8	no	—
228	D₄×C₆₇	—	4	8	no	—
229	D₄×C₆₇	—	4	8	no	—
230	D₄×C₆₇	—	4	8	no	—
231	D₄×C₆₇	—	4	4	no	—
232	D₄×C₆₇	—	4	4	no	—
233	C₂×C₄×C₆₇	—	2	4	no	—
234	C₂×C₄×C₆₇	—	2	4	no	—
235	C₂×C₄×C₆₇	—	2	4	no	—
236	C₂×C₄×C₆₇	—	2	2	no	—
237	C₂×C₄×C₆₇	—	2	4	no	—
238	C₂×C₄×C₆₇	—	2	4	no	—
239	C₂×C₄×C₆₇	—	2	4	no	—
240	C₂×C₄×C₆₇	—	2	4	no	—
241	C₂×C₄×C₆₇	—	4	1	no	—
242	C₂×C₄×C₆₇	—	4	2	no	—
243	C₂×C₄×C₆₇	—	4	2	no	—
244	C₂×C₄×C₆₇	—	4	2	no	—
245	C₂×C₄×C₆₇	—	4	2	no	—
246	C₂×C₄×C₆₇	—	4	8	no	—
247	C₂×C₄×C₆₇	—	4	8	no	—
248	C₂×C₄×C₆₇	—	4	8	no	—
249	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
251	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
252	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
256	C₈×C₆₇	—	2	2	no	—
257	C₈×C₆₇	—	2	4	no	—
258	C₈×C₆₇	—	2	2	no	—
259	C₂²×C₆₇	—	2	2	no	—
260	C₂²×C₆₇	—	2	1	no	—
261	C₂²×C₆₇	—	2	2	no	—
262	C₂²×C₆₇	—	2	2	no	—
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	4	4	no	—
265	C₂²×C₆₇	—	4	4	no	—
266	C₂²×C₆₇	—	4	4	no	—
267	C₂²×C₆₇	—	4	4	no	—
268	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
269	C₄×C₆₇	—	2	2	no	—
270	C₄×C₆₇	—	2	1	no	—
271	C₄×C₆₇	—	2	2	no	—
272	C₄×C₆₇	—	2	2	no	—
273	C₄×C₆₇	—	2	2	no	—
274	C₄×C₆₇	—	4	4	no	—
275	C₄×C₆₇	—	4	4	no	—
276	C₄×C₆₇	—	4	4	no	—
277	C₄×C₆₇	#575	1	4	yes	—
278	C₄×C₆₇	#585	1	2	yes	—
279	#60972	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₆₇	—	4	2	no	—
281	C₂×C₆₇	—	4	1	no	—
282	C₂×C₆₇	—	8	4	no	—
283	C₂×C₆₇	#3449	1	1	yes	characteristic
284	C₂×C₆₇	C₂×#575	1	2	yes	—
285	#3444	C₂×C₆₇	1	2	yes	complemented
286	#3256	C₂×C₆₇	1	4	yes	complemented
287	#3231	C₂×C₆₇	1	2	yes	complemented
288	#3228	C₂×C₆₇	1	2	yes	complemented
289	#3210	C₂×C₆₇	1	2	yes	complemented
290	#3207	C₂×C₆₇	1	2	yes	complemented
291	#3197	C₂×C₆₇	1	4	yes	complemented
292	#3185	C₂×C₆₇	1	1	yes	characteristic, complemented
293	#3184	C₂×C₆₇	1	1	yes	characteristic, complemented
294	#3182	C₂×C₆₇	1	2	yes	complemented
295	#3111	C₂×C₆₇	1	8	yes	complemented
296	#2920	C₂×C₆₇	1	1	yes	characteristic, complemented
297	C₆₇	#60972	1	1	yes	characteristic, direct product factor, Sylow
298	#533	C₂²×C₆₇	1	2	yes	—
299	#533	C₂²×C₆₇	1	2	yes	—
300	#533	C₂²×C₆₇	1	2	yes	—
301	#532	C₂²×C₆₇	1	2	yes	—
302	#532	C₂²×C₆₇	1	2	yes	—
303	#532	C₂²×C₆₇	1	2	yes	—
304	#529	—	2	2	no	—
305	#529	—	2	2	no	—
306	#529	—	2	2	no	—
307	#529	—	2	2	no	—
308	#529	C₂²×C₆₇	1	4	yes	—
309	#529	C₂²×C₆₇	1	4	yes	—
310	#525	—	2	2	no	—
311	#525	—	2	2	no	—
312	#525	C₂²×C₆₇	1	4	yes	—
313	C₄×#41	—	2	2	no	—
314	C₄×#41	—	2	2	no	—
315	C₄×#41	C₂²×C₆₇	1	4	yes	—
316	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
317	#500	C₂²×C₆₇	1	2	yes	complemented
318	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
319	#498	C₂²×C₆₇	1	1	yes	characteristic, complemented
320	#494	C₂²×C₆₇	1	1	yes	characteristic, complemented
321	#493	C₂²×C₆₇	1	1	yes	characteristic, complemented
322	#492	C₂²×C₆₇	1	2	yes	complemented
323	#487	C₂²×C₆₇	1	2	yes	complemented
324	#487	C₂²×C₆₇	1	2	yes	complemented
325	#486	C₂²×C₆₇	1	2	yes	complemented
326	#486	C₂²×C₆₇	1	2	yes	complemented
327	#485	C₂²×C₆₇	1	8	yes	—
328	#482	C₂²×C₆₇	1	8	yes	—
329	#479	C₂²×C₆₇	1	8	yes	complemented
330	#478	C₂²×C₆₇	1	8	yes	complemented
331	#470	C₂²×C₆₇	1	4	yes	—
332	#469	C₂²×C₆₇	1	4	yes	—
333	#468	C₂²×C₆₇	1	8	yes	—
334	#464	C₂²×C₆₇	1	8	yes	complemented
335	#463	C₂²×C₆₇	1	8	yes	complemented
336	#452	C₂²×C₆₇	1	8	yes	—
337	#449	C₂²×C₆₇	1	8	yes	—
338	#442	C₂²×C₆₇	1	4	yes	complemented
339	#441	C₂²×C₆₇	1	4	yes	complemented
340	#440	C₂²×C₆₇	1	8	yes	complemented
341	#432	C₂²×C₆₇	1	4	yes	—
342	#431	C₂²×C₆₇	1	2	yes	—
343	#431	C₂²×C₆₇	1	2	yes	—
344	#423	C₂²×C₆₇	1	2	yes	—
345	#407	C₂²×C₆₇	1	2	yes	—
346	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
347	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
348	C₂×#41	—	2	2	no	—
349	C₂×#41	—	2	2	no	—
350	C₂×#41	C₂³×C₆₇	1	4	yes	—
351	C₂³×C₄	D₄×C₆₇	1	2	yes	—
352	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
353	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
354	C₂×SD₁₆	C₂³×C₆₇	1	8	yes	—
355	C₂×D₈	C₂³×C₆₇	1	4	yes	—
356	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
357	C₂×M₁₆	C₂³×C₆₇	1	2	yes	—
358	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
359	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
360	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
361	#127	C₂³×C₆₇	1	2	yes	—
362	#126	—	2	4	no	—
363	#126	—	2	4	no	—
364	#125	—	2	2	no	—
365	#125	—	2	2	no	—
366	#125	C₂³×C₆₇	1	4	yes	—
367	#124	—	2	2	no	—
368	#124	—	2	2	no	—
369	#124	C₂³×C₆₇	1	4	yes	—
370	#123	—	2	4	no	—
371	#123	—	2	4	no	—
372	#123	—	2	4	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	C₂²⋊Q₈	—	2	2	no	—
375	C₂²⋊Q₈	—	2	2	no	—
376	C₂²⋊Q₈	—	2	2	no	—
377	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
378	C₂²⋊Q₈	C₂³×C₆₇	1	4	yes	—
379	#121	—	2	2	no	—
380	#121	—	2	2	no	—
381	#121	—	2	2	no	—
382	#121	—	2	2	no	—
383	#121	C₂³×C₆₇	1	4	yes	—
384	#121	C₂³×C₆₇	1	4	yes	—
385	#120	—	2	4	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
391	C₄×Q₈	C₂³×C₆₇	1	4	yes	—
392	C₄×D₄	—	2	4	no	—
393	C₄×D₄	—	2	4	no	—
394	C₄×D₄	—	2	4	no	—
395	C₄×D₄	—	2	4	no	—
396	C₄×D₄	—	2	4	no	—
397	C₄×D₄	—	2	4	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	—	2	2	no	—
400	C₄×D₄	—	2	2	no	—
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	C₂³×C₆₇	1	4	yes	—
403	C₄×D₄	C₂³×C₆₇	1	4	yes	—
404	#117	—	2	2	no	—
405	#117	—	2	2	no	—
406	#117	—	2	4	no	—
407	#117	—	2	4	no	—
408	#117	—	2	4	no	—
409	#117	C₂³×C₆₇	1	4	yes	—
410	C₂×C₄²	—	2	4	no	—
411	C₂×C₄²	D₄×C₆₇	1	2	yes	—
412	C₂×C₄²	D₄×C₆₇	1	2	yes	—
413	C₂×C₄²	D₄×C₆₇	1	2	yes	—
414	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
415	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
416	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
417	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
418	#107	C₂³×C₆₇	1	8	yes	—
419	#106	C₂³×C₆₇	1	4	yes	—
420	#106	C₂³×C₆₇	1	4	yes	—
421	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
422	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
423	#103	C₂³×C₆₇	1	8	yes	—
424	#103	C₂³×C₆₇	1	8	yes	—
425	#102	C₂³×C₆₇	1	8	yes	—
426	#102	C₂³×C₆₇	1	8	yes	—
427	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
428	C₂²⋊C₈	C₂³×C₆₇	1	4	yes	—
429	#97	C₂³×C₆₇	1	2	yes	—
430	#97	C₂³×C₆₇	1	2	yes	—
431	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
432	C₄×C₈	C₂³×C₆₇	1	2	yes	—
433	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
434	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
435	#41	—	4	8	no	—
436	#41	—	4	8	no	—
437	#41	—	4	8	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	2	2	no	—
440	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
441	C₂×D₄	—	2	4	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	C₂⁴×C₆₇	1	4	yes	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	2	4	no	—
449	C₂²×C₄	—	2	4	no	—
450	C₂²×C₄	—	4	4	no	—
451	C₂²×C₄	—	4	2	no	—
452	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
455	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
456	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
457	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
458	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
459	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
462	Q₁₆	—	4	4	no	—
463	SD₁₆	—	4	8	no	—
464	D₈	—	4	4	no	—
465	M₁₆	—	4	2	no	—
466	M₁₆	—	4	2	no	—
467	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
468	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
469	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	2	4	no	—
476	C₄⋊C₄	—	2	4	no	—
477	C₄⋊C₄	—	2	4	no	—
478	C₄⋊C₄	—	2	4	no	—
479	C₄⋊C₄	—	2	4	no	—
480	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
481	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	4	no	—
484	C₂²⋊C₄	—	2	4	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	4	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	—	2	4	no	—
489	C₂²⋊C₄	—	2	4	no	—
490	C₂²⋊C₄	—	2	4	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	4	no	—
493	C₄²	—	2	4	no	—
494	C₄²	C₂×D₄×C₆₇	1	2	yes	—
495	C₄²	C₂×D₄×C₆₇	1	2	yes	—
496	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
497	C₄²	C₂×D₄×C₆₇	1	2	yes	—
498	C₄²	C₂×D₄×C₆₇	1	2	yes	—
499	C₄²	C₂×D₄×C₆₇	1	2	yes	—
500	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
501	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
502	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
503	C₄²	C₂⁴×C₆₇	1	2	yes	—
504	C₂³	—	2	4	no	—
505	C₂³	—	4	2	no	—
506	C₂³	—	4	1	no	—
507	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
508	C₂³	C₂²×D₄×C₆₇	1	2	yes	—
509	Q₈	—	2	8	no	—
510	Q₈	—	4	4	no	—
511	Q₈	—	4	4	no	—
512	D₄	—	2	8	no	—
513	D₄	—	4	8	no	—
514	D₄	—	4	8	no	—
515	D₄	—	4	8	no	—
516	D₄	—	4	4	no	—
517	D₄	—	4	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	4	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	2	no	—
528	C₂×C₄	—	4	2	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	4	2	no	—
531	C₂×C₄	—	4	8	no	—
532	C₂×C₄	—	4	8	no	—
533	C₂×C₄	—	4	8	no	—
534	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
536	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
537	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
539	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
540	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
541	C₈	—	2	2	no	—
542	C₈	—	2	4	no	—
543	C₈	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	2	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	4	4	no	—
553	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
554	C₄	—	2	2	no	—
555	C₄	—	2	2	no	—
556	C₄	—	2	2	no	—
557	C₄	—	2	2	no	—
558	C₄	—	2	1	no	—
559	C₄	—	4	4	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	4	no	—
562	C₄	#575×C₆₇	1	4	yes	—
563	C₄	#585×C₆₇	1	2	yes	—
564	C₂	—	4	1	no	—
565	C₂	—	4	2	no	—
566	C₂	—	8	4	no	—
567	C₂	C₆₇×#3449	1	1	yes	characteristic
568	C₂	C₂×#575×C₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3444	C₂	semidirect
2. C₆₇×#3256	C₂	semidirect
3. C₆₇×#3231	C₂	semidirect
4. C₆₇×#3228	C₂	semidirect
5. C₆₇×#3210	C₂	semidirect
6. C₆₇×#3207	C₂	semidirect
7. C₆₇×#3197	C₂	semidirect
8. C₆₇×#3185	C₂	semidirect
9. C₆₇×#3184	C₂	semidirect
10. C₆₇×#3182	C₂	semidirect
11. C₆₇×#3111	C₂	semidirect
12. C₆₇×#2920	C₂	semidirect
32. #500×C₆₇	C₂²	semidirect
33. #498×C₆₇	C₂²	semidirect
34. #498×C₆₇	C₂²	semidirect
35. #494×C₆₇	C₂²	semidirect
36. #493×C₆₇	C₂²	semidirect
37. #492×C₆₇	C₂²	semidirect
38. #487×C₆₇	C₂²	semidirect
39. #487×C₆₇	C₂²	semidirect
40. #486×C₆₇	C₂²	semidirect
41. #486×C₆₇	C₂²	semidirect
44. #479×C₆₇	C₂²	semidirect
45. #478×C₆₇	C₂²	semidirect
49. #464×C₆₇	C₂²	semidirect
50. #463×C₆₇	C₂²	semidirect
53. #442×C₆₇	C₂²	semidirect
54. #441×C₆₇	C₂²	semidirect
55. #440×C₆₇	C₂²	semidirect
279. #60972	C₆₇	direct (N × H)
285. #3444	C₂×C₆₇	semidirect
286. #3256	C₂×C₆₇	semidirect
287. #3231	C₂×C₆₇	semidirect
288. #3228	C₂×C₆₇	semidirect
289. #3210	C₂×C₆₇	semidirect
290. #3207	C₂×C₆₇	semidirect
291. #3197	C₂×C₆₇	semidirect
292. #3185	C₂×C₆₇	semidirect
293. #3184	C₂×C₆₇	semidirect
294. #3182	C₂×C₆₇	semidirect
295. #3111	C₂×C₆₇	semidirect
296. #2920	C₂×C₆₇	semidirect
297. C₆₇	#60972	direct (N × H)
317. #500	C₂²×C₆₇	semidirect
318. #498	C₂²×C₆₇	semidirect
319. #498	C₂²×C₆₇	semidirect
320. #494	C₂²×C₆₇	semidirect
321. #493	C₂²×C₆₇	semidirect
322. #492	C₂²×C₆₇	semidirect
323. #487	C₂²×C₆₇	semidirect
324. #487	C₂²×C₆₇	semidirect
325. #486	C₂²×C₆₇	semidirect
326. #486	C₂²×C₆₇	semidirect
329. #479	C₂²×C₆₇	semidirect
330. #478	C₂²×C₆₇	semidirect
334. #464	C₂²×C₆₇	semidirect
335. #463	C₂²×C₆₇	semidirect
338. #442	C₂²×C₆₇	semidirect
339. #441	C₂²×C₆₇	semidirect
340. #440	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×#3444	2
C₆₇×#3256	2
C₆₇×#3231	2
C₆₇×#3228	2
C₆₇×#3210	2
C₆₇×#3207	2
C₆₇×#3197	2
C₆₇×#3185	2
C₆₇×#3184	2
C₆₇×#3182	2
C₆₇×#3111	2
C₆₇×#2920	2
#60972	67

Subgroups — C67×#60972 #30740223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#60972 #30740223