Subgroups — C₆₇×#33627 #30712878

Order 17152.

598 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2228	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2226	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2224	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2159	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2140	C₂	1	2	yes	maximal
9	C₂²×D₈×C₆₇	C₂²	1	1	yes	characteristic
10	#533×C₆₇	—	2	1	no	—
11	C₂×C₄×D₄×C₆₇	—	2	1	no	—
12	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
13	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
14	#500×C₆₇	C₂²	1	1	yes	characteristic, complemented
15	#499×C₆₇	—	2	1	no	—
16	#499×C₆₇	C₂²	1	2	yes	complemented
17	#498×C₆₇	—	2	1	no	—
18	C₄×D₈×C₆₇	—	2	1	no	—
19	C₄×D₈×C₆₇	—	2	1	no	—
20	C₄×D₈×C₆₇	C₂²	1	2	yes	—
21	C₄×D₈×C₆₇	C₂²	1	2	yes	—
22	#427×C₆₇	—	2	1	no	—
23	#427×C₆₇	C₂²	1	2	yes	—
24	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#102×C₆₇	—	2	1	no	—
26	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
27	#367×C₆₇	C₂²	1	4	yes	complemented
28	#366×C₆₇	C₂²	1	4	yes	complemented
29	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
30	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	complemented
31	#357×C₆₇	C₂²	1	4	yes	—
32	#357×C₆₇	C₂²	1	4	yes	—
33	C₂²×D₄×C₆₇	—	2	1	no	—
34	C₂³×C₄×C₆₇	—	4	1	no	—
35	C₂×D₈×C₆₇	—	2	1	no	—
36	C₂×D₈×C₆₇	—	2	2	no	—
37	C₂×D₈×C₆₇	Q₈	1	4	yes	complemented
38	C₂×D₈×C₆₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	—	2	1	no	—
42	C₄⋊Q₈×C₆₇	—	2	1	no	—
43	#125×C₆₇	—	2	2	no	—
44	C₂²⋊Q₈×C₆₇	—	4	1	no	—
45	C₂²⋊Q₈×C₆₇	—	4	1	no	—
46	C₄×Q₈×C₆₇	—	4	1	no	—
47	C₄×Q₈×C₆₇	—	4	1	no	—
48	C₄×D₄×C₆₇	—	2	2	no	—
49	C₄×D₄×C₆₇	—	2	2	no	—
50	C₄×D₄×C₆₇	—	4	1	no	—
51	C₄×D₄×C₆₇	—	4	1	no	—
52	#117×C₆₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₂²⋊C₄×C₆₇	—	4	1	no	—
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	M₃₂×C₆₇	—	2	4	no	—
57	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
58	#107×C₆₇	—	2	1	no	—
59	#107×C₆₇	—	2	1	no	—
60	#107×C₆₇	—	2	1	no	—
61	#107×C₆₇	C₂³	1	2	yes	—
62	#107×C₆₇	C₂³	1	2	yes	—
63	#107×C₆₇	C₂³	1	2	yes	—
64	#106×C₆₇	—	2	2	no	—
65	#106×C₆₇	—	2	2	no	—
66	#102×C₆₇	—	2	2	no	—
67	#102×C₆₇	—	2	2	no	—
68	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
69	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
72	C₂⁴×C₆₇	—	4	1	no	—
73	C₂×Q₈×C₆₇	—	4	1	no	—
74	C₂×Q₈×C₆₇	—	4	1	no	—
75	C₂×D₄×C₆₇	—	2	4	no	—
76	C₂×D₄×C₆₇	—	2	2	no	—
77	C₂×D₄×C₆₇	—	4	1	no	—
78	C₂×D₄×C₆₇	—	4	2	no	—
79	C₂²×C₄×C₆₇	—	4	1	no	—
80	C₂²×C₄×C₆₇	—	4	2	no	—
81	C₂²×C₄×C₆₇	—	4	2	no	—
82	C₂²×C₄×C₆₇	—	8	4	no	—
83	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
86	D₈×C₆₇	—	2	4	no	—
87	D₈×C₆₇	—	2	2	no	—
88	D₈×C₆₇	C₂×Q₈	1	4	yes	—
89	C₂×C₈×C₆₇	—	2	2	no	—
90	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂×C₈×C₆₇	#41	1	2	yes	—
95	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	—	2	2	no	—
98	C₄⋊C₄×C₆₇	—	2	2	no	—
99	C₄⋊C₄×C₆₇	—	4	1	no	—
100	C₄⋊C₄×C₆₇	—	4	2	no	—
101	C₄⋊C₄×C₆₇	—	4	2	no	—
102	C₄⋊C₄×C₆₇	—	4	1	no	—
103	C₄⋊C₄×C₆₇	—	4	1	no	—
104	C₂²⋊C₄×C₆₇	—	4	2	no	—
105	C₂²⋊C₄×C₆₇	—	4	2	no	—
106	C₂²⋊C₄×C₆₇	—	4	2	no	—
107	C₄²×C₆₇	—	4	2	no	—
108	C₄²×C₆₇	D₈	1	1	yes	characteristic
109	C₄²×C₆₇	D₈	1	1	yes	characteristic
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₆₇	—	2	4	no	—
113	C₂³×C₆₇	—	4	2	no	—
114	C₂³×C₆₇	—	4	4	no	—
115	C₂³×C₆₇	—	8	2	no	—
116	C₂³×C₆₇	—	8	2	no	—
117	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₆₇	—	8	2	no	—
119	Q₈×C₆₇	—	8	2	no	—
120	D₄×C₆₇	—	2	4	no	—
121	D₄×C₆₇	—	4	2	no	—
122	D₄×C₆₇	—	4	4	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	2	no	—
126	C₂×C₄×C₆₇	—	4	2	no	—
127	C₂×C₄×C₆₇	—	4	2	no	—
128	C₂×C₄×C₆₇	—	8	4	no	—
129	C₂×C₄×C₆₇	—	8	2	no	—
130	C₂×C₄×C₆₇	—	8	4	no	—
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
132	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₆₇	#137	1	2	yes	—
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₈×C₆₇	—	2	1	no	—
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
142	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
143	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	4	4	no	—
147	C₂²×C₆₇	—	8	8	no	—
148	C₂²×C₆₇	—	8	2	no	—
149	C₂²×C₆₇	—	8	2	no	—
150	C₂²×C₆₇	#474	1	2	yes	—
151	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	1	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	8	2	no	—
156	C₄×C₆₇	—	8	2	no	—
157	C₄×C₆₇	—	8	2	no	—
158	C₄×C₆₇	#474	1	2	yes	—
159	C₄×C₆₇	#509	1	2	yes	—
160	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
162	#33627	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₆₇	—	2	2	no	—
164	C₂×C₆₇	—	8	4	no	—
165	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
166	C₂×C₆₇	C₂×#509	1	1	yes	characteristic
167	C₂×C₆₇	#3414	1	1	yes	characteristic
168	#3160	C₂×C₆₇	1	1	yes	characteristic, complemented
169	C₂×C₄×D₈	C₂×C₆₇	1	1	yes	characteristic
170	#2228	C₂×C₆₇	1	4	yes	complemented
171	#2226	C₂×C₆₇	1	2	yes	complemented
172	#2224	C₂×C₆₇	1	2	yes	complemented
173	#2159	C₂×C₆₇	1	2	yes	complemented
174	#2150	C₂×C₆₇	1	1	yes	characteristic, complemented
175	#2140	C₂×C₆₇	1	2	yes	—
176	C₆₇	#33627	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²×D₈	C₂²×C₆₇	1	1	yes	characteristic
178	#533	—	2	1	no	—
179	C₂×C₄×D₄	—	2	1	no	—
180	C₂×M₃₂	C₂²×C₆₇	1	2	yes	—
181	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
182	#500	C₂²×C₆₇	1	1	yes	characteristic, complemented
183	#499	—	2	1	no	—
184	#499	C₂²×C₆₇	1	2	yes	complemented
185	#498	—	2	1	no	—
186	C₄×D₈	—	2	1	no	—
187	C₄×D₈	—	2	1	no	—
188	C₄×D₈	C₂²×C₆₇	1	2	yes	—
189	C₄×D₈	C₂²×C₆₇	1	2	yes	—
190	#427	—	2	1	no	—
191	#427	C₂²×C₆₇	1	2	yes	—
192	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
193	C₂×#102	—	2	1	no	—
194	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
195	#367	C₂²×C₆₇	1	4	yes	complemented
196	#366	C₂²×C₆₇	1	4	yes	complemented
197	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
198	C₄⋊C₁₆	C₂²×C₆₇	1	2	yes	complemented
199	#357	C₂²×C₆₇	1	4	yes	—
200	#357	C₂²×C₆₇	1	4	yes	—
201	C₂²×D₄	—	2	1	no	—
202	C₂³×C₄	—	4	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	2	no	—
205	C₂×D₈	Q₈×C₆₇	1	4	yes	complemented
206	C₂×D₈	C₂³×C₆₇	1	2	yes	—
207	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	2	2	no	—
212	C₂²⋊Q₈	—	4	1	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×Q₈	—	4	1	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	4	1	no	—
219	C₄×D₄	—	4	1	no	—
220	#117	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₂²⋊C₄	—	4	1	no	—
223	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
224	M₃₂	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
226	#107	—	2	1	no	—
227	#107	—	2	1	no	—
228	#107	—	2	1	no	—
229	#107	C₂³×C₆₇	1	2	yes	—
230	#107	C₂³×C₆₇	1	2	yes	—
231	#107	C₂³×C₆₇	1	2	yes	—
232	#106	—	2	2	no	—
233	#106	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
237	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
239	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
240	C₂⁴	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	2	4	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂²×C₄	—	4	1	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	8	4	no	—
251	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
252	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
254	D₈	—	2	4	no	—
255	D₈	—	2	2	no	—
256	D₈	C₂×Q₈×C₆₇	1	4	yes	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
261	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
262	C₂×C₈	#41×C₆₇	1	2	yes	—
263	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₄²	—	4	2	no	—
276	C₄²	D₈×C₆₇	1	1	yes	characteristic
277	C₄²	D₈×C₆₇	1	1	yes	characteristic
278	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
279	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₁₆	—	2	4	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	2	no	—
283	C₂³	—	8	2	no	—
284	C₂³	—	8	2	no	—
285	C₂³	C₂×D₈×C₆₇	1	1	yes	characteristic
286	Q₈	—	8	2	no	—
287	Q₈	—	8	2	no	—
288	D₄	—	2	4	no	—
289	D₄	—	4	4	no	—
290	D₄	—	4	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	4	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
300	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
301	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
302	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
305	C₂×C₄	#137×C₆₇	1	2	yes	—
306	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	C₂²⋊Q₈×C₆₇	1	2	yes	—
310	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
311	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	8	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	#474×C₆₇	1	2	yes	—
319	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#474×C₆₇	1	2	yes	—
327	C₄	#509×C₆₇	1	2	yes	—
328	C₄	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
329	C₄	C₂×#137×C₆₇	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	8	4	no	—
332	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
333	C₂	C₂×#509×C₆₇	1	1	yes	characteristic
334	C₂	C₆₇×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
prime radical	C₂²×D₈×C₆₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3160	C₂	semidirect
3. C₆₇×#2228	C₂	semidirect
4. C₆₇×#2226	C₂	semidirect
5. C₆₇×#2224	C₂	semidirect
6. C₆₇×#2159	C₂	semidirect
7. C₆₇×#2150	C₂	semidirect
13. #500×C₆₇	C₂²	semidirect
14. #500×C₆₇	C₂²	semidirect
16. #499×C₆₇	C₂²	semidirect
27. #367×C₆₇	C₂²	semidirect
28. #366×C₆₇	C₂²	semidirect
29. C₄⋊C₁₆×C₆₇	C₂²	semidirect
30. C₄⋊C₁₆×C₆₇	C₂²	semidirect
37. C₂×D₈×C₆₇	Q₈	semidirect
162. #33627	C₆₇	direct (N × H)
168. #3160	C₂×C₆₇	semidirect
170. #2228	C₂×C₆₇	semidirect
171. #2226	C₂×C₆₇	semidirect
172. #2224	C₂×C₆₇	semidirect
173. #2159	C₂×C₆₇	semidirect
174. #2150	C₂×C₆₇	semidirect
176. C₆₇	#33627	direct (N × H)
181. #500	C₂²×C₆₇	semidirect
182. #500	C₂²×C₆₇	semidirect
184. #499	C₂²×C₆₇	semidirect
195. #367	C₂²×C₆₇	semidirect
196. #366	C₂²×C₆₇	semidirect
197. C₄⋊C₁₆	C₂²×C₆₇	semidirect
198. C₄⋊C₁₆	C₂²×C₆₇	semidirect
205. C₂×D₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#3160	2
C₂×C₄×D₈×C₆₇	2
C₆₇×#2228	2
C₆₇×#2226	2
C₆₇×#2224	2
C₆₇×#2159	2
C₆₇×#2150	2
C₆₇×#2140	2
#33627	67

Subgroups — C67×#33627 #30712878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33627 #30712878