Subgroups — C₆₇×#8900 #30688151

Order 17152.

374 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2990	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1489	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1476	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1299	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
9	#446×C₆₇	—	2	1	no	—
10	#435×C₆₇	—	2	2	no	—
11	C₈×D₄×C₆₇	—	2	1	no	—
12	#432×C₆₇	—	2	1	no	—
13	#431×C₆₇	C₂²	1	1	yes	characteristic
14	#431×C₆₇	C₂²	1	1	yes	characteristic
15	#423×C₆₇	C₂²	1	1	yes	characteristic
16	#423×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₆₇	—	2	1	no	—
18	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
20	#341×C₆₇	—	2	2	no	—
21	#335×C₆₇	—	2	1	no	—
22	#334×C₆₇	—	2	1	no	—
23	Q₈⋊C₈×C₆₇	—	2	1	no	—
24	Q₈⋊C₈×C₆₇	C₄	1	1	yes	characteristic
25	Q₈⋊C₈×C₆₇	C₄	1	1	yes	characteristic
26	D₄⋊C₈×C₆₇	—	2	1	no	—
27	D₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
28	D₄⋊C₈×C₆₇	C₄	1	1	yes	characteristic
29	C₂×#41×C₆₇	Q₈	1	1	yes	characteristic
30	C₂×M₁₆×C₆₇	—	2	1	no	—
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂×M₁₆×C₆₇	D₄	1	2	yes	—
33	C₂²×C₈×C₆₇	—	2	1	no	—
34	C₂²×C₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
37	C₄×D₄×C₆₇	—	2	1	no	—
38	C₄×D₄×C₆₇	—	2	1	no	—
39	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
41	#117×C₆₇	—	2	1	no	—
42	#117×C₆₇	Q₈	1	1	yes	characteristic
43	C₂×C₄²×C₆₇	—	2	1	no	—
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₄⋊C₈×C₆₇	—	2	1	no	—
47	C₄⋊C₈×C₆₇	—	2	1	no	—
48	C₄⋊C₈×C₆₇	—	2	1	no	—
49	C₄⋊C₈×C₆₇	—	2	1	no	—
50	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
53	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
54	C₂²⋊C₈×C₆₇	—	2	2	no	—
55	C₂²⋊C₈×C₆₇	—	2	1	no	—
56	C₂²⋊C₈×C₆₇	—	2	1	no	—
57	#97×C₆₇	—	2	2	no	—
58	#97×C₆₇	—	2	1	no	—
59	C₄×C₈×C₆₇	—	2	1	no	—
60	C₄×C₈×C₆₇	—	2	1	no	—
61	C₄×C₈×C₆₇	—	2	1	no	—
62	C₄×C₈×C₆₇	—	2	1	no	—
63	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
65	#41×C₆₇	—	2	2	no	—
66	#41×C₆₇	—	4	1	no	—
67	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂×D₄×C₆₇	—	2	1	no	—
69	C₂×D₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	—	2	1	no	—
72	C₂²×C₄×C₆₇	—	2	1	no	—
73	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
76	M₁₆×C₆₇	—	2	2	no	—
77	M₁₆×C₆₇	—	4	1	no	—
78	M₁₆×C₆₇	—	4	2	no	—
79	M₁₆×C₆₇	—	4	2	no	—
80	C₂×C₈×C₆₇	—	2	1	no	—
81	C₂×C₈×C₆₇	—	2	1	no	—
82	C₂×C₈×C₆₇	—	2	1	no	—
83	C₂×C₈×C₆₇	—	2	1	no	—
84	C₂×C₈×C₆₇	—	2	2	no	—
85	C₂×C₈×C₆₇	—	2	1	no	—
86	C₂×C₈×C₆₇	—	2	1	no	—
87	C₂×C₈×C₆₇	—	4	1	no	—
88	C₂×C₈×C₆₇	—	4	1	no	—
89	C₂×C₈×C₆₇	—	4	1	no	—
90	C₂×C₈×C₆₇	—	4	1	no	—
91	C₂×C₈×C₆₇	—	8	2	no	—
92	C₄⋊C₄×C₆₇	—	2	1	no	—
93	C₄⋊C₄×C₆₇	—	2	1	no	—
94	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₆₇	—	2	1	no	—
97	C₂²⋊C₄×C₆₇	—	2	1	no	—
98	C₂²⋊C₄×C₆₇	—	2	1	no	—
99	C₄²×C₆₇	—	2	1	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₆₇	#41	1	1	yes	characteristic
106	C₂³×C₆₇	—	2	1	no	—
107	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
108	Q₈×C₆₇	—	2	1	no	—
109	Q₈×C₆₇	—	2	1	no	—
110	D₄×C₆₇	—	2	1	no	—
111	D₄×C₆₇	—	4	2	no	—
112	D₄×C₆₇	#108	1	2	yes	—
113	C₂×C₄×C₆₇	—	2	1	no	—
114	C₂×C₄×C₆₇	—	2	1	no	—
115	C₂×C₄×C₆₇	—	2	1	no	—
116	C₂×C₄×C₆₇	—	2	1	no	—
117	C₂×C₄×C₆₇	—	2	1	no	—
118	C₂×C₄×C₆₇	—	2	1	no	—
119	C₂×C₄×C₆₇	—	2	1	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	4	2	no	—
123	C₂×C₄×C₆₇	—	4	1	no	—
124	C₂×C₄×C₆₇	—	4	1	no	—
125	C₂×C₄×C₆₇	—	4	1	no	—
126	C₂×C₄×C₆₇	—	4	1	no	—
127	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
131	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
133	C₈×C₆₇	—	4	1	no	—
134	C₈×C₆₇	—	4	1	no	—
135	C₈×C₆₇	—	4	2	no	—
136	C₈×C₆₇	—	4	1	no	—
137	C₈×C₆₇	—	4	1	no	—
138	C₈×C₆₇	—	4	1	no	—
139	C₈×C₆₇	—	4	1	no	—
140	C₈×C₆₇	—	4	2	no	—
141	C₈×C₆₇	—	8	1	no	—
142	C₂²×C₆₇	—	2	1	no	—
143	C₂²×C₆₇	—	2	2	no	—
144	C₂²×C₆₇	—	2	1	no	—
145	C₂²×C₆₇	—	2	1	no	—
146	C₂²×C₆₇	—	4	1	no	—
147	C₂²×C₆₇	—	4	1	no	—
148	C₂²×C₆₇	#385	1	1	yes	characteristic
149	C₄×C₆₇	—	2	1	no	—
150	C₄×C₆₇	—	2	1	no	—
151	C₄×C₆₇	—	2	1	no	—
152	C₄×C₆₇	—	4	1	no	—
153	C₄×C₆₇	—	4	1	no	—
154	C₄×C₆₇	—	4	1	no	—
155	C₄×C₆₇	C₂×#108	1	1	yes	characteristic
156	C₄×C₆₇	#430	1	1	yes	characteristic
157	C₄×C₆₇	#443	1	1	yes	characteristic
158	C₄×C₆₇	#444	1	1	yes	characteristic
159	C₄×C₆₇	#454	1	1	yes	characteristic
160	C₄×C₆₇	#455	1	1	yes	characteristic
161	#8900	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₆₇	—	4	1	no	—
163	C₂×C₆₇	—	4	2	no	—
164	C₂×C₆₇	#1855	1	1	yes	characteristic
165	C₂×C₆₇	#1860	1	1	yes	characteristic
166	C₂×C₆₇	#1866	1	1	yes	characteristic
167	#2990	C₂×C₆₇	1	1	yes	characteristic
168	#1837	C₂×C₆₇	1	1	yes	characteristic, complemented
169	#1574	C₂×C₆₇	1	1	yes	characteristic, complemented
170	#1489	C₂×C₆₇	1	1	yes	characteristic
171	#1476	C₂×C₆₇	1	1	yes	characteristic
172	#1299	C₂×C₆₇	1	1	yes	characteristic, complemented
173	#1278	C₂×C₆₇	1	1	yes	characteristic, complemented
174	C₆₇	#8900	1	1	yes	characteristic, direct product factor, Sylow
175	C₄×#41	C₂²×C₆₇	1	1	yes	characteristic
176	#446	—	2	1	no	—
177	#435	—	2	2	no	—
178	C₈×D₄	—	2	1	no	—
179	#432	—	2	1	no	—
180	#431	C₂²×C₆₇	1	1	yes	characteristic
181	#431	C₂²×C₆₇	1	1	yes	characteristic
182	#423	C₂²×C₆₇	1	1	yes	characteristic
183	#423	C₂²×C₆₇	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	2	1	no	—
185	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
187	#341	—	2	2	no	—
188	#335	—	2	1	no	—
189	#334	—	2	1	no	—
190	Q₈⋊C₈	—	2	1	no	—
191	Q₈⋊C₈	C₄×C₆₇	1	1	yes	characteristic
192	Q₈⋊C₈	C₄×C₆₇	1	1	yes	characteristic
193	D₄⋊C₈	—	2	1	no	—
194	D₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
195	D₄⋊C₈	C₄×C₆₇	1	1	yes	characteristic
196	C₂×#41	Q₈×C₆₇	1	1	yes	characteristic
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	D₄×C₆₇	1	2	yes	—
200	C₂²×C₈	—	2	1	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₄×Q₈	C₂×C₄×C₆₇	1	1	yes	characteristic
203	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	—	2	1	no	—
206	C₄×D₄	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
208	#117	—	2	1	no	—
209	#117	Q₈×C₆₇	1	1	yes	characteristic
210	C₂×C₄²	—	2	1	no	—
211	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
212	C₄⋊C₈	—	2	2	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
218	C₄⋊C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
219	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
220	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
221	C₂²⋊C₈	—	2	2	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	#97	—	2	2	no	—
225	#97	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
231	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
232	#41	—	2	2	no	—
233	#41	—	4	1	no	—
234	C₂×Q₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
241	C₂²×C₄	C₂×Q₈×C₆₇	1	1	yes	characteristic
242	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
243	M₁₆	—	2	2	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	8	2	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
262	C₄⋊C₄	C₄⋊C₄×C₆₇	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	2	1	no	—
269	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
272	C₄²	#41×C₆₇	1	1	yes	characteristic
273	C₂³	—	2	1	no	—
274	C₂³	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	4	2	no	—
279	D₄	#108×C₆₇	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
296	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
297	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
298	C₂×C₄	C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
299	C₂×C₄	#123×C₆₇	1	1	yes	characteristic
300	C₈	—	4	1	no	—
301	C₈	—	4	1	no	—
302	C₈	—	4	2	no	—
303	C₈	—	4	1	no	—
304	C₈	—	4	1	no	—
305	C₈	—	4	2	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	—	8	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	#385×C₆₇	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	C₂×#108×C₆₇	1	1	yes	characteristic
323	C₄	#430×C₆₇	1	1	yes	characteristic
324	C₄	#443×C₆₇	1	1	yes	characteristic
325	C₄	#444×C₆₇	1	1	yes	characteristic
326	C₄	#454×C₆₇	1	1	yes	characteristic
327	C₄	#455×C₆₇	1	1	yes	characteristic
328	C₂	—	4	1	no	—
329	C₂	—	4	2	no	—
330	C₂	C₆₇×#1855	1	1	yes	characteristic
331	C₂	C₆₇×#1860	1	1	yes	characteristic
332	C₂	C₆₇×#1866	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	#120
commutator	C₄²	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#385
prime radical	C₂×#41×C₆₇	Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#1837	C₂	semidirect
3. C₆₇×#1574	C₂	semidirect
6. C₆₇×#1299	C₂	semidirect
7. C₆₇×#1278	C₂	semidirect
161. #8900	C₆₇	direct (N × H)
168. #1837	C₂×C₆₇	semidirect
169. #1574	C₂×C₆₇	semidirect
172. #1299	C₂×C₆₇	semidirect
173. #1278	C₂×C₆₇	semidirect
174. C₆₇	#8900	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2990	2
C₆₇×#1837	2
C₆₇×#1574	2
C₆₇×#1489	2
C₆₇×#1476	2
C₆₇×#1299	2
C₆₇×#1278	2
#8900	67

Subgroups — C67×#8900 #30688151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#8900 #30688151