Subgroups — #22600712

Order 9088.

354 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	#17986696	C₂	1	1	yes	maximal, characteristic
2	#17986688	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986400	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986341	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986336	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986311	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986309	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986301	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986227	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986226	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986090	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₇₁)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986035	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#482×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765853	C₂²	1	1	yes	characteristic
17	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
19	#14556121	—	2	1	no	—
20	#14556121	—	2	1	no	—
21	#14556120	—	2	1	no	—
22	#14556119	C₂²	1	1	yes	characteristic
23	#14556119	C₂²	1	1	yes	characteristic
24	#14556110	—	2	1	no	—
25	#14556109	—	2	1	no	—
26	#14556108	C₂²	1	1	yes	characteristic
27	#14556107	C₂²	1	1	yes	characteristic
28	#14556099	—	2	1	no	—
29	#14556099	C₂²	1	1	yes	characteristic
30	#14556093	C₂²	1	1	yes	characteristic
31	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
33	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
34	#14556087	C₂²	1	1	yes	characteristic
35	#14556085	—	2	1	no	—
36	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
38	#14556062	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556061	C₂²	1	1	yes	characteristic
41	#14556060	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	#14556059	C₂²	1	1	yes	characteristic
45	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556053	C₂²	1	1	yes	characteristic
48	#14556051	C₂²	1	1	yes	characteristic
49	#14556049	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	#121×C₇₁	C₂²	1	1	yes	characteristic
53	#117×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	#102×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₇₁	—	2	1	no	—
61	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
62	C₂²×Dic₇₁	—	2	1	no	—
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
66	#11765853	—	2	2	no	—
67	C₂×C₇₁⋊C₈	—	2	2	no	—
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₈×D₇₁	—	2	2	no	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	Dic₇₁⋊C₄	—	2	1	no	—
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₇₁	—	2	1	no	—
95	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	—	2	1	no	—
105	C₂³×C₇₁	—	2	1	no	—
106	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
108	D₄×C₇₁	—	2	1	no	—
109	D₄×C₇₁	—	2	1	no	—
110	D₄×C₇₁	—	8	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₇₁	—	2	2	no	—
119	C₂²⋊D₇₁	—	4	2	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	8	1	no	—
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₄×D₇₁	—	2	1	no	—
129	C₄×D₇₁	—	2	1	no	—
130	C₄×D₇₁	—	4	2	no	—
131	C₄×D₇₁	—	8	1	no	—
132	Dic₁₄₂	—	2	1	no	—
133	Dic₁₄₂	—	2	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₈×C₇₁	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₇₁⋊C₈	#41	1	2	yes	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	—	2	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	—	4	1	no	—
144	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₇₁	—	2	1	no	—
146	C₂×D₇₁	—	2	1	no	—
147	C₂×D₇₁	#135	1	2	yes	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	4	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	—	4	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	—	4	1	no	—
161	C₂×C₇₁	—	4	1	no	—
162	C₂×C₇₁	—	8	1	no	—
163	C₂×C₇₁	#547	1	1	yes	characteristic
164	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
165	C₂×C₇₁	#577	1	1	yes	characteristic
166	D₇₁	—	2	2	no	—
167	#3337	—	71	1	no	maximal, Sylow
168	C₇₁	#3337	1	1	yes	characteristic, complemented, Sylow
169	#553	—	71	1	no	—
170	#552	—	71	1	no	—
171	#499	—	71	1	no	—
172	#485	—	71	1	no	—
173	#484	—	71	1	no	—
174	#482	D₇₁	1	1	yes	characteristic, complemented
175	#480	—	71	1	no	—
176	#479	—	71	1	no	—
177	#478	—	71	1	no	—
178	#465	—	71	1	no	—
179	#465	—	71	1	no	—
180	C₄×SD₁₆	—	71	1	no	—
181	C₈×D₄	—	71	1	no	—
182	#427	—	71	1	no	—
183	C₂×#106	—	71	1	no	—
184	C₂×SD₁₆	—	71	1	no	—
185	C₂²×C₈	—	71	1	no	—
186	C₂²×C₈	—	71	1	no	—
187	#126	—	142	1	no	—
188	#126	—	142	1	no	—
189	#125	—	71	1	no	—
190	#125	—	71	1	no	—
191	#125	—	142	1	no	—
192	#123	—	142	1	no	—
193	#123	—	142	1	no	—
194	C₂²⋊Q₈	—	71	1	no	—
195	C₂²⋊Q₈	—	71	1	no	—
196	#121	—	71	1	no	—
197	#121	—	142	1	no	—
198	#121	C₂×D₇₁	1	1	yes	characteristic
199	C₄×Q₈	—	71	1	no	—
200	C₄×D₄	—	71	1	no	—
201	C₄×D₄	—	71	1	no	—
202	C₄×D₄	—	142	1	no	—
203	#117	—	71	1	no	—
204	#117	—	142	1	no	—
205	#117	C₂×D₇₁	1	1	yes	characteristic
206	C₂×C₄⋊C₄	—	71	1	no	—
207	C₂×C₄⋊C₄	—	71	1	no	—
208	#107	—	71	1	no	—
209	#107	—	71	1	no	—
210	#107	—	71	1	no	—
211	#107	C₂×D₇₁	1	1	yes	characteristic
212	#106	—	71	1	no	—
213	#106	—	71	1	no	—
214	#106	—	71	1	no	—
215	#106	—	71	1	no	—
216	#106	C₂×D₇₁	1	1	yes	characteristic
217	C₄⋊C₈	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	#103	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	C₂×D₇₁	1	1	yes	characteristic
223	#102	C₂×D₇₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	71	1	no	—
225	C₂²⋊C₈	C₂×D₇₁	1	1	yes	characteristic
226	C₄×C₈	—	71	1	no	—
227	C₂×Q₈	—	71	1	no	—
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	—	71	1	no	—
230	C₂×D₄	—	142	1	no	—
231	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
232	C₂²×C₄	—	71	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	C₂²×C₄	—	142	1	no	—
235	C₂²×C₄	—	142	1	no	—
236	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
237	SD₁₆	—	142	2	no	—
238	C₂×C₈	—	71	1	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	142	2	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₄⋊C₄	—	71	1	no	—
245	C₄⋊C₄	—	71	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₄⋊C₄	—	71	1	no	—
248	C₄⋊C₄	—	142	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	—	142	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	71	1	no	—
263	C₂²⋊C₄	—	71	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₂²⋊C₄	—	142	1	no	—
266	C₂²⋊C₄	—	142	1	no	—
267	C₂²⋊C₄	—	142	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	71	1	no	—
271	C₄²	—	142	1	no	—
272	C₄²	—	142	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	71	1	no	—
275	C₂³	D₄×D₇₁	1	1	yes	characteristic
276	Q₈	—	142	1	no	—
277	Q₈	—	142	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	8	1	no	—
281	D₄	—	284	2	no	—
282	D₄	—	142	2	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	—	284	2	no	—
289	C₂×C₄	—	568	1	no	—
290	C₂×C₄	—	568	1	no	—
291	C₂×C₄	—	71	1	no	—
292	C₂×C₄	—	71	1	no	—
293	C₂×C₄	—	71	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	—	142	1	no	—
296	C₂×C₄	—	142	1	no	—
297	C₂×C₄	—	142	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
301	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	71	2	no	—
305	C₈	—	142	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	71	2	no	—
313	C₂²	—	142	1	no	—
314	C₂²	—	142	1	no	—
315	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	284	1	no	—
321	C₄	—	284	1	no	—
322	C₄	—	284	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	142	1	no	—
325	C₄	—	142	1	no	—
326	C₄	—	142	1	no	—
327	C₄	#41×D₇₁	1	1	yes	characteristic
328	C₄	#14556175	1	1	yes	characteristic
329	C₂	—	4	1	no	—
330	C₂	—	8	1	no	—
331	C₂	—	142	2	no	—
332	C₂	#17986647	1	1	yes	characteristic
333	C₂	#135×D₇₁	1	1	yes	characteristic
334	C₂	#17986809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#482×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#482	D₇₁
hypocenter	C₇₁	#3337
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
2. #17986688	C₂	semidirect
3. #17986400	C₂	semidirect
4. #17986341	C₂	semidirect
5. #17986336	C₂	semidirect
6. #17986311	C₂	semidirect
7. #17986309	C₂	semidirect
8. #17986301	C₂	semidirect
9. #17986227	C₂	semidirect
10. #17986226	C₂	semidirect
11. #17986090	C₂	semidirect
12. (C₂²⋊D₇₁)⋊C₈	C₂	semidirect
13. #17986035	C₂	semidirect
14. #106×D₇₁	C₂	semidirect
15. #482×C₇₁	C₂	semidirect
168. C₇₁	#3337	semidirect
174. #482	D₇₁	semidirect