Subgroups — #22600790

Order 9088.

344 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986707	C₂	1	1	yes	maximal, characteristic
2	#17986683	C₂	1	1	yes	maximal, characteristic
3	#17986406	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986337	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986333	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986328	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986327	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986308	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986282	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986237	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986106	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986068	C₂	1	1	yes	maximal, characteristic
13	#17986041	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#470×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
17	#14556127	C₂²	1	1	yes	characteristic
18	#14556121	—	2	1	no	—
19	#14556119	C₂²	1	1	yes	characteristic
20	#14556118	—	2	1	no	—
21	#14556118	—	2	1	no	—
22	#14556110	—	2	1	no	—
23	#14556108	—	2	1	no	—
24	#14556105	—	2	1	no	—
25	#14556104	C₂²	1	1	yes	characteristic
26	#14556104	C₂²	1	1	yes	characteristic
27	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
28	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
30	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
31	#14556087	C₂²	1	1	yes	characteristic
32	#14556086	—	2	1	no	—
33	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556052	C₂²	1	1	yes	characteristic
48	#14556052	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#14556038	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#103×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
61	C₂²×Dic₇₁	—	2	1	no	—
62	C₂×C₄×D₇₁	—	2	1	no	—
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
65	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₈×D₇₁	—	2	2	no	—
68	C₄⋊Dic₇₁	—	2	1	no	—
69	C₄⋊Dic₇₁	—	2	1	no	—
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	Dic₇₁⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
95	Q₁₆×C₇₁	—	2	2	no	—
96	M₁₆×C₇₁	—	2	2	no	—
97	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	—	2	1	no	—
100	C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₂²⋊C₄×C₇₁	—	2	1	no	—
105	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	2	1	no	—
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	—	2	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	—	2	1	no	—
117	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₇₁	—	2	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	—	8	1	no	—
133	C₈×C₇₁	—	2	1	no	—
134	C₈×C₇₁	#41	1	2	yes	—
135	C₇₁⋊C₈	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	—	2	1	no	—
139	C₂²×C₇₁	—	2	1	no	—
140	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₇₁	—	2	1	no	—
142	C₂×D₇₁	—	2	1	no	—
143	C₂×D₇₁	#137	1	2	yes	—
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	—	4	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	2	1	no	—
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	C₂×C₇₁	—	4	1	no	—
159	C₂×C₇₁	#547	1	1	yes	characteristic
160	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
161	C₂×C₇₁	#577	1	1	yes	characteristic
162	D₇₁	—	2	2	no	—
163	#3344	—	71	1	no	maximal, Sylow
164	C₇₁	#3344	1	1	yes	characteristic, complemented, Sylow
165	#554	—	71	1	no	—
166	#552	—	71	1	no	—
167	#501	—	71	1	no	—
168	#485	—	71	1	no	—
169	#484	—	71	1	no	—
170	#483	—	71	1	no	—
171	#483	—	71	1	no	—
172	#479	—	71	1	no	—
173	#475	—	71	1	no	—
174	#470	D₇₁	1	1	yes	characteristic, complemented
175	#467	—	71	1	no	—
176	#441	—	71	1	no	—
177	#435	—	71	1	no	—
178	#428	—	71	1	no	—
179	C₂×#106	—	71	1	no	—
180	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
181	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
182	C₂²×C₈	—	71	1	no	—
183	C₄⋊Q₈	—	71	1	no	—
184	#126	—	142	1	no	—
185	#125	—	71	1	no	—
186	#125	—	142	1	no	—
187	#125	—	142	1	no	—
188	#123	—	142	1	no	—
189	C₂²⋊Q₈	—	71	1	no	—
190	C₂²⋊Q₈	—	71	1	no	—
191	C₂²⋊Q₈	—	142	1	no	—
192	C₂²⋊Q₈	—	142	1	no	—
193	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
194	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
195	C₄×Q₈	—	71	1	no	—
196	C₄×Q₈	—	71	1	no	—
197	C₄×D₄	—	71	1	no	—
198	C₄×D₄	—	142	1	no	—
199	#117	—	71	1	no	—
200	#117	—	142	1	no	—
201	C₂×C₄⋊C₄	—	71	1	no	—
202	C₂×C₄⋊C₄	—	71	1	no	—
203	C₂×C₄⋊C₄	—	71	1	no	—
204	#107	—	71	1	no	—
205	#107	—	71	1	no	—
206	#107	—	71	1	no	—
207	#106	—	71	1	no	—
208	#106	—	71	1	no	—
209	#106	—	71	1	no	—
210	#106	—	71	1	no	—
211	#106	—	71	1	no	—
212	#106	C₂×D₇₁	1	1	yes	characteristic
213	C₄⋊C₈	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	C₂×D₇₁	1	1	yes	characteristic
219	#103	C₂×D₇₁	1	1	yes	characteristic
220	C₂²⋊C₈	—	71	1	no	—
221	C₂²⋊C₈	—	71	1	no	—
222	#97	—	71	1	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
225	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
226	C₂×D₄	—	71	1	no	—
227	C₂²×C₄	—	71	1	no	—
228	C₂²×C₄	—	142	1	no	—
229	C₂²×C₄	—	142	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
232	Q₁₆	—	2	2	no	—
233	M₁₆	—	2	2	no	—
234	C₂×C₈	—	71	1	no	—
235	C₂×C₈	—	71	1	no	—
236	C₂×C₈	—	142	2	no	—
237	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	71	1	no	—
242	C₄⋊C₄	—	71	1	no	—
243	C₄⋊C₄	—	71	1	no	—
244	C₄⋊C₄	—	71	1	no	—
245	C₄⋊C₄	—	71	1	no	—
246	C₄⋊C₄	—	142	1	no	—
247	C₄⋊C₄	—	142	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	C₂²×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	71	1	no	—
261	C₂²⋊C₄	—	71	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₄²	—	71	1	no	—
266	C₄²	—	142	1	no	—
267	C₄²	—	142	1	no	—
268	C₄²	—	142	1	no	—
269	C₂³	—	71	1	no	—
270	C₂³	D₄×D₇₁	1	1	yes	characteristic
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	568	1	no	—
276	D₄	—	142	2	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	284	2	no	—
284	C₂×C₄	—	568	1	no	—
285	C₂×C₄	—	568	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	71	1	no	—
288	C₂×C₄	—	71	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	—	142	1	no	—
292	C₂×C₄	—	142	1	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	142	1	no	—
299	C₈	—	142	1	no	—
300	C₈	#11765864	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	71	2	no	—
305	C₂²	—	142	1	no	—
306	C₂²	—	142	1	no	—
307	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	284	1	no	—
314	C₄	—	284	1	no	—
315	C₄	—	284	1	no	—
316	C₄	—	284	1	no	—
317	C₄	—	142	1	no	—
318	C₄	—	142	1	no	—
319	C₄	—	142	1	no	—
320	C₄	C₂×#11765864	1	1	yes	characteristic
321	C₄	#14556175	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	142	2	no	—
324	C₂	#17986649	1	1	yes	characteristic
325	C₂	#137×D₇₁	1	1	yes	characteristic
326	C₂	#17986808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#470×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#470	D₇₁
hypocenter	C₇₁	#3344
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #17986406	C₂	semidirect
4. #17986337	C₂	semidirect
5. #17986333	C₂	semidirect
6. #17986328	C₂	semidirect
7. #17986327	C₂	semidirect
8. #17986308	C₂	semidirect
9. #17986282	C₂	semidirect
10. #17986237	C₂	semidirect
11. #17986106	C₂	semidirect
13. #17986041	C₂	semidirect
14. #106×D₇₁	C₂	semidirect
15. #470×C₇₁	C₂	semidirect
164. C₇₁	#3344	semidirect
174. #470	D₇₁	semidirect