Subgroups — #22599520

Order 9088.

332 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986555	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986534	C₂	1	1	yes	maximal, characteristic
3	#17986398	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986356	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986354	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986343	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986342	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986326	C₂	1	1	yes	maximal, characteristic
9	#17986325	C₂	1	1	yes	maximal, characteristic
10	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986288	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986283	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986278	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986050	C₂	1	1	yes	maximal, characteristic, complemented
15	#499×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	#14556127	C₂²	1	1	yes	characteristic
17	#14556127	C₂²	1	1	yes	characteristic
18	#14556123	—	2	1	no	—
19	#14556120	C₂²	1	1	yes	characteristic
20	#14556116	C₂²	1	1	yes	characteristic
21	#14556113	—	2	1	no	—
22	#14556108	C₂²	1	1	yes	characteristic
23	#14556106	—	2	1	no	—
24	#14556106	—	2	1	no	—
25	#14556103	C₂²	1	1	yes	characteristic
26	#14556093	—	2	1	no	—
27	#14556093	—	2	1	no	—
28	#14556093	C₂²	1	1	yes	characteristic
29	#14556090	—	2	1	no	—
30	#14556090	C₂²	1	1	yes	characteristic
31	#14556086	—	2	1	no	—
32	#14556086	C₂²	1	1	yes	characteristic
33	#14556086	C₂²	1	1	yes	characteristic
34	C₄²×D₇₁	C₂²	1	1	yes	characteristic
35	#14556061	C₂²	1	1	yes	characteristic
36	#14556061	C₂²	1	1	yes	characteristic
37	#14556059	C₂²	1	1	yes	characteristic
38	#14556059	C₂²	1	1	yes	characteristic
39	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	#14556053	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556051	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556050	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
52	#125×C₇₁	C₂²	1	1	yes	characteristic
53	#125×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#106×C₇₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	—	2	1	no	—
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
66	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
67	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
73	Dic₇₁⋊C₄	—	2	1	no	—
74	Dic₇₁⋊C₄	—	2	1	no	—
75	Dic₇₁⋊C₄	—	2	1	no	—
76	Dic₇₁⋊C₄	—	2	1	no	—
77	Dic₇₁⋊C₄	—	2	1	no	—
78	(C₂×D₇₁)⋊C₄	—	2	1	no	—
79	(C₂×D₇₁)⋊C₄	—	2	1	no	—
80	(C₂×D₇₁)⋊C₄	—	2	1	no	—
81	(C₂×D₇₁)⋊C₄	—	2	1	no	—
82	(C₂×D₇₁)⋊C₄	—	2	1	no	—
83	(C₂×D₇₁)⋊C₄	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	C₄×Dic₇₁	—	2	1	no	—
86	C₄×Dic₇₁	—	2	1	no	—
87	C₄×Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	D₄	1	1	yes	characteristic
89	C₄×Dic₇₁	D₄	1	1	yes	characteristic
90	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
93	C₄⋊C₄×C₇₁	—	2	1	no	—
94	C₄⋊C₄×C₇₁	—	2	1	no	—
95	C₄⋊C₄×C₇₁	—	2	1	no	—
96	C₄⋊C₄×C₇₁	—	2	1	no	—
97	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×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	C₄²×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²×D₇₁	—	2	1	no	—
103	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇₁	—	2	1	no	—
105	C₂×C₄×C₇₁	—	2	1	no	—
106	C₂×C₄×C₇₁	—	2	1	no	—
107	C₂×C₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
111	C₂×Dic₇₁	—	2	1	no	—
112	C₂×Dic₇₁	—	2	1	no	—
113	C₂×Dic₇₁	—	2	1	no	—
114	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
117	D₂₈₄	—	2	1	no	—
118	D₂₈₄	—	2	1	no	—
119	D₂₈₄	—	4	2	no	—
120	C₄×D₇₁	—	2	1	no	—
121	C₄×D₇₁	—	2	1	no	—
122	C₄×D₇₁	—	2	2	no	—
123	C₄×D₇₁	—	4	1	no	—
124	C₄×D₇₁	—	4	2	no	—
125	Dic₁₄₂	—	2	1	no	—
126	Dic₁₄₂	—	2	1	no	—
127	Dic₁₄₂	—	4	2	no	—
128	Dic₁₄₂	—	4	2	no	—
129	C₈×C₇₁	—	2	1	no	—
130	C₈×C₇₁	—	2	1	no	—
131	C₇₁⋊C₈	—	2	1	no	—
132	C₇₁⋊C₈	—	2	1	no	—
133	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
134	C₂×D₇₁	—	2	1	no	—
135	C₂×D₇₁	—	2	1	no	—
136	C₂×D₇₁	—	2	1	no	—
137	C₂×D₇₁	—	4	1	no	—
138	C₂×D₇₁	—	4	1	no	—
139	C₂×D₇₁	—	4	1	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
147	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₇₁	—	2	1	no	—
149	Dic₇₁	—	2	1	no	—
150	Dic₇₁	—	4	1	no	—
151	Dic₇₁	—	4	1	no	—
152	Dic₇₁	—	4	1	no	—
153	Dic₇₁	#135	1	2	yes	—
154	C₂×C₇₁	#536	1	1	yes	characteristic
155	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
156	C₂×C₇₁	#575	1	1	yes	characteristic
157	D₇₁	—	4	1	no	—
158	D₇₁	—	8	1	no	—
159	#3233	—	71	1	no	maximal, Sylow
160	C₇₁	#3233	1	1	yes	characteristic, complemented, Sylow
161	#533	—	71	1	no	—
162	#529	—	71	1	no	—
163	#499	—	71	1	no	—
164	#499	D₇₁	1	1	yes	characteristic, complemented
165	#488	—	71	1	no	—
166	#488	—	71	1	no	—
167	#485	—	71	1	no	—
168	#485	—	71	1	no	—
169	#482	—	71	1	no	—
170	#482	—	71	1	no	—
171	#477	—	71	1	no	—
172	#476	—	71	1	no	—
173	#475	—	71	1	no	—
174	#474	—	71	1	no	—
175	#431	—	71	1	no	—
176	C₄⋊Q₈	—	71	1	no	—
177	C₄⋊Q₈	—	71	1	no	—
178	#126	—	142	1	no	—
179	#125	—	71	1	no	—
180	#125	C₂×D₇₁	1	1	yes	characteristic
181	#125	C₂×D₇₁	1	1	yes	characteristic
182	#124	—	71	1	no	—
183	#123	—	142	1	no	—
184	C₂²⋊Q₈	—	71	1	no	—
185	C₂²⋊Q₈	—	142	1	no	—
186	C₂²⋊Q₈	—	142	1	no	—
187	#121	—	71	1	no	—
188	C₄×Q₈	—	71	1	no	—
189	C₄×Q₈	—	142	1	no	—
190	C₄×Q₈	—	142	1	no	—
191	C₄×D₄	—	71	1	no	—
192	C₄×D₄	—	142	1	no	—
193	#117	—	71	1	no	—
194	#117	—	71	1	no	—
195	#117	—	142	1	no	—
196	C₂×C₄²	—	71	1	no	—
197	#107	—	71	1	no	—
198	#107	—	71	1	no	—
199	#107	C₂×D₇₁	1	1	yes	characteristic
200	#107	C₂×D₇₁	1	1	yes	characteristic
201	#106	—	71	1	no	—
202	#106	—	71	1	no	—
203	#106	C₂×D₇₁	1	1	yes	characteristic
204	#106	C₂×D₇₁	1	1	yes	characteristic
205	C₄⋊C₈	—	71	1	no	—
206	C₄⋊C₈	—	71	1	no	—
207	#103	—	71	1	no	—
208	#103	—	71	1	no	—
209	#103	—	71	1	no	—
210	#103	—	71	1	no	—
211	#102	—	71	1	no	—
212	#102	—	71	1	no	—
213	#102	—	71	1	no	—
214	#102	—	71	1	no	—
215	C₂²⋊C₈	—	71	1	no	—
216	C₂²⋊C₈	—	71	1	no	—
217	C₄×C₈	—	71	1	no	—
218	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
219	C₂×Q₈	—	71	1	no	—
220	C₂×Q₈	—	142	1	no	—
221	C₂×Q₈	—	142	1	no	—
222	C₂×D₄	—	71	1	no	—
223	C₂×D₄	—	142	1	no	—
224	C₂²×C₄	—	71	1	no	—
225	C₂²×C₄	—	71	1	no	—
226	C₂²×C₄	—	71	1	no	—
227	C₂²×C₄	—	142	1	no	—
228	C₂×C₈	—	71	1	no	—
229	C₂×C₈	—	71	1	no	—
230	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	1	no	—
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	71	1	no	—
237	C₄⋊C₄	—	71	1	no	—
238	C₄⋊C₄	—	142	1	no	—
239	C₄⋊C₄	—	142	1	no	—
240	C₄⋊C₄	—	142	1	no	—
241	C₄⋊C₄	—	142	1	no	—
242	C₄⋊C₄	—	142	1	no	—
243	C₄⋊C₄	—	142	1	no	—
244	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
246	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
248	C₂²⋊C₄	—	142	1	no	—
249	C₂²⋊C₄	—	142	1	no	—
250	C₂²⋊C₄	—	142	1	no	—
251	C₂²⋊C₄	—	142	1	no	—
252	C₂²⋊C₄	—	142	1	no	—
253	C₂²⋊C₄	—	142	1	no	—
254	C₂²⋊C₄	—	142	1	no	—
255	C₄²	—	71	1	no	—
256	C₄²	—	71	1	no	—
257	C₄²	—	71	1	no	—
258	C₄²	—	142	1	no	—
259	C₄²	—	142	1	no	—
260	C₄²	—	142	1	no	—
261	C₄²	C₂²×D₇₁	1	1	yes	characteristic
262	C₂³	—	71	1	no	—
263	C₂³	—	142	1	no	—
264	Q₈	—	284	2	no	—
265	Q₈	—	284	2	no	—
266	Q₈	—	142	1	no	—
267	Q₈	—	142	1	no	—
268	D₄	—	284	2	no	—
269	D₄	—	142	1	no	—
270	D₄	—	142	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	284	1	no	—
276	C₂×C₄	—	284	2	no	—
277	C₂×C₄	—	71	1	no	—
278	C₂×C₄	—	71	1	no	—
279	C₂×C₄	—	71	1	no	—
280	C₂×C₄	—	142	2	no	—
281	C₂×C₄	—	142	1	no	—
282	C₂×C₄	—	142	1	no	—
283	C₂×C₄	—	142	1	no	—
284	C₂×C₄	—	142	1	no	—
285	C₂×C₄	—	142	1	no	—
286	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
287	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	142	1	no	—
292	C₈	—	142	1	no	—
293	C₂²	—	284	1	no	—
294	C₂²	—	284	1	no	—
295	C₂²	—	284	1	no	—
296	C₂²	—	142	1	no	—
297	C₂²	—	142	1	no	—
298	C₂²	—	142	1	no	—
299	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	284	1	no	—
307	C₄	—	284	1	no	—
308	C₄	—	284	1	no	—
309	C₄	—	71	2	no	—
310	C₄	—	142	1	no	—
311	C₄	—	142	1	no	—
312	C₄	#14556176	1	1	yes	characteristic
313	C₄	#14556176	1	1	yes	characteristic
314	C₂	—	284	1	no	—
315	C₂	—	568	1	no	—
316	C₂	#17986574	1	1	yes	characteristic
317	C₂	#135×D₇₁	1	1	yes	characteristic
318	C₂	#17986796	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#499×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#499	D₇₁
hypocenter	C₇₁	#3233
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #17986555	C₂	semidirect
3. #17986398	C₂	semidirect
4. #17986356	C₂	semidirect
5. #17986354	C₂	semidirect
6. #17986343	C₂	semidirect
7. #17986342	C₂	semidirect
10. #17986293	C₂	semidirect
11. #17986288	C₂	semidirect
12. #17986283	C₂	semidirect
13. #17986278	C₂	semidirect
14. #17986050	C₂	semidirect
15. #499×C₇₁	C₂	semidirect
160. C₇₁	#3233	semidirect
164. #499	D₇₁	semidirect