Subgroups — #22599503

Order 9088.

334 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	#17986528	C₂	1	1	yes	maximal, characteristic
2	#125×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986401	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986398	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986353	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986333	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986328	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986316	C₂	1	1	yes	maximal, characteristic
9	#17986311	C₂	1	1	yes	maximal, characteristic
10	#17986308	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986304	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986301	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986297	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986050	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	#14556121	—	2	1	no	—
17	#14556120	C₂²	1	1	yes	characteristic
18	#14556120	C₂²	1	1	yes	characteristic
19	#14556119	C₂²	1	1	yes	characteristic
20	#14556119	C₂²	1	1	yes	characteristic
21	#14556118	—	2	1	no	—
22	#14556118	—	2	1	no	—
23	#14556110	—	2	1	no	—
24	#14556104	C₂²	1	1	yes	characteristic
25	#14556099	C₂²	1	1	yes	characteristic
26	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
27	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
28	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
29	#14556084	—	2	1	no	—
30	C₄⋊C₄×D₇₁	—	2	1	no	—
31	C₄⋊C₄×D₇₁	—	2	1	no	—
32	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
34	C₄²×D₇₁	C₂²	1	1	yes	characteristic
35	#14556062	C₂²	1	1	yes	characteristic
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556061	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	Dic₇₁⋊C₈	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	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	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	#124×C₇₁	C₂²	1	1	yes	characteristic
54	#103×C₇₁	C₂²	1	1	yes	characteristic
55	#103×C₇₁	C₂²	1	1	yes	characteristic
56	#102×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×C₂²⋊D₇₁	—	2	1	no	—
60	C₂²×Dic₇₁	—	2	1	no	—
61	C₂×C₄×D₇₁	—	2	1	no	—
62	C₂×C₄×D₇₁	—	2	1	no	—
63	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
64	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
65	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
66	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₇₁	—	2	1	no	—
69	C₄⋊Dic₇₁	—	2	1	no	—
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	Dic₇₁⋊C₄	—	2	1	no	—
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	—	2	1	no	—
83	Dic₇₁⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	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×Q₈×C₇₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
100	C₂²⋊C₄×C₇₁	—	2	1	no	—
101	C₄²×C₇₁	C₂³	1	1	yes	characteristic
102	C₂³×C₇₁	—	2	1	no	—
103	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
104	Q₈×C₇₁	—	2	1	no	—
105	Q₈×C₇₁	—	2	1	no	—
106	D₄×C₇₁	—	2	1	no	—
107	D₄×C₇₁	—	2	1	no	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₇₁	—	4	2	no	—
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	4	2	no	—
120	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
123	C₄×D₇₁	—	2	2	no	—
124	C₄×D₇₁	—	2	1	no	—
125	C₄×D₇₁	—	2	1	no	—
126	C₄×D₇₁	—	4	1	no	—
127	C₄×D₇₁	—	4	2	no	—
128	C₄×D₇₁	—	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	—	4	1	no	—
134	C₂²×C₇₁	—	4	1	no	—
135	C₂²×C₇₁	—	4	1	no	—
136	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₇₁	—	2	1	no	—
138	C₂×D₇₁	—	2	1	no	—
139	C₂×D₇₁	#135	1	2	yes	—
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	2₋¹⁺⁴	1	1	yes	characteristic
146	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₇₁	—	2	1	no	—
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	—	4	1	no	—
154	C₂×C₇₁	—	8	1	no	—
155	C₂×C₇₁	#536	1	1	yes	characteristic
156	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
157	C₂×C₇₁	#575	1	1	yes	characteristic
158	D₇₁	—	2	2	no	—
159	#3232	—	71	1	no	maximal, Sylow
160	C₇₁	#3232	1	1	yes	characteristic, complemented, Sylow
161	#529	—	71	1	no	—
162	C₂×#125	—	71	1	no	—
163	#499	—	71	1	no	—
164	#499	—	71	1	no	—
165	#488	—	71	1	no	—
166	#488	D₇₁	1	1	yes	characteristic, complemented
167	#484	—	71	1	no	—
168	#483	—	71	1	no	—
169	#481	—	71	1	no	—
170	#480	—	71	1	no	—
171	#479	—	71	1	no	—
172	#479	—	71	1	no	—
173	#478	—	71	1	no	—
174	#478	—	71	1	no	—
175	#431	—	71	1	no	—
176	#126	—	142	1	no	—
177	#125	—	71	1	no	—
178	#125	—	71	1	no	—
179	#125	—	71	1	no	—
180	#125	—	71	1	no	—
181	#125	—	142	1	no	—
182	#125	—	142	1	no	—
183	#125	C₂×D₇₁	1	1	yes	characteristic
184	#124	C₂×D₇₁	1	1	yes	characteristic
185	#123	—	142	1	no	—
186	C₂²⋊Q₈	—	71	1	no	—
187	#121	—	71	1	no	—
188	C₄×Q₈	—	71	1	no	—
189	C₄×D₄	—	71	1	no	—
190	C₄×D₄	—	142	1	no	—
191	#117	—	142	1	no	—
192	C₂×C₄⋊C₄	—	71	1	no	—
193	C₂×C₄⋊C₄	—	71	1	no	—
194	C₂×C₄⋊C₄	—	142	1	no	—
195	C₂×C₄⋊C₄	—	142	1	no	—
196	C₂×C₄²	—	71	1	no	—
197	#107	—	71	1	no	—
198	#107	—	71	1	no	—
199	#107	—	71	1	no	—
200	#107	—	71	1	no	—
201	#106	—	71	1	no	—
202	#106	—	71	1	no	—
203	#106	—	71	1	no	—
204	#106	—	71	1	no	—
205	C₄⋊C₈	—	71	1	no	—
206	C₄⋊C₈	—	71	1	no	—
207	#103	—	71	1	no	—
208	#103	—	71	1	no	—
209	#103	C₂×D₇₁	1	1	yes	characteristic
210	#103	C₂×D₇₁	1	1	yes	characteristic
211	#102	—	71	1	no	—
212	#102	—	71	1	no	—
213	#102	C₂×D₇₁	1	1	yes	characteristic
214	#102	C₂×D₇₁	1	1	yes	characteristic
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	C₂²×D₇₁	1	1	yes	characteristic
220	C₂×D₄	—	142	1	no	—
221	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
222	C₂²×C₄	—	71	1	no	—
223	C₂²×C₄	—	71	1	no	—
224	C₂²×C₄	—	71	1	no	—
225	C₂²×C₄	—	142	1	no	—
226	C₂²×C₄	—	142	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	—	71	1	no	—
235	C₄⋊C₄	—	71	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	—	142	1	no	—
245	C₄⋊C₄	—	142	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	C₂²×D₇₁	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	142	1	no	—
255	C₂²⋊C₄	—	142	1	no	—
256	C₂²⋊C₄	—	142	1	no	—
257	C₄²	—	71	1	no	—
258	C₄²	—	71	1	no	—
259	C₄²	—	71	1	no	—
260	C₄²	—	142	1	no	—
261	C₄²	C₂²×D₇₁	1	1	yes	characteristic
262	C₂³	—	2	1	no	—
263	C₂³	—	71	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	2	1	no	—
266	D₄	—	2	1	no	—
267	D₄	—	2	1	no	—
268	D₄	—	284	2	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	284	1	no	—
273	C₂×C₄	—	284	2	no	—
274	C₂×C₄	—	284	2	no	—
275	C₂×C₄	—	284	2	no	—
276	C₂×C₄	—	71	1	no	—
277	C₂×C₄	—	71	1	no	—
278	C₂×C₄	—	71	1	no	—
279	C₂×C₄	—	142	2	no	—
280	C₂×C₄	—	142	1	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	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	71	2	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	—	284	1	no	—
306	C₄	—	284	1	no	—
307	C₄	—	284	1	no	—
308	C₄	—	284	1	no	—
309	C₄	—	142	1	no	—
310	C₄	—	142	1	no	—
311	C₄	—	142	1	no	—
312	C₄	#14556175	1	1	yes	characteristic
313	C₄	#14556175	1	1	yes	characteristic
314	C₂	—	8	1	no	—
315	C₂	—	142	2	no	—
316	C₂	#17986569	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	#488×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#488	D₇₁
hypocenter	C₇₁	#3232
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
2. #125×D₇₁	C₂	semidirect
3. #17986401	C₂	semidirect
4. #17986398	C₂	semidirect
5. #17986353	C₂	semidirect
6. #17986333	C₂	semidirect
7. #17986328	C₂	semidirect
10. #17986308	C₂	semidirect
11. #17986304	C₂	semidirect
12. #17986301	C₂	semidirect
13. #17986297	C₂	semidirect
14. #17986050	C₂	semidirect
15. #488×C₇₁	C₂	semidirect
160. C₇₁	#3232	semidirect
166. #488	D₇₁	semidirect