Subgroups — #22599611

Order 9088.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986629	C₂	1	1	yes	maximal, characteristic
2	#17986612	C₂	1	1	yes	maximal, characteristic
3	#17986409	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986369	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986350	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986343	C₂	1	1	yes	maximal, characteristic
7	#17986342	C₂	1	1	yes	maximal, characteristic
8	#17986336	C₂	1	1	yes	maximal, characteristic
9	#17986331	C₂	1	1	yes	maximal, characteristic
10	#17986309	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986283	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986058	C₂	1	1	yes	maximal, characteristic
15	#499×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	#14556128	C₂²	1	1	yes	characteristic
17	#14556128	C₂²	1	1	yes	characteristic
18	#14556127	—	2	1	no	—
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556123	—	2	1	no	—
21	#14556123	—	2	1	no	—
22	#14556119	—	2	1	no	—
23	#14556119	—	2	1	no	—
24	#14556119	C₂²	1	1	yes	characteristic
25	#14556118	—	2	1	no	—
26	#14556108	—	2	1	no	—
27	#14556108	C₂²	1	1	yes	characteristic
28	#14556108	C₂²	1	1	yes	characteristic
29	#14556106	—	2	1	no	—
30	#14556093	C₂²	1	1	yes	characteristic
31	#14556093	C₂²	1	1	yes	characteristic
32	#14556086	C₂²	1	1	yes	characteristic
33	#14556085	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×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	#14556053	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556051	C₂²	1	1	yes	characteristic
46	#14556051	C₂²	1	1	yes	characteristic
47	#14556051	C₂²	1	1	yes	characteristic
48	#14556051	C₂²	1	1	yes	characteristic
49	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#14556039	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×Dic₁₄₂	C₂³	1	1	yes	characteristic
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	C₄⋊Dic₇₁	—	2	1	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	—	2	1	no	—
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	D₄	1	1	yes	characteristic
81	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
82	(C₂×D₇₁)⋊C₄	—	2	1	no	—
83	(C₂×D₇₁)⋊C₄	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
86	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
87	C₄×Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇₁	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₇₁	—	2	1	no	—
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	C₄×D₇₁	—	2	1	no	—
118	C₄×D₇₁	—	2	1	no	—
119	C₄×D₇₁	—	8	1	no	—
120	Dic₁₄₂	—	2	1	no	—
121	Dic₁₄₂	—	2	1	no	—
122	Dic₁₄₂	—	2	1	no	—
123	Dic₁₄₂	—	2	1	no	—
124	Dic₁₄₂	—	8	1	no	—
125	Dic₁₄₂	—	8	1	no	—
126	C₈×C₇₁	—	2	1	no	—
127	C₈×C₇₁	—	2	1	no	—
128	C₇₁⋊C₈	—	2	1	no	—
129	C₇₁⋊C₈	—	2	1	no	—
130	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₇₁	—	2	1	no	—
132	C₂×D₇₁	—	2	1	no	—
133	C₂×D₇₁	—	2	1	no	—
134	C₄×C₇₁	—	2	1	no	—
135	C₄×C₇₁	—	2	1	no	—
136	C₄×C₇₁	—	4	1	no	—
137	C₄×C₇₁	—	4	1	no	—
138	C₄×C₇₁	—	4	1	no	—
139	C₄×C₇₁	—	4	1	no	—
140	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
141	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₇₁	—	2	1	no	—
143	Dic₇₁	—	2	1	no	—
144	Dic₇₁	—	2	1	no	—
145	Dic₇₁	—	4	1	no	—
146	Dic₇₁	—	4	1	no	—
147	Dic₇₁	—	4	1	no	—
148	Dic₇₁	—	4	1	no	—
149	C₂×C₇₁	#536	1	1	yes	characteristic
150	C₂×C₇₁	#577	1	1	yes	characteristic
151	C₂×C₇₁	#578	1	1	yes	characteristic
152	D₇₁	—	4	1	no	—
153	#3242	—	71	1	no	maximal, Sylow
154	C₇₁	#3242	1	1	yes	characteristic, complemented, Sylow
155	#544	—	71	1	no	—
156	#541	—	71	1	no	—
157	#501	—	71	1	no	—
158	#499	D₇₁	1	1	yes	characteristic, complemented
159	#491	—	71	1	no	—
160	#487	—	71	1	no	—
161	#485	—	71	1	no	—
162	#485	—	71	1	no	—
163	#484	—	71	1	no	—
164	#483	—	71	1	no	—
165	#479	—	71	1	no	—
166	#479	—	71	1	no	—
167	#477	—	71	1	no	—
168	#475	—	71	1	no	—
169	#433	—	71	1	no	—
170	C₄⋊Q₈	—	71	1	no	—
171	C₄⋊Q₈	—	71	1	no	—
172	C₄⋊Q₈	—	71	1	no	—
173	C₄⋊Q₈	—	142	1	no	—
174	#126	—	142	1	no	—
175	#126	—	142	1	no	—
176	#125	—	71	1	no	—
177	#125	—	142	1	no	—
178	#125	—	142	1	no	—
179	#125	—	142	1	no	—
180	#125	C₂×D₇₁	1	1	yes	characteristic
181	#125	C₂×D₇₁	1	1	yes	characteristic
182	C₂²⋊Q₈	—	71	1	no	—
183	C₂²⋊Q₈	—	71	1	no	—
184	C₂²⋊Q₈	—	142	1	no	—
185	C₂²⋊Q₈	—	142	1	no	—
186	C₄×Q₈	—	71	1	no	—
187	C₄×Q₈	—	71	1	no	—
188	#117	—	71	1	no	—
189	#117	—	71	1	no	—
190	C₂×C₄⋊C₄	—	71	1	no	—
191	#107	—	71	1	no	—
192	#107	—	71	1	no	—
193	#107	C₂×D₇₁	1	1	yes	characteristic
194	#107	C₂×D₇₁	1	1	yes	characteristic
195	#106	—	71	1	no	—
196	#106	—	71	1	no	—
197	#106	C₂×D₇₁	1	1	yes	characteristic
198	#106	C₂×D₇₁	1	1	yes	characteristic
199	C₄⋊C₈	—	71	1	no	—
200	C₄⋊C₈	—	71	1	no	—
201	#103	—	71	1	no	—
202	#103	—	71	1	no	—
203	#103	—	71	1	no	—
204	#103	—	71	1	no	—
205	#103	—	71	1	no	—
206	#103	—	71	1	no	—
207	#103	—	71	1	no	—
208	#103	—	71	1	no	—
209	C₂²⋊C₈	—	71	1	no	—
210	C₂²⋊C₈	—	71	1	no	—
211	#97	—	71	1	no	—
212	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
213	C₂×Q₈	—	71	1	no	—
214	C₂×Q₈	—	71	1	no	—
215	C₂×Q₈	—	142	1	no	—
216	C₂×Q₈	—	142	1	no	—
217	C₂²×C₄	—	71	1	no	—
218	C₂²×C₄	—	142	1	no	—
219	C₂×C₈	—	71	1	no	—
220	C₂×C₈	—	71	1	no	—
221	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	71	1	no	—
228	C₄⋊C₄	—	71	1	no	—
229	C₄⋊C₄	—	71	1	no	—
230	C₄⋊C₄	—	71	1	no	—
231	C₄⋊C₄	—	142	1	no	—
232	C₄⋊C₄	—	142	1	no	—
233	C₄⋊C₄	—	142	1	no	—
234	C₄⋊C₄	—	142	1	no	—
235	C₄⋊C₄	—	142	1	no	—
236	C₄⋊C₄	—	142	1	no	—
237	C₄⋊C₄	—	142	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	C₂²×D₇₁	1	1	yes	characteristic
243	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
246	C₂²⋊C₄	—	71	1	no	—
247	C₂²⋊C₄	—	71	1	no	—
248	C₂²⋊C₄	—	142	1	no	—
249	C₂²⋊C₄	—	142	1	no	—
250	C₂²⋊C₄	—	142	1	no	—
251	C₄²	—	71	1	no	—
252	C₄²	—	142	1	no	—
253	C₄²	—	142	1	no	—
254	C₄²	—	142	1	no	—
255	C₄²	C₂²×D₇₁	1	1	yes	characteristic
256	C₂³	—	71	1	no	—
257	Q₈	—	568	1	no	—
258	Q₈	—	568	1	no	—
259	Q₈	—	142	1	no	—
260	Q₈	—	142	1	no	—
261	Q₈	—	142	1	no	—
262	Q₈	—	142	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	568	1	no	—
268	C₂×C₄	—	71	1	no	—
269	C₂×C₄	—	71	1	no	—
270	C₂×C₄	—	71	1	no	—
271	C₂×C₄	—	142	1	no	—
272	C₂×C₄	—	142	1	no	—
273	C₂×C₄	—	142	1	no	—
274	C₂×C₄	—	142	1	no	—
275	C₂×C₄	—	142	1	no	—
276	C₂×C₄	—	142	1	no	—
277	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
278	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	142	1	no	—
283	C₈	—	142	1	no	—
284	C₂²	—	142	1	no	—
285	C₂²	—	142	1	no	—
286	C₂²	—	142	1	no	—
287	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	284	1	no	—
295	C₄	—	284	1	no	—
296	C₄	—	284	1	no	—
297	C₄	—	284	1	no	—
298	C₄	—	142	1	no	—
299	C₄	—	142	1	no	—
300	C₄	—	142	1	no	—
301	C₄	#14556176	1	1	yes	characteristic
302	C₄	#14556176	1	1	yes	characteristic
303	C₂	—	284	1	no	—
304	C₂	#17986574	1	1	yes	characteristic
305	C₂	#17986806	1	1	yes	characteristic
306	C₂	#17986813	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	#3242
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
3. #17986409	C₂	semidirect
4. #17986369	C₂	semidirect
5. #17986350	C₂	semidirect
10. #17986309	C₂	semidirect
11. #17986305	C₂	semidirect
12. #17986293	C₂	semidirect
13. #17986283	C₂	semidirect
15. #499×C₇₁	C₂	semidirect
154. C₇₁	#3242	semidirect
158. #499	D₇₁	semidirect