Subgroups — #22600866

Order 9088.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986717	C₂	1	1	yes	maximal, characteristic
2	#17986688	C₂	1	1	yes	maximal, characteristic
3	#17986406	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986342	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986340	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986340	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986336	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986258	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986240	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986105	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986070	C₂	1	1	yes	maximal, characteristic
14	#17986038	C₂	1	1	yes	maximal, characteristic, complemented
15	#427×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
17	C₂×#11765848	C₂²	1	1	yes	characteristic
18	#14556127	C₂²	1	1	yes	characteristic
19	#14556122	—	2	1	no	—
20	#14556121	—	2	1	no	—
21	#14556121	—	2	1	no	—
22	#14556120	—	2	1	no	—
23	#14556119	C₂²	1	1	yes	characteristic
24	#14556114	—	2	1	no	—
25	#14556110	—	2	1	no	—
26	#14556108	C₂²	1	1	yes	characteristic
27	#14556108	C₂²	1	1	yes	characteristic
28	#14556107	C₂²	1	1	yes	characteristic
29	#14556107	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₄₂	—	2	1	no	—
31	#14556093	C₂²	1	1	yes	characteristic
32	#14556093	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
34	#14556086	C₂²	1	1	yes	characteristic
35	#14556085	—	2	1	no	—
36	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	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	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	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
53	#117×C₇₁	C₂²	1	1	yes	characteristic
54	#117×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#107×C₇₁	C₂²	1	1	yes	characteristic
57	#106×C₇₁	C₂²	1	1	yes	characteristic
58	#106×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
63	C₂×C₄×D₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
65	Dic₂₈₄	—	2	2	no	—
66	#11765848	—	2	2	no	—
67	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
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	Dic₇₁⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₇₁	—	2	1	no	—
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	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×C₈×C₇₁	—	2	2	no	—
95	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	—	2	1	no	—
105	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×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	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₇₁	—	4	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	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	1	no	—
124	C₄×D₇₁	—	2	1	no	—
125	C₄×D₇₁	—	4	2	no	—
126	Dic₁₄₂	—	2	1	no	—
127	Dic₁₄₂	—	2	1	no	—
128	Dic₁₄₂	—	2	1	no	—
129	Dic₁₄₂	—	2	1	no	—
130	Dic₁₄₂	—	4	2	no	—
131	C₈×C₇₁	—	2	1	no	—
132	C₈×C₇₁	#41	1	2	yes	—
133	C₇₁⋊C₈	—	2	1	no	—
134	C₇₁⋊C₈	—	2	1	no	—
135	C₂²×C₇₁	—	2	1	no	—
136	C₂²×C₇₁	—	2	1	no	—
137	C₂²×C₇₁	—	2	1	no	—
138	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₇₁	—	2	1	no	—
140	C₂×D₇₁	—	2	1	no	—
141	C₂×D₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
148	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₇₁	—	2	1	no	—
150	Dic₇₁	—	2	1	no	—
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	4	1	no	—
153	Dic₇₁	—	4	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	—	4	1	no	—
156	C₂×C₇₁	—	4	1	no	—
157	C₂×C₇₁	#547	1	1	yes	characteristic
158	C₂×C₇₁	#575	1	1	yes	characteristic
159	C₂×C₇₁	#578	1	1	yes	characteristic
160	D₇₁	—	4	1	no	—
161	#3349	—	71	1	no	maximal, Sylow
162	C₇₁	#3349	1	1	yes	characteristic, complemented, Sylow
163	#556	—	71	1	no	—
164	#552	—	71	1	no	—
165	#501	—	71	1	no	—
166	#485	—	71	1	no	—
167	#485	—	71	1	no	—
168	#485	—	71	1	no	—
169	#484	—	71	1	no	—
170	#479	—	71	1	no	—
171	#477	—	71	1	no	—
172	#470	—	71	1	no	—
173	#467	—	71	1	no	—
174	#441	—	71	1	no	—
175	#435	—	71	1	no	—
176	#428	—	71	1	no	—
177	#427	D₇₁	1	1	yes	characteristic, complemented
178	C₂×Q₁₆	—	71	1	no	—
179	C₂×M₁₆	—	71	1	no	—
180	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
181	C₄⋊Q₈	—	71	1	no	—
182	#126	—	142	1	no	—
183	#126	—	142	1	no	—
184	#126	—	142	1	no	—
185	#125	—	71	1	no	—
186	#125	—	142	1	no	—
187	#124	—	142	1	no	—
188	#123	—	142	1	no	—
189	C₂²⋊Q₈	—	71	1	no	—
190	C₂²⋊Q₈	—	71	1	no	—
191	C₂²⋊Q₈	—	71	1	no	—
192	C₂²⋊Q₈	—	71	1	no	—
193	C₄×Q₈	—	71	1	no	—
194	C₄×Q₈	—	71	1	no	—
195	C₄×Q₈	—	142	1	no	—
196	C₄×D₄	—	71	1	no	—
197	#117	—	71	1	no	—
198	#117	—	142	1	no	—
199	#117	C₂×D₇₁	1	1	yes	characteristic
200	#117	C₂×D₇₁	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	71	1	no	—
202	#107	—	71	1	no	—
203	#107	—	71	1	no	—
204	#107	—	71	1	no	—
205	#107	C₂×D₇₁	1	1	yes	characteristic
206	#107	C₂×D₇₁	1	1	yes	characteristic
207	#106	—	71	1	no	—
208	#106	—	71	1	no	—
209	#106	C₂×D₇₁	1	1	yes	characteristic
210	#106	C₂×D₇₁	1	1	yes	characteristic
211	C₄⋊C₈	—	71	1	no	—
212	#103	—	71	1	no	—
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	C₂²⋊C₈	—	71	1	no	—
219	C₂²⋊C₈	—	71	1	no	—
220	#97	—	71	1	no	—
221	C₂×Q₈	—	71	1	no	—
222	C₂×Q₈	—	71	1	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂×D₄	—	71	1	no	—
225	C₂²×C₄	—	71	1	no	—
226	C₂²×C₄	—	142	1	no	—
227	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
228	Q₁₆	—	142	2	no	—
229	M₁₆	—	142	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	71	1	no	—
232	C₂×C₈	—	71	1	no	—
233	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
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	C₂²×D₇₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
255	C₂²⋊C₄	—	71	1	no	—
256	C₂²⋊C₄	—	142	1	no	—
257	C₂²⋊C₄	—	142	1	no	—
258	C₂²⋊C₄	—	142	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	142	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	71	1	no	—
264	C₄²	—	142	1	no	—
265	C₄²	—	142	1	no	—
266	C₄²	—	142	1	no	—
267	C₂³	—	71	1	no	—
268	C₂³	D₄×D₇₁	1	1	yes	characteristic
269	Q₈	—	284	2	no	—
270	Q₈	—	142	1	no	—
271	Q₈	—	142	1	no	—
272	Q₈	—	142	1	no	—
273	Q₈	—	142	1	no	—
274	D₄	—	284	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	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	—	284	2	no	—
282	C₂×C₄	—	71	1	no	—
283	C₂×C₄	—	71	1	no	—
284	C₂×C₄	—	71	1	no	—
285	C₂×C₄	—	142	1	no	—
286	C₂×C₄	—	142	1	no	—
287	C₂×C₄	—	142	1	no	—
288	C₂×C₄	—	142	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	142	1	no	—
295	C₈	—	142	1	no	—
296	C₈	#11765865	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	142	1	no	—
301	C₂²	—	142	1	no	—
302	C₂²	—	142	1	no	—
303	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	284	1	no	—
310	C₄	—	284	1	no	—
311	C₄	—	284	1	no	—
312	C₄	—	284	1	no	—
313	C₄	—	142	1	no	—
314	C₄	—	142	1	no	—
315	C₄	—	142	1	no	—
316	C₄	C₂×#11765865	1	1	yes	characteristic
317	C₄	#14556176	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	284	1	no	—
320	C₂	#17986650	1	1	yes	characteristic
321	C₂	#17986796	1	1	yes	characteristic
322	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	#427×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#427	D₇₁
hypocenter	C₇₁	#3349
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #427

Complements

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

N	H ≅ G/N	type
3. #17986406	C₂	semidirect
4. #17986342	C₂	semidirect
5. #17986340	C₂	semidirect
6. #17986340	C₂	semidirect
7. #17986336	C₂	semidirect
8. #17986305	C₂	semidirect
9. #17986293	C₂	semidirect
10. #17986258	C₂	semidirect
11. #17986240	C₂	semidirect
12. #17986105	C₂	semidirect
14. #17986038	C₂	semidirect
15. #427×C₇₁	C₂	semidirect
162. C₇₁	#3349	semidirect
177. #427	D₇₁	semidirect