Subgroups — #22601122

Order 9088.

340 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986727	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986708	C₂	1	1	yes	maximal, characteristic
3	#17986368	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986366	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986348	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986306	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986204	C₂	1	1	yes	maximal, characteristic
10	#17986202	C₂	1	1	yes	maximal, characteristic
11	#17986202	C₂	1	1	yes	maximal, characteristic
12	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986096	C₂	1	1	yes	maximal, characteristic
14	#17986090	C₂	1	1	yes	maximal, characteristic
15	#440×C₇₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#11765853	C₂²	1	1	yes	characteristic
17	C₂×#11765853	C₂²	1	1	yes	characteristic
18	#14556128	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556126	C₂²	1	1	yes	characteristic
21	#14556126	C₂²	1	1	yes	characteristic
22	#14556120	—	2	1	no	—
23	#14556120	C₂²	1	1	yes	characteristic
24	#14556119	C₂²	1	1	yes	characteristic
25	#14556118	—	2	1	no	—
26	#14556118	—	2	1	no	—
27	#14556107	—	2	1	no	—
28	#14556105	—	2	1	no	—
29	#14556105	—	2	1	no	—
30	C₄×Dic₁₄₂	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	#14556093	C₂²	1	1	yes	characteristic
33	#14556093	C₂²	1	1	yes	characteristic
34	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
35	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
36	C₂×Dic₇₁⋊C₄	—	2	1	no	—
37	#14556061	C₂²	1	1	yes	characteristic
38	#14556059	C₂²	1	1	yes	characteristic
39	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₇₁⋊C₈	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	#14556053	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556052	C₂²	1	1	yes	characteristic
48	#14556051	C₂²	1	1	yes	characteristic
49	#14556051	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	C₂²×Dic₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	2	1	no	—
65	#11765853	—	2	2	no	—
66	#11765853	—	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	D₄	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	Dic₇₁⋊C₄	—	2	1	no	—
85	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
86	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	C₂²⋊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	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₇₁	—	2	1	no	—
96	SD₁₆×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	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	C₂³	1	1	yes	characteristic
105	C₂³×C₇₁	—	2	1	no	—
106	Q₈×C₇₁	—	2	1	no	—
107	Q₈×C₇₁	—	2	1	no	—
108	D₄×C₇₁	—	2	1	no	—
109	D₄×C₇₁	#41	1	2	yes	—
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	—	8	1	no	—
114	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	4	2	no	—
122	C₂×Dic₇₁	—	8	1	no	—
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	Dic₁₄₂	—	2	2	no	—
128	Dic₁₄₂	—	2	1	no	—
129	Dic₁₄₂	—	2	1	no	—
130	Dic₁₄₂	—	2	1	no	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	8	1	no	—
133	C₈×C₇₁	—	2	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₇₁⋊C₈	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₂²×C₇₁	—	2	2	no	—
138	C₂²×C₇₁	—	4	1	no	—
139	C₂²×C₇₁	—	4	1	no	—
140	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₇₁	—	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	C₂×#41	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	—	2	1	no	—
151	Dic₇₁	—	4	1	no	—
152	Dic₇₁	—	4	1	no	—
153	Dic₇₁	—	4	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	#137	1	2	yes	—
156	C₂×C₇₁	—	4	2	no	—
157	C₂×C₇₁	#548	1	1	yes	characteristic
158	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
159	C₂×C₇₁	#577	1	1	yes	characteristic
160	#3372	—	71	1	no	maximal, Sylow
161	C₇₁	#3372	1	1	yes	characteristic, complemented, Sylow
162	#557	—	71	1	no	—
163	#554	—	71	1	no	—
164	#491	—	71	1	no	—
165	#491	—	71	1	no	—
166	#487	—	71	1	no	—
167	#479	—	71	1	no	—
168	#479	—	71	1	no	—
169	#477	—	71	1	no	—
170	#461	—	71	1	no	—
171	#461	—	71	1	no	—
172	#461	—	71	1	no	—
173	#446	—	71	1	no	—
174	#440	—	71	1	no	—
175	#440	D₇₁	1	1	yes	characteristic
176	C₄×SD₁₆	—	71	1	no	—
177	C₂×SD₁₆	—	71	1	no	—
178	C₂×SD₁₆	—	71	1	no	—
179	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
180	C₄⋊Q₈	—	71	1	no	—
181	C₄⋊Q₈	—	71	1	no	—
182	C₄⋊Q₈	—	71	1	no	—
183	C₄⋊Q₈	—	71	1	no	—
184	#125	—	71	1	no	—
185	#125	—	71	1	no	—
186	#125	—	142	1	no	—
187	#125	—	142	1	no	—
188	#125	—	142	1	no	—
189	C₂²⋊Q₈	—	142	1	no	—
190	C₂²⋊Q₈	—	142	1	no	—
191	C₂²⋊Q₈	—	142	1	no	—
192	C₄×Q₈	—	71	1	no	—
193	C₄×Q₈	—	71	1	no	—
194	C₄×Q₈	—	71	1	no	—
195	C₄×Q₈	—	142	1	no	—
196	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
197	C₄×D₄	—	71	1	no	—
198	C₄×D₄	—	71	1	no	—
199	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
200	C₂×C₄⋊C₄	—	142	1	no	—
201	#107	—	71	1	no	—
202	#107	C₂×D₇₁	1	1	yes	characteristic
203	#106	—	71	1	no	—
204	C₄⋊C₈	—	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	—	71	1	no	—
210	#103	—	71	1	no	—
211	#103	—	71	1	no	—
212	#103	—	71	1	no	—
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	C₂×D₇₁	1	1	yes	characteristic
216	#102	C₂×D₇₁	1	1	yes	characteristic
217	#97	—	71	1	no	—
218	#97	C₂×D₇₁	1	1	yes	characteristic
219	C₄×C₈	—	71	1	no	—
220	C₂×Q₈	—	71	1	no	—
221	C₂×Q₈	—	71	1	no	—
222	C₂×Q₈	—	71	1	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
225	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	142	1	no	—
228	C₂²×C₄	—	142	1	no	—
229	SD₁₆	—	2	2	no	—
230	SD₁₆	—	142	2	no	—
231	SD₁₆	—	142	2	no	—
232	C₂×C₈	—	71	1	no	—
233	C₂×C₈	—	71	1	no	—
234	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	71	1	no	—
238	C₄⋊C₄	—	71	1	no	—
239	C₄⋊C₄	—	71	1	no	—
240	C₄⋊C₄	—	71	1	no	—
241	C₄⋊C₄	—	71	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	—	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	C₂²×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	142	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	71	1	no	—
262	C₄²	—	71	1	no	—
263	C₄²	—	142	1	no	—
264	C₄²	—	142	1	no	—
265	C₄²	C₂²×D₇₁	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	568	1	no	—
270	Q₈	—	142	2	no	—
271	Q₈	—	142	1	no	—
272	Q₈	—	142	1	no	—
273	Q₈	—	142	1	no	—
274	Q₈	—	142	1	no	—
275	D₄	—	2	1	no	—
276	D₄	#11765865	1	2	yes	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	8	1	no	—
281	C₂×C₄	—	284	2	no	—
282	C₂×C₄	—	568	1	no	—
283	C₂×C₄	—	71	1	no	—
284	C₂×C₄	—	71	1	no	—
285	C₂×C₄	—	71	1	no	—
286	C₂×C₄	—	71	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	D₄×D₇₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	142	1	no	—
297	C₈	—	142	1	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	284	1	no	—
308	C₄	—	284	1	no	—
309	C₄	—	284	1	no	—
310	C₄	—	284	1	no	—
311	C₄	—	71	2	no	—
312	C₄	—	142	1	no	—
313	C₄	—	142	1	no	—
314	C₄	—	142	1	no	—
315	C₄	C₂×#11765865	1	1	yes	characteristic
316	C₄	#14556175	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	#17986658	1	1	yes	characteristic
319	C₂	#137×D₇₁	1	1	yes	characteristic
320	C₂	#17986809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#440×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#440	D₇₁
hypocenter	C₇₁	#3372
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #17986727	C₂	semidirect
3. #17986368	C₂	semidirect
4. #17986366	C₂	semidirect
5. #17986348	C₂	semidirect
6. #17986306	C₂	semidirect
7. #17986305	C₂	semidirect
8. #17986293	C₂	semidirect
12. #17986127	C₂	semidirect
161. C₇₁	#3372	semidirect