Subgroups — #22597448

Order 9088.

432 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#11765865	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₁₄₂	C₂	1	2	yes	maximal, complemented
3	C₈×D₂₈₄	C₂	1	2	yes	maximal, complemented
4	(C₂²⋊D₇₁)⋊C₈	C₂	1	4	yes	maximal, complemented
5	#17986051	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×D₇₁	C₂	1	2	yes	maximal, complemented
8	#431×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#11765865	C₄	1	1	yes	characteristic
10	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₂×C₈×D₇₁	C₂²	1	4	yes	—
12	C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
13	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
14	C₄×D₂₈₄	C₄	1	1	yes	characteristic
15	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
16	C₄×C₂²⋊D₇₁	C₄	1	2	yes	—
17	C₄×C₂²⋊D₇₁	C₂²	1	2	yes	—
18	#14556087	C₄	1	1	yes	characteristic
19	#14556085	C₄	1	2	yes	—
20	C₄²×D₇₁	C₂²	1	2	yes	—
21	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
22	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
23	Dic₇₁⋊C₈	C₂²	1	4	yes	—
24	(C₂×D₇₁)⋊C₈	C₂²	1	4	yes	—
25	C₈×Dic₇₁	C₂²	1	4	yes	—
26	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
29	C₂²⋊C₈×C₇₁	C₂²	1	2	yes	—
30	C₄×C₈×C₇₁	C₂²	1	2	yes	—
31	#11765865	C₈	1	8	yes	complemented
32	C₂×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
33	C₂×D₂₈₄	C₂×C₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₇₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₇₁	C₂×C₄	1	2	yes	—
37	C₂×C₄×D₇₁	C₂³	1	2	yes	—
38	C₂×C₇₁⋊C₈	—	4	2	no	—
39	C₂×C₇₁⋊C₈	C₂³	1	4	yes	—
40	C₈×D₇₁	—	4	4	no	—
41	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
42	C₄⋊Dic₇₁	C₂×C₄	1	1	yes	characteristic
43	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
44	Dic₇₁⋊C₄	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₇₁	C₂×C₄	1	2	yes	—
46	(C₂×D₇₁)⋊C₄	C₂×C₄	1	2	yes	—
47	(C₂×D₇₁)⋊C₄	C₂×C₄	1	2	yes	—
48	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₇₁	C₂×C₄	1	2	yes	—
50	C₄×Dic₇₁	C₂³	1	2	yes	—
51	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₇₁	C₂³	1	2	yes	—
55	C₂×C₈×C₇₁	C₂³	1	2	yes	—
56	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
58	C₄²×C₇₁	C₂³	1	1	yes	characteristic
59	C₄²×C₇₁	C₂³	1	1	yes	characteristic
60	C₂³×C₇₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×D₇₁	C₂²×C₄	1	2	yes	—
62	C₂×C₄×C₇₁	—	2	1	no	—
63	C₂×C₄×C₇₁	—	2	1	no	—
64	C₂×C₄×C₇₁	—	4	1	no	—
65	C₂×C₄×C₇₁	C₂×C₈	1	4	yes	complemented
66	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
72	C₂²⋊D₇₁	C₂×C₈	1	8	yes	complemented
73	C₂×Dic₇₁	C₂²×C₄	1	2	yes	—
74	C₂×Dic₇₁	C₂²×C₄	1	2	yes	—
75	C₂×Dic₇₁	C₂²×C₄	1	2	yes	—
76	D₂₈₄	C₂×C₈	1	4	yes	complemented
77	C₄×D₇₁	—	2	2	no	—
78	C₄×D₇₁	—	2	2	no	—
79	C₄×D₇₁	—	4	2	no	—
80	C₄×D₇₁	C₂×C₈	1	8	yes	complemented
81	Dic₁₄₂	C₂×C₈	1	4	yes	complemented
82	C₈×C₇₁	—	2	2	no	—
83	C₈×C₇₁	—	2	2	no	—
84	C₇₁⋊C₈	—	2	4	no	—
85	C₂²×C₇₁	—	2	1	no	—
86	C₂²×C₇₁	—	2	1	no	—
87	C₂²×C₇₁	C₂²×C₈	1	2	yes	—
88	C₂²×C₇₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₇₁	—	2	2	no	—
90	C₂×D₇₁	—	2	2	no	—
91	C₂×D₇₁	C₂²×C₈	1	4	yes	—
92	C₄×C₇₁	—	2	1	no	—
93	C₄×C₇₁	—	2	1	no	—
94	C₄×C₇₁	—	2	1	no	—
95	C₄×C₇₁	C₂²×C₈	1	2	yes	—
96	C₄×C₇₁	C₂²×C₈	1	2	yes	—
97	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
98	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
99	Dic₇₁	—	2	2	no	—
100	Dic₇₁	—	2	2	no	—
101	Dic₇₁	C₂²×C₈	1	4	yes	—
102	C₂×C₇₁	—	2	2	no	—
103	C₂×C₇₁	#520	1	1	yes	characteristic
104	C₂×C₇₁	C₂³×C₈	1	1	yes	characteristic
105	C₂×C₇₁	#568	1	1	yes	characteristic
106	D₇₁	—	2	4	no	—
107	#2988	—	71	1	no	maximal, Sylow
108	C₇₁	#2988	1	1	yes	characteristic, complemented, Sylow
109	C₄×#41	—	71	1	no	—
110	C₈×Q₈	—	71	2	no	—
111	C₈×D₄	—	71	4	no	—
112	C₈×D₄	—	71	2	no	—
113	#431	—	71	2	no	—
114	#431	D₇₁	1	1	yes	characteristic, complemented
115	C₂×C₄⋊C₈	—	71	1	no	—
116	C₂×C₄⋊C₈	—	71	2	no	—
117	C₂×#41	—	71	1	no	—
118	C₂²×C₈	—	71	2	no	—
119	C₂²×C₈	—	71	4	no	—
120	C₄×Q₈	—	71	1	no	—
121	C₄×Q₈	—	71	1	no	—
122	C₄×D₄	—	71	2	no	—
123	C₄×D₄	—	71	2	no	—
124	C₄×D₄	—	71	1	no	—
125	C₄×D₄	—	71	1	no	—
126	#117	—	71	1	no	—
127	#117	—	71	2	no	—
128	C₂×C₄²	—	71	2	no	—
129	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
130	C₄⋊C₈	—	71	4	no	—
131	C₄⋊C₈	—	71	2	no	—
132	C₄⋊C₈	—	71	2	no	—
133	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
134	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
135	C₂²⋊C₈	—	71	4	no	—
136	C₂²⋊C₈	C₂×D₇₁	1	2	yes	—
137	C₄×C₈	—	71	4	no	—
138	C₄×C₈	C₂×D₇₁	1	2	yes	—
139	#41	—	71	8	no	—
140	C₂×Q₈	—	71	1	no	—
141	C₂×D₄	—	71	2	no	—
142	C₂×D₄	—	71	1	no	—
143	C₂²×C₄	—	71	2	no	—
144	C₂²×C₄	—	71	2	no	—
145	C₂²×C₄	—	71	2	no	—
146	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
148	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
149	C₂×C₈	—	284	4	no	—
150	C₂×C₈	—	284	2	no	—
151	C₂×C₈	—	71	4	no	—
152	C₂×C₈	C₂²×D₇₁	1	2	yes	—
153	C₂×C₈	C₂²×D₇₁	1	2	yes	—
154	C₄⋊C₄	—	71	2	no	—
155	C₄⋊C₄	—	71	2	no	—
156	C₄⋊C₄	—	71	1	no	—
157	C₄⋊C₄	—	71	1	no	—
158	C₂²⋊C₄	—	71	2	no	—
159	C₂²⋊C₄	—	71	2	no	—
160	C₂²⋊C₄	—	71	2	no	—
161	C₄²	—	71	2	no	—
162	C₄²	—	71	2	no	—
163	C₄²	—	71	2	no	—
164	C₄²	C₄×D₇₁	1	1	yes	characteristic
165	C₄²	C₄×D₇₁	1	1	yes	characteristic
166	C₄²	C₂²×D₇₁	1	1	yes	characteristic
167	C₄²	C₂²×D₇₁	1	1	yes	characteristic
168	C₂³	—	71	2	no	—
169	C₂³	C₂×C₄×D₇₁	1	1	yes	characteristic
170	Q₈	—	71	4	no	—
171	D₄	—	71	8	no	—
172	D₄	—	71	4	no	—
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	4	1	no	—
176	C₂×C₄	—	284	2	no	—
177	C₂×C₄	—	71	2	no	—
178	C₂×C₄	—	71	8	no	—
179	C₂×C₄	—	71	2	no	—
180	C₂×C₄	—	71	2	no	—
181	C₂×C₄	—	142	2	no	—
182	C₂×C₄	—	142	2	no	—
183	C₂×C₄	C₈×D₇₁	1	4	yes	complemented
184	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
185	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
186	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
187	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
188	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
190	C₈	—	2	2	no	—
191	C₈	—	2	2	no	—
192	C₈	—	142	4	no	—
193	C₂²	—	2	1	no	—
194	C₂²	—	2	1	no	—
195	C₂²	—	71	4	no	—
196	C₂²	—	142	2	no	—
197	C₂²	—	142	2	no	—
198	C₂²	C₂×C₈×D₇₁	1	2	yes	—
199	C₂²	C₂²×C₄×D₇₁	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	2	1	no	—
203	C₄	—	71	4	no	—
204	C₄	—	142	2	no	—
205	C₄	—	142	2	no	—
206	C₄	C₂×C₈×D₇₁	1	2	yes	—
207	C₄	C₂×C₈×D₇₁	1	2	yes	—
208	C₄	#14556174	1	1	yes	characteristic
209	C₄	#14556175	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	—	142	4	no	—
212	C₂	#17986479	1	1	yes	characteristic
213	C₂	C₂²×C₈×D₇₁	1	1	yes	characteristic
214	C₂	#17986765	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₁
commutator	C₂×C₇₁	C₂³×C₈
Fitting	#431×C₇₁	C₂
socle	C₂²×C₇₁	C₂³×C₄
hypercenter	#431	D₇₁
hypocenter	C₇₁	#2988
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
2. C₈×Dic₁₄₂	C₂	semidirect
3. C₈×D₂₈₄	C₂	semidirect
4. (C₂²⋊D₇₁)⋊C₈	C₂	semidirect
5. #17986051	C₂	semidirect
6. C₂×C₄⋊(C₇₁⋊C₈)	C₂	semidirect
7. C₄⋊C₈×D₇₁	C₂	semidirect
8. #431×C₇₁	C₂	semidirect
31. #11765865	C₈	semidirect
65. C₂×C₄×C₇₁	C₂×C₈	semidirect
72. C₂²⋊D₇₁	C₂×C₈	semidirect
76. D₂₈₄	C₂×C₈	semidirect
80. C₄×D₇₁	C₂×C₈	semidirect
81. Dic₁₄₂	C₂×C₈	semidirect
108. C₇₁	#2988	semidirect
114. #431	D₇₁	semidirect
183. C₂×C₄	C₈×D₇₁	semidirect