Subgroups — #22553389

Order 8384.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523496	C₂	1	2	yes	maximal, complemented
2	C₄×D₅₂₄	C₂	1	1	yes	maximal, characteristic
3	(C₂×D₁₃₁)⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₂₄	C₄	1	1	yes	characteristic
7	C₂×C₄×D₁₃₁	C₂²	1	2	yes	—
8	#14523496	—	2	4	no	—
9	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₃₁	C₄	1	1	yes	characteristic
12	(C₂×D₁₃₁)⋊C₄	C₄	1	2	yes	—
13	C₂×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂²×D₁₃₁	C₂×C₄	1	2	yes	—
16	D₅₂₄	—	2	2	no	—
17	C₂×Dic₁₃₁	C₂×C₄	1	2	yes	—
18	C₄×D₁₃₁	—	2	2	no	—
19	C₄×D₁₃₁	—	2	2	no	—
20	C₁₃₁⋊C₈	—	2	1	no	—
21	C₁₃₁⋊C₈	D₄	1	2	yes	complemented
22	C₂×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₈×C₁₃₁	—	2	2	no	—
26	C₂²×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₁₃₁	—	2	2	no	—
28	C₂×D₁₃₁	—	2	2	no	—
29	C₂×D₁₃₁	—	2	2	no	—
30	C₄×C₁₃₁	—	2	1	no	—
31	C₄×C₁₃₁	M₁₆	1	2	yes	complemented
32	C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₃₁	#41	1	1	yes	characteristic
34	Dic₁₃₁	—	2	2	no	—
35	C₂×C₁₃₁	C₄×D₄	1	1	yes	characteristic
36	C₂×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
37	C₂×C₁₃₁	#131	1	1	yes	characteristic
38	D₁₃₁	—	4	2	no	—
39	C₁₃₁	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	131	1	no	maximal, Sylow
41	C₂×M₁₆	—	131	2	no	—
42	C₄×D₄	—	131	1	no	—
43	C₄⋊C₈	D₁₃₁	1	1	yes	characteristic, complemented
44	C₂²⋊C₈	—	131	2	no	—
45	C₄×C₈	—	131	1	no	—
46	C₂×D₄	—	131	1	no	—
47	C₂²×C₄	—	131	2	no	—
48	M₁₆	—	262	4	no	—
49	C₂×C₈	—	131	1	no	—
50	C₂×C₈	—	131	1	no	—
51	C₂×C₈	C₂×D₁₃₁	1	2	yes	—
52	C₄⋊C₄	—	131	1	no	—
53	C₂²⋊C₄	—	131	2	no	—
54	C₄²	C₂×D₁₃₁	1	1	yes	characteristic
55	C₂³	—	131	2	no	—
56	D₄	—	262	2	no	—
57	C₂×C₄	—	262	2	no	—
58	C₂×C₄	—	262	2	no	—
59	C₂×C₄	—	131	2	no	—
60	C₂×C₄	C₄×D₁₃₁	1	1	yes	characteristic
61	C₂×C₄	C₄×D₁₃₁	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
63	C₈	—	2	2	no	—
64	C₈	—	262	1	no	—
65	C₈	—	131	2	no	—
66	C₂²	—	262	2	no	—
67	C₂²	—	262	2	no	—
68	C₂²	—	262	2	no	—
69	C₂²	C₂×C₄×D₁₃₁	1	1	yes	characteristic
70	C₄	—	2	1	no	—
71	C₄	—	262	2	no	—
72	C₄	#14523496	1	2	yes	complemented
73	C₄	D₄×D₁₃₁	1	1	yes	characteristic
74	C₄	#14523514	1	1	yes	characteristic
75	C₂	—	524	2	no	—
76	C₂	#16862208	1	1	yes	characteristic
77	C₂	C₂×#14523496	1	1	yes	characteristic
78	C₂	#16862254	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂²×C₄
Fitting	C₄⋊C₈×C₁₃₁	C₂
hypercenter	C₄⋊C₈	D₁₃₁
hypocenter	C₁₃₁	#436
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#14523496	C₂	semidirect
3. (C₂×D₁₃₁)⋊C₈	C₂	semidirect
4. C₄×C₁₃₁⋊C₈	C₂	semidirect
5. C₄⋊C₈×C₁₃₁	C₂	semidirect
21. C₁₃₁⋊C₈	D₄	semidirect
31. C₄×C₁₃₁	M₁₆	semidirect
39. C₁₃₁	#436	semidirect
43. C₄⋊C₈	D₁₃₁	semidirect
72. C₄	#14523496	semidirect