Subgroups — #20299481

Order 7232.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268657	C₂	1	1	yes	maximal, characteristic
2	#16813701	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₁₁₃	C₂	1	1	yes	maximal, characteristic
4	#14268657	—	2	1	no	—
5	#14268657	—	2	1	no	—
6	C₂×#11744753	C₄	1	2	yes	complemented
7	C₄⋊#90673	—	2	1	no	—
8	C₂²⋊#90673	C₄	1	2	yes	—
9	C₄×#90673	—	2	1	no	—
10	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
11	#14268631	C₄	1	1	yes	characteristic, complemented
12	#14268630	—	2	1	no	—
13	C₈×D₁₁₃	—	2	1	no	—
14	M₁₆×C₁₁₃	C₄	1	1	yes	characteristic, complemented
15	C₂²×D₁₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×#90673	—	2	2	no	—
17	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×D₁₁₃	D₄	1	1	yes	characteristic
20	C₄×D₁₁₃	D₄	1	1	yes	characteristic
21	C₄×D₁₁₃	D₄	1	1	yes	characteristic
22	C₄×D₁₁₃	Q₈	1	1	yes	characteristic
23	#11744753	—	2	2	no	—
24	C₈×C₁₁₃	—	2	1	no	—
25	#11744751	—	2	1	no	—
26	C₂²×C₁₁₃	C₄²	1	1	yes	characteristic
27	C₂×D₁₁₃	—	4	1	no	—
28	C₂×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×D₁₁₃	C₄⋊C₄	1	1	yes	characteristic
30	#90673	—	4	2	no	—
31	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
33	Dic₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
34	Dic₁₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₁₁₃	—	2	1	no	—
36	C₂×C₁₁₃	#95	1	1	yes	characteristic
37	D₁₁₃	—	2	1	no	—
38	D₁₁₃	—	2	1	no	—
39	C₁₁₃	#344	1	1	yes	characteristic, complemented, Sylow
40	#344	—	113	1	no	maximal, Sylow
41	C₂×M₁₆	—	113	1	no	—
42	C₂×M₁₆	—	113	1	no	—
43	#117	—	113	1	no	—
44	C₂²×C₄	—	113	1	no	—
45	M₁₆	—	113	1	no	—
46	M₁₆	—	226	1	no	—
47	M₁₆	—	226	1	no	—
48	M₁₆	—	226	1	no	—
49	M₁₆	#90673	1	1	yes	characteristic, complemented
50	C₂×C₈	—	113	2	no	—
51	C₂×C₈	—	226	1	no	—
52	C₄⋊C₄	—	226	1	no	—
53	C₂²⋊C₄	—	113	2	no	—
54	C₄²	—	226	1	no	—
55	C₂³	—	113	1	no	—
56	C₂×C₄	—	113	1	no	—
57	C₂×C₄	—	113	1	no	—
58	C₂×C₄	—	113	1	no	—
59	C₂×C₄	—	113	1	no	—
60	C₂×C₄	—	113	1	no	—
61	C₂×C₄	—	226	2	no	—
62	C₂×C₄	C₂×#90673	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	226	2	no	—
65	C₈	—	226	1	no	—
66	C₂²	—	113	1	no	—
67	C₂²	—	113	1	no	—
68	C₂²	—	452	1	no	—
69	C₂²	C₄×#90673	1	1	yes	characteristic
70	C₄	—	113	1	no	—
71	C₄	—	113	1	no	—
72	C₄	—	452	2	no	—
73	C₄	C₂²⋊#90673	1	1	yes	characteristic
74	C₄	C₄⋊#90673	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	226	1	no	—
77	C₂	—	226	1	no	—
78	C₂	(C₂×#90673)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	(C₂×#90673)⋊C₄
commutator	C₂²×C₁₁₃	C₄²
Frattini	C₂×C₄	C₂×#90673
Fitting	M₁₆×C₁₁₃	C₄
socle	C₂×C₁₁₃	#95
hypercenter	M₁₆	#90673
hypocenter	C₁₁₃	#344
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
6. C₂×#11744753	C₄	semidirect
11. #14268631	C₄	semidirect
14. M₁₆×C₁₁₃	C₄	semidirect
39. C₁₁₃	#344	semidirect
49. M₁₆	#90673	semidirect