Subgroups — #20299633

Order 7232.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14268656	C₂	1	2	yes	maximal, complemented
2	C₄⋊#11744753	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#11744753	C₂	1	2	yes	maximal, complemented
4	C₄×#11744753	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic
6	#14268656	—	2	4	no	—
7	C₂×#11744753	C₂²	1	1	yes	characteristic
8	C₂×#11744753	C₂²	1	2	yes	—
9	C₂×#11744753	C₂²	1	1	yes	characteristic
10	C₂²×Dic₁₁₃	C₂²	1	2	yes	—
11	C₄⋊Dic₁₁₃	C₄	1	1	yes	characteristic
12	C₂²⋊Dic₁₁₃	C₄	1	2	yes	—
13	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
14	C₂×D₄×C₁₁₃	C₄	1	1	yes	characteristic
15	C₂³×C₁₁₃	C₂×C₄	1	2	yes	—
16	D₄×C₁₁₃	—	2	2	no	—
17	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₁₁₃	—	2	2	no	—
19	C₂×Dic₁₁₃	—	2	2	no	—
20	C₂×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₁₁₃	C₂×C₄	1	2	yes	—
22	C₂×Dic₁₁₃	C₂³	1	1	yes	characteristic
23	#11744753	—	2	2	no	—
24	#11744753	—	2	1	no	—
25	#11744753	D₄	1	2	yes	complemented
26	C₂²×C₁₁₃	—	2	2	no	—
27	C₂²×C₁₁₃	—	2	2	no	—
28	C₂²×C₁₁₃	—	2	2	no	—
29	C₂²×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₁₁₃	M₁₆	1	2	yes	complemented
31	Dic₁₁₃	—	2	2	no	—
32	Dic₁₁₃	—	2	1	no	—
33	Dic₁₁₃	C₂×D₄	1	1	yes	characteristic
34	Dic₁₁₃	#41	1	1	yes	characteristic
35	C₂×C₁₁₃	—	4	2	no	—
36	C₂×C₁₁₃	C₄×D₄	1	1	yes	characteristic
37	C₂×C₁₁₃	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₁₁₃	#131	1	1	yes	characteristic
39	C₁₁₃	#436	1	1	yes	characteristic, complemented, Sylow
40	#436	—	113	1	no	maximal, Sylow
41	C₂×M₁₆	—	113	2	no	—
42	C₄×D₄	—	113	1	no	—
43	C₄⋊C₈	—	113	1	no	—
44	C₂²⋊C₈	—	113	2	no	—
45	C₄×C₈	—	113	1	no	—
46	C₂×D₄	#90673	1	1	yes	characteristic
47	C₂²×C₄	—	113	2	no	—
48	M₁₆	—	226	4	no	—
49	C₂×C₈	—	113	1	no	—
50	C₂×C₈	—	113	1	no	—
51	C₂×C₈	—	113	2	no	—
52	C₄⋊C₄	—	113	1	no	—
53	C₂²⋊C₄	—	113	2	no	—
54	C₄²	—	113	1	no	—
55	C₂³	C₂×#90673	1	2	yes	—
56	D₄	—	2	2	no	—
57	C₂×C₄	—	113	1	no	—
58	C₂×C₄	—	113	1	no	—
59	C₂×C₄	—	113	2	no	—
60	C₂×C₄	—	226	2	no	—
61	C₂×C₄	—	226	2	no	—
62	C₂×C₄	C₂×#90673	1	1	yes	characteristic
63	C₈	—	113	2	no	—
64	C₈	—	226	2	no	—
65	C₈	—	226	1	no	—
66	C₂²	—	2	2	no	—
67	C₂²	—	2	2	no	—
68	C₂²	—	2	2	no	—
69	C₂²	C₂²×#90673	1	1	yes	characteristic
70	C₄	—	113	1	no	—
71	C₄	—	113	1	no	—
72	C₄	—	226	2	no	—
73	C₄	—	226	1	no	—
74	C₄	#14268656	1	2	yes	complemented
75	C₂	—	4	2	no	—
76	C₂	D₄×#90673	1	1	yes	characteristic
77	C₂	C₂×#14268656	1	1	yes	characteristic
78	C₂	#16813709	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#90673
commutator, socle	C₂²×C₁₁₃	C₂²×C₄
Fitting	C₂×D₄×C₁₁₃	C₄
hypercenter	C₂×D₄	#90673
hypocenter	C₁₁₃	#436
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#14268656	C₂	semidirect
2. C₄⋊#11744753	C₂	semidirect
3. C₂²⋊#11744753	C₂	semidirect
4. C₄×#11744753	C₂	semidirect
25. #11744753	D₄	semidirect
30. C₄×C₁₁₃	M₁₆	semidirect
39. C₁₁₃	#436	semidirect
74. C₄	#14268656	semidirect