Subgroups — #20299684

Order 7232.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊#11744762	C₂	1	2	yes	maximal
2	C₂×C₄⋊#90673	C₂	1	1	yes	maximal, characteristic
3	C₂×#11744762	—	2	2	no	—
4	C₂²×#90673	C₄	1	1	yes	characteristic
5	C₂²×#90673	C₂²	1	1	yes	characteristic
6	C₄⋊#90673	—	2	2	no	—
7	C₂×C₄×D₁₁₃	C₄	1	1	yes	characteristic
8	C₂²×D₁₁₃	C₂×C₄	1	1	yes	characteristic
9	C₂×#90673	—	2	1	no	—
10	C₂×#90673	—	4	1	no	—
11	C₂×#90673	—	4	1	no	—
12	C₂×#90673	D₄	1	2	yes	—
13	#11744762	—	4	2	no	—
14	C₂×C₄×C₁₁₃	C₈	1	1	yes	characteristic, complemented
15	C₂×Dic₁₁₃	C₈	1	1	yes	characteristic, complemented
16	C₄×D₁₁₃	—	2	1	no	—
17	C₄×D₁₁₃	—	2	1	no	—
18	C₂²×C₁₁₃	C₂×C₈	1	1	yes	characteristic
19	C₂×D₁₁₃	—	2	1	no	—
20	C₂×D₁₁₃	—	2	1	no	—
21	C₂×D₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×D₁₁₃	M₁₆	1	1	yes	characteristic
23	#90673	—	2	2	no	—
24	#90673	—	4	1	no	—
25	C₄×C₁₁₃	—	2	1	no	—
26	Dic₁₁₃	—	2	1	no	—
27	C₂×C₁₁₃	—	2	1	no	—
28	C₂×C₁₁₃	C₂²⋊C₈	1	1	yes	characteristic
29	D₁₁₃	—	2	1	no	—
30	D₁₁₃	#99	1	1	yes	characteristic
31	D₁₁₃	#101	1	1	yes	characteristic
32	C₁₁₃	#324	1	1	yes	characteristic, complemented, Sylow
33	#324	—	113	1	no	maximal, Sylow
34	C₂×C₄⋊C₄	—	113	1	no	—
35	C₂²⋊C₈	—	113	2	no	—
36	C₂²×C₄	—	113	1	no	—
37	C₂²×C₄	—	113	1	no	—
38	C₂²×C₄	—	113	1	no	—
39	C₂×C₈	—	226	2	no	—
40	C₄⋊C₄	—	226	2	no	—
41	C₂³	—	113	1	no	—
42	C₂×C₄	—	113	2	no	—
43	C₂×C₄	—	113	1	no	—
44	C₂×C₄	—	452	1	no	—
45	C₂×C₄	—	452	1	no	—
46	C₂×C₄	—	226	1	no	—
47	C₂×C₄	—	226	1	no	—
48	C₂×C₄	—	226	1	no	—
49	C₂×C₄	#11744762	1	1	yes	characteristic, complemented
50	C₈	—	452	2	no	—
51	C₂²	—	113	1	no	—
52	C₂²	—	113	1	no	—
53	C₂²	—	226	1	no	—
54	C₂²	—	226	1	no	—
55	C₂²	C₂×#11744762	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	452	1	no	—
58	C₄	—	226	2	no	—
59	C₄	—	226	1	no	—
60	C₂	—	2	1	no	—
61	C₂	—	113	1	no	—
62	C₂	—	113	1	no	—
63	C₂	—	226	1	no	—
64	C₂	C₂²⋊#11744762	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#11744762
commutator	C₂²×C₁₁₃	C₂×C₈
Frattini	C₂²	C₂×#11744762
Fitting	C₂×C₄×C₁₁₃	C₈
socle	C₂×C₁₁₃	C₂²⋊C₈
hypercenter	C₂×C₄	#11744762
hypocenter	C₁₁₃	#324
LCS	G ▷ C₂²×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
14. C₂×C₄×C₁₁₃	C₈	semidirect
15. C₂×Dic₁₁₃	C₈	semidirect
32. C₁₁₃	#324	semidirect
49. C₂×C₄	#11744762	semidirect