Subgroups — #20299472

Order 7232.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#90673	C₂	1	2	yes	maximal
2	C₂×C₈×D₁₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#90673	—	2	2	no	—
4	C₄⋊#90673	—	2	2	no	—
5	C₄⋊#90673	C₄	1	4	yes	complemented
6	C₂×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
7	C₂×#11744751	C₄	1	1	yes	characteristic, complemented
8	C₈×D₁₁₃	—	2	2	no	—
9	C₂×C₈×C₁₁₃	C₄	1	1	yes	characteristic, complemented
10	C₂²×D₁₁₃	D₄	1	1	yes	characteristic
11	C₂×#90673	—	2	4	no	—
12	C₂×#90673	—	4	2	no	—
13	C₂×#90673	—	4	2	no	—
14	C₂×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×Dic₁₁₃	Q₈	1	1	yes	characteristic
16	C₄×D₁₁₃	C₂×C₄	1	2	yes	—
17	C₄×D₁₁₃	D₄	1	2	yes	—
18	C₈×C₁₁₃	—	2	1	no	—
19	#11744751	—	2	1	no	—
20	C₂²×C₁₁₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₁₁₃	C₂²⋊C₄	1	2	yes	—
22	C₂×D₁₁₃	D₈	1	1	yes	characteristic
23	C₂×D₁₁₃	SD₁₆	1	2	yes	—
24	C₂×D₁₁₃	Q₁₆	1	1	yes	characteristic
25	#90673	—	4	4	no	—
26	C₄×C₁₁₃	C₄²	1	1	yes	characteristic
27	C₄×C₁₁₃	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₁₁₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₁₁₃	#95	1	1	yes	characteristic
30	C₂×C₁₁₃	#106	1	1	yes	characteristic
31	C₂×C₁₁₃	#107	1	1	yes	characteristic
32	D₁₁₃	#102	1	2	yes	—
33	D₁₁₃	#103	1	2	yes	—
34	C₁₁₃	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	113	1	no	maximal, Sylow
36	C₂²×C₈	—	113	1	no	—
37	C₂×C₄⋊C₄	—	113	2	no	—
38	C₂²×C₄	—	113	1	no	—
39	C₂²×C₄	—	226	2	no	—
40	C₂×C₈	—	113	1	no	—
41	C₂×C₈	—	226	2	no	—
42	C₂×C₈	#90673	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	113	4	no	—
44	C₄⋊C₄	—	226	2	no	—
45	C₂³	—	113	1	no	—
46	C₂×C₄	—	113	1	no	—
47	C₂×C₄	—	113	2	no	—
48	C₂×C₄	—	113	2	no	—
49	C₂×C₄	—	452	2	no	—
50	C₂×C₄	—	452	2	no	—
51	C₂×C₄	—	226	4	no	—
52	C₂×C₄	C₂×#90673	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	226	1	no	—
55	C₂²	—	113	2	no	—
56	C₂²	—	113	2	no	—
57	C₂²	—	113	1	no	—
58	C₂²	—	113	1	no	—
59	C₂²	C₄⋊#90673	1	1	yes	characteristic
60	C₄	—	113	2	no	—
61	C₄	—	452	4	no	—
62	C₄	C₄×#90673	1	1	yes	characteristic
63	C₄	C₂²⋊#90673	1	1	yes	characteristic
64	C₂	—	113	2	no	—
65	C₂	—	113	2	no	—
66	C₂	(C₂×#90673)⋊C₄	1	1	yes	characteristic
67	C₂	#16813690	1	1	yes	characteristic
68	C₂	#16813691	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#90673
commutator	C₄×C₁₁₃	C₄²
Frattini	C₂×C₄	C₂×#90673
Fitting	C₂×C₈×C₁₁₃	C₄
socle	C₂²×C₁₁₃	C₄⋊C₄
hypercenter	C₂×C₈	#90673
hypocenter	C₁₁₃	#340
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#90673	C₄	semidirect
7. C₂×#11744751	C₄	semidirect
9. C₂×C₈×C₁₁₃	C₄	semidirect
34. C₁₁₃	#340	semidirect
42. C₂×C₈	#90673	semidirect