Subgroups — #20298709

Order 7232.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₉₀₄	C₂	1	1	yes	maximal, characteristic
2	#16813584	C₂	1	2	yes	maximal, complemented
3	#16813542	C₂	1	2	yes	maximal, complemented
4	#16813532	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₅₂	C₂²	1	2	yes	—
7	C₂×C₄×D₁₁₃	—	2	2	no	—
8	D₉₀₄	C₄	1	4	yes	complemented
9	C₂×#11744751	C₂²	1	1	yes	characteristic
10	#14268624	—	2	2	no	—
11	C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₁₃	C₂²	1	2	yes	—
14	C₂²×D₁₁₃	—	2	2	no	—
15	C₂×C₄×C₁₁₃	—	2	2	no	—
16	C₂×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₁₃	D₄	1	2	yes	—
18	D₄₅₂	—	2	2	no	—
19	D₄₅₂	C₂×C₄	1	4	yes	complemented
20	C₄×D₁₁₃	—	4	4	no	—
21	C₈×C₁₁₃	C₂×C₄	1	2	yes	complemented
22	#11744751	—	2	1	no	—
23	C₂²×C₁₁₃	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₁₃	—	2	4	no	—
25	C₂×D₁₁₃	—	4	2	no	—
26	C₂×D₁₁₃	—	4	2	no	—
27	C₄×C₁₁₃	—	4	2	no	—
28	C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₁₁₃	#41	1	1	yes	characteristic
30	Dic₁₁₃	—	2	2	no	—
31	C₂×C₁₁₃	C₄×D₄	1	1	yes	characteristic
32	C₂×C₁₁₃	AGL(1,ℤ/8)	1	2	yes	—
33	D₁₁₃	—	4	4	no	—
34	C₁₁₃	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	113	1	no	maximal, Sylow
36	C₂×D₈	—	113	1	no	—
37	C₄×D₄	—	113	2	no	—
38	#106	D₁₁₃	1	1	yes	characteristic, complemented
39	#102	—	113	2	no	—
40	#97	—	113	1	no	—
41	C₂×D₄	—	113	2	no	—
42	C₂²×C₄	—	226	2	no	—
43	D₈	—	113	4	no	—
44	C₂×C₈	—	113	1	no	—
45	C₂×C₈	C₂×D₁₁₃	1	1	yes	characteristic
46	C₄⋊C₄	C₂×D₁₁₃	1	2	yes	—
47	C₂²⋊C₄	—	226	2	no	—
48	C₄²	—	113	1	no	—
49	C₂³	—	226	2	no	—
50	D₄	—	113	4	no	—
51	D₄	—	226	2	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	113	2	no	—
54	C₂×C₄	—	452	4	no	—
55	C₂×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
56	C₈	—	226	1	no	—
57	C₈	C₄×D₁₁₃	1	2	yes	complemented
58	C₂²	—	452	2	no	—
59	C₂²	—	452	2	no	—
60	C₂²	—	226	4	no	—
61	C₂²	D₄×D₁₁₃	1	1	yes	characteristic
62	C₄	—	4	2	no	—
63	C₄	—	226	2	no	—
64	C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
65	C₄	#14268648	1	1	yes	characteristic
66	C₂	—	452	4	no	—
67	C₂	#16813584	1	1	yes	characteristic
68	C₂	#16813649	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₁₃
commutator	C₄×C₁₁₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁₃
Fitting	#106×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₄
hypercenter	#106	D₁₁₃
hypocenter	C₁₁₃	#442
LCS	G ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
2. #16813584	C₂	semidirect
3. #16813542	C₂	semidirect
4. #16813532	C₂	semidirect
5. #106×C₁₁₃	C₂	semidirect
8. D₉₀₄	C₄	semidirect
19. D₄₅₂	C₂×C₄	semidirect
21. C₈×C₁₁₃	C₂×C₄	semidirect
34. C₁₁₃	#442	semidirect
38. #106	D₁₁₃	semidirect
57. C₈	C₄×D₁₁₃	semidirect