Subgroups — #28392824

Order 14464.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₈₀₈	C₂	1	4	yes	maximal, complemented
2	#20298975	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₉₀₄	—	2	2	no	—
5	C₂×D₉₀₄	C₂²	1	4	yes	—
6	#16813619	—	2	2	no	—
7	D₁₈₀₈	—	2	8	no	—
8	C₂×C₁₆×C₁₁₃	C₂²	1	2	yes	—
9	C₄×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
10	C₂×D₄₅₂	—	2	4	no	—
11	C₂×D₄₅₂	—	4	4	no	—
12	D₉₀₄	—	2	8	no	—
13	D₉₀₄	—	4	4	no	—
14	C₂×C₈×C₁₁₃	D₄	1	1	yes	characteristic
15	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
16	C₄²×C₁₁₃	D₄	1	1	yes	characteristic
17	C₁₆×C₁₁₃	D₄	1	4	yes	complemented
18	C₂²×D₁₁₃	—	4	4	no	—
19	C₂×C₄×C₁₁₃	D₈	1	2	yes	—
20	C₂×C₄×C₁₁₃	C₂×D₄	1	1	yes	characteristic
21	D₄₅₂	—	4	4	no	—
22	D₄₅₂	—	4	4	no	—
23	D₄₅₂	—	8	8	no	—
24	C₈×C₁₁₃	D₈	1	2	yes	—
25	C₈×C₁₁₃	C₂×D₄	1	2	yes	—
26	C₂²×C₁₁₃	C₂×D₈	1	1	yes	characteristic
27	C₂×D₁₁₃	—	8	4	no	—
28	C₂×D₁₁₃	—	8	8	no	—
29	C₄×C₁₁₃	D₁₆	1	4	yes	complemented
30	C₄×C₁₁₃	#127	1	1	yes	characteristic
31	C₄×C₁₁₃	C₂×D₈	1	1	yes	characteristic
32	C₂×C₁₁₃	#493	1	1	yes	characteristic
33	C₂×C₁₁₃	C₂×D₁₆	1	2	yes	—
34	D₁₁₃	—	16	4	no	—
35	#2246	—	113	1	no	maximal, Sylow
36	C₁₁₃	#2246	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₁₆	—	113	4	no	—
38	#493	—	113	2	no	—
39	C₄×C₁₆	D₁₁₃	1	1	yes	characteristic, complemented
40	C₂×D₈	—	113	4	no	—
41	C₂×D₈	—	226	2	no	—
42	#127	—	226	2	no	—
43	D₁₆	—	226	8	no	—
44	C₂×C₁₆	C₂×D₁₁₃	1	2	yes	—
45	C₄×C₈	C₂×D₁₁₃	1	1	yes	characteristic
46	C₂×D₄	—	452	4	no	—
47	C₂×D₄	—	226	4	no	—
48	D₈	—	452	4	no	—
49	D₈	—	226	8	no	—
50	C₂×C₈	D₄₅₂	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
52	C₄²	D₄₅₂	1	1	yes	characteristic
53	C₁₆	D₄₅₂	1	4	yes	complemented
54	C₂³	—	452	4	no	—
55	D₄	—	904	8	no	—
56	D₄	—	452	4	no	—
57	D₄	—	452	4	no	—
58	C₂×C₄	D₉₀₄	1	2	yes	—
59	C₂×C₄	C₂×D₄₅₂	1	1	yes	characteristic
60	C₈	D₉₀₄	1	2	yes	—
61	C₈	C₂×D₄₅₂	1	2	yes	—
62	C₂²	—	904	8	no	—
63	C₂²	—	904	4	no	—
64	C₂²	C₂×D₉₀₄	1	1	yes	characteristic
65	C₄	D₁₈₀₈	1	4	yes	complemented
66	C₄	#16813619	1	1	yes	characteristic
67	C₄	C₂×D₉₀₄	1	1	yes	characteristic
68	C₂	—	1808	4	no	—
69	C₂	#20298975	1	1	yes	characteristic
70	C₂	C₂×D₁₈₀₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₉₀₄
commutator, cosocle	C₂×C₈×C₁₁₃	C₂³
Frattini	C₂×C₈	C₂²×D₁₁₃
Fitting	C₄×C₁₆×C₁₁₃	C₂
socle	C₂²×C₁₁₃	C₂×D₈
hypercenter	C₄×C₁₆	D₁₁₃
hypocenter	C₁₁₃	#2246
LCS	G ▷ C₂×C₈×C₁₁₃ ▷ C₄×C₁₁₃ ▷ C₂×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₈₀₈	C₂	semidirect
2. #20298975	C₂	semidirect
3. C₄×C₁₆×C₁₁₃	C₂	semidirect
17. C₁₆×C₁₁₃	D₄	semidirect
29. C₄×C₁₁₃	D₁₆	semidirect
36. C₁₁₃	#2246	semidirect
39. C₄×C₁₆	D₁₁₃	semidirect
53. C₁₆	D₄₅₂	semidirect
65. C₄	D₁₈₀₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₁₈₀₈	2
#20298975	2
C₄×C₁₆×C₁₁₃	2
#2246	113