Subgroups — #27298804

Order 13568.

398 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276965	C₂	1	4	yes	maximal, complemented
2	#20274330	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₄₈	—	2	8	no	—
5	#16811000	—	2	4	no	—
6	#16811000	C₂²	1	4	yes	—
7	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
8	C₈²×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×D₄₂₄	—	2	8	no	—
10	C₂×D₄₂₄	—	4	8	no	—
11	#13133214	—	2	4	no	—
12	D₈₄₈	—	4	16	no	—
13	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
14	C₄×C₈×C₅₃	D₄	1	2	yes	—
15	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×D₂₁₂	—	4	4	no	—
17	C₂×D₂₁₂	—	4	8	no	—
18	C₂×C₈×C₅₃	D₈	1	4	yes	—
19	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
20	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
21	D₄₂₄	—	4	16	no	—
22	D₄₂₄	—	8	16	no	—
23	C₁₆×C₅₃	D₈	1	8	yes	complemented
24	C₂²×D₅₃	—	8	4	no	—
25	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
26	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
27	D₂₁₂	—	8	8	no	—
28	D₂₁₂	—	8	16	no	—
29	C₈×C₅₃	D₁₆	1	8	yes	complemented
30	C₈×C₅₃	C₂×D₈	1	4	yes	—
31	#12515	—	53	1	no	maximal, Sylow
32	C₂²×C₅₃	#493	1	1	yes	characteristic
33	C₂×D₅₃	—	16	4	no	—
34	C₂×D₅₃	—	16	8	no	—
35	C₄×C₅₃	#493	1	2	yes	—
36	C₄×C₅₃	C₂×D₁₆	1	4	yes	—
37	#2246	—	53	4	no	—
38	#1712	—	53	2	no	—
39	C₈×C₁₆	D₅₃	1	1	yes	characteristic, complemented
40	C₂×C₅₃	#1712	1	1	yes	characteristic
41	C₂×C₅₃	#2246	1	2	yes	—
42	D₅₃	—	32	4	no	—
43	C₂×D₁₆	—	106	8	no	—
44	#493	—	53	4	no	—
45	#493	—	106	4	no	—
46	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
47	C₈²	C₂×D₅₃	1	1	yes	characteristic
48	C₅₃	#12515	1	1	yes	characteristic, complemented, Sylow
49	C₂×D₈	—	106	8	no	—
50	C₂×D₈	—	212	8	no	—
51	#127	—	106	4	no	—
52	D₁₆	—	212	16	no	—
53	C₂×C₁₆	D₂₁₂	1	4	yes	—
54	C₄×C₈	D₂₁₂	1	2	yes	—
55	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
56	C₂×D₄	—	212	4	no	—
57	C₂×D₄	—	212	8	no	—
58	D₈	—	424	16	no	—
59	D₈	—	212	16	no	—
60	C₂×C₈	D₄₂₄	1	4	yes	—
61	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
62	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
63	C₁₆	D₄₂₄	1	8	yes	complemented
64	C₂³	—	424	4	no	—
65	D₄	—	424	16	no	—
66	D₄	—	424	8	no	—
67	C₂×C₄	#13133214	1	1	yes	characteristic
68	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
69	C₈	D₈₄₈	1	8	yes	complemented
70	C₈	C₂×D₄₂₄	1	4	yes	—
71	C₂²	—	848	4	no	—
72	C₂²	—	848	8	no	—
73	C₂²	#16811000	1	1	yes	characteristic
74	C₄	#16811000	1	2	yes	—
75	C₄	C₂×D₈₄₈	1	4	yes	—
76	C₂	—	1696	4	no	—
77	C₂	#20274330	1	1	yes	characteristic
78	C₂	#20276965	1	2	yes	—

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#493
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12515
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
1. #20276965	C₂	semidirect
2. #20274330	C₂	semidirect
3. C₈×C₁₆×C₅₃	C₂	semidirect
23. C₁₆×C₅₃	D₈	semidirect
29. C₈×C₅₃	D₁₆	semidirect
39. C₈×C₁₆	D₅₃	semidirect
48. C₅₃	#12515	semidirect
63. C₁₆	D₄₂₄	semidirect
69. C₈	D₈₄₈	semidirect

Maximal subgroups

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

subgroup	index
#20276965	2
#20274330	2
C₈×C₁₆×C₅₃	2
#12515	53