Subgroups — #28299365

Order 13568.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289853	C₂	1	2	yes	maximal, complemented
2	#20274321	C₂	1	1	yes	maximal, characteristic
3	#16811534	—	2	2	no	—
4	#16811003	C₂²	1	1	yes	characteristic
5	#16811000	C₄	1	1	yes	characteristic
6	#16810970	—	2	2	no	—
7	C₈²×C₅₃	C₄	1	1	yes	characteristic
8	#13133278	—	4	2	no	—
9	C₂×Dic₂₁₂	—	4	1	no	—
10	C₂×D₄₂₄	—	4	1	no	—
11	#13133217	D₄	1	2	yes	—
12	#13133214	—	2	1	no	—
13	#13133139	—	4	4	no	—
14	C₄×C₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×D₂₁₂	—	4	2	no	—
17	C₂×D₂₁₂	—	4	1	no	—
18	C₂×Dic₁₀₆	—	2	2	no	—
19	#11719940	—	8	2	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₂×C₈×C₅₃	—	2	2	no	—
22	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	4	2	no	—
24	Dic₂₁₂	—	8	2	no	—
25	D₄₂₄	—	8	2	no	—
26	C₂²×D₅₃	—	8	1	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
29	C₂×Dic₅₃	—	4	2	no	—
30	D₂₁₂	—	8	4	no	—
31	D₂₁₂	—	8	2	no	—
32	Dic₁₀₆	—	8	2	no	—
33	#88324	—	16	2	no	—
34	C₈×C₅₃	—	2	2	no	—
35	C₈×C₅₃	—	4	2	no	—
36	#7531	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	#353	1	1	yes	characteristic
38	C₂×D₅₃	—	16	2	no	—
39	C₂×D₅₃	—	16	1	no	—
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	—	2	1	no	—
42	Dic₅₃	—	8	2	no	—
43	#1709	—	53	1	no	—
44	#1411	—	53	2	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#1408	1	1	yes	characteristic
47	D₅₃	—	32	1	no	—
48	#494	—	53	1	no	—
49	#493	—	53	1	no	—
50	#486	—	106	2	no	—
51	#356	—	106	2	no	—
52	C₈²	#6157	1	1	yes	characteristic
53	C₅₃	#7531	1	1	yes	characteristic, complemented, Sylow
54	C₂×Q₁₆	—	212	1	no	—
55	C₂×D₈	—	212	1	no	—
56	C₄⋊Q₈	—	53	2	no	—
57	#127	—	106	1	no	—
58	#102	—	212	4	no	—
59	#101	—	212	2	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	C₂×#6157	1	1	yes	characteristic
62	C₂×Q₈	—	106	2	no	—
63	C₂×D₄	—	212	2	no	—
64	C₂×D₄	—	212	1	no	—
65	Q₁₆	—	424	2	no	—
66	D₈	—	424	2	no	—
67	M₁₆	—	424	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	2	no	—
70	C₄⋊C₄	—	212	2	no	—
71	C₄²	C₂²⋊#6157	1	1	yes	characteristic
72	C₂³	—	424	1	no	—
73	Q₈	—	424	2	no	—
74	D₄	—	424	2	no	—
75	D₄	—	424	4	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	212	2	no	—
78	C₂×C₄	#13133276	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	4	2	no	—
81	C₈	—	848	2	no	—
82	C₂²	—	848	2	no	—
83	C₂²	—	848	1	no	—
84	C₂²	#16811528	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	2	1	no	—
87	C₄	—	424	2	no	—
88	C₂	—	2	1	no	—
89	C₂	—	1696	1	no	—
90	C₂	#20289847	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289847
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₈²×C₅₃	C₄
socle	C₂×C₅₃	#1408
prime radical	#16811000	C₄
cosocle	#16811003	C₂²
hypercenter	C₈²	#6157
hypocenter	C₅₃	#7531
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₈²

Complements

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

N	H ≅ G/N	type
1. #20289853	C₂	semidirect
53. C₅₃	#7531	semidirect

Maximal subgroups

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

subgroup	index
#20289853	2
#20274321	2
#7531	53