Subgroups — #28299366

Order 13568.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20289853	C₂	1	2	yes	maximal
2	#20274331	C₂	1	1	yes	maximal, characteristic
3	#16811534	—	2	2	no	—
4	#16811026	—	2	2	no	—
5	#16811003	C₂²	1	1	yes	characteristic
6	C₈×#88322	C₄	1	1	yes	characteristic
7	C₅₃×#494	C₄	1	1	yes	characteristic
8	#13133278	—	4	2	no	—
9	C₂×Dic₂₁₂	—	4	1	no	—
10	#13133217	D₄	1	2	yes	—
11	#13133150	—	4	4	no	—
12	C₄×#88322	—	2	1	no	—
13	C₂×Q₁₆×C₅₃	—	4	1	no	—
14	C₄⋊Q₈×C₅₃	—	2	1	no	—
15	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈×C₅₃	—	4	1	no	—
17	C₂×Dic₁₀₆	—	2	2	no	—
18	#11719940	—	8	2	no	—
19	Q₁₆×C₅₃	—	8	2	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₄⋊C₄×C₅₃	—	4	2	no	—
22	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
23	C₄⋊Dic₅₃	—	4	2	no	—
24	C₂×#88322	—	2	2	no	—
25	Dic₂₁₂	—	8	2	no	—
26	Q₈×C₅₃	—	8	2	no	—
27	C₂×C₄×C₅₃	—	2	1	no	—
28	C₂×C₄×C₅₃	—	8	1	no	—
29	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
30	C₂×Dic₅₃	—	4	2	no	—
31	Dic₁₀₆	—	8	2	no	—
32	#88324	—	16	2	no	—
33	C₈×C₅₃	—	2	2	no	—
34	#88322	—	2	4	no	—
35	#7532	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	#353	1	1	yes	characteristic
37	C₄×C₅₃	—	2	2	no	—
38	C₄×C₅₃	—	2	1	no	—
39	C₄×C₅₃	—	16	1	no	—
40	Dic₅₃	—	8	2	no	—
41	#1713	—	53	1	no	—
42	#1411	—	53	2	no	—
43	C₂×C₅₃	—	2	1	no	—
44	C₂×C₅₃	#1408	1	1	yes	characteristic
45	#500	—	106	2	no	—
46	#494	—	53	1	no	—
47	#494	#6157	1	1	yes	characteristic
48	#356	—	106	2	no	—
49	C₈²	—	53	1	no	—
50	C₅₃	#7532	1	1	yes	characteristic, complemented, Sylow
51	C₂×Q₁₆	—	4	1	no	—
52	C₂×Q₁₆	—	212	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	53	2	no	—
55	#107	—	212	4	no	—
56	#101	—	212	2	no	—
57	C₄×C₈	—	106	1	no	—
58	C₄×C₈	C₂×#6157	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	106	2	no	—
61	Q₁₆	—	8	2	no	—
62	Q₁₆	—	424	2	no	—
63	M₁₆	—	424	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	106	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	212	2	no	—
68	C₄²	C₂²⋊#6157	1	1	yes	characteristic
69	Q₈	—	8	2	no	—
70	Q₈	—	424	2	no	—
71	C₂×C₄	—	2	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	212	2	no	—
74	C₂×C₄	#13133275	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	848	2	no	—
77	C₈	—	106	4	no	—
78	C₂²	#16811527	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	424	2	no	—
83	C₂	—	2	1	no	—
84	C₂	#20289846	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289846
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₄×C₈	C₂×#6157
Fitting	C₅₃×#494	C₄
socle	C₂×C₅₃	#1408
prime radical	C₂²×C₅₃	#353
cosocle	#16811003	C₂²
hypercenter	#494	#6157
hypocenter	C₅₃	#7532
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
50. C₅₃	#7532	semidirect

Maximal subgroups

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

subgroup	index
#20289853	2
#20274331	2
#7532	53