Subgroups — #27271751

Order 13568.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20276997	C₂	1	1	yes	maximal, characteristic
2	#20272791	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1422	C₂	1	1	yes	maximal, characteristic
4	#16811028	—	2	1	no	—
5	#16810451	—	2	1	no	—
6	#16810451	C₄	1	2	yes	—
7	#16810444	—	2	1	no	—
8	C₅₃×M₆₄	—	2	1	no	—
9	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
10	#13133217	—	4	1	no	—
11	#13133157	—	4	1	no	—
12	#13133151	—	2	2	no	—
13	#13133151	—	4	1	no	—
14	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
16	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
17	C₃₂×C₅₃	—	4	1	no	—
18	C₂×Dic₁₀₆	—	8	1	no	—
19	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
20	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
21	C₄²×C₅₃	SD₁₆	1	1	yes	characteristic
22	C₄⋊Dic₅₃	—	4	2	no	—
23	C₄⋊Dic₅₃	—	8	1	no	—
24	C₁₆×C₅₃	—	2	1	no	—
25	C₁₆×C₅₃	SD₁₆	1	1	yes	characteristic
26	C₁₆×C₅₃	Q₁₆	1	1	yes	characteristic
27	#11719908	—	8	1	no	—
28	C₂×C₄×C₅₃	—	2	1	no	—
29	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
30	C₂×Dic₅₃	—	8	2	no	—
31	Dic₁₀₆	—	16	2	no	—
32	C₈×C₅₃	#101	1	1	yes	characteristic
33	C₈×C₅₃	#103	1	1	yes	characteristic
34	C₈×C₅₃	SD₃₂	1	1	yes	characteristic
35	C₈×C₅₃	Q₃₂	1	1	yes	characteristic
36	#7453	—	53	1	no	maximal, Sylow
37	C₂²×C₅₃	#361	1	1	yes	characteristic
38	C₄×C₅₃	—	2	2	no	—
39	C₄×C₅₃	#332	1	1	yes	characteristic
40	C₄×C₅₃	#358	1	1	yes	characteristic
41	Dic₅₃	—	16	2	no	—
42	#2254	—	53	1	no	—
43	#1422	D₅₃	1	1	yes	characteristic
44	#1373	—	53	1	no	—
45	C₂×C₅₃	—	2	1	no	—
46	C₂×C₅₃	#1365	1	1	yes	characteristic
47	#500	—	106	1	no	—
48	M₆₄	—	2	1	no	—
49	#367	—	53	2	no	—
50	#367	—	106	1	no	—
51	#364	—	106	1	no	—
52	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
53	C₅₃	#7453	1	1	yes	characteristic, complemented, Sylow
54	C₄⋊Q₈	—	212	1	no	—
55	M₃₂	—	212	1	no	—
56	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
58	#107	—	106	2	no	—
59	#107	—	212	1	no	—
60	C₄×C₈	D₂₁₂	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₂×Q₈	—	424	1	no	—
63	C₂×C₈	D₄₂₄	1	1	yes	characteristic
64	C₂×C₈	#11719921	1	1	yes	characteristic
65	C₄⋊C₄	—	424	1	no	—
66	C₄⋊C₄	—	212	2	no	—
67	C₄²	#11719913	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	424	1	no	—
70	C₁₆	#11719923	1	1	yes	characteristic
71	C₁₆	#11719925	1	1	yes	characteristic
72	Q₈	—	848	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	424	2	no	—
75	C₂×C₄	#13133139	1	1	yes	characteristic
76	C₈	#13133135	1	1	yes	characteristic
77	C₈	#13133140	1	1	yes	characteristic
78	C₈	#13133162	1	1	yes	characteristic
79	C₈	Dic₄₂₄	1	1	yes	characteristic
80	C₂²	#16810437	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	848	2	no	—
83	C₄	#16810368	1	1	yes	characteristic
84	C₄	#16810428	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	#20272772	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20272772
commutator	C₂×C₁₆×C₅₃	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₅₃
Fitting	C₅₃×#1422	C₂
socle	C₂×C₅₃	#1365
prime radical	C₂²×C₅₃	#361
cosocle	C₄×C₁₆×C₅₃	C₂²
hypercenter	#1422	D₅₃
hypocenter	C₅₃	#7453
LCS	G ▷ C₂×C₁₆×C₅₃ ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
53. C₅₃	#7453	semidirect

Maximal subgroups

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

subgroup	index
#20276997	2
#20272791	2
C₅₃×#1422	2
#7453	53