Subgroups — #27270823

Order 13568.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272785	C₂	1	2	yes	maximal, complemented
2	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11719908	—	2	2	no	—
4	C₅₃×#493	—	2	2	no	—
5	C₅₃×#493	C₄	1	2	yes	—
6	C₈²×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#11719908	—	4	2	no	—
8	C₂×D₈×C₅₃	—	4	2	no	—
9	C₂×D₈×C₅₃	—	4	2	no	—
10	C₂×D₈×C₅₃	—	4	2	no	—
11	#127×C₅₃	—	2	2	no	—
12	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
13	C₄×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×D₄×C₅₃	—	4	2	no	—
15	C₂×D₄×C₅₃	—	4	4	no	—
16	D₈×C₅₃	—	8	4	no	—
17	D₈×C₅₃	—	8	4	no	—
18	D₈×C₅₃	—	8	2	no	—
19	D₈×C₅₃	—	8	2	no	—
20	C₂×C₈×C₅₃	—	2	1	no	—
21	C₂×C₈×C₅₃	D₈	1	2	yes	—
22	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
23	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	#11719908	—	8	2	no	—
25	C₂³×C₅₃	—	8	2	no	—
26	D₄×C₅₃	—	8	4	no	—
27	D₄×C₅₃	—	8	4	no	—
28	D₄×C₅₃	—	8	4	no	—
29	C₂×C₄×C₅₃	#100	1	1	yes	characteristic
30	C₂×C₄×C₅₃	#102	1	2	yes	—
31	C₈×C₅₃	—	2	2	no	—
32	C₈×C₅₃	—	2	2	no	—
33	C₈×C₅₃	D₁₆	1	2	yes	—
34	C₈×C₅₃	SD₃₂	1	2	yes	—
35	#7096	—	53	1	no	maximal, Sylow
36	C₂²×C₅₃	—	16	2	no	—
37	C₂²×C₅₃	—	16	4	no	—
38	C₂²×C₅₃	#331	1	1	yes	characteristic
39	C₄×C₅₃	—	2	1	no	—
40	C₄×C₅₃	#357	1	2	yes	—
41	C₄×C₅₃	#361	1	2	yes	—
42	#1712	D₅₃	1	1	yes	characteristic
43	#1370	—	53	2	no	—
44	C₂×C₅₃	—	32	2	no	—
45	C₂×C₅₃	#1357	1	1	yes	characteristic
46	C₂×C₅₃	#1361	1	2	yes	—
47	#493	—	2	2	no	—
48	#493	Dic₅₃	1	2	yes	—
49	C₄⋊C₁₆	—	106	2	no	—
50	C₈²	C₂×D₅₃	1	1	yes	characteristic
51	C₅₃	#7096	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₈	—	4	2	no	—
53	C₂×D₈	—	4	2	no	—
54	C₂×D₈	—	4	2	no	—
55	#127	—	2	2	no	—
56	C₂×C₁₆	—	212	2	no	—
57	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
58	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
59	C₂×D₄	—	4	2	no	—
60	C₂×D₄	—	4	4	no	—
61	D₈	—	8	4	no	—
62	D₈	—	8	4	no	—
63	D₈	—	8	2	no	—
64	D₈	—	8	2	no	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	#11719922	1	2	yes	—
67	C₂×C₈	#11719923	1	2	yes	—
68	C₄²	C₂²⋊Dic₅₃	1	1	yes	characteristic
69	C₁₆	—	424	2	no	—
70	C₂³	—	8	2	no	—
71	D₄	—	8	4	no	—
72	D₄	—	8	4	no	—
73	D₄	—	8	4	no	—
74	C₂×C₄	#13133134	1	1	yes	characteristic
75	C₂×C₄	#13133138	1	2	yes	—
76	C₈	—	2	2	no	—
77	C₈	—	2	2	no	—
78	C₈	#13133158	1	2	yes	—
79	C₈	#13133161	1	2	yes	—
80	C₂²	—	16	2	no	—
81	C₂²	—	16	4	no	—
82	C₂²	#16810367	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#16810424	1	2	yes	—
85	C₄	#16810436	1	2	yes	—
86	C₂	—	32	2	no	—
87	C₂	#20272752	1	1	yes	characteristic
88	C₂	#20272761	1	2	yes	—

	subgroup	G/H
center	C₂²	#16810367
commutator	C₄×C₈×C₅₃	C₂×C₄
Frattini	C₈²	C₂×D₅₃
Fitting, prime radical	C₅₃×#1712	C₂
socle	C₂²×C₅₃	#331
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1712	D₅₃
hypocenter	C₅₃	#7096
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

Complements

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

N	H ≅ G/N	type
1. #20272785	C₂	semidirect
51. C₅₃	#7096	semidirect

Maximal subgroups

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

subgroup	index
#20272785	2
C₅₃×#1712	2
#7096	53