Group #27349655 — subgroups

Subgroups — #27349655

Order 13568.

1034 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 1034 conjugacy classes of subgroups — too many to draw. Only its 251 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20272986	C₂	1	2	yes	maximal
2	#20272971	C₂	1	12	yes	maximal
3	C₅₃×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₅₃	C₂²	1	4	yes	—
5	C₂×#13133124	C₂²	1	12	yes	—
6	C₂×#13133124	C₂²	1	12	yes	—
7	C₂³×Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#95×C₅₃	C₂²	1	6	yes	—
9	C₂³×Dic₅₃	C₂³	1	8	yes	—
10	C₂×C₄⋊Dic₅₃	—	4	12	no	—
11	#13133124	—	2	24	no	—
12	#13133124	—	4	12	no	—
13	C₂²×Q₈×C₅₃	—	4	6	no	—
14	C₂²×Q₈×C₅₃	Q₈	1	4	yes	—
15	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
17	#95×C₅₃	—	4	6	no	—
18	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	4	12	no	—
20	C₂×Q₈×C₅₃	—	4	16	no	—
21	C₂²×C₄×C₅₃	—	2	6	no	—
22	C₂²×C₄×C₅₃	—	2	12	no	—
23	C₂²×C₄×C₅₃	—	2	12	no	—
24	C₂²×C₄×C₅₃	—	2	12	no	—
25	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
26	C₂²×C₄×C₅₃	#41	1	8	yes	—
27	C₂²×Dic₅₃	—	2	24	no	—
28	C₂²×Dic₅₃	—	2	24	no	—
29	C₂²×Dic₅₃	—	2	8	no	—
30	C₄⋊Dic₅₃	—	4	16	no	—
31	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
32	C₂³×C₅₃	C₂×#41	1	4	yes	—
33	C₂³×C₅₃	H(5,𝔽₂)	1	6	yes	—
34	C₂³×C₅₃	2₋¹⁺⁴	1	4	yes	—
35	Q₈×C₅₃	—	4	16	no	—
36	C₂×C₄×C₅₃	—	2	24	no	—
37	C₂×C₄×C₅₃	—	2	6	no	—
38	C₂×C₄×C₅₃	—	2	12	no	—
39	C₂×C₄×C₅₃	—	4	12	no	—
40	C₂×C₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	—	4	12	no	—
42	C₂×Dic₅₃	—	4	24	no	—
43	C₂×Dic₅₃	—	4	24	no	—
44	C₂×Dic₅₃	—	4	8	no	—
45	#17308	—	53	1	no	maximal, Sylow
46	C₂²×C₅₃	#539	1	12	yes	—
47	C₂²×C₅₃	#543	1	3	yes	—
48	C₂²×C₅₃	#555	1	6	yes	—
49	C₂²×C₅₃	#557	1	4	yes	—
50	C₂²×C₅₃	H(3,𝔽₄)	1	4	yes	—
51	C₂²×C₅₃	#563	1	6	yes	—
52	C₄×C₅₃	—	4	8	no	—
53	C₄×C₅₃	—	4	6	no	—
54	Dic₅₃	—	8	8	no	—
55	#1446	—	53	2	no	—
56	#1446	D₅₃	1	1	yes	characteristic
57	#1442	—	53	12	no	—
58	C₂×C₅₃	#2785	1	4	yes	—
59	C₂×C₅₃	#2809	1	6	yes	—
60	C₂×C₅₃	#2835	1	4	yes	—
61	C₂×C₅₃	#2840	1	1	yes	characteristic
62	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
63	C₂²×C₄⋊C₄	—	53	4	no	—
64	C₂×#95	—	53	12	no	—
65	C₂×#95	—	53	12	no	—
66	C₂×#95	C₂×D₅₃	1	6	yes	—
67	C₅₃	#17308	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	—	4	6	no	—
69	C₂²×Q₈	Dic₁₀₆	1	4	yes	—
70	C₂³×C₄	—	53	8	no	—
71	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
72	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
73	C₂×C₄⋊C₄	—	212	12	no	—
74	#95	—	4	6	no	—
75	#95	—	106	24	no	—
76	#95	—	212	12	no	—
77	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
78	C₂×Q₈	—	4	12	no	—
79	C₂×Q₈	—	4	16	no	—
80	C₂²×C₄	—	2	12	no	—
81	C₂²×C₄	—	2	6	no	—
82	C₂²×C₄	—	2	12	no	—
83	C₂²×C₄	—	2	12	no	—
84	C₂²×C₄	—	106	24	no	—
85	C₂²×C₄	—	106	24	no	—
86	C₂²×C₄	—	106	8	no	—
87	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
88	C₂²×C₄	#11719949	1	8	yes	—
89	C₄⋊C₄	—	212	16	no	—
90	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
91	C₂³	C₂×#11719949	1	4	yes	—
92	C₂³	#13133262	1	6	yes	—
93	C₂³	#13133264	1	4	yes	—
94	Q₈	—	4	16	no	—
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	2	6	no	—
97	C₂×C₄	—	2	12	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	4	12	no	—
100	C₂×C₄	—	4	12	no	—
101	C₂×C₄	—	212	24	no	—
102	C₂×C₄	—	212	8	no	—
103	C₂×C₄	—	212	24	no	—
104	C₂²	#16811222	1	12	yes	—
105	C₂²	#16811249	1	3	yes	—
106	C₂²	#16811339	1	6	yes	—
107	C₂²	#16811352	1	4	yes	—
108	C₂²	#16811366	1	4	yes	—
109	C₂²	#16811373	1	6	yes	—
110	C₄	—	4	8	no	—
111	C₄	—	4	6	no	—
112	C₄	—	424	8	no	—
113	C₂	#20282423	1	4	yes	—
114	C₂	#20282703	1	6	yes	—
115	C₂	#20282906	1	4	yes	—
116	C₂	#20282968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272986

C₂

yes

maximal

#20272971

C₂

yes

maximal

C₅₃×#1446

C₂

yes

maximal, characteristic

C₂²×C₄⋊Dic₅₃

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂³×Q₈×C₅₃

C₂²

yes

characteristic

C₂×#95×C₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂³

yes

—

C₂×C₄⋊Dic₅₃

—

#13133124

—

#13133124

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

Q₈

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

#95×C₅₃

—

C₂⁴×C₅₃

C₂⁴

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂²×C₄×C₅₃

#41

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₄⋊Dic₅₃

—

C₂³×C₅₃

C₂²×Q₈

yes

characteristic

C₂³×C₅₃

C₂×#41

yes

—

C₂³×C₅₃

H(5,𝔽₂)

yes

—

C₂³×C₅₃

2₋¹⁺⁴

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#17308

—

maximal, Sylow

C₂²×C₅₃

#539

yes

—

C₂²×C₅₃

#543

yes

—

C₂²×C₅₃

#555

yes

—

C₂²×C₅₃

#557

yes

—

C₂²×C₅₃

H(3,𝔽₄)

yes

—

C₂²×C₅₃

#563

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1446

—

#1446

D₅₃

yes

characteristic

#1442

—

C₂×C₅₃

#2785

yes

—

C₂×C₅₃

#2809

yes

—

C₂×C₅₃

#2835

yes

—

C₂×C₅₃

#2840

yes

characteristic

C₂³×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₄⋊C₄

—

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₅₃

yes

—

C₅₃

#17308

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

Dic₁₀₆

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×C₄⋊C₄

—

212

—

#95

—

#95

—

106

—

#95

—

212

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×Dic₁₀₆

yes

—

C₂²×C₄

#11719949

yes

—

C₄⋊C₄

—

212

—

C₂³

C₂²×Dic₁₀₆

yes

characteristic

C₂³

C₂×#11719949

yes

—

C₂³

#13133262

yes

—

C₂³

#13133264

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

212

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂²

#16811222

yes

—

105

C₂²

#16811249

yes

—

106

C₂²

#16811339

yes

—

107

C₂²

#16811352

yes

—

108

C₂²

#16811366

yes

—

109

C₂²

#16811373

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

424

—

113

C₂

#20282423

yes

—

114

C₂

#20282703

yes

—

115

C₂

#20282906

yes

—

116

C₂

#20282968

yes

characteristic

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator, socle, prime radical, cosocle	C₂⁴×C₅₃	C₂⁴
Fitting	C₅₃×#1446	C₂
hypercenter	#1446	D₅₃
hypocenter	C₅₃	#17308
LCS	G ▷ C₂⁴×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ #1446

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₃

commutator, socle, prime radical, cosocle

C₂⁴×C₅₃

C₂⁴

Fitting

C₅₃×#1446

C₂

D₅₃

C₅₃

G ▷ C₂⁴×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ #1446

N	H ≅ G/N	type
67. C₅₃	#17308	semidirect

H ≅ G/N

type

67. C₅₃

#17308

semidirect

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

subgroup	index
#20272986	2
#20272971	2
C₅₃×#1446	2
#17308	53

subgroup

index

#20272986

#20272971

C₅₃×#1446

#17308

Subgroups — #27349655

Conjugacy classes of subgroups

Complements

Maximal subgroups