Group #27349656 — subgroups

Subgroups — C₅₃⋊#17309 #27349656

Order 13568.

1466 conjugacy classes of subgroups; only the 283 normal subgroups are shown.

normal subgroups only

This group has 1466 conjugacy classes of subgroups — too many to draw. Only its 283 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	#20272985	C₂	1	12	yes	maximal
3	C₅₃×#1446	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₀₆	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	C₂²×Dic₁₀₆	—	4	12	no	—
10	C₂²×Dic₁₀₆	—	4	12	no	—
11	C₂²×Dic₁₀₆	Q₈	1	16	yes	complemented
12	C₂³×Dic₅₃	C₂³	1	8	yes	—
13	#13133124	—	4	12	no	—
14	#13133124	—	4	12	no	—
15	C₂²×Q₈×C₅₃	—	4	6	no	—
16	C₂²×Q₈×C₅₃	Q₈	1	4	yes	complemented
17	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
18	C₂³×C₄×C₅₃	C₂³	1	4	yes	—
19	#95×C₅₃	—	4	6	no	—
20	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	4	12	no	—
22	C₂×Q₈×C₅₃	—	4	16	no	—
23	C₂²×C₄×C₅₃	—	2	6	no	—
24	C₂²×C₄×C₅₃	—	2	12	no	—
25	C₂²×C₄×C₅₃	—	2	12	no	—
26	C₂²×C₄×C₅₃	—	2	12	no	—
27	C₂²×C₄×C₅₃	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₅₃	C₂×Q₈	1	8	yes	—
29	C₂²×Dic₅₃	—	2	24	no	—
30	C₂²×Dic₅₃	—	2	24	no	—
31	C₂²×Dic₅₃	C₂×Q₈	1	16	yes	—
32	C₂×Dic₁₀₆	—	4	48	no	—
33	C₂×Dic₁₀₆	—	4	48	no	—
34	C₂×Dic₁₀₆	—	4	16	no	—
35	C₂³×C₅₃	C₂²×Q₈	1	4	yes	—
36	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
37	C₂³×C₅₃	H(5,𝔽₂)	1	6	yes	—
38	C₂³×C₅₃	H(5,𝔽₂)	1	4	yes	—
39	Q₈×C₅₃	—	4	16	no	—
40	C₂×C₄×C₅₃	—	2	24	no	—
41	C₂×C₄×C₅₃	—	2	6	no	—
42	C₂×C₄×C₅₃	—	2	12	no	—
43	C₂×C₄×C₅₃	—	4	12	no	—
44	C₂×C₄×C₅₃	—	4	4	no	—
45	C₂×C₄×C₅₃	—	4	12	no	—
46	C₂×Dic₅₃	—	2	48	no	—
47	C₂×Dic₅₃	—	4	24	no	—
48	C₂×Dic₅₃	—	4	8	no	—
49	Dic₁₀₆	—	4	64	no	—
50	#17309	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	#543	1	12	yes	—
52	C₂²×C₅₃	#543	1	3	yes	—
53	C₂²×C₅₃	Q₈²	1	6	yes	—
54	C₂²×C₅₃	Q₈²	1	4	yes	—
55	C₂²×C₅₃	H(3,𝔽₄)	1	6	yes	—
56	C₂²×C₅₃	H(3,𝔽₄)	1	4	yes	—
57	C₄×C₅₃	—	4	8	no	—
58	C₄×C₅₃	—	4	6	no	—
59	Dic₅₃	—	4	16	no	—
60	#1446	—	53	12	no	—
61	#1446	—	53	2	no	—
62	#1446	D₅₃	1	1	yes	characteristic
63	C₂×C₅₃	#2811	1	4	yes	—
64	C₂×C₅₃	#2811	1	6	yes	—
65	C₂×C₅₃	#2840	1	4	yes	—
66	C₂×C₅₃	#2840	1	1	yes	characteristic
67	C₂³×Q₈	—	53	4	no	—
68	C₂³×Q₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×#95	—	53	12	no	—
70	C₂×#95	—	53	12	no	—
71	C₂×#95	C₂×D₅₃	1	6	yes	—
72	C₅₃	#17309	1	1	yes	characteristic, complemented, Sylow
73	C₂²×Q₈	—	4	6	no	—
74	C₂²×Q₈	—	53	16	no	—
75	C₂²×Q₈	—	212	12	no	—
76	C₂²×Q₈	—	212	12	no	—
77	C₂²×Q₈	Dic₁₀₆	1	4	yes	complemented
78	C₂³×C₄	—	53	8	no	—
79	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
80	C₂³×C₄	C₂²×D₅₃	1	4	yes	—
81	#95	—	4	6	no	—
82	#95	—	212	12	no	—
83	#95	—	212	12	no	—
84	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
85	C₂×Q₈	—	4	12	no	—
86	C₂×Q₈	—	4	16	no	—
87	C₂×Q₈	—	212	48	no	—
88	C₂×Q₈	—	212	48	no	—
89	C₂×Q₈	—	212	16	no	—
90	C₂²×C₄	—	2	6	no	—
91	C₂²×C₄	—	2	12	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	12	no	—
94	C₂²×C₄	—	53	16	no	—
95	C₂²×C₄	—	106	24	no	—
96	C₂²×C₄	—	106	24	no	—
97	C₂²×C₄	C₂×Dic₁₀₆	1	6	yes	—
98	C₂²×C₄	Q₈×D₅₃	1	8	yes	—
99	C₂³	C₂×Q₈×D₅₃	1	4	yes	—
100	C₂³	C₂²×Dic₁₀₆	1	1	yes	characteristic
101	C₂³	#13133262	1	6	yes	—
102	C₂³	#13133263	1	4	yes	—
103	Q₈	—	4	16	no	—
104	Q₈	—	212	64	no	—
105	C₂×C₄	—	2	24	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	12	no	—
108	C₂×C₄	—	4	12	no	—
109	C₂×C₄	—	4	12	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	—	106	48	no	—
112	C₂×C₄	—	212	24	no	—
113	C₂×C₄	—	212	8	no	—
114	C₂²	#16811248	1	12	yes	—
115	C₂²	#16811249	1	3	yes	—
116	C₂²	#16811353	1	6	yes	—
117	C₂²	Q₈×Dic₁₀₆	1	4	yes	—
118	C₂²	#16811365	1	6	yes	—
119	C₂²	#16811366	1	4	yes	—
120	C₄	—	4	8	no	—
121	C₄	—	4	6	no	—
122	C₄	—	212	16	no	—
123	C₂	#20282707	1	4	yes	—
124	C₂	#20282708	1	6	yes	—
125	C₂	#20282967	1	4	yes	—
126	C₂	#20282968	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20272986

C₂

yes

maximal

#20272985

C₂

yes

maximal

C₅₃×#1446

C₂

yes

maximal, characteristic

C₂³×Dic₁₀₆

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂³×Q₈×C₅₃

C₂²

yes

characteristic

C₂×#95×C₅₃

C₂²

yes

—

C₂²×Dic₁₀₆

—

C₂²×Dic₁₀₆

—

C₂²×Dic₁₀₆

Q₈

yes

complemented

C₂³×Dic₅₃

C₂³

yes

—

#13133124

—

#13133124

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

Q₈

yes

complemented

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

C₂×Q₈

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

C₂×Q₈

yes

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂³×C₅₃

C₂²×Q₈

yes

—

C₂³×C₅₃

C₂²×Q₈

yes

characteristic

C₂³×C₅₃

H(5,𝔽₂)

yes

—

C₂³×C₅₃

H(5,𝔽₂)

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

—

Dic₁₀₆

—

#17309

—

maximal, Sylow

C₂²×C₅₃

#543

yes

—

C₂²×C₅₃

#543

yes

—

C₂²×C₅₃

Q₈²

yes

—

C₂²×C₅₃

Q₈²

yes

—

C₂²×C₅₃

H(3,𝔽₄)

yes

—

C₂²×C₅₃

H(3,𝔽₄)

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

#1446

—

#1446

—

#1446

D₅₃

yes

characteristic

C₂×C₅₃

#2811

yes

—

C₂×C₅₃

#2811

yes

—

C₂×C₅₃

#2840

yes

—

C₂×C₅₃

#2840

yes

characteristic

C₂³×Q₈

—

C₂³×Q₈

C₂×D₅₃

yes

characteristic

C₂×#95

—

C₂×#95

—

C₂×#95

C₂×D₅₃

yes

—

C₅₃

#17309

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

—

212

—

C₂²×Q₈

—

212

—

C₂²×Q₈

Dic₁₀₆

yes

complemented

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

—

#95

—

#95

—

212

—

#95

—

212

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂×Dic₁₀₆

yes

—

C₂²×C₄

Q₈×D₅₃

yes

—

C₂³

C₂×Q₈×D₅₃

yes

—

100

C₂³

C₂²×Dic₁₀₆

yes

characteristic

101

C₂³

#13133262

yes

—

102

C₂³

#13133263

yes

—

103

Q₈

—

104

Q₈

—

212

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

—

212

—

113

C₂×C₄

—

212

—

114

C₂²

#16811248

yes

—

115

C₂²

#16811249

yes

—

116

C₂²

#16811353

yes

—

117

C₂²

Q₈×Dic₁₀₆

yes

—

118

C₂²

#16811365

yes

—

119

C₂²

#16811366

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

212

—

123

C₂

#20282707

yes

—

124

C₂

#20282708

yes

—

125

C₂

#20282967

yes

—

126

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	#17309
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
11. C₂²×Dic₁₀₆	Q₈	semidirect
16. C₂²×Q₈×C₅₃	Q₈	semidirect
72. C₅₃	#17309	semidirect
77. C₂²×Q₈	Dic₁₀₆	semidirect

H ≅ G/N

type

11. C₂²×Dic₁₀₆

Q₈

semidirect

16. C₂²×Q₈×C₅₃

Q₈

semidirect

72. C₅₃

#17309

semidirect

77. C₂²×Q₈

Dic₁₀₆

semidirect

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

subgroup	index
#20272986	2
#20272985	2
C₅₃×#1446	2
#17309	53

subgroup

index

#20272986

#20272985

C₅₃×#1446

#17309

Subgroups — C₅₃⋊#17309 #27349656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53⋊#17309 #27349656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃⋊#17309 #27349656