Group #27311323 — subgroups

Subgroups — #27311323

Order 13568.

1638 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 411 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	#20272977	C₂	1	8	yes	maximal, complemented
2	#20272938	C₂	1	6	yes	maximal
3	C₂²×#95×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
5	C₂×#13133124	C₂²	1	24	yes	—
6	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
7	C₂×#95×C₅₃	C₄	1	8	yes	complemented
8	C₂×#95×C₅₃	C₂²	1	4	yes	—
9	C₂³×Dic₅₃	C₂³	1	8	yes	—
10	C₂×C₂²⋊Dic₅₃	—	2	24	no	—
11	#13133124	—	4	24	no	—
12	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
13	C₂³×C₄×C₅₃	—	2	12	no	—
14	C₂³×C₄×C₅₃	—	4	12	no	—
15	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
16	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
17	#95×C₅₃	C₂×C₄	1	16	yes	—
18	C₂⁴×C₅₃	—	2	6	no	—
19	C₂⁴×C₅₃	—	2	2	no	—
20	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
21	C₂⁴×C₅₃	#41	1	12	yes	—
22	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	—	2	24	no	—
24	C₂²×C₄×C₅₃	—	2	12	no	—
25	C₂²×C₄×C₅₃	—	2	24	no	—
26	C₂²×C₄×C₅₃	—	4	24	no	—
27	C₂²×C₄×C₅₃	—	4	48	no	—
28	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
29	C₂²×Dic₅₃	—	2	24	no	—
30	C₂²×Dic₅₃	—	2	24	no	—
31	C₂²×Dic₅₃	—	2	8	no	—
32	C₂²⋊Dic₅₃	—	4	16	no	—
33	C₂³×C₅₃	—	2	6	no	—
34	C₂³×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	—	2	24	no	—
36	C₂³×C₅₃	—	2	24	no	—
37	C₂³×C₅₃	#117	1	12	yes	—
38	C₂³×C₅₃	#126	1	16	yes	—
39	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
40	C₂³×C₅₃	C₂×#41	1	6	yes	—
41	C₂³×C₅₃	H(5,𝔽₂)	1	6	yes	—
42	C₂³×C₅₃	2₋¹⁺⁴	1	2	yes	—
43	C₂×C₄×C₅₃	—	2	12	no	—
44	C₂×C₄×C₅₃	—	2	24	no	—
45	C₂×C₄×C₅₃	—	4	48	no	—
46	C₂×C₄×C₅₃	—	4	24	no	—
47	C₂×Dic₅₃	—	4	8	no	—
48	C₂×Dic₅₃	—	4	24	no	—
49	C₂×Dic₅₃	—	4	24	no	—
50	#14349	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	12	no	—
52	C₂²×C₅₃	—	2	12	no	—
53	C₂²×C₅₃	—	2	32	no	—
54	C₂²×C₅₃	#383	1	8	yes	—
55	C₂²×C₅₃	C₂×#117	1	3	yes	—
56	C₂²×C₅₃	#518	1	3	yes	—
57	C₂²×C₅₃	#519	1	1	yes	characteristic
58	C₂²×C₅₃	C₂×#126	1	4	yes	—
59	C₂²×C₅₃	#538	1	12	yes	—
60	C₂²×C₅₃	#539	1	12	yes	—
61	C₄×C₅₃	—	4	12	no	—
62	Dic₅₃	—	8	8	no	—
63	C₂²×#95	D₅₃	1	1	yes	characteristic
64	#1443	—	53	8	no	—
65	C₂²⋊#95	—	53	6	no	—
66	C₂×C₅₃	—	2	8	no	—
67	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
68	C₂×C₅₃	#2312	1	3	yes	—
69	C₂×C₅₃	#2313	1	3	yes	—
70	C₂×C₅₃	#2643	1	8	yes	—
71	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
72	C₂²×C₂²⋊C₄	—	53	4	no	—
73	C₂×#95	—	53	24	no	—
74	C₂×#95	Dic₅₃	1	8	yes	complemented
75	C₂×#95	C₂×D₅₃	1	4	yes	—
76	C₅₃	#14349	1	1	yes	characteristic, complemented, Sylow
77	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
78	C₂³×C₄	—	2	12	no	—
79	C₂³×C₄	—	4	12	no	—
80	C₂³×C₄	—	53	8	no	—
81	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
82	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
83	C₂×C₂²⋊C₄	—	106	24	no	—
84	#95	—	212	24	no	—
85	#95	C₂×Dic₅₃	1	16	yes	—
86	C₂⁴	—	2	6	no	—
87	C₂⁴	—	2	2	no	—
88	C₂⁴	#11719945	1	12	yes	—
89	C₂⁴	C₂²×Dic₅₃	1	2	yes	—
90	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
91	C₂²×C₄	—	2	24	no	—
92	C₂²×C₄	—	2	12	no	—
93	C₂²×C₄	—	2	24	no	—
94	C₂²×C₄	—	4	24	no	—
95	C₂²×C₄	—	4	48	no	—
96	C₂²×C₄	—	106	24	no	—
97	C₂²×C₄	—	106	24	no	—
98	C₂²×C₄	—	106	8	no	—
99	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
100	C₂²⋊C₄	—	212	16	no	—
101	C₂³	—	2	6	no	—
102	C₂³	—	2	2	no	—
103	C₂³	—	2	24	no	—
104	C₂³	—	2	24	no	—
105	C₂³	#13133176	1	12	yes	—
106	C₂³	#13133211	1	16	yes	—
107	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
108	C₂³	C₂×#11719945	1	6	yes	—
109	C₂³	#13133262	1	6	yes	—
110	C₂³	#13133265	1	2	yes	—
111	C₂×C₄	—	2	12	no	—
112	C₂×C₄	—	2	24	no	—
113	C₂×C₄	—	4	48	no	—
114	C₂×C₄	—	4	24	no	—
115	C₂×C₄	—	212	8	no	—
116	C₂×C₄	—	212	24	no	—
117	C₂×C₄	—	212	24	no	—
118	C₂²	—	2	12	no	—
119	C₂²	—	2	12	no	—
120	C₂²	—	2	32	no	—
121	C₂²	#16810491	1	8	yes	—
122	C₂²	C₂×#13133176	1	3	yes	—
123	C₂²	#16811098	1	3	yes	—
124	C₂²	#16811101	1	1	yes	characteristic
125	C₂²	C₂×#13133211	1	4	yes	—
126	C₂²	#16811211	1	12	yes	—
127	C₂²	#16811220	1	12	yes	—
128	C₄	—	4	12	no	—
129	C₄	—	424	8	no	—
130	C₂	—	2	8	no	—
131	C₂	C₂×#16810491	1	1	yes	characteristic
132	C₂	#20277313	1	3	yes	—
133	C₂	#20277318	1	3	yes	—
134	C₂	#20280668	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272977

C₂

yes

maximal, complemented

#20272938

C₂

yes

maximal

C₂²×#95×C₅₃

C₂

yes

maximal, characteristic

C₂²×C₂²⋊Dic₅₃

C₂²

yes

—

C₂×#13133124

C₂²

yes

—

C₂⁴×C₄×C₅₃

C₂²

yes

—

C₂×#95×C₅₃

C₄

yes

complemented

C₂×#95×C₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂³

yes

—

C₂×C₂²⋊Dic₅₃

—

#13133124

—

C₂⁵×C₅₃

C₂³

yes

characteristic

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

#95×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂⁴×C₅₃

#41

yes

—

C₂⁴×C₅₃

C₂⁴

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#117

yes

—

C₂³×C₅₃

#126

yes

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂³×C₅₃

C₂×#41

yes

—

C₂³×C₅₃

H(5,𝔽₂)

yes

—

C₂³×C₅₃

2₋¹⁺⁴

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#14349

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#383

yes

—

C₂²×C₅₃

C₂×#117

yes

—

C₂²×C₅₃

#518

yes

—

C₂²×C₅₃

#519

yes

characteristic

C₂²×C₅₃

C₂×#126

yes

—

C₂²×C₅₃

#538

yes

—

C₂²×C₅₃

#539

yes

—

C₄×C₅₃

—

Dic₅₃

—

C₂²×#95

D₅₃

yes

characteristic

#1443

—

C₂²⋊#95

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#383

yes

characteristic

C₂×C₅₃

#2312

yes

—

C₂×C₅₃

#2313

yes

—

C₂×C₅₃

#2643

yes

—

C₂⁴×C₄

C₂×D₅₃

yes

—

C₂²×C₂²⋊C₄

—

C₂×#95

—

C₂×#95

Dic₅₃

yes

complemented

C₂×#95

C₂×D₅₃

yes

—

C₅₃

#14349

yes

characteristic, complemented, Sylow

C₂⁵

C₂²×D₅₃

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×Dic₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

#95

—

212

—

#95

C₂×Dic₅₃

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

#11719945

yes

—

C₂⁴

C₂²×Dic₅₃

yes

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×Dic₅₃

yes

—

100

C₂²⋊C₄

—

212

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

#13133176

yes

—

106

C₂³

#13133211

yes

—

107

C₂³

C₂³×Dic₅₃

yes

characteristic

108

C₂³

C₂×#11719945

yes

—

109

C₂³

#13133262

yes

—

110

C₂³

#13133265

yes

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

—

212

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#16810491

yes

—

122

C₂²

C₂×#13133176

yes

—

123

C₂²

#16811098

yes

—

124

C₂²

#16811101

yes

characteristic

125

C₂²

C₂×#13133211

yes

—

126

C₂²

#16811211

yes

—

127

C₂²

#16811220

yes

—

128

C₄

—

129

C₄

—

424

—

130

C₂

—

131

C₂

C₂×#16810491

yes

characteristic

132

C₂

#20277313

yes

—

133

C₂

#20277318

yes

—

134

C₂

#20280668

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂²×#95×C₅₃	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂²×#95	D₅₃
hypocenter	C₅₃	#14349
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#95

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₃

commutator

C₂³×C₅₃

C₂³×C₄

Fitting

C₂²×#95×C₅₃

C₂

socle, cosocle

C₂⁴×C₅₃

C₂⁴

prime radical

C₂⁵×C₅₃

C₂³

hypercenter

C₂²×#95

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂²×#95

N	H ≅ G/N	type
1. #20272977	C₂	semidirect
7. C₂×#95×C₅₃	C₄	semidirect
74. C₂×#95	Dic₅₃	semidirect
76. C₅₃	#14349	semidirect

H ≅ G/N

type

1. #20272977

C₂

semidirect

7. C₂×#95×C₅₃

C₄

semidirect

74. C₂×#95

Dic₅₃

semidirect

76. C₅₃

#14349

semidirect

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

subgroup	index
#20272977	2
#20272938	2
C₂²×#95×C₅₃	2
#14349	53

subgroup

index

#20272977

#20272938

C₂²×#95×C₅₃

#14349

Subgroups — #27311323

Conjugacy classes of subgroups

Complements

Maximal subgroups