Group #27539594 — subgroups

Subgroups — #27539594

Order 13568.

482 conjugacy classes of subgroups; only the 219 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 219 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	#20285229	C₂	1	2	yes	maximal
2	#20277001	C₂	1	8	yes	maximal, complemented
3	#20276499	C₂	1	4	yes	maximal
4	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
5	#16811180	—	2	2	no	—
6	#16811028	C₂²	1	4	yes	—
7	#16811025	—	2	2	no	—
8	#16811025	C₂²	1	4	yes	—
9	#16811021	—	2	2	no	—
10	#16810661	—	2	2	no	—
11	#16810661	C₂²	1	4	yes	—
12	#16810451	C₂²	1	8	yes	—
13	#16810449	C₂²	1	8	yes	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₅₃×#346	C₂²	1	4	yes	—
17	#13133217	—	2	4	no	—
18	#13133209	—	2	4	no	—
19	#13133196	—	4	4	no	—
20	C₄×Dic₁₀₆	—	4	4	no	—
21	#13133176	—	2	4	no	—
22	#13133151	—	2	4	no	—
23	#13133151	C₂³	1	8	yes	—
24	#13133149	—	2	4	no	—
25	#13133149	—	2	4	no	—
26	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	Q₈	1	8	yes	complemented
30	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×C₈×C₅₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₅₃	D₈	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₁₀₆	—	4	4	no	—
36	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
40	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₄²×C₅₃	D₈	1	2	yes	—
42	C₄²×C₅₃	C₂×D₄	1	2	yes	—
43	C₂²⋊Dic₅₃	—	4	4	no	—
44	C₄⋊Dic₅₃	—	2	8	no	—
45	C₄⋊Dic₅₃	—	4	4	no	—
46	#11719919	—	4	8	no	—
47	C₄×Dic₅₃	—	4	4	no	—
48	C₁₆×C₅₃	C₂×Q₈	1	8	yes	—
49	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	—	4	2	no	—
51	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₅₃	C₂×D₈	1	4	yes	—
53	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
55	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
56	C₂×Dic₅₃	—	4	8	no	—
57	Dic₁₀₆	—	8	8	no	—
58	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
59	C₈×C₅₃	C₂×Q₁₆	1	4	yes	—
60	C₈×C₅₃	C₂²×Q₈	1	2	yes	—
61	#33728	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	2	no	—
63	C₂²×C₅₃	#500	1	2	yes	—
64	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
65	C₄×C₅₃	—	2	4	no	—
66	C₄×C₅₃	#500	1	2	yes	—
67	C₄×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
68	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
69	Dic₅₃	—	8	8	no	—
70	#3160	—	53	2	no	—
71	#2255	—	53	8	no	—
72	#2158	—	53	4	no	—
73	C₂×#346	D₅₃	1	1	yes	characteristic
74	C₂×C₅₃	—	2	2	no	—
75	C₂×C₅₃	C₂×#500	1	1	yes	characteristic
76	C₂×C₅₃	#3414	1	2	yes	—
77	#533	—	106	2	no	—
78	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
79	#500	—	53	4	no	—
80	#499	—	53	4	no	—
81	#499	—	106	2	no	—
82	#498	—	106	2	no	—
83	#427	—	53	4	no	—
84	#427	—	106	2	no	—
85	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
86	#367	—	53	8	no	—
87	#366	—	53	8	no	—
88	#346	C₂×D₅₃	1	4	yes	—
89	C₅₃	#33728	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
91	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₄⋊Q₈	—	106	4	no	—
93	#125	—	106	4	no	—
94	C₂²⋊Q₈	—	212	4	no	—
95	C₄×Q₈	—	212	4	no	—
96	#117	—	106	4	no	—
97	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
98	C₂×C₁₆	Dic₁₀₆	1	8	yes	complemented
99	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
100	#107	—	53	8	no	—
101	#107	—	106	4	no	—
102	#106	—	106	4	no	—
103	#106	—	106	4	no	—
104	C₄×C₈	D₂₁₂	1	2	yes	—
105	C₄×C₈	C₂²×D₅₃	1	2	yes	—
106	C₂×Q₈	—	212	4	no	—
107	C₂²×C₄	D₄₂₄	1	2	yes	—
108	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
109	C₂×C₈	Dic₂₁₂	1	4	yes	—
110	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
111	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
112	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
113	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
114	C₄⋊C₄	—	106	8	no	—
115	C₄⋊C₄	—	212	8	no	—
116	C₄⋊C₄	—	212	4	no	—
117	C₂²⋊C₄	—	212	4	no	—
118	C₄²	—	212	4	no	—
119	C₄²	D₄₂₄	1	2	yes	—
120	C₄²	C₂×D₂₁₂	1	2	yes	—
121	C₁₆	C₂×Dic₁₀₆	1	8	yes	—
122	C₂³	C₂×D₄₂₄	1	1	yes	characteristic
123	Q₈	—	424	8	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	212	8	no	—
126	C₂×C₄	#13133217	1	2	yes	—
127	C₂×C₄	C₂×D₄₂₄	1	4	yes	—
128	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
129	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
130	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
131	C₈	#13133217	1	2	yes	—
132	C₈	C₂×Dic₂₁₂	1	4	yes	—
133	C₈	C₂²×Dic₁₀₆	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#16811028	1	2	yes	—
136	C₂²	C₂²×D₄₂₄	1	1	yes	characteristic
137	C₄	—	2	4	no	—
138	C₄	—	424	8	no	—
139	C₄	#16811028	1	2	yes	—
140	C₄	C₂×#13133217	1	1	yes	characteristic
141	C₄	C₂²×Dic₂₁₂	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×#16811028	1	1	yes	characteristic
144	C₂	#20287851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285229

C₂

yes

maximal

#20277001

C₂

yes

maximal, complemented

#20276499

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

#16811180

—

#16811028

C₂²

yes

—

#16811025

—

#16811025

C₂²

yes

—

#16811021

—

#16810661

—

#16810661

C₂²

yes

—

#16810451

C₂²

yes

—

#16810449

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

—

#13133217

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

#13133176

—

#13133151

—

#13133151

C₂³

yes

—

#13133149

—

#13133149

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

Q₈

yes

complemented

C₂×C₁₆×C₅₃

C₂³

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂⁴

yes

characteristic

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₁₆×C₅₃

C₂×Q₈

yes

—

C₂³×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₄⋊Q₈

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

C₂²×Q₈

yes

—

#33728

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#500

yes

—

C₂²×C₅₃

C₂²×D₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#500

yes

—

C₄×C₅₃

C₂×C₄⋊Q₈

yes

characteristic

C₄×C₅₃

C₂²×Q₁₆

yes

characteristic

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

C₂×#500

yes

characteristic

C₂×C₅₃

#3414

yes

—

#533

—

106

—

C₂²×C₁₆

C₂×D₅₃

yes

—

#500

—

#499

—

#499

—

106

—

#498

—

106

—

#427

—

#427

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

—

C₂×D₅₃

yes

—

C₅₃

#33728

yes

characteristic, complemented, Sylow

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊Q₈

—

106

—

#125

—

106

—

C₂²⋊Q₈

—

212

—

C₄×Q₈

—

212

—

#117

—

106

—

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

Dic₁₀₆

yes

complemented

C₂×C₁₆

C₂²×D₅₃

yes

—

100

#107

—

101

#107

—

106

—

102

#106

—

106

—

103

#106

—

106

—

104

C₄×C₈

D₂₁₂

yes

—

105

C₄×C₈

C₂²×D₅₃

yes

—

106

C₂×Q₈

—

212

—

107

C₂²×C₄

D₄₂₄

yes

—

108

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

109

C₂×C₈

Dic₂₁₂

yes

—

110

C₂×C₈

C₂×Dic₁₀₆

yes

—

111

C₂×C₈

C₂×D₂₁₂

yes

characteristic

112

C₂×C₈

C₂×D₂₁₂

yes

—

113

C₂×C₈

C₂³×D₅₃

yes

characteristic

114

C₄⋊C₄

—

106

—

115

C₄⋊C₄

—

212

—

116

C₄⋊C₄

—

212

—

117

C₂²⋊C₄

—

212

—

118

C₄²

—

212

—

119

C₄²

D₄₂₄

yes

—

120

C₄²

C₂×D₂₁₂

yes

—

121

C₁₆

C₂×Dic₁₀₆

yes

—

122

C₂³

C₂×D₄₂₄

yes

characteristic

123

Q₈

—

424

—

124

C₂×C₄

—

125

C₂×C₄

—

212

—

126

C₂×C₄

#13133217

yes

—

127

C₂×C₄

C₂×D₄₂₄

yes

—

128

C₂×C₄

C₂×D₄₂₄

yes

characteristic

129

C₂×C₄

C₂×Dic₂₁₂

yes

—

130

C₂×C₄

C₂²×D₂₁₂

yes

characteristic

131

C₈

#13133217

yes

—

132

C₈

C₂×Dic₂₁₂

yes

—

133

C₈

C₂²×Dic₁₀₆

yes

—

134

C₂²

—

135

C₂²

#16811028

yes

—

136

C₂²

C₂²×D₄₂₄

yes

characteristic

137

C₄

—

138

C₄

—

424

—

139

C₄

#16811028

yes

—

140

C₄

C₂×#13133217

yes

characteristic

141

C₄

C₂²×Dic₂₁₂

yes

characteristic

142

C₂

—

143

C₂

C₂×#16811028

yes

characteristic

144

C₂

#20287851

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂²×D₈
prime radical	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#33728
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator, cosocle

C₂×C₈×C₅₃

C₂⁴

Frattini

C₂×C₈

C₂³×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂²×D₈

prime radical

C₂³×C₅₃

C₂×D₈

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
2. #20277001	C₂	semidirect
29. C₂×C₁₆×C₅₃	Q₈	semidirect
89. C₅₃	#33728	semidirect
98. C₂×C₁₆	Dic₁₀₆	semidirect

H ≅ G/N

type

2. #20277001

C₂

semidirect

29. C₂×C₁₆×C₅₃

Q₈

semidirect

89. C₅₃

#33728

semidirect

98. C₂×C₁₆

Dic₁₀₆

semidirect

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

subgroup	index
#20285229	2
#20277001	2
#20276499	2
C₂×C₅₃×#346	2
#33728	53

subgroup

index

C₂×C₅₃×#346

Subgroups — #27539594

Conjugacy classes of subgroups

Complements

Maximal subgroups