Group #28328557 — subgroups

Subgroups — #28328557

Order 13568.

444 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 210 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	#20291132	C₂	1	8	yes	maximal, complemented
2	#20291081	C₂	1	4	yes	maximal
3	#20291076	C₂	1	2	yes	maximal
4	#20283645	C₂	1	1	yes	maximal, characteristic
5	#16811544	C₂²	1	8	yes	—
6	C₄⋊#11719910	C₂²	1	4	yes	—
7	C₂²⋊#11719910	C₂²	1	4	yes	—
8	#16811512	C₂²	1	4	yes	—
9	#16811511	C₂²	1	4	yes	—
10	#16811509	C₂²	1	4	yes	—
11	#16811180	C₄	1	1	yes	characteristic
12	#16810750	C₄	1	4	yes	—
13	#16810747	C₂²	1	4	yes	—
14	#16810651	C₄	1	2	yes	—
15	#16810581	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	C₅₃×#423	C₄	1	1	yes	characteristic
18	C₂×#11719910	C₂³	1	8	yes	—
19	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
20	C₂²×#88322	C₂³	1	1	yes	characteristic
21	#13133217	C₈	1	2	yes	—
22	#13133209	C₈	1	2	yes	—
23	#13133196	C₈	1	4	yes	—
24	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
25	#13133176	C₂×C₄	1	2	yes	—
26	#13133146	C₂×C₄	1	8	yes	—
27	#13133127	C₂×C₄	1	4	yes	—
28	C₄×#88322	C₂×C₄	1	2	yes	—
29	C₄×#88322	C₂³	1	2	yes	—
30	C₈×Dic₅₃	C₂³	1	4	yes	—
31	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
39	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
43	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
44	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
45	#11719919	C₂×C₈	1	8	yes	—
46	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
47	C₂×#88322	—	2	2	no	—
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
50	C₂×#88322	C₂²×C₄	1	2	yes	—
51	C₂×#88322	C₂⁴	1	1	yes	characteristic
52	#11719910	—	2	8	no	—
53	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	1	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
62	Dic₁₀₆	—	4	4	no	—
63	C₈×C₅₃	—	2	4	no	—
64	#88322	—	2	1	no	—
65	#88322	—	2	2	no	—
66	#88322	2₋¹⁺⁴	1	2	yes	—
67	#33501	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	#519	1	1	yes	characteristic
75	C₄×C₅₃	#568	1	1	yes	characteristic
76	Dic₅₃	—	2	8	no	—
77	#2960	—	53	1	no	—
78	#2183	—	53	8	no	—
79	#2164	—	53	4	no	—
80	#2163	—	53	2	no	—
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	#2982	1	1	yes	characteristic
83	C₂×C₅₃	#3407	1	2	yes	—
84	#533	—	53	1	no	—
85	#446	—	53	4	no	—
86	C₈×Q₈	—	53	4	no	—
87	#424	—	53	2	no	—
88	#423	#6157	1	1	yes	characteristic
89	#405	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄⋊C₁₆	—	53	4	no	—
92	C₄⋊C₁₆	—	53	8	no	—
93	C₂²⋊C₁₆	—	53	4	no	—
94	#346	—	53	4	no	—
95	#346	—	53	4	no	—
96	C₄×C₁₆	—	53	4	no	—
97	C₅₃	#33501	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	C₂×#6157	1	2	yes	—
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₄⋊Q₈	—	53	2	no	—
102	#125	—	53	2	no	—
103	C₂²⋊Q₈	—	53	4	no	—
104	C₄×Q₈	—	53	4	no	—
105	#117	—	53	2	no	—
106	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
107	C₂×C₁₆	—	53	8	no	—
108	C₄⋊C₈	—	53	8	no	—
109	C₄⋊C₈	C₂×#6157	1	4	yes	—
110	#97	—	53	4	no	—
111	C₄×C₈	—	53	4	no	—
112	C₄×C₈	—	53	2	no	—
113	C₄×C₈	—	53	2	no	—
114	C₂×Q₈	—	53	4	no	—
115	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
116	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
117	M₁₆	—	4	2	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	2	no	—
121	C₂×C₈	—	106	2	no	—
122	C₂×C₈	—	212	1	no	—
123	C₂×C₈	C₂²×#6157	1	4	yes	—
124	C₄⋊C₄	—	53	4	no	—
125	C₄⋊C₄	—	53	8	no	—
126	C₄⋊C₄	—	53	4	no	—
127	C₂²⋊C₄	—	53	4	no	—
128	C₄²	—	53	4	no	—
129	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
130	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
131	C₄²	C₂²×#6157	1	2	yes	—
132	C₁₆	—	106	8	no	—
133	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
134	Q₈	—	212	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	53	8	no	—
139	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
141	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
142	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₈	—	53	2	no	—
145	C₈	—	106	1	no	—
146	C₈	—	106	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	106	8	no	—
154	C₄	#16811691	1	1	yes	characteristic
155	C₄	#16811701	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#20291610	1	1	yes	characteristic
158	C₂	#20291684	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291132

C₂

yes

maximal, complemented

#20291081

C₂

yes

maximal

#20291076

C₂

yes

maximal

#20283645

C₂

yes

maximal, characteristic

#16811544

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

C₂²⋊#11719910

C₂²

yes

—

#16811512

C₂²

yes

—

#16811511

C₂²

yes

—

#16811509

C₂²

yes

—

#16811180

C₄

yes

characteristic

#16810750

C₄

yes

—

#16810747

C₂²

yes

—

#16810651

C₄

yes

—

#16810581

C₂²

yes

—

C₂×C₄×#88322

C₂²

yes

characteristic

C₅₃×#423

C₄

yes

characteristic

C₂×#11719910

C₂³

yes

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂³

yes

characteristic

#13133217

C₈

yes

—

#13133209

C₈

yes

—

#13133196

C₈

yes

—

C₄×Dic₁₀₆

C₂×C₄

yes

—

#13133176

C₂×C₄

yes

—

#13133146

C₂×C₄

yes

—

#13133127

C₂×C₄

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

—

C₈×Dic₅₃

C₂³

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₆

C₂×C₈

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

—

C₂²⋊Dic₅₃

C₂×C₈

yes

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

C₄⋊Dic₅₃

C₂×C₈

yes

—

#11719919

C₂×C₈

yes

—

C₄×Dic₅₃

C₂²×C₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

—

C₂×#88322

C₂⁴

yes

characteristic

#11719910

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₈

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

2₋¹⁺⁴

yes

—

#33501

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#2982

yes

characteristic

C₂×C₅₃

#3407

yes

—

#533

—

#446

—

C₈×Q₈

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#346

—

C₄×C₁₆

—

C₅₃

#33501

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×#6157

yes

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₄⋊Q₈

—

102

#125

—

103

C₂²⋊Q₈

—

104

C₄×Q₈

—

105

#117

—

106

C₂×C₄²

C₂×#6157

yes

characteristic

107

C₂×C₁₆

—

108

C₄⋊C₈

—

109

C₄⋊C₈

C₂×#6157

yes

—

110

#97

—

111

C₄×C₈

—

112

C₄×C₈

—

113

C₄×C₈

—

114

C₂×Q₈

—

115

C₂²×C₄

D₅₃⋊C₈

yes

—

116

C₂²×C₄

C₂²×#6157

yes

characteristic

117

M₁₆

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

106

—

122

C₂×C₈

—

212

—

123

C₂×C₈

C₂²×#6157

yes

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

D₅₃⋊C₈

yes

characteristic

130

C₄²

D₅₃⋊C₈

yes

characteristic

131

C₄²

C₂²×#6157

yes

—

132

C₁₆

—

106

—

133

C₂³

C₂×D₅₃⋊C₈

yes

characteristic

134

Q₈

—

212

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

140

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

141

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

142

C₂×C₄

C₂³×#6157

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

106

—

146

C₈

—

106

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂²×D₅₃⋊C₈

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

106

—

154

C₄

#16811691

yes

characteristic

155

C₄

#16811701

yes

characteristic

156

C₂

—

157

C₂

#20291610

yes

characteristic

158

C₂

#20291684

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#423	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#423	#6157
hypocenter	C₅₃	#33501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₈

Fitting

C₅₃×#423

C₄

prime radical

C₂³×C₅₃

C₂²×C₈

C₂⁴

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
1. #20291132	C₂	semidirect
97. C₅₃	#33501	semidirect

H ≅ G/N

type

1. #20291132

C₂

semidirect

97. C₅₃

#33501

semidirect

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

subgroup	index
#20291132	2
#20291081	2
#20291076	2
#20283645	2
#33501	53

subgroup

index

Subgroups — #28328557

Conjugacy classes of subgroups

Complements

Maximal subgroups