Group #28328553 — subgroups

Subgroups — #28328553

Order 13568.

460 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 218 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	#20291131	C₂	1	4	yes	maximal, complemented
2	#20291129	C₂	1	4	yes	maximal, complemented
3	#20291081	C₂	1	4	yes	maximal
4	#20291072	C₂	1	2	yes	maximal
5	#20283645	C₂	1	1	yes	maximal, characteristic
6	#16811544	C₂²	1	8	yes	—
7	C₄⋊#11719910	C₂²	1	2	yes	—
8	C₄⋊#11719910	C₂²	1	2	yes	—
9	C₂²⋊#11719910	C₂²	1	4	yes	—
10	#16811512	C₂²	1	4	yes	—
11	#16811509	C₂²	1	4	yes	—
12	C₄×#11719910	C₂²	1	4	yes	—
13	#16811180	C₄	1	1	yes	characteristic
14	#16810750	C₄	1	4	yes	—
15	#16810747	C₂²	1	4	yes	—
16	#16810651	C₄	1	2	yes	—
17	#16810581	C₂²	1	2	yes	—
18	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
19	C₅₃×#423	C₄	1	1	yes	characteristic
20	C₂×#11719910	C₂³	1	4	yes	—
21	C₂×#11719910	C₂³	1	4	yes	—
22	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
23	C₂²×#88322	C₂³	1	1	yes	characteristic
24	#13133217	C₈	1	2	yes	—
25	#13133209	C₈	1	2	yes	—
26	#13133196	C₈	1	4	yes	—
27	C₄×Dic₁₀₆	C₂×C₄	1	4	yes	—
28	#13133176	C₂×C₄	1	2	yes	—
29	#13133146	C₂×C₄	1	8	yes	—
30	#13133127	C₂×C₄	1	4	yes	—
31	C₄×#88322	C₂×C₄	1	2	yes	—
32	C₄×#88322	C₂³	1	2	yes	—
33	C₈×Dic₅₃	C₂³	1	4	yes	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×Dic₁₀₆	C₂×C₈	1	4	yes	—
40	M₁₆×C₅₃	—	4	2	no	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²⋊Dic₅₃	C₂×C₈	1	4	yes	—
46	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
47	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
48	#11719919	C₂×C₈	1	8	yes	—
49	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
50	C₂×#88322	—	2	2	no	—
51	C₂×#88322	—	4	1	no	—
52	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
53	C₂×#88322	C₂²×C₄	1	2	yes	—
54	C₂×#88322	C₂⁴	1	1	yes	characteristic
55	#11719910	—	2	4	no	—
56	#11719910	—	2	4	no	—
57	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	1	no	—
60	C₂×C₄×C₅₃	—	2	4	no	—
61	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×Dic₅₃	C₂²×C₈	1	8	yes	—
66	Dic₁₀₆	—	2	8	no	—
67	C₈×C₅₃	—	2	4	no	—
68	#88322	—	2	1	no	—
69	#88322	—	2	2	no	—
70	#88322	2₋¹⁺⁴	1	2	yes	—
71	#33500	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	—	2	1	no	—
74	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	1	no	—
77	C₄×C₅₃	#504	1	4	yes	—
78	C₄×C₅₃	#519	1	1	yes	characteristic
79	C₄×C₅₃	#568	1	1	yes	characteristic
80	Dic₅₃	—	2	8	no	—
81	#2960	—	53	1	no	—
82	#2183	—	53	4	no	—
83	Q₈×C₁₆	—	53	4	no	—
84	#2164	—	53	4	no	—
85	#2162	—	53	2	no	—
86	C₂×C₅₃	—	4	1	no	—
87	C₂×C₅₃	#2982	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#504	1	2	yes	—
89	#533	—	53	1	no	—
90	#446	—	53	4	no	—
91	C₈×Q₈	—	53	4	no	—
92	#424	—	53	2	no	—
93	#423	#6157	1	1	yes	characteristic
94	#405	—	53	2	no	—
95	C₂×C₄×C₈	—	53	1	no	—
96	C₄⋊C₁₆	—	53	2	no	—
97	C₄⋊C₁₆	—	53	2	no	—
98	C₄⋊C₁₆	—	53	8	no	—
99	C₂²⋊C₁₆	—	53	4	no	—
100	#346	—	53	4	no	—
101	C₄×C₁₆	—	53	4	no	—
102	C₄×C₁₆	—	53	4	no	—
103	C₅₃	#33500	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	C₂×#6157	1	2	yes	—
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	—	53	1	no	—
107	C₄⋊Q₈	—	53	2	no	—
108	#125	—	53	2	no	—
109	C₂²⋊Q₈	—	53	4	no	—
110	C₄×Q₈	—	53	4	no	—
111	#117	—	53	2	no	—
112	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
113	C₂×C₁₆	—	53	4	no	—
114	C₂×C₁₆	—	53	4	no	—
115	C₄⋊C₈	—	53	8	no	—
116	C₄⋊C₈	C₂×#6157	1	4	yes	—
117	#97	—	53	4	no	—
118	C₄×C₈	—	53	4	no	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	2	no	—
121	C₂×Q₈	—	53	4	no	—
122	C₂²×C₄	D₅₃⋊C₈	1	2	yes	—
123	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	53	1	no	—
127	C₂×C₈	—	53	2	no	—
128	C₂×C₈	—	106	2	no	—
129	C₂×C₈	—	212	1	no	—
130	C₂×C₈	C₂²×#6157	1	4	yes	—
131	C₄⋊C₄	—	53	4	no	—
132	C₄⋊C₄	—	53	8	no	—
133	C₄⋊C₄	—	53	4	no	—
134	C₂²⋊C₄	—	53	4	no	—
135	C₄²	—	53	4	no	—
136	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
137	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
138	C₄²	C₂²×#6157	1	2	yes	—
139	C₁₆	—	106	4	no	—
140	C₁₆	—	106	4	no	—
141	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
142	Q₈	—	106	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	53	8	no	—
147	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
149	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
150	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	53	2	no	—
153	C₈	—	106	1	no	—
154	C₈	—	106	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	2	2	no	—
160	C₄	—	106	8	no	—
161	C₄	#16811682	1	4	yes	—
162	C₄	#16811691	1	1	yes	characteristic
163	C₄	#16811701	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	#20291610	1	1	yes	characteristic
166	C₂	C₂×#16811682	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20291131

C₂

yes

maximal, complemented

#20291129

C₂

yes

maximal, complemented

#20291081

C₂

yes

maximal

#20291072

C₂

yes

maximal

#20283645

C₂

yes

maximal, characteristic

#16811544

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

C₄⋊#11719910

C₂²

yes

—

C₂²⋊#11719910

C₂²

yes

—

#16811512

C₂²

yes

—

#16811509

C₂²

yes

—

C₄×#11719910

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

—

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

—

#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

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×C₄×C₅₃

C₂³×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₈

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

#88322

2₋¹⁺⁴

yes

—

#33500

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#504

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2960

—

#2183

—

Q₈×C₁₆

—

#2164

—

#2162

—

C₂×C₅₃

—

C₂×C₅₃

#2982

yes

characteristic

C₂×C₅₃

C₂×#504

yes

—

#533

—

#446

—

C₈×Q₈

—

—

yes

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

100

#346

—

101

C₄×C₁₆

—

102

C₄×C₁₆

—

103

C₅₃

#33500

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

C₂×#6157

yes

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

C₂×C₄²

C₂×#6157

yes

characteristic

113

C₂×C₁₆

—

114

C₂×C₁₆

—

115

C₄⋊C₈

—

116

C₄⋊C₈

C₂×#6157

yes

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

—

120

C₄×C₈

—

121

C₂×Q₈

—

122

C₂²×C₄

D₅₃⋊C₈

yes

—

123

C₂²×C₄

C₂²×#6157

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

212

—

130

C₂×C₈

C₂²×#6157

yes

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

D₅₃⋊C₈

yes

characteristic

137

C₄²

D₅₃⋊C₈

yes

characteristic

138

C₄²

C₂²×#6157

yes

—

139

C₁₆

—

106

—

140

C₁₆

—

106

—

141

C₂³

C₂×D₅₃⋊C₈

yes

characteristic

142

Q₈

—

106

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

148

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

149

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

150

C₂×C₄

C₂³×#6157

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

106

—

154

C₈

—

106

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂²×D₅₃⋊C₈

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

106

—

161

C₄

#16811682

yes

—

162

C₄

#16811691

yes

characteristic

163

C₄

#16811701

yes

characteristic

164

C₂

—

165

C₂

#20291610

yes

characteristic

166

C₂

C₂×#16811682

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	#33500
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. #20291131	C₂	semidirect
2. #20291129	C₂	semidirect
103. C₅₃	#33500	semidirect

H ≅ G/N

type

1. #20291131

C₂

semidirect

2. #20291129

C₂

semidirect

103. C₅₃

#33500

semidirect

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

subgroup	index
#20291131	2
#20291129	2
#20291081	2
#20291072	2
#20283645	2
#33500	53

subgroup

index

Subgroups — #28328553

Conjugacy classes of subgroups

Complements

Maximal subgroups