Group #20290952 — subgroups

Subgroups — #20290952

Order 6784.

220 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 72 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	D₅₃⋊M₃₂	C₂	1	2	yes	maximal, complemented
2	#16811520	C₂	1	2	yes	maximal, complemented
3	#16811519	C₂	1	2	yes	maximal, complemented
4	C₂×#13133223	C₂	1	1	yes	maximal, characteristic
5	#13133292	—	2	2	no	—
6	#13133291	C₂²	1	4	yes	—
7	D₅₃⋊C₁₆	—	2	2	no	—
8	C₂×#11719945	C₄	1	1	yes	characteristic
9	#13133223	—	2	2	no	—
10	#13133223	—	2	2	no	—
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	C₂×#11719912	C₄	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	C₄	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₅₃	C₈	1	1	yes	characteristic
20	#11719945	—	2	2	no	—
21	#11719945	—	2	2	no	—
22	C₂×D₂₁₂	C₈	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₀₆	C₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	1	no	—
27	C₂×C₈×C₅₃	—	2	1	no	—
28	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#11719917	—	2	2	no	—
31	C₂×#88322	C₂×C₄	1	1	yes	characteristic
32	C₂×#88322	C₂³	1	1	yes	characteristic
33	#11719912	—	2	2	no	—
34	#11719912	—	2	2	no	—
35	C₈×D₅₃	—	2	1	no	—
36	C₈×D₅₃	—	2	1	no	—
37	C₈×D₅₃	D₄	1	2	yes	—
38	C₈×D₅₃	D₄	1	2	yes	—
39	#11719910	—	2	4	no	—
40	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	—	2	1	no	—
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂²⋊D₅₃	—	2	2	no	—
48	C₂²⋊D₅₃	—	2	2	no	—
49	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
51	D₂₁₂	—	2	2	no	—
52	C₄×D₅₃	—	2	1	no	—
53	C₄×D₅₃	—	2	1	no	—
54	C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
55	C₄×D₅₃	M₁₆	1	2	yes	—
56	Dic₁₀₆	—	2	2	no	—
57	C₈×C₅₃	—	2	1	no	—
58	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	#88322	—	2	1	no	—
61	#88322	C₂×D₄	1	2	yes	—
62	C₂²×C₅₃	—	2	1	no	—
63	C₂²×C₅₃	—	4	1	no	—
64	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×D₅₃	—	2	1	no	—
66	C₂×D₅₃	—	4	1	no	—
67	C₂×D₅₃	—	4	1	no	—
68	C₂×D₅₃	C₂²⋊C₈	1	2	yes	—
69	C₄×C₅₃	—	2	1	no	—
70	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	Dic₅₃	—	2	1	no	—
73	Dic₅₃	C₂²⋊C₈	1	2	yes	—
74	#2115	—	53	1	no	maximal, Sylow
75	C₂×C₅₃	—	2	1	no	—
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
78	D₅₃	—	2	2	no	—
79	D₅₃	—	4	1	no	—
80	C₂×#131	—	53	1	no	—
81	C₂×M₃₂	—	53	2	no	—
82	#349	—	53	2	no	—
83	#349	—	53	2	no	—
84	C₅₃	#2115	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	53	1	no	—
86	#131	—	106	2	no	—
87	#131	—	106	2	no	—
88	C₂×M₁₆	—	53	1	no	—
89	C₂×M₁₆	—	53	1	no	—
90	C₂×M₁₆	—	53	1	no	—
91	C₂²×C₈	—	53	1	no	—
92	C₂²×C₈	—	53	1	no	—
93	C₂²×C₈	#6157	1	1	yes	characteristic
94	M₃₂	—	53	4	no	—
95	M₃₂	—	106	2	no	—
96	C₂×C₁₆	—	106	2	no	—
97	#41	—	106	2	no	—
98	#41	—	106	2	no	—
99	C₂×Q₈	—	53	1	no	—
100	C₂×D₄	—	53	1	no	—
101	C₂×D₄	—	53	1	no	—
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	53	1	no	—
104	C₂²×C₄	—	53	1	no	—
105	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
106	M₁₆	—	106	2	no	—
107	M₁₆	—	106	2	no	—
108	M₁₆	—	106	2	no	—
109	C₂×C₈	—	2	1	no	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	—	106	1	no	—
117	C₂×C₈	C₂×#6157	1	1	yes	characteristic
118	C₂×C₈	C₂×#6157	1	1	yes	characteristic
119	C₁₆	—	106	4	no	—
120	C₂³	—	53	1	no	—
121	C₂³	—	53	1	no	—
122	C₂³	D₅₃⋊C₈	1	1	yes	characteristic
123	Q₈	—	106	2	no	—
124	D₄	—	106	2	no	—
125	D₄	—	106	2	no	—
126	D₄	—	106	2	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	106	1	no	—
134	C₂×C₄	—	106	1	no	—
135	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
136	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	53	2	no	—
139	C₈	—	106	1	no	—
140	C₈	C₂²⋊#6157	1	1	yes	characteristic
141	C₈	C₂²⋊#6157	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	53	2	no	—
145	C₂²	—	106	1	no	—
146	C₂²	—	212	1	no	—
147	C₂²	—	212	1	no	—
148	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	53	2	no	—
151	C₄	—	106	1	no	—
152	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
153	C₄	D₅₃⋊M₁₆	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	4	1	no	—
156	C₂	—	106	2	no	—
157	C₂	—	212	1	no	—
158	C₂	D₅₃⋊(C₂²⋊C₈)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₅₃⋊M₃₂

C₂

yes

maximal, complemented

#16811520

C₂

yes

maximal, complemented

#16811519

C₂

yes

maximal, complemented

C₂×#13133223

C₂

yes

maximal, characteristic

#13133292

—

#13133291

C₂²

yes

—

D₅₃⋊C₁₆

—

C₂×#11719945

C₄

yes

—

—

C₂²

yes

characteristic

C₂×#11719912

C₄

yes

characteristic

C₂×#11719912

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₄

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

characteristic

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

C₈

yes

characteristic

C₂×C₂²⋊D₅₃

C₈

yes

characteristic

#11719945

—

#11719945

—

C₂×D₂₁₂

C₈

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×Dic₁₀₆

C₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

#11719917

—

C₂×#88322

C₂×C₄

yes

characteristic

C₂×#88322

C₂³

yes

characteristic

#11719912

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₈×D₅₃

D₄

yes

—

C₈×D₅₃

D₄

yes

—

#11719910

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂×C₈

yes

characteristic

C₂×Dic₅₃

C₂×C₈

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂²⋊C₄

yes

—

C₄×D₅₃

M₁₆

yes

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂²⋊C₄

yes

characteristic

#88322

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²⋊C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

C₂²⋊C₈

yes

—

#2115

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

D₅₃

—

D₅₃

—

C₂×#131

—

C₂×M₃₂

—

#349

—

#349

—

C₅₃

#2115

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

106

—

#131

—

106

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

#6157

yes

characteristic

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂×#6157

yes

characteristic

106

M₁₆

—

106

—

107

M₁₆

—

106

—

108

M₁₆

—

106

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

106

—

116

C₂×C₈

—

106

—

117

C₂×C₈

C₂×#6157

yes

characteristic

118

C₂×C₈

C₂×#6157

yes

characteristic

119

C₁₆

—

106

—

120

C₂³

—

121

C₂³

—

122

C₂³

D₅₃⋊C₈

yes

characteristic

123

Q₈

—

106

—

124

D₄

—

106

—

125

D₄

—

106

—

126

D₄

—

106

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

D₅₃⋊C₈

yes

characteristic

136

C₂×C₄

C₂²×#6157

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

106

—

140

C₈

C₂²⋊#6157

yes

characteristic

141

C₈

C₂²⋊#6157

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

106

—

146

C₂²

—

212

—

147

C₂²

—

212

—

148

C₂²

C₂×D₅₃⋊C₈

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

106

—

152

C₄

C₂×C₂²⋊#6157

yes

characteristic

153

C₄

D₅₃⋊M₁₆

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

—

106

—

157

C₂

—

212

—

158

C₂

D₅₃⋊(C₂²⋊C₈)

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊#6157
commutator	C₂²×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	C₂²×C₈×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₂²⋊C₈
hypercenter	C₂²×C₈	#6157
hypocenter	C₅₃	#2115
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

subgroup

G/H

center

C₈

C₂²⋊#6157

commutator

C₂²×C₅₃

C₂²×C₈

Frattini

C₂×C₄

C₂²×#6157

Fitting

C₂²×C₈×C₅₃

C₄

socle

C₂×C₅₃

C₂×C₂²⋊C₈

C₂²×C₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈

N	H ≅ G/N	type
1. D₅₃⋊M₃₂	C₂	semidirect
2. #16811520	C₂	semidirect
3. #16811519	C₂	semidirect
84. C₅₃	#2115	semidirect

H ≅ G/N

type

1. D₅₃⋊M₃₂

C₂

semidirect

2. #16811520

C₂

semidirect

3. #16811519

C₂

semidirect

84. C₅₃

#2115

semidirect

Subgroups — #20290952

Conjugacy classes of subgroups

Complements