Group #27300306 — subgroups

Subgroups — #27300306

Order 13568.

454 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 171 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	C₂×C₂²⋊#11719908	C₂	1	2	yes	maximal
2	#20272793	C₂	1	4	yes	maximal, complemented
3	C₂×C₅₃×#405	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719908	C₂²	1	4	yes	—
5	C₂²⋊#11719908	—	2	4	no	—
6	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₅₃	C₄	1	1	yes	characteristic
9	C₅₃×#405	—	2	4	no	—
10	C₂×#97×C₅₃	C₄	1	2	yes	—
11	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
12	C₂×#11719908	—	2	4	no	—
13	C₂×#11719908	—	2	8	no	—
14	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₅₃	—	2	4	no	—
16	C₂×M₁₆×C₅₃	—	2	2	no	—
17	C₂²×C₈×C₅₃	—	2	4	no	—
18	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	D₄	1	4	yes	—
21	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
22	#117×C₅₃	—	2	2	no	—
23	#117×C₅₃	—	2	2	no	—
24	C₂×C₄⋊C₄×C₅₃	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₅₃	C₈	1	2	yes	—
26	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
27	#97×C₅₃	—	2	4	no	—
28	C₄×C₈×C₅₃	—	2	4	no	—
29	C₂⁴×C₅₃	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	—	2	2	no	—
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₅₃	—	2	8	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	2	no	—
41	C₂×C₈×C₅₃	—	2	8	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	M₁₆	1	4	yes	—
44	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₅₃	#41	1	4	yes	—
46	C₄⋊C₄×C₅₃	—	2	4	no	—
47	C₂²⋊C₄×C₅₃	—	2	4	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	—	2	2	no	—
50	#11719908	—	4	4	no	—
51	C₂³×C₅₃	—	2	2	no	—
52	C₂³×C₅₃	—	2	4	no	—
53	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	—	2	4	no	—
60	C₂×C₄×C₅₃	—	2	8	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#117	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	—	2	4	no	—
68	C₈×C₅₃	#123	1	4	yes	—
69	#12845	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	2	no	—
71	C₂²×C₅₃	—	2	4	no	—
72	C₂²×C₅₃	—	4	4	no	—
73	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
74	C₂²×C₅₃	#431	1	2	yes	—
75	C₂²×C₅₃	#504	1	4	yes	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	4	no	—
78	C₄×C₅₃	#381	1	1	yes	characteristic
79	C₄×C₅₃	#407	1	1	yes	characteristic
80	C₄×C₅₃	#432	1	2	yes	—
81	C₂×#405	D₅₃	1	1	yes	characteristic
82	C₂×C₂²⋊C₁₆	—	53	2	no	—
83	#1374	—	53	4	no	—
84	C₂×C₅₃	—	4	2	no	—
85	C₂×C₅₃	#1817	1	1	yes	characteristic
86	C₂×C₅₃	#2113	1	2	yes	—
87	C₂×C₅₃	#2164	1	4	yes	—
88	C₂²×M₁₆	Dic₅₃	1	1	yes	characteristic
89	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂×#117	Dic₅₃	1	1	yes	characteristic
91	C₂²×C₁₆	—	53	4	no	—
92	#405	—	2	4	no	—
93	C₂×#97	Dic₅₃	1	2	yes	—
94	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
95	C₂²⋊C₁₆	—	106	4	no	—
96	C₅₃	#12845	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
102	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₅₃	1	4	yes	—
104	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	#88322	1	2	yes	—
108	C₂×C₂²⋊C₄	#88322	1	2	yes	—
109	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
110	C₂×C₁₆	—	106	8	no	—
111	C₂×C₁₆	—	106	4	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×#88322	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×#88322	1	4	yes	—
118	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
119	C₂²×C₄	#11719917	1	2	yes	—
120	C₂²×C₄	C₂²⋊Dic₅₃	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	#11719917	1	4	yes	—
128	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
129	C₂×C₈	#11719945	1	4	yes	—
130	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	212	4	no	—
136	C₂³	—	2	4	no	—
137	C₂³	—	2	2	no	—
138	C₂³	C₂²⋊#88322	1	2	yes	—
139	C₂³	C₂²×#88322	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	4	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	8	no	—
145	C₂×C₄	—	2	4	no	—
146	C₂×C₄	C₂²⋊#88322	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₄	#13133176	1	2	yes	—
149	C₂×C₄	C₂×#11719917	1	2	yes	—
150	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
151	C₈	—	2	4	no	—
152	C₈	—	2	2	no	—
153	C₈	#13133200	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	4	4	no	—
157	C₂²	C₂×C₂²⋊#88322	1	1	yes	characteristic
158	C₂²	#16810676	1	2	yes	—
159	C₂²	#16811040	1	4	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#16810486	1	1	yes	characteristic
163	C₄	#16810587	1	1	yes	characteristic
164	C₄	#16810680	1	2	yes	—
165	C₂	—	4	2	no	—
166	C₂	#20274783	1	1	yes	characteristic
167	C₂	#20276329	1	2	yes	—
168	C₂	#20276522	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#11719908

C₂

yes

maximal

#20272793

C₂

yes

maximal, complemented

C₂×C₅₃×#405

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂²⋊#11719908

—

C₂²×M₁₆×C₅₃

C₄

yes

characteristic

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×#117×C₅₃

C₄

yes

characteristic

C₅₃×#405

—

C₂×#97×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×#11719908

—

C₂×#11719908

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#117×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₈

yes

—

C₂×C₂²⋊C₄×C₅₃

C₈

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#123

yes

—

#12845

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#405

D₅₃

yes

characteristic

C₂×C₂²⋊C₁₆

—

#1374

—

C₂×C₅₃

—

C₂×C₅₃

#1817

yes

characteristic

C₂×C₅₃

#2113

yes

—

C₂×C₅₃

#2164

yes

—

C₂²×M₁₆

Dic₅₃

yes

characteristic

C₂³×C₈

C₂×D₅₃

yes

characteristic

C₂×#117

Dic₅₃

yes

characteristic

C₂²×C₁₆

—

#405

—

C₂×#97

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₂²⋊C₁₆

—

106

—

C₅₃

#12845

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₅₃

yes

—

102

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

103

C₂²×C₈

C₂²⋊D₅₃

yes

—

104

C₂²×C₈

C₂²×D₅₃

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

#88322

yes

—

108

C₂×C₂²⋊C₄

#88322

yes

—

109

C₂×C₄²

C₂×Dic₅₃

yes

—

110

C₂×C₁₆

—

106

—

111

C₂×C₁₆

—

106

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×#88322

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×#88322

yes

—

118

C₂²×C₄

C₂×#88322

yes

characteristic

119

C₂²×C₄

#11719917

yes

—

120

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

121

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#11719917

yes

—

128

C₂×C₈

C₂²⋊Dic₅₃

yes

—

129

C₂×C₈

#11719945

yes

—

130

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

212

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊#88322

yes

—

139

C₂³

C₂²×#88322

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊#88322

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

148

C₂×C₄

#13133176

yes

—

149

C₂×C₄

C₂×#11719917

yes

—

150

C₂×C₄

C₂×#11719917

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#13133200

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

C₂×C₂²⋊#88322

yes

characteristic

158

C₂²

#16810676

yes

—

159

C₂²

#16811040

yes

—

160

C₄

—

161

C₄

—

162

C₄

#16810486

yes

characteristic

163

C₄

#16810587

yes

characteristic

164

C₄

#16810680

yes

—

165

C₂

—

166

C₂

#20274783

yes

characteristic

167

C₂

#20276329

yes

—

168

C₂

#20276522

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×#405	C₂
prime radical	C₂⁴×C₅₃	C₂×C₈
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#405	D₅₃
hypocenter	C₅₃	#12845
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#405

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂²×C₈

Fitting

C₂×C₅₃×#405

C₂

prime radical

C₂⁴×C₅₃

C₂×C₈

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #20272793	C₂	semidirect
96. C₅₃	#12845	semidirect

H ≅ G/N

type

2. #20272793

C₂

semidirect

96. C₅₃

#12845

semidirect

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

subgroup	index
C₂×C₂²⋊#11719908	2
#20272793	2
C₂×C₅₃×#405	2
#12845	53

subgroup

index

C₂×C₂²⋊#11719908

#20272793

C₂×C₅₃×#405

#12845

Subgroups — #27300306

Conjugacy classes of subgroups

Complements

Maximal subgroups