Group #27302686 — subgroups

Subgroups — #27302686

Order 13568.

334 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20276471	C₂	1	1	yes	maximal, characteristic
2	C₄×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20272823	C₂	1	4	yes	maximal
4	C₂×C₅₃×#499	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×#13133153	C₂²	1	2	yes	—
7	C₄×#11719917	—	2	1	no	—
8	C₄⋊#11719908	—	2	2	no	—
9	#16810388	—	2	1	no	—
10	C₂×#125×C₅₃	—	2	1	no	—
11	C₅₃×#499	—	2	2	no	—
12	C₅₃×#499	C₄	1	4	yes	—
13	C₂×#107×C₅₃	C₂²	1	2	yes	—
14	C₂×#106×C₅₃	—	2	1	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719917	—	2	2	no	—
17	#13133157	—	2	4	no	—
18	C₂×#11719908	—	2	2	no	—
19	#13133153	—	2	2	no	—
20	#13133153	Q₈	1	4	yes	—
21	#13133128	—	4	1	no	—
22	C₄×#88322	—	4	1	no	—
23	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	#125×C₅₃	—	2	4	no	—
26	#125×C₅₃	—	4	1	no	—
27	#125×C₅₃	—	4	1	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	4	2	no	—
30	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
31	#107×C₅₃	—	2	2	no	—
32	#107×C₅₃	C₂×C₄	1	4	yes	—
33	#106×C₅₃	—	2	4	no	—
34	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₅₃	D₄	1	2	yes	—
36	C₂²×C₄×C₅₃	—	4	2	no	—
37	C₂²×C₄×C₅₃	SD₁₆	1	2	yes	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₅₃	#41	1	2	yes	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	—	4	4	no	—
47	C₄⋊C₄×C₅₃	—	4	4	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₄²×C₅₃	D₈	1	1	yes	characteristic
51	C₄²×C₅₃	D₈	1	1	yes	characteristic
52	#11719917	—	2	4	no	—
53	#11719917	—	4	2	no	—
54	C₂×#88322	—	4	2	no	—
55	#11719908	—	4	2	no	—
56	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	—	4	4	no	—
59	C₂×C₄×C₅₃	—	8	4	no	—
60	C₂×C₄×C₅₃	#102	1	2	yes	—
61	C₂×C₄×C₅₃	#102	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	#137	1	2	yes	—
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	C₄×Q₈	1	2	yes	—
68	C₈×C₅₃	#124	1	1	yes	characteristic
69	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
70	#88322	—	4	4	no	—
71	#13280	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
74	C₂²×C₅₃	#475	1	2	yes	—
75	C₄×C₅₃	—	2	2	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	8	4	no	—
78	C₄×C₅₃	#391	1	1	yes	characteristic
79	C₄×C₅₃	#419	1	1	yes	characteristic
80	C₄×C₅₃	#474	1	2	yes	—
81	C₂×#499	D₅₃	1	1	yes	characteristic
82	#2150	—	53	1	no	—
83	C₄×#108	—	53	1	no	—
84	#1387	—	53	4	no	—
85	C₂×C₅₃	—	2	2	no	—
86	C₂×C₅₃	#1988	1	1	yes	characteristic
87	C₂×C₅₃	#2148	1	2	yes	—
88	C₂×#125	—	2	1	no	—
89	C₂×M₃₂	—	53	2	no	—
90	#499	—	2	2	no	—
91	#499	Dic₅₃	1	4	yes	—
92	C₂×#108	—	53	2	no	—
93	C₂×#107	C₂×D₅₃	1	2	yes	—
94	C₂×#106	—	2	1	no	—
95	C₄×M₁₆	—	106	1	no	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₁₆	—	106	2	no	—
98	#341	—	106	1	no	—
99	C₅₃	#13280	1	1	yes	characteristic, complemented, Sylow
100	C₂×M₁₆	—	106	2	no	—
101	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	#125	—	2	4	no	—
104	#125	—	4	1	no	—
105	#125	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	4	2	no	—
108	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
109	M₃₂	—	106	4	no	—
110	C₂×C₁₆	—	106	2	no	—
111	#108	—	53	4	no	—
112	#108	—	106	2	no	—
113	#107	—	2	2	no	—
114	#107	C₂×Dic₅₃	1	4	yes	—
115	#106	—	2	4	no	—
116	#97	—	212	1	no	—
117	C₄×C₈	—	212	1	no	—
118	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
119	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	#11719923	1	2	yes	—
122	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
123	M₁₆	—	106	4	no	—
124	M₁₆	—	212	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	212	2	no	—
127	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
128	C₂×C₈	Q₈×D₅₃	1	2	yes	—
129	C₂×C₈	#11719949	1	2	yes	—
130	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
131	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	—	4	4	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	2	no	—
136	C₄²	#11719922	1	1	yes	characteristic
137	C₄²	#11719922	1	1	yes	characteristic
138	C₄²	C₂²⋊Dic₅₃	1	2	yes	—
139	C₁₆	—	212	2	no	—
140	C₂³	C₂×#11719923	1	1	yes	characteristic
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	4	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	#13133138	1	2	yes	—
145	C₂×C₄	#13133138	1	2	yes	—
146	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
147	C₂×C₄	#13133193	1	2	yes	—
148	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
149	C₂×C₄	#13133248	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	212	4	no	—
152	C₈	Q₈×Dic₅₃	1	2	yes	—
153	C₈	#13133206	1	1	yes	characteristic
154	C₈	#13133215	1	1	yes	characteristic
155	C₂²	—	2	2	no	—
156	C₂²	C₂×#13133138	1	1	yes	characteristic
157	C₂²	#16810905	1	2	yes	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	8	4	no	—
161	C₄	#16810530	1	1	yes	characteristic
162	C₄	#16810632	1	1	yes	characteristic
163	C₄	#16810900	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#20275668	1	1	yes	characteristic
166	C₂	#20276464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276471

C₂

yes

maximal, characteristic

C₄×#13133153

C₂

yes

maximal, characteristic

#20272823

C₂

yes

maximal

C₂×C₅₃×#499

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×#13133153

C₂²

yes

—

C₄×#11719917

—

C₄⋊#11719908

—

#16810388

—

C₂×#125×C₅₃

—

C₅₃×#499

—

C₅₃×#499

C₄

yes

—

C₂×#107×C₅₃

C₂²

yes

—

C₂×#106×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

—

Q₈

yes

—

#13133128

—

C₄×#88322

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

characteristic

#107×C₅₃

—

#107×C₅₃

C₂×C₄

yes

—

#106×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

characteristic

C₄²×C₅₃

D₈

yes

—

—

—

—

C₂³×C₅₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄×Q₈

yes

—

C₈×C₅₃

#124

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#102

yes

characteristic

C₂²×C₅₃

#475

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×#499

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1988

yes

characteristic

C₂×C₅₃

#2148

yes

—

C₂×#125

—

C₂×M₃₂

—

#499

—

#499

Dic₅₃

yes

—

C₂×#108

—

C₂×#107

C₂×D₅₃

yes

—

C₂×#106

—

C₄×M₁₆

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

#341

—

106

—

C₅₃

#13280

yes

characteristic, complemented, Sylow

100

C₂×M₁₆

—

106

—

101

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

102

C₂²×C₈

C₂²×D₅₃

yes

characteristic

103

#125

—

104

#125

—

105

#125

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄²

C₂²⋊D₅₃

yes

characteristic

109

M₃₂

—

106

—

110

C₂×C₁₆

—

106

—

111

#108

—

112

#108

—

106

—

113

#107

—

114

#107

C₂×Dic₅₃

yes

—

115

#106

—

116

#97

—

212

—

117

C₄×C₈

—

212

—

118

C₄×C₈

C₂×Dic₅₃

yes

—

119

C₄×C₈

C₂²⋊D₅₃

yes

—

120

C₂²×C₄

—

121

C₂²×C₄

#11719923

yes

—

122

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

123

M₁₆

—

106

—

124

M₁₆

—

212

—

125

C₂×C₈

—

126

C₂×C₈

—

212

—

127

C₂×C₈

C₂²⋊Dic₅₃

yes

—

128

C₂×C₈

Q₈×D₅₃

yes

—

129

C₂×C₈

#11719949

yes

—

130

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

131

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄²

#11719922

yes

characteristic

137

C₄²

#11719922

yes

characteristic

138

C₄²

C₂²⋊Dic₅₃

yes

—

139

C₁₆

—

212

—

140

C₂³

C₂×#11719923

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#13133138

yes

—

145

C₂×C₄

#13133138

yes

—

146

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

147

C₂×C₄

#13133193

yes

—

148

C₂×C₄

C₂×#11719922

yes

characteristic

149

C₂×C₄

#13133248

yes

—

150

C₈

—

151

C₈

—

212

—

152

C₈

Q₈×Dic₅₃

yes

—

153

C₈

#13133206

yes

characteristic

154

C₈

#13133215

yes

characteristic

155

C₂²

—

156

C₂²

C₂×#13133138

yes

characteristic

157

C₂²

#16810905

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#16810530

yes

characteristic

162

C₄

#16810632

yes

characteristic

163

C₄

#16810900

yes

—

164

C₂

—

165

C₂

#20275668

yes

characteristic

166

C₂

#20276464

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133138
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#499	C₂
socle	C₂²×C₅₃	C₂×#102
prime radical	C₂³×C₅₃	C₂×SD₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#499	D₅₃
hypocenter	C₅₃	#13280
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

subgroup

G/H

center

C₂²

C₂×#13133138

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#499

C₂

socle

C₂²×C₅₃

C₂×#102

prime radical

C₂³×C₅₃

C₂×SD₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

N	H ≅ G/N	type
99. C₅₃	#13280	semidirect

H ≅ G/N

type

99. C₅₃

#13280

semidirect

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

subgroup	index
#20276471	2
C₄×#13133153	2
#20272823	2
C₂×C₅₃×#499	2
#13280	53

subgroup

index

C₂×C₅₃×#499

Subgroups — #27302686

Conjugacy classes of subgroups

Complements

Maximal subgroups