Group #27277122 — subgroups

Subgroups — #27277122

Order 13568.

336 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20273474	C₂	1	2	yes	maximal
2	#20273453	C₂	1	2	yes	maximal
3	#20272628	C₂	1	2	yes	maximal
4	C₅₃×#2960	C₂	1	1	yes	maximal, characteristic
5	#16810647	C₂²	1	4	yes	—
6	#16810376	—	2	2	no	—
7	#16810376	—	2	2	no	—
8	#16810369	—	2	4	no	—
9	C₅₃×#533	C₂²	1	1	yes	characteristic
10	C₅₃×#446	—	2	2	no	—
11	C₈×Q₈×C₅₃	—	2	2	no	—
12	C₅₃×#424	C₄	1	2	yes	—
13	C₅₃×#423	C₂²	1	1	yes	characteristic
14	C₅₃×#405	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719917	—	2	4	no	—
17	C₄⋊#88322	—	2	4	no	—
18	C₄⋊#88322	—	2	4	no	—
19	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
23	#125×C₅₃	Q₈	1	2	yes	—
24	C₂²⋊Q₈×C₅₃	—	2	2	no	—
25	C₄×Q₈×C₅₃	—	2	2	no	—
26	#117×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₅₃	—	2	4	no	—
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	#97×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	—	2	2	no	—
32	C₄×C₈×C₅₃	D₄	1	2	yes	—
33	C₄×C₈×C₅₃	D₄	1	2	yes	—
34	C₂×Q₈×C₅₃	—	2	2	no	—
35	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	#41	1	2	yes	—
37	M₁₆×C₅₃	—	4	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	4	1	no	—
41	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄⋊C₄×C₅₃	—	2	2	no	—
45	C₄⋊C₄×C₅₃	—	2	4	no	—
46	C₄⋊C₄×C₅₃	C₄⋊C₄	1	4	yes	—
47	C₂²⋊C₄×C₅₃	—	2	2	no	—
48	C₄²×C₅₃	—	2	2	no	—
49	C₄²×C₅₃	C₂×D₄	1	2	yes	—
50	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
52	#11719917	—	8	4	no	—
53	C₂×#88322	—	4	4	no	—
54	C₂³×C₅₃	#117	1	1	yes	characteristic
55	Q₈×C₅₃	—	4	2	no	—
56	Q₈×C₅₃	—	4	2	no	—
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	—	2	1	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	—	2	4	no	—
62	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	#121	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
66	C₈×C₅₃	—	2	1	no	—
67	C₈×C₅₃	—	2	2	no	—
68	C₈×C₅₃	—	4	2	no	—
69	C₈×C₅₃	#101	1	2	yes	—
70	#88322	—	8	4	no	—
71	#8639	—	53	1	no	maximal, Sylow
72	C₂²×C₅₃	—	2	1	no	—
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#380	1	1	yes	characteristic
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	—	4	4	no	—
78	C₄×C₅₃	C₂×#101	1	1	yes	characteristic
79	C₄×C₅₃	#413	1	1	yes	characteristic
80	C₄×C₅₃	#472	1	2	yes	—
81	C₄×C₅₃	#473	1	2	yes	—
82	#2960	D₅₃	1	1	yes	characteristic
83	#1567	—	53	2	no	—
84	#1561	—	53	2	no	—
85	#1309	—	53	2	no	—
86	C₂×C₅₃	—	4	1	no	—
87	C₂×C₅₃	#1801	1	1	yes	characteristic
88	C₂×C₅₃	#1816	1	2	yes	—
89	#533	C₂×D₅₃	1	1	yes	characteristic
90	#446	—	2	2	no	—
91	C₈×Q₈	—	2	2	no	—
92	#424	Dic₅₃	1	2	yes	—
93	#423	—	53	4	no	—
94	#423	C₂×D₅₃	1	1	yes	characteristic
95	#405	Dic₅₃	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
97	#335	—	106	2	no	—
98	#335	—	106	2	no	—
99	#332	—	106	4	no	—
100	C₅₃	#8639	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	106	4	no	—
102	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
103	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
104	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
105	C₄⋊Q₈	D₂₁₂	1	2	yes	—
106	#125	Dic₁₀₆	1	2	yes	—
107	C₂²⋊Q₈	—	2	2	no	—
108	C₄×Q₈	—	2	2	no	—
109	#117	C₂×Dic₅₃	1	2	yes	—
110	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
111	C₄⋊C₈	—	2	4	no	—
112	C₄⋊C₈	—	2	2	no	—
113	C₄⋊C₈	—	106	4	no	—
114	C₄⋊C₈	—	106	4	no	—
115	#97	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
118	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
119	C₂×Q₈	—	2	2	no	—
120	C₂²×C₄	#11719945	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
122	M₁₆	—	4	2	no	—
123	M₁₆	—	424	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	212	4	no	—
128	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
129	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
130	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	4	no	—
133	C₄⋊C₄	C₄⋊Dic₅₃	1	4	yes	—
134	C₂²⋊C₄	—	2	2	no	—
135	C₄²	—	2	2	no	—
136	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
137	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
138	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
139	C₂³	#13133176	1	1	yes	characteristic
140	Q₈	—	4	2	no	—
141	Q₈	—	4	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	4	no	—
147	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
148	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
149	C₂×C₄	#13133191	1	2	yes	—
150	C₂×C₄	#13133196	1	2	yes	—
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	2	no	—
154	C₈	—	424	4	no	—
155	C₈	#13133136	1	2	yes	—
156	C₂²	—	2	1	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#16810483	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	C₂×#13133136	1	1	yes	characteristic
163	C₄	#16810607	1	1	yes	characteristic
164	C₄	#16810893	1	2	yes	—
165	C₄	#16810898	1	2	yes	—
166	C₂	—	4	1	no	—
167	C₂	#20274711	1	1	yes	characteristic
168	C₂	#20274779	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20273474

C₂

yes

maximal

#20273453

C₂

yes

maximal

#20272628

C₂

yes

maximal

C₅₃×#2960

C₂

yes

maximal, characteristic

#16810647

C₂²

yes

—

#16810376

—

#16810376

—

#16810369

—

C₅₃×#533

C₂²

yes

characteristic

C₅₃×#446

—

C₈×Q₈×C₅₃

—

C₅₃×#424

C₄

yes

—

C₅₃×#423

C₂²

yes

characteristic

C₅₃×#405

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄⋊Q₈×C₅₃

D₄

yes

—

#125×C₅₃

Q₈

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₄⋊C₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#117

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#101

yes

—

#88322

—

#8639

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#380

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#473

yes

—

#2960

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1801

yes

characteristic

C₂×C₅₃

#1816

yes

—

#533

C₂×D₅₃

yes

characteristic

#446

—

C₈×Q₈

—

#424

Dic₅₃

yes

—

#423

—

#423

C₂×D₅₃

yes

characteristic

#405

Dic₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#335

—

106

—

#335

—

106

—

#332

—

106

—

100

C₅₃

#8639

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

106

—

102

C₂×M₁₆

C₂×Dic₅₃

yes

—

103

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

104

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

105

C₄⋊Q₈

D₂₁₂

yes

—

106

#125

Dic₁₀₆

yes

—

107

C₂²⋊Q₈

—

108

C₄×Q₈

—

109

#117

C₂×Dic₅₃

yes

—

110

C₂×C₄²

C₂²×D₅₃

yes

characteristic

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₄⋊C₈

—

106

—

114

C₄⋊C₈

—

106

—

115

#97

—

116

C₄×C₈

—

117

C₄×C₈

C₂²⋊D₅₃

yes

—

118

C₄×C₈

C₂²⋊D₅₃

yes

—

119

C₂×Q₈

—

120

C₂²×C₄

#11719945

yes

—

121

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

122

M₁₆

—

123

M₁₆

—

424

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

212

—

128

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

129

C₂×C₈

C₂²⋊Dic₅₃

yes

—

130

C₂×C₈

C₂²⋊Dic₅₃

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

C₄⋊Dic₅₃

yes

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×Dic₁₀₆

yes

characteristic

137

C₄²

C₂×D₂₁₂

yes

characteristic

138

C₄²

C₂×C₂²⋊D₅₃

yes

—

139

C₂³

#13133176

yes

characteristic

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

148

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

149

C₂×C₄

#13133191

yes

—

150

C₂×C₄

#13133196

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

424

—

155

C₈

#13133136

yes

—

156

C₂²

—

157

C₂²

—

158

C₂²

#16810483

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

C₂×#13133136

yes

characteristic

163

C₄

#16810607

yes

characteristic

164

C₄

#16810893

yes

—

165

C₄

#16810898

yes

—

166

C₂

—

167

C₂

#20274711

yes

characteristic

168

C₂

#20274779

yes

—

	subgroup	G/H
center	C₂²	#16810483
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#2960	C₂
socle	C₂²×C₅₃	#380
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#2960	D₅₃
hypocenter	C₅₃	#8639
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

subgroup

G/H

center

C₂²

#16810483

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#2960

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960

N	H ≅ G/N	type
100. C₅₃	#8639	semidirect

H ≅ G/N

type

100. C₅₃

#8639

semidirect

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

subgroup	index
#20273474	2
#20273453	2
#20272628	2
C₅₃×#2960	2
#8639	53

subgroup

index

C₅₃×#2960

Subgroups — #27277122

Conjugacy classes of subgroups

Complements

Maximal subgroups