Group #27293045 — subgroups

Subgroups — #27293045

Order 13568.

842 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 235 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	#20274454	C₂	1	3	yes	maximal
2	#20274412	C₂	1	3	yes	maximal
3	C₅₃×#3458	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	3	yes	—
5	C₂³×#88322	C₂²	1	1	yes	characteristic
6	#16810392	—	2	6	no	—
7	#16810378	—	2	6	no	—
8	C₅₃×#544	—	2	6	no	—
9	C₅₃×#543	C₄	1	4	yes	—
10	C₅₃×#533	—	2	6	no	—
11	C₂×#117×C₅₃	C₂²	1	3	yes	—
12	C₂×#11719917	—	2	6	no	—
13	C₂×#11719917	—	2	3	no	—
14	C₂×#11719917	—	2	6	no	—
15	C₂×#11719917	Q₈	1	6	yes	complemented
16	C₂²×#88322	—	2	3	no	—
17	C₂²×#88322	—	2	3	no	—
18	C₂²×#88322	—	2	3	no	—
19	C₂²×#88322	Q₈	1	2	yes	complemented
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₅₃	—	2	12	no	—
22	#125×C₅₃	—	2	12	no	—
23	C₂²⋊Q₈×C₅₃	—	2	24	no	—
24	C₄×Q₈×C₅₃	—	2	8	no	—
25	#117×C₅₃	—	2	6	no	—
26	#117×C₅₃	—	2	6	no	—
27	C₂×C₄⋊C₄×C₅₃	D₄	1	6	yes	—
28	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₅₃	D₄	1	6	yes	—
30	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×Q₈×C₅₃	—	2	8	no	—
32	C₂²×C₄×C₅₃	C₂²⋊C₄	1	12	yes	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
34	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
37	C₂²×C₄×C₅₃	#41	1	3	yes	—
38	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₅₃	—	2	24	no	—
40	C₄⋊C₄×C₅₃	—	2	12	no	—
41	C₂²⋊C₄×C₅₃	—	2	12	no	—
42	C₄²×C₅₃	—	2	6	no	—
43	C₄²×C₅₃	—	2	6	no	—
44	#11719917	—	2	12	no	—
45	#11719917	—	4	6	no	—
46	C₂×#88322	—	2	6	no	—
47	C₂×#88322	—	2	6	no	—
48	C₂×#88322	—	4	1	no	—
49	C₂×#88322	—	4	3	no	—
50	C₂×#88322	—	4	3	no	—
51	C₂×#88322	—	4	3	no	—
52	C₂³×C₅₃	—	4	2	no	—
53	C₂³×C₅₃	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
55	C₂³×C₅₃	C₄×Q₈	1	3	yes	—
56	Q₈×C₅₃	—	8	8	no	—
57	C₂×C₄×C₅₃	—	2	12	no	—
58	C₂×C₄×C₅₃	—	4	12	no	—
59	C₂×C₄×C₅₃	#120	1	3	yes	—
60	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₅₃	C₂²⋊Q₈	1	3	yes	—
63	C₂×C₄×C₅₃	C₂²⋊Q₈	1	6	yes	—
64	C₂×C₄×C₅₃	#124	1	3	yes	—
65	C₂×C₄×C₅₃	#124	1	3	yes	—
66	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
67	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
68	#88322	—	4	2	no	—
69	#88322	—	4	6	no	—
70	#11391	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	6	no	—
72	C₂²×C₅₃	—	4	1	no	—
73	C₂²×C₅₃	—	4	3	no	—
74	C₂²×C₅₃	#379	1	1	yes	characteristic
75	C₂²×C₅₃	#391	1	3	yes	—
76	C₂²×C₅₃	#391	1	3	yes	—
77	C₄×C₅₃	—	4	12	no	—
78	C₄×C₅₃	#393	1	3	yes	—
79	C₄×C₅₃	#393	1	1	yes	characteristic
80	C₄×C₅₃	#395	1	3	yes	—
81	C₄×C₅₃	#395	1	1	yes	characteristic
82	#3458	D₅₃	1	1	yes	characteristic
83	#1742	—	53	3	no	—
84	#1729	—	53	3	no	—
85	C₂×C₅₃	—	2	3	no	—
86	C₂×C₅₃	—	2	3	no	—
87	C₂×C₅₃	#1446	1	1	yes	characteristic
88	C₂×C₅₃	#1791	1	2	yes	—
89	C₂²×M₁₆	—	53	3	no	—
90	C₂³×C₈	—	53	1	no	—
91	#544	—	2	6	no	—
92	#543	Dic₅₃	1	4	yes	—
93	#533	—	2	6	no	—
94	C₂×#117	C₂×D₅₃	1	3	yes	—
95	#343	—	106	6	no	—
96	#336	—	106	6	no	—
97	C₅₃	#11391	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×M₁₆	—	53	6	no	—
100	C₂×M₁₆	—	106	3	no	—
101	C₂×M₁₆	—	106	6	no	—
102	C₂×M₁₆	—	106	6	no	—
103	C₂²×C₈	—	53	2	no	—
104	C₂²×C₈	—	106	3	no	—
105	C₂²×C₈	—	106	3	no	—
106	C₂²×C₈	—	106	3	no	—
107	C₄⋊Q₈	—	2	12	no	—
108	#125	—	2	12	no	—
109	C₂²⋊Q₈	—	2	24	no	—
110	C₄×Q₈	—	2	8	no	—
111	#117	—	2	6	no	—
112	#117	—	2	6	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₅₃	1	6	yes	—
114	C₂×C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
115	C₂×C₄²	C₂²⋊D₅₃	1	6	yes	—
116	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
117	C₂×Q₈	—	2	8	no	—
118	C₂²×C₄	C₂²⋊Dic₅₃	1	12	yes	—
119	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
120	C₂²×C₄	Q₈×D₅₃	1	3	yes	—
121	C₂²×C₄	#11719949	1	3	yes	—
122	C₂²×C₄	#11719949	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
125	M₁₆	—	106	12	no	—
126	M₁₆	—	212	6	no	—
127	C₂×C₈	—	106	6	no	—
128	C₂×C₈	—	106	6	no	—
129	C₂×C₈	—	212	3	no	—
130	C₂×C₈	—	212	1	no	—
131	C₂×C₈	—	212	3	no	—
132	C₂×C₈	—	212	3	no	—
133	C₄⋊C₄	—	2	24	no	—
134	C₄⋊C₄	—	2	12	no	—
135	C₂²⋊C₄	—	2	12	no	—
136	C₄²	—	2	6	no	—
137	C₄²	—	2	6	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂×C₂²⋊Dic₅₃	1	3	yes	—
140	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
141	C₂³	Q₈×Dic₅₃	1	3	yes	—
142	Q₈	—	8	8	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	#13133186	1	3	yes	—
146	C₂×C₄	#13133186	1	1	yes	characteristic
147	C₂×C₄	#13133193	1	3	yes	—
148	C₂×C₄	#13133193	1	3	yes	—
149	C₂×C₄	#13133193	1	6	yes	—
150	C₂×C₄	#13133206	1	3	yes	—
151	C₂×C₄	#13133206	1	3	yes	—
152	C₂×C₄	#13133215	1	3	yes	—
153	C₂×C₄	#13133215	1	3	yes	—
154	C₈	—	212	2	no	—
155	C₈	—	212	6	no	—
156	C₂²	—	2	6	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	3	no	—
159	C₂²	#16810479	1	1	yes	characteristic
160	C₂²	#16810530	1	3	yes	—
161	C₂²	#16810530	1	3	yes	—
162	C₄	—	4	12	no	—
163	C₄	#16810534	1	3	yes	—
164	C₄	#16810534	1	1	yes	characteristic
165	C₄	#16810542	1	3	yes	—
166	C₄	#16810542	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	2	3	no	—
169	C₂	#20272986	1	1	yes	characteristic
170	C₂	#20274666	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274454

C₂

yes

maximal

#20274412

C₂

yes

maximal

C₅₃×#3458

C₂

yes

maximal, characteristic

C₂²×#11719917

C₂²

yes

—

C₂³×#88322

C₂²

yes

characteristic

#16810392

—

#16810378

—

C₅₃×#544

—

C₅₃×#543

C₄

yes

—

C₅₃×#533

—

C₂×#117×C₅₃

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂²×#88322

—

C₂²×#88322

—

C₂²×#88322

—

C₂²×#88322

Q₈

yes

complemented

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊Q₈×C₅₃

—

#125×C₅₃

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

—

#117×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂⁴×C₅₃

C₂²×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂²×C₄×C₅₃

#41

yes

—

C₂²×C₄×C₅₃

#41

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

—

—

—

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂³×C₅₃

C₄×Q₈

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#379

yes

characteristic

C₂²×C₅₃

#391

yes

—

C₂²×C₅₃

#391

yes

—

C₄×C₅₃

—

C₄×C₅₃

#393

yes

—

C₄×C₅₃

#393

yes

characteristic

C₄×C₅₃

#395

yes

—

C₄×C₅₃

#395

yes

characteristic

#3458

D₅₃

yes

characteristic

#1742

—

#1729

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1446

yes

characteristic

C₂×C₅₃

#1791

yes

—

C₂²×M₁₆

—

C₂³×C₈

—

#544

—

#543

Dic₅₃

yes

—

#533

—

C₂×#117

C₂×D₅₃

yes

—

#343

—

106

—

#336

—

106

—

C₅₃

#11391

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂²×D₅₃

yes

characteristic

C₂×M₁₆

—

100

C₂×M₁₆

—

106

—

101

C₂×M₁₆

—

106

—

102

C₂×M₁₆

—

106

—

103

C₂²×C₈

—

104

C₂²×C₈

—

106

—

105

C₂²×C₈

—

106

—

106

C₂²×C₈

—

106

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

C₂²⋊D₅₃

yes

—

114

C₂×C₂²⋊C₄

C₂×Dic₅₃

yes

—

115

C₂×C₄²

C₂²⋊D₅₃

yes

—

116

C₂⁴

C₂²×Dic₅₃

yes

characteristic

117

C₂×Q₈

—

118

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

119

C₂²×C₄

Q₈×D₅₃

yes

characteristic

120

C₂²×C₄

Q₈×D₅₃

yes

—

121

C₂²×C₄

#11719949

yes

—

122

C₂²×C₄

#11719949

yes

characteristic

123

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

125

M₁₆

—

106

—

126

M₁₆

—

212

—

127

C₂×C₈

—

106

—

128

C₂×C₈

—

106

—

129

C₂×C₈

—

212

—

130

C₂×C₈

—

212

—

131

C₂×C₈

—

212

—

132

C₂×C₈

—

212

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

C₂×C₂²⋊Dic₅₃

yes

—

140

C₂³

Q₈×Dic₅₃

yes

characteristic

141

C₂³

Q₈×Dic₅₃

yes

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#13133186

yes

—

146

C₂×C₄

#13133186

yes

characteristic

147

C₂×C₄

#13133193

yes

—

148

C₂×C₄

#13133193

yes

—

149

C₂×C₄

#13133193

yes

—

150

C₂×C₄

#13133206

yes

—

151

C₂×C₄

#13133206

yes

—

152

C₂×C₄

#13133215

yes

—

153

C₂×C₄

#13133215

yes

—

154

C₈

—

212

—

155

C₈

—

212

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#16810479

yes

characteristic

160

C₂²

#16810530

yes

—

161

C₂²

#16810530

yes

—

162

C₄

—

163

C₄

#16810534

yes

—

164

C₄

#16810534

yes

characteristic

165

C₄

#16810542

yes

—

166

C₄

#16810542

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#20272986

yes

characteristic

170

C₂

#20274666

yes

—

	subgroup	G/H
center	C₂×C₄	#13133186
commutator, prime radical	C₂⁴×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₅₃×#3458	C₂
socle	C₂²×C₅₃	#379
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	#3458	D₅₃
hypocenter	C₅₃	#11391
LCS	G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

subgroup

G/H

center

C₂×C₄

#13133186

commutator, prime radical

C₂⁴×C₅₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₅₃

Fitting

C₅₃×#3458

C₂

socle

C₂²×C₅₃

#379

cosocle

C₂³×C₄×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂⁴×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

N	H ≅ G/N	type
15. C₂×#11719917	Q₈	semidirect
19. C₂²×#88322	Q₈	semidirect
97. C₅₃	#11391	semidirect

H ≅ G/N

type

15. C₂×#11719917

Q₈

semidirect

19. C₂²×#88322

Q₈

semidirect

97. C₅₃

#11391

semidirect

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

subgroup	index
#20274454	2
#20274412	2
C₅₃×#3458	2
#11391	53

subgroup

index

#20274454

#20274412

C₅₃×#3458

#11391

Subgroups — #27293045

Conjugacy classes of subgroups

Complements

Maximal subgroups