Group #27292778 — subgroups

Subgroups — #27292778

Order 13568.

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

normal subgroups only

This group has 702 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	C₂×#16810672	C₂	1	3	yes	maximal
2	#20274408	C₂	1	3	yes	maximal
3	C₂×C₅₃×#338	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	3	yes	—
5	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
6	#16810672	—	2	12	no	—
7	C₂×#13133153	—	2	6	no	—
8	#16810377	—	2	6	no	—
9	C₂²×M₁₆×C₅₃	C₂²	1	3	yes	—
10	C₅₃×#338	C₄	1	4	yes	complemented
11	C₂³×Dic₅₃	Q₈	1	2	yes	—
12	C₂×#11719917	—	2	6	no	—
13	C₂×#11719917	—	2	6	no	—
14	C₂×#11719917	—	2	3	no	—
15	C₂×#11719917	D₄	1	6	yes	—
16	C₂²×#88322	—	2	3	no	—
17	C₂×C₄×Dic₅₃	—	2	3	no	—
18	C₂×C₄×Dic₅₃	—	2	3	no	—
19	#13133153	—	2	24	no	—
20	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂×M₁₆×C₅₃	—	2	6	no	—
22	C₂×M₁₆×C₅₃	—	2	6	no	—
23	C₂×M₁₆×C₅₃	C₂×C₄	1	6	yes	—
24	C₂×M₁₆×C₅₃	Q₈	1	6	yes	—
25	C₂²×C₈×C₅₃	D₄	1	6	yes	—
26	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
29	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
30	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
31	C₂²×C₄×C₅₃	#41	1	3	yes	—
32	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
33	C₂²×Dic₅₃	—	2	6	no	—
34	C₂²×Dic₅₃	—	4	4	no	—
35	M₁₆×C₅₃	—	2	12	no	—
36	M₁₆×C₅₃	—	2	12	no	—
37	C₂×C₈×C₅₃	—	2	6	no	—
38	C₂×C₈×C₅₃	—	2	6	no	—
39	C₂×C₈×C₅₃	C₄⋊C₄	1	12	yes	—
40	C₄×Dic₅₃	—	4	3	no	—
41	C₄×Dic₅₃	—	4	1	no	—
42	#11719917	—	2	12	no	—
43	#11719917	—	4	6	no	—
44	C₂×#88322	—	2	6	no	—
45	C₂×#88322	—	4	3	no	—
46	C₂×#88322	—	4	3	no	—
47	C₂³×C₅₃	—	4	2	no	—
48	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
49	C₂³×C₅₃	C₂²⋊Q₈	1	3	yes	—
50	C₂³×C₅₃	C₄⋊Q₈	1	3	yes	—
51	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
52	C₂×C₄×C₅₃	C₄×D₄	1	3	yes	—
53	C₂×C₄×C₅₃	#120	1	3	yes	—
54	C₂×C₄×C₅₃	#121	1	3	yes	—
55	C₂×C₄×C₅₃	C₂²⋊Q₈	1	6	yes	—
56	C₂×C₄×C₅₃	#123	1	3	yes	—
57	C₂×C₄×C₅₃	#125	1	3	yes	—
58	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄⋊Q₈	1	3	yes	—
60	C₂×Dic₅₃	—	2	12	no	—
61	C₂×Dic₅₃	—	4	6	no	—
62	C₂×Dic₅₃	—	4	2	no	—
63	C₈×C₅₃	—	2	12	no	—
64	#88322	—	4	6	no	—
65	#11338	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	6	no	—
67	C₂²×C₅₃	—	4	1	no	—
68	C₂²×C₅₃	—	4	3	no	—
69	C₂²×C₅₃	#389	1	3	yes	—
70	C₂²×C₅₃	#395	1	3	yes	—
71	C₂²×C₅₃	#399	1	1	yes	characteristic
72	C₄×C₅₃	#384	1	1	yes	characteristic
73	C₄×C₅₃	#385	1	3	yes	—
74	C₄×C₅₃	#392	1	1	yes	characteristic
75	C₄×C₅₃	#398	1	3	yes	—
76	Dic₅₃	—	4	4	no	—
77	C₂×#430	—	53	3	no	—
78	C₂×#338	D₅₃	1	1	yes	characteristic
79	#1728	—	53	3	no	—
80	C₂×C₅₃	—	2	3	no	—
81	C₂×C₅₃	—	2	3	no	—
82	C₂×C₅₃	#1444	1	1	yes	characteristic
83	C₂×C₅₃	#2091	1	2	yes	—
84	C₂²×M₁₆	—	53	3	no	—
85	C₂²×M₁₆	C₂×D₅₃	1	3	yes	—
86	C₂²×C₄²	—	53	1	no	—
87	#430	—	106	12	no	—
88	C₂×#108	—	106	6	no	—
89	#338	Dic₅₃	1	4	yes	complemented
90	#336	—	106	6	no	—
91	C₅₃	#11338	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	—	53	2	no	—
93	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
94	C₂×M₁₆	—	2	6	no	—
95	C₂×M₁₆	—	2	6	no	—
96	C₂×M₁₆	—	53	6	no	—
97	C₂×M₁₆	—	106	3	no	—
98	C₂×M₁₆	—	106	6	no	—
99	C₂×M₁₆	—	106	6	no	—
100	C₂×M₁₆	Dic₁₀₆	1	6	yes	—
101	C₂×M₁₆	C₂×Dic₅₃	1	6	yes	—
102	C₂²×C₈	—	106	3	no	—
103	C₂²×C₈	D₂₁₂	1	6	yes	—
104	C₂×C₄²	—	106	3	no	—
105	C₂×C₄²	—	106	3	no	—
106	#108	—	106	24	no	—
107	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
108	C₂²×C₄	—	106	6	no	—
109	C₂²×C₄	—	212	4	no	—
110	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
111	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
112	C₂²×C₄	D₄×D₅₃	1	3	yes	—
113	C₂²×C₄	#11719947	1	3	yes	—
114	C₂²×C₄	#11719949	1	1	yes	characteristic
115	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
116	M₁₆	—	2	12	no	—
117	M₁₆	—	2	12	no	—
118	M₁₆	—	106	12	no	—
119	M₁₆	—	212	6	no	—
120	C₂×C₈	—	2	6	no	—
121	C₂×C₈	—	2	6	no	—
122	C₂×C₈	—	106	6	no	—
123	C₂×C₈	—	212	3	no	—
124	C₂×C₈	—	212	3	no	—
125	C₂×C₈	C₄⋊Dic₅₃	1	12	yes	—
126	C₄²	—	212	3	no	—
127	C₄²	—	212	1	no	—
128	C₂³	—	4	2	no	—
129	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
130	C₂³	#13133197	1	3	yes	—
131	C₂³	#13133216	1	3	yes	—
132	C₂×C₄	—	106	12	no	—
133	C₂×C₄	—	212	6	no	—
134	C₂×C₄	—	212	2	no	—
135	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
136	C₂×C₄	D₄×Dic₅₃	1	3	yes	—
137	C₂×C₄	#13133187	1	3	yes	—
138	C₂×C₄	#13133192	1	3	yes	—
139	C₂×C₄	#13133194	1	6	yes	—
140	C₂×C₄	#13133201	1	3	yes	—
141	C₂×C₄	#13133208	1	3	yes	—
142	C₂×C₄	#13133214	1	1	yes	characteristic
143	C₂×C₄	#13133217	1	3	yes	—
144	C₈	—	2	12	no	—
145	C₈	—	212	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	3	no	—
149	C₂²	#16810522	1	3	yes	—
150	C₂²	#16810544	1	3	yes	—
151	C₂²	#16810560	1	1	yes	characteristic
152	C₄	—	212	4	no	—
153	C₄	#16810493	1	1	yes	characteristic
154	C₄	#16810498	1	3	yes	—
155	C₄	#16810533	1	1	yes	characteristic
156	C₄	#16810555	1	3	yes	—
157	C₂	—	2	3	no	—
158	C₂	—	2	3	no	—
159	C₂	#20272980	1	1	yes	characteristic
160	C₂	#20276232	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810672

C₂

yes

maximal

#20274408

C₂

yes

maximal

C₂×C₅₃×#338

C₂

yes

maximal, characteristic

C₂²×#11719917

C₂²

yes

—

C₂²×C₄×Dic₅₃

C₂²

yes

—

—

—

C₂²×M₁₆×C₅₃

C₂²

yes

—

C₅₃×#338

C₄

yes

complemented

C₂³×Dic₅₃

Q₈

yes

—

—

—

—

D₄

yes

—

C₂²×#88322

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

#13133153

—

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂⁴×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂²×C₄×C₅₃

#41

yes

—

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄×Q₈

yes

characteristic

C₂³×C₅₃

C₂²⋊Q₈

yes

—

C₂³×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

#11338

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#389

yes

—

C₂²×C₅₃

#395

yes

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₄×C₅₃

#392

yes

characteristic

C₄×C₅₃

#398

yes

—

Dic₅₃

—

C₂×#430

—

C₂×#338

D₅₃

yes

characteristic

#1728

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1444

yes

characteristic

C₂×C₅₃

#2091

yes

—

C₂²×M₁₆

—

C₂²×M₁₆

C₂×D₅₃

yes

—

C₂²×C₄²

—

#430

—

106

—

C₂×#108

—

106

—

#338

Dic₅₃

yes

complemented

#336

—

106

—

C₅₃

#11338

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

C₂²×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

C₂×M₁₆

—

106

—

100

C₂×M₁₆

Dic₁₀₆

yes

—

101

C₂×M₁₆

C₂×Dic₅₃

yes

—

102

C₂²×C₈

—

106

—

103

C₂²×C₈

D₂₁₂

yes

—

104

C₂×C₄²

—

106

—

105

C₂×C₄²

—

106

—

106

#108

—

106

—

107

C₂⁴

Q₈×D₅₃

yes

characteristic

108

C₂²×C₄

—

106

—

109

C₂²×C₄

—

212

—

110

C₂²×C₄

C₂×Dic₁₀₆

yes

—

111

C₂²×C₄

C₂×D₂₁₂

yes

—

112

C₂²×C₄

D₄×D₅₃

yes

—

113

C₂²×C₄

#11719947

yes

—

114

C₂²×C₄

#11719949

yes

characteristic

115

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

106

—

119

M₁₆

—

212

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

106

—

123

C₂×C₈

—

212

—

124

C₂×C₈

—

212

—

125

C₂×C₈

C₄⋊Dic₅₃

yes

—

126

C₄²

—

212

—

127

C₄²

—

212

—

128

C₂³

—

129

C₂³

Q₈×Dic₅₃

yes

characteristic

130

C₂³

#13133197

yes

—

131

C₂³

#13133216

yes

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

136

C₂×C₄

D₄×Dic₅₃

yes

—

137

C₂×C₄

#13133187

yes

—

138

C₂×C₄

#13133192

yes

—

139

C₂×C₄

#13133194

yes

—

140

C₂×C₄

#13133201

yes

—

141

C₂×C₄

#13133208

yes

—

142

C₂×C₄

#13133214

yes

characteristic

143

C₂×C₄

#13133217

yes

—

144

C₈

—

145

C₈

—

212

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#16810522

yes

—

150

C₂²

#16810544

yes

—

151

C₂²

#16810560

yes

characteristic

152

C₄

—

212

—

153

C₄

#16810493

yes

characteristic

154

C₄

#16810498

yes

—

155

C₄

#16810533

yes

characteristic

156

C₄

#16810555

yes

—

157

C₂

—

158

C₂

—

159

C₂

#20272980

yes

characteristic

160

C₂

#20276232

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₃
Fitting	C₂×C₅₃×#338	C₂
socle	C₂²×C₅₃	#399
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂³×C₄×C₅₃	C₂³
hypercenter	C₂×#338	D₅₃
hypocenter	C₅₃	#11338
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

subgroup

G/H

center

C₂×C₄

#13133214

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₅₃

Fitting

C₂×C₅₃×#338

C₂

socle

C₂²×C₅₃

#399

prime radical

C₂⁴×C₅₃

C₂×Q₈

cosocle

C₂³×C₄×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#338

N	H ≅ G/N	type
10. C₅₃×#338	C₄	semidirect
89. #338	Dic₅₃	semidirect
91. C₅₃	#11338	semidirect

H ≅ G/N

type

10. C₅₃×#338

C₄

semidirect

89. #338

Dic₅₃

semidirect

91. C₅₃

#11338

semidirect

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

subgroup	index
C₂×#16810672	2
#20274408	2
C₂×C₅₃×#338	2
#11338	53

subgroup

index

C₂×#16810672

#20274408

C₂×C₅₃×#338

#11338

Subgroups — #27292778

Conjugacy classes of subgroups

Complements

Maximal subgroups