Group #27304687 — subgroups

Subgroups — #27304687

Order 13568.

188 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 65 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×#16810463	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	#20276573	C₂	1	1	yes	maximal, characteristic
4	#20272916	C₂	1	1	yes	maximal, characteristic
5	#20272897	C₂	1	1	yes	maximal, characteristic
6	#20272848	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₄₂₄	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₅₃	—	2	1	no	—
10	#16810717	—	2	1	no	—
11	#16810463	C₄	1	4	yes	complemented
12	C₂×#13133123	C₂²	1	1	yes	characteristic
13	#16810449	C₂²	1	1	yes	characteristic
14	#16810427	—	2	1	no	—
15	#16810426	—	2	1	no	—
16	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
19	C₅₃×#366	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₁₂	—	2	1	no	—
21	Q₈×Dic₅₃	—	4	1	no	—
22	#13133182	—	4	1	no	—
23	Dic₄₂₄	—	2	1	no	—
24	Dic₄₂₄	C₂×C₄	1	2	yes	—
25	C₂×#11719908	D₄	1	1	yes	characteristic
26	#13133151	—	2	1	no	—
27	#13133141	—	4	1	no	—
28	#13133140	—	4	1	no	—
29	C₈×Dic₅₃	D₄	1	1	yes	characteristic
30	#13133123	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₅₃	—	2	1	no	—
32	Q₃₂×C₅₃	—	2	1	no	—
33	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
35	#107×C₅₃	—	2	1	no	—
36	C₃₂×C₅₃	—	2	1	no	—
37	C₂×Q₈×C₅₃	—	4	1	no	—
38	C₂×Dic₁₀₆	—	4	1	no	—
39	Q₁₆×C₅₃	—	2	2	no	—
40	Q₁₆×C₅₃	—	4	1	no	—
41	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₅₃	—	4	1	no	—
43	C₄⋊Dic₅₃	—	4	1	no	—
44	C₄⋊Dic₅₃	—	8	1	no	—
45	#11719919	—	8	1	no	—
46	C₄×Dic₅₃	—	8	1	no	—
47	C₄×Dic₅₃	—	8	1	no	—
48	C₄×Dic₅₃	D₈	1	1	yes	characteristic
49	C₂×#88322	D₈	1	1	yes	characteristic
50	Dic₂₁₂	—	2	2	no	—
51	Dic₂₁₂	—	4	1	no	—
52	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₅₃	#41	1	1	yes	characteristic
54	#11719908	—	2	1	no	—
55	Q₈×C₅₃	—	4	2	no	—
56	Q₈×C₅₃	—	8	1	no	—
57	C₂×C₄×C₅₃	—	8	1	no	—
58	C₂×C₄×C₅₃	—	8	1	no	—
59	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₅₃	—	8	1	no	—
61	C₂×Dic₅₃	—	8	1	no	—
62	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
63	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
64	Dic₁₀₆	—	4	2	no	—
65	Dic₁₀₆	—	8	1	no	—
66	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₈×C₅₃	#135	1	1	yes	characteristic
68	#88322	—	2	1	no	—
69	#13653	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₅₃	—	8	2	no	—
72	C₄×C₅₃	—	16	1	no	—
73	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
74	C₄×C₅₃	#508	1	1	yes	characteristic
75	Dic₅₃	—	2	1	no	—
76	Dic₅₃	—	2	1	no	—
77	Dic₅₃	—	8	2	no	—
78	Dic₅₃	—	16	1	no	—
79	C₂×Q₆₄	—	53	1	no	—
80	C₄×Q₃₂	—	53	1	no	—
81	C₄×Q₃₂	—	53	1	no	—
82	#1424	—	53	1	no	—
83	#1416	—	53	1	no	—
84	#1416	D₅₃	1	1	yes	characteristic
85	#1397	—	53	1	no	—
86	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
87	C₂×C₅₃	#2264	1	1	yes	characteristic
88	C₂×C₅₃	#2264	1	1	yes	characteristic
89	C₂×Q₃₂	—	53	1	no	—
90	C₂×Q₃₂	C₂×D₅₃	1	1	yes	characteristic
91	C₄×Q₁₆	—	106	1	no	—
92	C₄×Q₁₆	—	106	1	no	—
93	Q₆₄	—	53	4	no	—
94	C₂×C₃₂	—	53	1	no	—
95	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
96	#366	—	53	1	no	—
97	#366	C₂×D₅₃	1	1	yes	characteristic
98	#358	—	106	1	no	—
99	#358	—	106	1	no	—
100	C₄×C₁₆	—	53	1	no	—
101	C₅₃	#13653	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	106	1	no	—
104	C₄×Q₈	—	212	1	no	—
105	C₄×Q₈	—	212	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	53	2	no	—
108	Q₃₂	—	106	1	no	—
109	Q₃₂	C₄×D₅₃	1	2	yes	—
110	C₂×C₁₆	—	53	1	no	—
111	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	106	1	no	—
114	#103	—	212	1	no	—
115	#103	—	212	1	no	—
116	C₄×C₈	—	53	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	53	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	212	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	106	2	no	—
124	Q₁₆	—	212	1	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	424	1	no	—
129	C₄⋊C₄	—	424	1	no	—
130	C₄⋊C₄	—	212	1	no	—
131	C₄²	—	53	1	no	—
132	C₄²	—	424	1	no	—
133	C₄²	—	424	1	no	—
134	C₁₆	—	106	1	no	—
135	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
136	C₁₆	#11719947	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	424	1	no	—
140	Q₈	—	212	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	53	1	no	—
144	C₂×C₄	—	53	1	no	—
145	C₂×C₄	—	424	1	no	—
146	C₂×C₄	—	424	1	no	—
147	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
148	C₈	—	106	1	no	—
149	C₈	#13133177	1	1	yes	characteristic
150	C₈	#13133238	1	1	yes	characteristic
151	C₂²	D₁₆×D₅₃	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	848	1	no	—
155	C₄	—	106	1	no	—
156	C₄	—	106	1	no	—
157	C₄	—	424	2	no	—
158	C₄	#16810703	1	1	yes	characteristic
159	C₄	#16811054	1	1	yes	characteristic
160	C₂	#20276559	1	1	yes	characteristic
161	C₂	#20277031	1	1	yes	characteristic
162	C₂	#20277034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810463

C₂

yes

maximal, characteristic

Q₃₂×Dic₅₃

C₂

yes

maximal, characteristic

#20276573

C₂

yes

maximal, characteristic

#20272916

C₂

yes

maximal, characteristic

#20272897

C₂

yes

maximal, characteristic

#20272848

C₂

yes

maximal, characteristic

C₅₃×#1416

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

C₂²

yes

characteristic

Q₁₆×Dic₅₃

—

#16810717

—

#16810463

C₄

yes

complemented

C₂×#13133123

C₂²

yes

characteristic

#16810449

C₂²

yes

characteristic

#16810427

—

#16810426

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₂²

yes

characteristic

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×#366

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

Q₈×Dic₅₃

—

#13133182

—

Dic₄₂₄

—

Dic₄₂₄

C₂×C₄

yes

—

C₂×#11719908

D₄

yes

—

—

—

C₈×Dic₅₃

D₄

yes

characteristic

#13133123

C₂×C₄

yes

—

C₂×Q₁₆×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₃₂×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₈

yes

characteristic

C₂×#88322

D₈

yes

characteristic

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

C₂²×C₄

yes

C₁₆×C₅₃

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₁₆

yes

characteristic

C₂×Dic₅₃

D₁₆

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₈

yes

characteristic

C₄×C₅₃

#508

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1416

D₅₃

yes

characteristic

#1397

—

C₂×C₅₃

C₄×D₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₅₃

yes

characteristic

C₄×Q₁₆

—

106

—

C₄×Q₁₆

—

106

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

characteristic

#366

—

#366

C₂×D₅₃

yes

characteristic

#358

—

106

—

#358

—

106

—

100

C₄×C₁₆

—

101

C₅₃

#13653

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

106

—

104

C₄×Q₈

—

212

—

105

C₄×Q₈

—

212

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

106

—

109

Q₃₂

C₄×D₅₃

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

112

#107

—

113

#107

—

106

—

114

#103

—

212

—

115

#103

—

212

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

212

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

106

—

124

Q₁₆

—

212

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₅₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

424

—

129

C₄⋊C₄

—

424

—

130

C₄⋊C₄

—

212

—

131

C₄²

—

132

C₄²

—

424

—

133

C₄²

—

424

—

134

C₁₆

—

106

—

135

C₁₆

C₂×C₄×D₅₃

yes

characteristic

136

C₁₆

#11719947

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

424

—

140

Q₈

—

212

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

424

—

146

C₂×C₄

—

424

—

147

C₂×C₄

D₈×D₅₃

yes

characteristic

148

C₈

—

106

—

149

C₈

#13133177

yes

characteristic

150

C₈

#13133238

yes

characteristic

151

C₂²

D₁₆×D₅₃

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

848

—

155

C₄

—

106

—

156

C₄

—

106

—

157

C₄

—

424

—

158

C₄

#16810703

yes

characteristic

159

C₄

#16811054

yes

characteristic

160

C₂

#20276559

yes

characteristic

161

C₂

#20277031

yes

characteristic

162

C₂

#20277034

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₅₃
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#1416	C₂
socle, prime radical	C₂²×C₅₃	C₂×D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#1416	D₅₃
hypocenter	C₅₃	#13653
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₅₃

commutator

C₁₆×C₅₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#1416

C₂

socle, prime radical

C₂²×C₅₃

C₂×D₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
11. #16810463	C₄	semidirect
101. C₅₃	#13653	semidirect

H ≅ G/N

type

11. #16810463

C₄

semidirect

101. C₅₃

#13653

semidirect

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

subgroup	index
C₂×#16810463	2
Q₃₂×Dic₅₃	2
#20276573	2
#20272916	2
#20272897	2
#20272848	2
C₅₃×#1416	2
#13653	53

subgroup

index

Q₃₂×Dic₅₃

C₅₃×#1416

Subgroups — #27304687

Conjugacy classes of subgroups

Complements

Maximal subgroups