Group #27300273 — subgroups

Subgroups — #27300273

Order 13568.

394 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 155 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×#16810453	C₂	1	2	yes	maximal, complemented
2	#20274753	C₂	1	2	yes	maximal
3	#20272809	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810587	—	2	2	no	—
7	#16810453	—	2	4	no	—
8	#16810388	—	2	2	no	—
9	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₅₃	—	2	4	no	—
11	C₂²×C₁₆×C₅₃	C₄	1	4	yes	—
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#11719917	—	2	4	no	—
15	#13133153	—	2	4	no	—
16	#13133153	—	2	4	no	—
17	C₂²⋊#88322	—	4	4	no	—
18	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	—	2	2	no	—
26	C₂×C₁₆×C₅₃	—	2	8	no	—
27	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
30	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	—	2	2	no	—
32	C₂²×C₄×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	D₈	1	2	yes	—
35	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₅₃	—	2	2	no	—
38	C₂×C₈×C₅₃	—	2	2	no	—
39	C₂×C₈×C₅₃	—	2	2	no	—
40	C₂×C₈×C₅₃	—	2	4	no	—
41	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₈×C₅₃	#41	1	2	yes	—
46	#11719917	—	4	4	no	—
47	#11719917	—	4	4	no	—
48	C₂×#88322	—	4	4	no	—
49	C₁₆×C₅₃	—	2	2	no	—
50	C₁₆×C₅₃	C₂²⋊C₄	1	4	yes	—
51	C₂³×C₅₃	—	2	4	no	—
52	C₂³×C₅₃	—	2	2	no	—
53	C₂³×C₅₃	#107	1	2	yes	—
54	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
55	C₂×C₄×C₅₃	—	2	2	no	—
56	C₂×C₄×C₅₃	—	2	2	no	—
57	C₂×C₄×C₅₃	—	2	2	no	—
58	C₂×C₄×C₅₃	—	2	4	no	—
59	C₂×C₄×C₅₃	#107	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₅₃	#121	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄×C₅₃	#135	1	2	yes	—
64	C₈×C₅₃	—	2	2	no	—
65	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₈×C₅₃	#117	1	1	yes	characteristic
67	C₈×C₅₃	C₂²⋊Q₈	1	2	yes	—
68	#88322	—	8	4	no	—
69	#12837	—	53	1	no	maximal, Sylow
70	C₂²×C₅₃	—	2	8	no	—
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	—	2	2	no	—
74	C₂²×C₅₃	#368	1	4	yes	complemented
75	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
76	C₂²×C₅₃	#467	1	2	yes	—
77	C₄×C₅₃	—	2	2	no	—
78	C₄×C₅₃	#380	1	1	yes	characteristic
79	C₄×C₅₃	#427	1	1	yes	characteristic
80	C₄×C₅₃	#466	1	2	yes	—
81	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
82	C₂×#368	—	53	2	no	—
83	#1810	—	53	2	no	—
84	#1381	—	53	2	no	—
85	C₂×C₅₃	—	2	4	no	—
86	C₂×C₅₃	—	2	2	no	—
87	C₂×C₅₃	#1808	1	1	yes	characteristic
88	C₂×C₅₃	C₂×#368	1	2	yes	—
89	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂²×C₁₆	—	2	4	no	—
91	C₂²×C₁₆	Dic₅₃	1	4	yes	—
92	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
93	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
94	C₂×#108	—	53	4	no	—
95	#407	—	106	2	no	—
96	#368	—	106	4	no	—
97	#341	—	106	2	no	—
98	C₅₃	#12837	1	1	yes	characteristic, complemented, Sylow
99	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
100	C₂×M₁₆	—	106	4	no	—
101	C₂²×C₈	—	2	2	no	—
102	C₂²×C₈	—	2	2	no	—
103	C₂²×C₈	Dic₁₀₆	1	2	yes	—
104	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
105	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
106	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₂×C₁₆	—	2	8	no	—
109	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
110	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
111	C₂×C₁₆	C₂²⋊D₅₃	1	4	yes	—
112	#108	—	106	4	no	—
113	#108	—	106	4	no	—
114	C₂²⋊C₈	—	212	4	no	—
115	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	D₄₂₄	1	2	yes	—
119	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
120	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
121	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
122	M₁₆	—	212	4	no	—
123	M₁₆	—	212	4	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	212	4	no	—
129	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
130	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
131	C₂×C₈	#11719945	1	2	yes	—
132	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
133	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₅₃	1	4	yes	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	4	no	—
138	C₂³	#13133151	1	2	yes	—
139	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	#13133151	1	2	yes	—
145	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
146	C₂×C₄	#13133191	1	2	yes	—
147	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
148	C₂×C₄	#13133239	1	2	yes	—
149	C₈	—	2	2	no	—
150	C₈	—	424	4	no	—
151	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
152	C₈	#13133176	1	1	yes	characteristic
153	C₈	#13133196	1	2	yes	—
154	C₂²	—	2	8	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	#16810453	1	4	yes	complemented
159	C₂²	C₂×#13133151	1	1	yes	characteristic
160	C₂²	#16810864	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	#16810483	1	1	yes	characteristic
163	C₄	#16810661	1	1	yes	characteristic
164	C₄	#16810859	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	2	4	no	—
167	C₂	#20274744	1	1	yes	characteristic
168	C₂	C₂×#16810453	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal, complemented

#20274753

C₂

yes

maximal

#20272809

C₂

yes

maximal, complemented

C₂³×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133153

C₂²

yes

—

#16810587

—

#16810453

—

#16810388

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

—

C₂²×C₁₆×C₅₃

C₄

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

—

—

—

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂⁴×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#107

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

#117

yes

characteristic

C₈×C₅₃

C₂²⋊Q₈

yes

—

#88322

—

#12837

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂³×C₁₆

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1808

yes

characteristic

C₂×C₅₃

C₂×#368

yes

—

C₂³×C₈

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

Dic₅₃

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

C₂×#108

—

#407

—

106

—

#368

—

106

—

#341

—

106

—

C₅₃

#12837

yes

characteristic, complemented, Sylow

C₂³×C₄

D₂₁₂

yes

characteristic

100

C₂×M₁₆

—

106

—

101

C₂²×C₈

—

102

C₂²×C₈

—

103

C₂²×C₈

Dic₁₀₆

yes

—

104

C₂²×C₈

D₂₁₂

yes

characteristic

105

C₂²×C₈

C₂×Dic₅₃

yes

—

106

C₂²×C₈

C₂²×D₅₃

yes

characteristic

107

C₂×C₁₆

—

108

C₂×C₁₆

—

109

C₂×C₁₆

C₂×Dic₅₃

yes

—

110

C₂×C₁₆

C₂×Dic₅₃

yes

—

111

C₂×C₁₆

C₂²⋊D₅₃

yes

—

112

#108

—

106

—

113

#108

—

106

—

114

C₂²⋊C₈

—

212

—

115

C₂⁴

Dic₂₁₂

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

D₄₂₄

yes

—

119

C₂²×C₄

Dic₂₁₂

yes

characteristic

120

C₂²×C₄

C₄⋊Dic₅₃

yes

—

121

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

122

M₁₆

—

212

—

123

M₁₆

—

212

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

212

—

129

C₂×C₈

C₄⋊Dic₅₃

yes

—

130

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

131

C₂×C₈

#11719945

yes

—

132

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

133

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

134

C₁₆

—

135

C₁₆

C₂²⋊Dic₅₃

yes

—

136

C₂³

—

137

C₂³

—

138

C₂³

#13133151

yes

—

139

C₂³

C₂×Dic₂₁₂

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

#13133151

yes

—

145

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

146

C₂×C₄

#13133191

yes

—

147

C₂×C₄

C₂×D₄₂₄

yes

characteristic

148

C₂×C₄

#13133239

yes

—

149

C₈

—

150

C₈

—

424

—

151

C₈

C₂×C₂²⋊Dic₅₃

yes

characteristic

152

C₈

#13133176

yes

characteristic

153

C₈

#13133196

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

#16810453

yes

complemented

159

C₂²

C₂×#13133151

yes

characteristic

160

C₂²

#16810864

yes

—

161

C₄

—

162

C₄

#16810483

yes

characteristic

163

C₄

#16810661

yes

characteristic

164

C₄

#16810859

yes

—

165

C₂

—

166

C₂

—

167

C₂

#20274744

yes

characteristic

168

C₂

C₂×#16810453

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	#12837
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂³×C₁₆×C₅₃

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂⁴×C₅₃

Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂³×C₁₆

D₅₃

hypocenter

C₅₃

#12837

LCS

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

N	H ≅ G/N	type
1. C₂×#16810453	C₂	semidirect
3. #20272809	C₂	semidirect
74. C₂²×C₅₃	#368	semidirect
98. C₅₃	#12837	semidirect
158. C₂²	#16810453	semidirect

H ≅ G/N

type

1. C₂×#16810453

C₂

semidirect

3. #20272809

C₂

semidirect

74. C₂²×C₅₃

#368

semidirect

98. C₅₃

#12837

semidirect

158. C₂²

#16810453

semidirect

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

subgroup	index
C₂×#16810453	2
#20274753	2
#20272809	2
C₂³×C₁₆×C₅₃	2
#12837	53

subgroup

index

C₂³×C₁₆×C₅₃

Subgroups — #27300273

Conjugacy classes of subgroups

Complements

Maximal subgroups