Group #26134809 — subgroups

Subgroups — #26134809

Order 12032.

342 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 147 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	#20252322	C₂	1	1	yes	maximal, characteristic
2	#20245144	C₂	1	1	yes	maximal, characteristic
3	#20241476	C₂	1	4	yes	maximal
4	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#15689680	C₄	1	1	yes	characteristic
6	C₂×#13093964	C₂²	1	2	yes	—
7	#15689253	—	2	2	no	—
8	C₈×Dic₉₄	—	2	2	no	—
9	#15689149	C₄	1	2	yes	—
10	#15689147	C₄	1	1	yes	characteristic
11	#15689078	C₂²	1	2	yes	—
12	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₇	—	2	2	no	—
16	C₂×#10625329	C₂×C₄	1	2	yes	—
17	#13094024	C₈	1	2	yes	—
18	#13094016	C₈	1	2	yes	—
19	#13094003	—	2	2	no	—
20	C₄×Dic₉₄	—	2	2	no	—
21	#13093983	C₂×C₄	1	2	yes	—
22	#13093964	—	2	4	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
25	Dic₄₇⋊C₈	—	2	4	no	—
26	#13093934	—	2	2	no	—
27	C₈×Dic₄₇	—	2	2	no	—
28	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₄₇	—	2	2	no	—
32	C₂×C₁₆×C₄₇	—	2	2	no	—
33	C₂×C₁₆×C₄₇	Q₈	1	4	yes	complemented
34	C₄×C₈×C₄₇	D₄	1	2	yes	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×Dic₉₄	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂²⋊Dic₄₇	—	2	2	no	—
49	C₄⋊Dic₄₇	—	2	2	no	—
50	C₄⋊Dic₄₇	C₂×C₈	1	4	yes	—
51	Dic₄₇⋊C₄	—	2	4	no	—
52	C₄×Dic₄₇	—	2	2	no	—
53	#10625329	—	2	4	no	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₁₆×C₄₇	—	2	4	no	—
56	C₄₇⋊C₁₆	—	4	2	no	—
57	C₂³×C₄₇	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#131	1	2	yes	—
65	C₂×Dic₄₇	—	2	4	no	—
66	Dic₉₄	—	4	4	no	—
67	C₈×C₄₇	—	2	2	no	—
68	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
69	C₈×C₄₇	#124	1	1	yes	characteristic
70	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
71	C₄₇⋊C₈	—	4	2	no	—
72	#13266	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₄₇	#446	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	#350	1	4	yes	—
78	C₄×C₄₇	#391	1	1	yes	characteristic
79	C₄×C₄₇	#408	1	1	yes	characteristic
80	C₄×C₄₇	C₈×Q₈	1	2	yes	—
81	Dic₄₇	—	4	4	no	—
82	#2960	—	47	1	no	—
83	#2150	—	47	1	no	—
84	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
85	#1379	—	47	4	no	—
86	C₂×C₄₇	—	2	2	no	—
87	C₂×C₄₇	#1984	1	1	yes	characteristic
88	C₂×C₄₇	C₂×#350	1	2	yes	—
89	#533	—	47	1	no	—
90	C₂×M₃₂	—	47	2	no	—
91	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
92	#446	—	94	2	no	—
93	C₈×Q₈	—	94	2	no	—
94	#424	—	47	2	no	—
95	#423	—	47	1	no	—
96	#405	—	47	2	no	—
97	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
98	C₄⋊C₁₆	—	94	2	no	—
99	C₄×C₁₆	—	2	2	no	—
100	C₄₇	#13266	1	1	yes	characteristic, complemented, Sylow
101	C₂×M₁₆	—	47	2	no	—
102	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	C₄⋊Q₈	—	47	2	no	—
105	#125	—	47	2	no	—
106	C₂²⋊Q₈	—	94	2	no	—
107	C₄×Q₈	—	94	2	no	—
108	#117	—	47	2	no	—
109	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
110	M₃₂	—	94	4	no	—
111	C₂×C₁₆	—	2	2	no	—
112	C₂×C₁₆	—	2	2	no	—
113	C₂×C₁₆	—	94	2	no	—
114	C₂×C₁₆	Dic₉₄	1	4	yes	complemented
115	C₄⋊C₈	—	94	4	no	—
116	C₄⋊C₈	—	94	2	no	—
117	#97	—	94	2	no	—
118	C₄×C₈	—	94	2	no	—
119	C₄×C₈	D₁₈₈	1	2	yes	—
120	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
121	C₂×Q₈	—	94	2	no	—
122	C₂²×C₄	#10625324	1	2	yes	—
123	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
124	M₁₆	—	94	4	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	94	2	no	—
127	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
128	C₂×C₈	C₂×Dic₉₄	1	2	yes	—
129	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂×C₈	#10625350	1	2	yes	—
131	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₄⋊C₄	—	47	4	no	—
133	C₄⋊C₄	—	94	4	no	—
134	C₄⋊C₄	—	94	2	no	—
135	C₂²⋊C₄	—	94	2	no	—
136	C₄²	—	94	2	no	—
137	C₄²	C₈×D₄₇	1	1	yes	characteristic
138	C₄²	C₈×D₄₇	1	1	yes	characteristic
139	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
140	C₁₆	—	2	4	no	—
141	C₁₆	—	188	2	no	—
142	C₂³	C₂×#10625324	1	1	yes	characteristic
143	Q₈	—	188	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	94	4	no	—
146	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
147	C₂×C₄	(C₂×D₄₇)⋊C₈	1	2	yes	—
148	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×Dic₉₄	1	2	yes	—
150	C₂×C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
151	C₂×C₄	#13094030	1	2	yes	—
152	C₈	—	2	2	no	—
153	C₈	—	188	2	no	—
154	C₈	#13094003	1	2	yes	—
155	C₈	#13094012	1	1	yes	characteristic
156	C₈	#13094024	1	1	yes	characteristic
157	C₂²	—	2	2	no	—
158	C₂²	C₂×(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
159	C₂²	#15689253	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	188	4	no	—
162	C₄	#15688906	1	4	yes	—
163	C₄	#15689029	1	1	yes	characteristic
164	C₄	#15689089	1	1	yes	characteristic
165	C₄	C₈×Dic₉₄	1	2	yes	—
166	C₂	—	2	2	no	—
167	C₂	#20244324	1	1	yes	characteristic
168	C₂	C₂×#15688906	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20252322

C₂

yes

maximal, characteristic

#20245144

C₂

yes

maximal, characteristic

#20241476

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₇

C₂

yes

maximal, characteristic

#15689680

C₄

yes

characteristic

C₂×#13093964

C₂²

yes

—

#15689253

—

C₈×Dic₉₄

—

#15689149

C₄

yes

—

#15689147

C₄

yes

characteristic

#15689078

C₂²

yes

—

C₄⋊(C₄₇⋊C₁₆)

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

—

C₂×#10625329

C₂×C₄

yes

—

#13094024

C₈

yes

—

#13094016

C₈

yes

—

#13094003

—

C₄×Dic₉₄

—

#13093983

C₂×C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₄⋊(C₄₇⋊C₈)

—

Dic₄₇⋊C₈

—

#13093934

—

C₈×Dic₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

Q₈

yes

complemented

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₈

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×M₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×Q₈

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₈×C₄₇

#124

yes

characteristic

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₄₇⋊C₈

—

#13266

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

#446

yes

—

C₄×C₄₇

—

C₄×C₄₇

#350

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₈×Q₈

yes

—

Dic₄₇

—

#2960

—

#2150

—

C₂×C₄×C₁₆

D₄₇

yes

characteristic

#1379

—

C₂×C₄₇

—

C₂×C₄₇

#1984

yes

characteristic

C₂×C₄₇

C₂×#350

yes

—

#533

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₄₇

yes

—

#446

—

C₈×Q₈

—

#424

—

#423

—

#405

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

100

C₄₇

#13266

yes

characteristic, complemented, Sylow

101

C₂×M₁₆

—

102

C₂²×C₈

C₄×D₄₇

yes

characteristic

103

C₂²×C₈

C₂²×D₄₇

yes

characteristic

104

C₄⋊Q₈

—

105

#125

—

106

C₂²⋊Q₈

—

107

C₄×Q₈

—

108

#117

—

109

C₂×C₄²

C₄×D₄₇

yes

characteristic

110

M₃₂

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

C₂×C₁₆

—

114

C₂×C₁₆

Dic₉₄

yes

complemented

115

C₄⋊C₈

—

116

C₄⋊C₈

—

117

#97

—

118

C₄×C₈

—

119

C₄×C₈

D₁₈₈

yes

—

120

C₄×C₈

C₂²⋊D₄₇

yes

—

121

C₂×Q₈

—

122

C₂²×C₄

#10625324

yes

—

123

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

128

C₂×C₈

C₂×Dic₉₄

yes

—

129

C₂×C₈

C₂×D₁₈₈

yes

characteristic

130

C₂×C₈

#10625350

yes

—

131

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

C₈×D₄₇

yes

characteristic

138

C₄²

C₈×D₄₇

yes

characteristic

139

C₄²

(C₂×D₄₇)⋊C₄

yes

—

140

C₁₆

—

141

C₁₆

—

188

—

142

C₂³

C₂×#10625324

yes

characteristic

143

Q₈

—

188

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

147

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

—

148

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

149

C₂×C₄

C₄×Dic₉₄

yes

—

150

C₂×C₄

C₂×C₈×D₄₇

yes

characteristic

151

C₂×C₄

#13094030

yes

—

152

C₈

—

153

C₈

—

188

—

154

C₈

#13094003

yes

—

155

C₈

#13094012

yes

characteristic

156

C₈

#13094024

yes

characteristic

157

C₂²

—

158

C₂²

C₂×(C₂×D₄₇)⋊C₈

yes

characteristic

159

C₂²

#15689253

yes

—

160

C₄

—

161

C₄

—

188

—

162

C₄

#15688906

yes

—

163

C₄

#15689029

yes

characteristic

164

C₄

#15689089

yes

characteristic

165

C₄

C₈×Dic₉₄

yes

—

166

C₂

—

167

C₂

#20244324

yes

characteristic

168

C₂

C₂×#15688906

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂³×C₄₇	C₂×M₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#13266
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂×C₈

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄×C₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₈

prime radical

C₂³×C₄₇

C₂×M₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₂×C₄×C₁₆

D₄₇

hypocenter

C₄₇

#13266

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
33. C₂×C₁₆×C₄₇	Q₈	semidirect
100. C₄₇	#13266	semidirect
114. C₂×C₁₆	Dic₉₄	semidirect

H ≅ G/N

type

33. C₂×C₁₆×C₄₇

Q₈

semidirect

100. C₄₇

#13266

semidirect

114. C₂×C₁₆

Dic₉₄

semidirect

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

subgroup	index
#20252322	2
#20245144	2
#20241476	2
C₂×C₄×C₁₆×C₄₇	2
#13266	47

subgroup

index

C₂×C₄×C₁₆×C₄₇

Subgroups — #26134809

Conjugacy classes of subgroups

Complements

Maximal subgroups