Group #26132498 — subgroups

Subgroups — #26132498

Order 12032.

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

normal subgroups only

This group has 434 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	#20244999	C₂	1	2	yes	maximal
2	#20241472	C₂	1	4	yes	maximal, complemented
3	C₂×C₄₇×#405	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	C₂²	1	4	yes	—
5	C₂²⋊(C₄₇⋊C₁₆)	—	2	4	no	—
6	C₂²×M₁₆×C₄₇	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×#117×C₄₇	C₄	1	1	yes	characteristic
9	C₄₇×#405	—	2	4	no	—
10	C₂×#97×C₄₇	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
12	#13093964	—	2	8	no	—
13	C₂×C₄₇⋊C₁₆	—	2	4	no	—
14	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂×M₁₆×C₄₇	—	2	2	no	—
16	C₂×M₁₆×C₄₇	—	2	4	no	—
17	C₂²×C₈×C₄₇	—	2	4	no	—
18	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₄₇	D₄	1	4	yes	—
21	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	#117×C₄₇	—	2	2	no	—
23	#117×C₄₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₇	C₈	1	2	yes	—
25	C₂×C₂²⋊C₄×C₄₇	C₈	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
27	#97×C₄₇	—	2	4	no	—
28	C₄×C₈×C₄₇	—	2	4	no	—
29	C₂⁴×C₄₇	C₂×C₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
37	M₁₆×C₄₇	—	2	8	no	—
38	C₂×C₈×C₄₇	—	2	2	no	—
39	C₂×C₈×C₄₇	—	2	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	8	no	—
42	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
44	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	4	yes	—
46	C₄⋊C₄×C₄₇	—	2	4	no	—
47	C₂²⋊C₄×C₄₇	—	2	4	no	—
48	C₄²×C₄₇	—	2	2	no	—
49	C₄²×C₄₇	—	2	2	no	—
50	C₄₇⋊C₁₆	—	4	4	no	—
51	C₂³×C₄₇	—	2	2	no	—
52	C₂³×C₄₇	—	4	2	no	—
53	C₂³×C₄₇	C₂²⋊C₈	1	2	yes	—
54	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	4	no	—
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#117	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
65	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
66	C₈×C₄₇	—	2	2	no	—
67	C₈×C₄₇	—	2	4	no	—
68	C₈×C₄₇	#123	1	4	yes	—
69	#12847	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	—	2	4	no	—
73	C₂²×C₄₇	—	4	4	no	—
74	C₂²×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
75	C₂²×C₄₇	#431	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	4	4	no	—
78	C₄×C₄₇	#381	1	1	yes	characteristic
79	C₄×C₄₇	#407	1	1	yes	characteristic
80	C₄×C₄₇	#432	1	2	yes	—
81	C₂×#405	D₄₇	1	1	yes	characteristic
82	#2112	—	47	2	no	—
83	#1377	—	47	4	no	—
84	C₂×C₄₇	—	2	2	no	—
85	C₂×C₄₇	—	4	2	no	—
86	C₂×C₄₇	#1817	1	1	yes	characteristic
87	C₂×C₄₇	#2115	1	2	yes	—
88	C₂²×M₁₆	Dic₄₇	1	1	yes	characteristic
89	C₂³×C₈	C₂×D₄₇	1	1	yes	characteristic
90	C₂×#117	Dic₄₇	1	1	yes	characteristic
91	C₂×M₃₂	—	47	4	no	—
92	#405	—	2	4	no	—
93	C₂×#97	C₂×D₄₇	1	2	yes	—
94	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
95	C₂²⋊C₁₆	—	94	4	no	—
96	C₄₇	#12847	1	1	yes	characteristic, complemented, Sylow
97	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	2	4	no	—
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
102	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²⋊D₄₇	1	4	yes	—
104	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
105	#117	—	2	2	no	—
106	#117	—	2	2	no	—
107	C₂×C₄⋊C₄	C₄₇⋊C₈	1	2	yes	—
108	C₂×C₂²⋊C₄	C₄₇⋊C₈	1	2	yes	—
109	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
110	M₃₂	—	94	8	no	—
111	C₂×C₁₆	—	94	4	no	—
112	#97	—	2	4	no	—
113	C₄×C₈	—	2	4	no	—
114	C₂⁴	C₂×C₄₇⋊C₈	1	1	yes	characteristic
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
118	C₂²×C₄	C₂×C₄₇⋊C₈	1	4	yes	—
119	C₂²×C₄	#10625329	1	2	yes	—
120	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
121	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
122	M₁₆	—	2	8	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	2	8	no	—
127	C₂×C₈	#10625329	1	4	yes	—
128	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
129	C₂×C₈	#10625350	1	4	yes	—
130	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
131	C₄⋊C₄	—	2	4	no	—
132	C₂²⋊C₄	—	2	4	no	—
133	C₄²	—	2	2	no	—
134	C₄²	—	2	2	no	—
135	C₁₆	—	188	4	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	4	2	no	—
138	C₂³	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
139	C₂³	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
148	C₂×C₄	#13093983	1	2	yes	—
149	C₂×C₄	C₂×#10625329	1	2	yes	—
150	C₂×C₄	C₂×#10625329	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	4	no	—
153	C₈	#13094007	1	4	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
159	C₂²	#15689176	1	2	yes	—
160	C₄	—	2	2	no	—
161	C₄	—	4	4	no	—
162	C₄	#15688986	1	1	yes	characteristic
163	C₄	#15689087	1	1	yes	characteristic
164	C₄	#15689180	1	2	yes	—
165	C₂	—	2	2	no	—
166	C₂	—	4	2	no	—
167	C₂	#20243456	1	1	yes	characteristic
168	C₂	#20245009	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244999

C₂

yes

maximal

#20241472

C₂

yes

maximal, complemented

C₂×C₄₇×#405

C₂

yes

maximal, characteristic

C₂×#13093964

C₂²

yes

—

C₂²⋊(C₄₇⋊C₁₆)

—

C₂²×M₁₆×C₄₇

C₄

yes

characteristic

C₂³×C₈×C₄₇

C₂²

yes

characteristic

C₂×#117×C₄₇

C₄

yes

characteristic

C₄₇×#405

—

C₂×#97×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×M₁₆×C₄₇

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₈

yes

—

C₂×C₂²⋊C₄×C₄₇

C₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₂⁴×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

—

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₈

yes

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#123

yes

—

#12847

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×#405

D₄₇

yes

characteristic

#2112

—

#1377

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1817

yes

characteristic

C₂×C₄₇

#2115

yes

—

C₂²×M₁₆

Dic₄₇

yes

characteristic

C₂³×C₈

C₂×D₄₇

yes

characteristic

C₂×#117

Dic₄₇

yes

characteristic

C₂×M₃₂

—

#405

—

C₂×#97

C₂×D₄₇

yes

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂²⋊C₁₆

—

C₄₇

#12847

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

C₂×Dic₄₇

yes

—

102

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

103

C₂²×C₈

C₂²⋊D₄₇

yes

—

104

C₂²×C₈

C₂²×D₄₇

yes

characteristic

105

#117

—

106

#117

—

107

C₂×C₄⋊C₄

C₄₇⋊C₈

yes

—

108

C₂×C₂²⋊C₄

C₄₇⋊C₈

yes

—

109

C₂×C₄²

C₂×Dic₄₇

yes

—

110

M₃₂

—

111

C₂×C₁₆

—

112

#97

—

113

C₄×C₈

—

114

C₂⁴

C₂×C₄₇⋊C₈

yes

characteristic

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×C₄₇⋊C₈

yes

characteristic

118

C₂²×C₄

C₂×C₄₇⋊C₈

yes

—

119

C₂²×C₄

#10625329

yes

—

120

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

121

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

122

M₁₆

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#10625329

yes

—

128

C₂×C₈

C₂²⋊Dic₄₇

yes

—

129

C₂×C₈

#10625350

yes

—

130

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

131

C₄⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₄²

—

135

C₁₆

—

188

—

136

C₂³

—

137

C₂³

—

138

C₂³

C₂²⋊(C₄₇⋊C₈)

yes

—

139

C₂³

C₂²×C₄₇⋊C₈

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

147

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

148

C₂×C₄

#13093983

yes

—

149

C₂×C₄

C₂×#10625329

yes

—

150

C₂×C₄

C₂×#10625329

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

#13094007

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

159

C₂²

#15689176

yes

—

160

C₄

—

161

C₄

—

162

C₄

#15688986

yes

characteristic

163

C₄

#15689087

yes

characteristic

164

C₄

#15689180

yes

—

165

C₂

—

166

C₂

—

167

C₂

#20243456

yes

characteristic

168

C₂

#20245009

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₄₇
commutator	C₂³×C₄₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#405	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₄₇	C₂×C₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#405	D₄₇
hypocenter	C₄₇	#12847
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊Dic₄₇

commutator

C₂³×C₄₇

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#405

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₈

prime radical

C₂⁴×C₄₇

C₂×C₈

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂×#405

N	H ≅ G/N	type
2. #20241472	C₂	semidirect
96. C₄₇	#12847	semidirect

H ≅ G/N

type

2. #20241472

C₂

semidirect

96. C₄₇

#12847

semidirect

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

subgroup	index
#20244999	2
#20241472	2
C₂×C₄₇×#405	2
#12847	47

subgroup

index

#20244999

#20241472

C₂×C₄₇×#405

#12847

Subgroups — #26132498

Conjugacy classes of subgroups

Complements

Maximal subgroups