Group #26114897 — subgroups

Subgroups — #26114897

Order 12032.

294 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 119 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	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
2	#20242138	C₂	1	2	yes	maximal
3	#20241680	C₂	1	1	yes	maximal, characteristic
4	#20241238	C₂	1	2	yes	maximal
5	C₂×C₄₇×#334	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₄	1	2	yes	—
9	C₄×#10625329	C₂²	1	2	yes	—
10	#15688888	C₄	1	1	yes	characteristic
11	#15688888	C₄	1	1	yes	characteristic
12	#15688875	—	2	2	no	—
13	#15688873	—	2	2	no	—
14	#15688845	—	2	2	no	—
15	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#334	—	2	2	no	—
18	C₂×#10625329	C₂×C₄	1	2	yes	—
19	C₂×#10625329	C₂×C₄	1	2	yes	—
20	#13093960	—	2	4	no	—
21	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
22	C₄⋊(C₄₇⋊C₈)	D₄	1	2	yes	—
23	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
24	C₂²⋊(C₄₇⋊C₈)	Q₈	1	2	yes	—
25	#13093935	—	2	2	no	—
26	C₄×C₄₇⋊C₈	—	2	2	no	—
27	C₄×C₄₇⋊C₈	—	2	1	no	—
28	C₄×C₄₇⋊C₈	—	2	1	no	—
29	C₂²×C₈×C₄₇	—	2	2	no	—
30	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₄₇	—	2	2	no	—
34	C₄⋊C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	D₄	1	2	yes	—
36	C₄×C₈×C₄₇	Q₈	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	C₂×C₈×C₄₇	—	8	2	no	—
44	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
46	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	1	yes	characteristic
50	#10625329	—	2	2	no	—
51	#10625329	—	2	2	no	—
52	#10625329	—	2	2	no	—
53	#10625329	M₁₆	1	4	yes	—
54	C₂×C₄₇⋊C₈	—	2	2	no	—
55	C₂×C₄₇⋊C₈	—	2	2	no	—
56	C₂×C₄₇⋊C₈	—	2	2	no	—
57	C₂×C₄₇⋊C₈	—	2	2	no	—
58	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	4	1	no	—
61	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#125	1	1	yes	characteristic
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₄₇	#131	1	2	yes	—
69	C₈×C₄₇	—	2	2	no	—
70	C₈×C₄₇	—	2	2	no	—
71	C₈×C₄₇	—	8	2	no	—
72	C₄₇⋊C₈	—	2	4	no	—
73	C₄₇⋊C₈	—	4	2	no	—
74	C₄₇⋊C₈	—	4	2	no	—
75	C₄₇⋊C₈	—	4	2	no	—
76	#9609	—	47	1	no	maximal, Sylow
77	C₂²×C₄₇	—	2	1	no	—
78	C₂²×C₄₇	—	2	1	no	—
79	C₂²×C₄₇	#389	1	1	yes	characteristic
80	C₂²×C₄₇	#446	1	2	yes	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	2	1	no	—
83	C₄×C₄₇	—	2	1	no	—
84	C₄×C₄₇	#423	1	1	yes	characteristic
85	C₄×C₄₇	#424	1	1	yes	characteristic
86	C₄×C₄₇	#435	1	2	yes	—
87	C₄×C₄₇	#471	1	2	yes	—
88	C₄×#108	—	47	1	no	—
89	#1565	—	47	2	no	—
90	C₂×#334	D₄₇	1	1	yes	characteristic
91	#1452	—	47	1	no	—
92	#1283	—	47	2	no	—
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	#1940	1	1	yes	characteristic
95	C₂×C₄₇	#1951	1	1	yes	characteristic
96	C₂×C₄₇	#1953	1	1	yes	characteristic
97	#431	—	47	1	no	—
98	#431	—	47	1	no	—
99	C₂×#108	—	47	2	no	—
100	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
101	C₄×M₁₆	—	47	2	no	—
102	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
103	#341	—	47	1	no	—
104	#341	—	47	1	no	—
105	#335	—	94	2	no	—
106	#334	—	2	2	no	—
107	#334	—	94	2	no	—
108	#322	—	94	2	no	—
109	C₄₇	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	47	2	no	—
111	C₂×M₁₆	—	47	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	#108	—	94	4	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	2	2	no	—
119	C₄⋊C₈	—	47	2	no	—
120	C₄⋊C₈	—	94	1	no	—
121	C₂²⋊C₈	—	47	2	no	—
122	C₂²⋊C₈	—	94	1	no	—
123	#97	—	94	2	no	—
124	C₄×C₈	—	94	2	no	—
125	C₄×C₈	—	94	1	no	—
126	C₄×C₈	—	94	1	no	—
127	C₄×C₈	Dic₉₄	1	2	yes	—
128	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
129	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂²×C₄	#10625352	1	1	yes	characteristic
131	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
132	M₁₆	—	47	4	no	—
133	M₁₆	—	94	2	no	—
134	M₁₆	—	94	2	no	—
135	M₁₆	—	94	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	8	2	no	—
140	C₂×C₈	—	94	2	no	—
141	C₂×C₈	—	94	2	no	—
142	C₂×C₈	—	94	2	no	—
143	C₂×C₈	—	94	2	no	—
144	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
145	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
146	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
147	C₄²	D₄×D₄₇	1	1	yes	characteristic
148	C₄²	#10625354	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
150	C₂³	#13093989	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
154	C₂×C₄	#13093986	1	1	yes	characteristic
155	C₂×C₄	#13093995	1	1	yes	characteristic
156	C₂×C₄	#13094000	1	1	yes	characteristic
157	C₂×C₄	#13094015	1	1	yes	characteristic
158	C₂×C₄	#13094023	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
160	C₂×C₄	#13094030	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	8	2	no	—
164	C₈	—	94	4	no	—
165	C₈	—	188	2	no	—
166	C₈	—	188	2	no	—
167	C₈	—	188	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#15689019	1	1	yes	characteristic
171	C₂²	#15689252	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#15689146	1	1	yes	characteristic
176	C₄	#15689149	1	1	yes	characteristic
177	C₄	#15689193	1	2	yes	—
178	C₄	#15689383	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#20244109	1	1	yes	characteristic
181	C₂	#20244166	1	1	yes	characteristic
182	C₂	#20244176	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093960

C₂

yes

maximal, characteristic

#20242138

C₂

yes

maximal

#20241680

C₂

yes

maximal, characteristic

#20241238

C₂

yes

maximal

C₂×C₄₇×#334

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

characteristic

#15689176

C₂²

yes

characteristic

C₂×#13093960

C₄

yes

—

C₄×#10625329

C₂²

yes

—

#15688888

C₄

yes

characteristic

#15688888

C₄

yes

—

—

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#334

—

C₂×#10625329

C₂×C₄

yes

—

C₂×#10625329

C₂×C₄

yes

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

Q₈

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

C₄²×C₄₇

yes

—

—

—

M₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄×Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#125

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9609

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#389

yes

characteristic

C₂²×C₄₇

#446

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#471

yes

—

C₄×#108

—

#1565

—

C₂×#334

D₄₇

yes

characteristic

#1452

—

#1283

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

100

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

103

#341

—

104

#341

—

105

#335

—

106

#334

—

107

#334

—

108

#322

—

109

C₄₇

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₄₇

yes

characteristic

114

C₂²×C₈

C₄×D₄₇

yes

characteristic

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

#108

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

—

123

#97

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

Dic₉₄

yes

—

128

C₄×C₈

C₂²⋊D₄₇

yes

—

129

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

130

C₂²×C₄

#10625352

yes

characteristic

131

C₂²×C₄

Q₈×D₄₇

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

Dic₄₇⋊C₄

yes

—

145

C₂×C₈

Dic₄₇⋊C₄

yes

—

146

C₄²

C₂×Dic₉₄

yes

characteristic

147

C₄²

D₄×D₄₇

yes

characteristic

148

C₄²

#10625354

yes

characteristic

149

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

150

C₂³

#13093989

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

154

C₂×C₄

#13093986

yes

characteristic

155

C₂×C₄

#13093995

yes

characteristic

156

C₂×C₄

#13094000

yes

characteristic

157

C₂×C₄

#13094015

yes

characteristic

158

C₂×C₄

#13094023

yes

characteristic

159

C₂×C₄

M₁₆×D₄₇

yes

—

160

C₂×C₄

#13094030

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

188

—

166

C₈

—

188

—

167

C₈

—

188

—

168

C₂²

—

169

C₂²

—

170

C₂²

#15689019

yes

characteristic

171

C₂²

#15689252

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#15689146

yes

characteristic

176

C₄

#15689149

yes

characteristic

177

C₄

#15689193

yes

—

178

C₄

#15689383

yes

—

179

C₂

—

180

C₂

#20244109

yes

characteristic

181

C₂

#20244166

yes

characteristic

182

C₂

#20244176

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13093995
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#334	C₂
socle	C₂²×C₄₇	#389
prime radical	C₂³×C₄₇	C₄×Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#334	D₄₇
hypocenter	C₄₇	#9609
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#13093995

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#334

C₂

socle

C₂²×C₄₇

#389

prime radical

C₂³×C₄₇

C₄×Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#334

N	H ≅ G/N	type
109. C₄₇	#9609	semidirect

H ≅ G/N

type

109. C₄₇

#9609

semidirect

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

subgroup	index
C₄×#13093960	2
#20242138	2
#20241680	2
#20241238	2
C₂×C₄₇×#334	2
#9609	47

subgroup

index

C₂×C₄₇×#334

Subgroups — #26114897

Conjugacy classes of subgroups

Complements

Maximal subgroups