Group #26136175 — subgroups

Subgroups — #26136175

Order 12032.

276 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20245183	C₂	1	1	yes	maximal, characteristic, complemented
2	#20245003	C₂	1	1	yes	maximal, characteristic
3	#20243439	C₂	1	2	yes	maximal
4	#20241483	C₂	1	2	yes	maximal, complemented
5	C₄₇×#1384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094030	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₂²	1	2	yes	—
9	#15689141	—	2	2	no	—
10	#15688953	—	2	2	no	—
11	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
12	#15688893	—	2	2	no	—
13	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#427	C₂²	1	2	yes	—
15	C₂×#10625350	D₄	1	1	yes	characteristic
16	#13094030	—	2	2	no	—
17	#13094030	—	2	2	no	—
18	C₂×#10625329	—	2	2	no	—
19	C₂×#10625329	D₄	1	1	yes	characteristic
20	C₂×#10625324	—	2	1	no	—
21	C₂×C₈×D₄₇	—	2	1	no	—
22	C₂×C₄₇⋊C₁₆	—	2	1	no	—
23	C₂×C₄₇⋊C₁₆	—	2	2	no	—
24	#13093960	—	2	2	no	—
25	#13093960	—	2	2	no	—
26	#13093952	—	4	4	no	—
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	#117×C₄₇	—	2	2	no	—
29	M₃₂×C₄₇	—	2	2	no	—
30	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
31	#107×C₄₇	—	2	2	no	—
32	#106×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₂²⋊D₄₇	—	2	1	no	—
35	#10625350	—	2	2	no	—
36	#10625350	—	2	2	no	—
37	C₂×D₁₈₈	Q₁₆	1	1	yes	characteristic
38	C₂×C₄×D₄₇	—	2	1	no	—
39	C₂×Dic₉₄	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	#41	1	2	yes	—
42	C₂×C₈×C₄₇	#41	1	2	yes	—
43	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
44	C₄⋊C₄×C₄₇	—	4	2	no	—
45	C₂²⋊C₄×C₄₇	—	4	2	no	—
46	C₄²×C₄₇	—	4	2	no	—
47	#10625329	—	2	2	no	—
48	#10625329	—	4	2	no	—
49	#10625329	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	2	1	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	#10625324	—	4	2	no	—
53	C₈×D₄₇	—	4	1	no	—
54	C₈×D₄₇	—	4	1	no	—
55	C₁₆×C₄₇	—	2	2	no	—
56	C₄₇⋊C₁₆	—	4	1	no	—
57	C₂³×C₄₇	#135	1	1	yes	characteristic
58	C₂²×D₄₇	—	2	1	no	—
59	C₂×C₄×C₄₇	—	8	2	no	—
60	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#123	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
64	C₂²⋊D₄₇	—	4	2	no	—
65	C₂×Dic₄₇	—	2	1	no	—
66	D₁₈₈	—	2	2	no	—
67	C₄×D₄₇	—	4	1	no	—
68	C₄×D₄₇	—	4	1	no	—
69	Dic₉₄	—	2	2	no	—
70	C₈×C₄₇	#123	1	1	yes	characteristic
71	C₈×C₄₇	#124	1	1	yes	characteristic
72	C₈×C₄₇	#126	1	2	yes	—
73	C₄₇⋊C₈	—	4	1	no	—
74	C₄₇⋊C₈	—	8	2	no	—
75	#13499	—	47	1	no	maximal, Sylow
76	C₂²×C₄₇	—	4	1	no	—
77	C₂²×C₄₇	#467	1	1	yes	characteristic
78	C₂²×C₄₇	#482	1	2	yes	—
79	C₂×D₄₇	—	4	2	no	—
80	C₂×D₄₇	—	4	1	no	—
81	C₄×C₄₇	—	16	2	no	—
82	C₄×C₄₇	#397	1	1	yes	characteristic
83	C₄×C₄₇	#469	1	1	yes	characteristic
84	C₄×C₄₇	#484	1	2	yes	—
85	Dic₄₇	—	4	1	no	—
86	#2161	—	47	1	no	—
87	#2113	—	47	1	no	—
88	#1814	—	47	2	no	—
89	#1384	D₄₇	1	1	yes	characteristic, complemented
90	#1382	—	47	2	no	—
91	C₂×C₄₇	—	2	2	no	—
92	C₂×C₄₇	—	2	1	no	—
93	C₂×C₄₇	#2076	1	1	yes	characteristic
94	D₄₇	—	8	1	no	—
95	C₂×#131	—	47	1	no	—
96	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
97	C₂²×C₁₆	—	47	1	no	—
98	C₂×#108	—	47	2	no	—
99	#427	C₂×D₄₇	1	2	yes	—
100	#421	—	94	2	no	—
101	#368	—	94	2	no	—
102	C₂²⋊C₁₆	—	94	2	no	—
103	#344	—	94	2	no	—
104	C₄₇	#13499	1	1	yes	characteristic, complemented, Sylow
105	C₂×#41	—	47	1	no	—
106	#131	—	94	2	no	—
107	#131	—	94	2	no	—
108	C₂×M₁₆	—	47	1	no	—
109	C₂×M₁₆	—	94	1	no	—
110	C₂×M₁₆	—	94	2	no	—
111	C₂²×C₈	—	94	1	no	—
112	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	#117	—	2	2	no	—
114	M₃₂	—	2	2	no	—
115	C₂×C₁₆	—	94	1	no	—
116	C₂×C₁₆	—	94	2	no	—
117	C₂×C₁₆	D₁₈₈	1	2	yes	—
118	#108	—	94	2	no	—
119	#108	—	94	2	no	—
120	#107	—	2	2	no	—
121	#106	—	2	2	no	—
122	#104	—	188	4	no	—
123	#41	—	94	2	no	—
124	#41	—	94	2	no	—
125	C₂×Q₈	—	47	1	no	—
126	C₂×D₄	—	47	1	no	—
127	C₂×D₄	—	94	1	no	—
128	C₂²×C₄	—	94	1	no	—
129	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
130	M₁₆	—	94	2	no	—
131	M₁₆	—	188	2	no	—
132	M₁₆	—	188	2	no	—
133	M₁₆	—	188	2	no	—
134	C₂×C₈	—	94	1	no	—
135	C₂×C₈	—	188	2	no	—
136	C₂×C₈	—	188	1	no	—
137	C₂×C₈	—	188	1	no	—
138	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
139	C₂×C₈	#10625350	1	2	yes	—
140	C₂×C₈	#10625352	1	2	yes	—
141	C₂×C₈	#10625354	1	1	yes	characteristic
142	C₄⋊C₄	—	4	2	no	—
143	C₂²⋊C₄	—	4	2	no	—
144	C₄²	—	4	2	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	188	1	no	—
147	C₂³	—	94	1	no	—
148	C₂³	#13094043	1	1	yes	characteristic
149	Q₈	—	94	2	no	—
150	D₄	—	94	2	no	—
151	D₄	—	188	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	188	1	no	—
155	C₂×C₄	—	188	1	no	—
156	C₂×C₄	#13093999	1	1	yes	characteristic
157	C₂×C₄	#13094006	1	2	yes	—
158	C₂×C₄	Q₁₆×D₄₇	1	1	yes	characteristic
159	C₂×C₄	#13094051	1	2	yes	—
160	C₈	—	376	2	no	—
161	C₈	—	188	1	no	—
162	C₈	#13094008	1	1	yes	characteristic
163	C₈	#13094012	1	1	yes	characteristic
164	C₈	#13094019	1	2	yes	—
165	C₂²	—	4	1	no	—
166	C₂²	—	188	2	no	—
167	C₂²	—	188	1	no	—
168	C₂²	#15689365	1	1	yes	characteristic
169	C₂²	#15689445	1	2	yes	—
170	C₄	—	16	2	no	—
171	C₄	—	188	1	no	—
172	C₄	#15689050	1	1	yes	characteristic
173	C₄	#15689377	1	1	yes	characteristic
174	C₄	#15689457	1	2	yes	—
175	C₂	—	2	2	no	—
176	C₂	—	2	1	no	—
177	C₂	—	376	1	no	—
178	C₂	#20244818	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245183

C₂

yes

maximal, characteristic, complemented

#20245003

C₂

yes

maximal, characteristic

#20243439

C₂

yes

maximal

#20241483

C₂

yes

maximal, complemented

C₄₇×#1384

C₂

yes

maximal, characteristic, complemented

C₂×#13094030

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

#15689141

—

#15688953

—

(C₂×D₄₇)⋊C₁₆

—

#15688893

—

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#427

C₂²

yes

—

C₂×#10625350

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#10625324

—

C₂×C₈×D₄₇

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093960

—

#13093952

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

#117×C₄₇

—

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#106×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

Q₁₆

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625329

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#135

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#13499

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#467

yes

characteristic

C₂²×C₄₇

#482

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#1382

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2076

yes

characteristic

D₄₇

—

C₂×#131

—

C₂×M₃₂

C₂×D₄₇

yes

characteristic

C₂²×C₁₆

—

C₂×#108

—

#427

C₂×D₄₇

yes

—

100

#421

—

101

#368

—

102

C₂²⋊C₁₆

—

103

#344

—

104

C₄₇

#13499

yes

characteristic, complemented, Sylow

105

C₂×#41

—

106

#131

—

107

#131

—

108

C₂×M₁₆

—

109

C₂×M₁₆

—

110

C₂×M₁₆

—

111

C₂²×C₈

—

112

C₂²×C₈

C₂²×D₄₇

yes

characteristic

113

#117

—

114

M₃₂

—

115

C₂×C₁₆

—

116

C₂×C₁₆

—

117

C₂×C₁₆

D₁₈₈

yes

—

118

#108

—

119

#108

—

120

#107

—

121

#106

—

122

#104

—

188

—

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂²×C₄

—

129

C₂²×C₄

D₄×D₄₇

yes

characteristic

130

M₁₆

—

131

M₁₆

—

188

—

132

M₁₆

—

188

—

133

M₁₆

—

188

—

134

C₂×C₈

—

135

C₂×C₈

—

188

—

136

C₂×C₈

—

188

—

137

C₂×C₈

—

188

—

138

C₂×C₈

C₂×D₁₈₈

yes

characteristic

139

C₂×C₈

#10625350

yes

—

140

C₂×C₈

#10625352

yes

—

141

C₂×C₈

#10625354

yes

characteristic

142

C₄⋊C₄

—

143

C₂²⋊C₄

—

144

C₄²

—

145

C₁₆

—

146

C₁₆

—

188

—

147

C₂³

—

148

C₂³

#13094043

yes

characteristic

149

Q₈

—

150

D₄

—

151

D₄

—

188

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

188

—

155

C₂×C₄

—

188

—

156

C₂×C₄

#13093999

yes

characteristic

157

C₂×C₄

#13094006

yes

—

158

C₂×C₄

Q₁₆×D₄₇

yes

characteristic

159

C₂×C₄

#13094051

yes

—

160

C₈

—

376

—

161

C₈

—

188

—

162

C₈

#13094008

yes

characteristic

163

C₈

#13094012

yes

characteristic

164

C₈

#13094019

yes

—

165

C₂²

—

166

C₂²

—

188

—

167

C₂²

—

188

—

168

C₂²

#15689365

yes

characteristic

169

C₂²

#15689445

yes

—

170

C₄

—

171

C₄

—

188

—

172

C₄

#15689050

yes

characteristic

173

C₄

#15689377

yes

characteristic

174

C₄

#15689457

yes

—

175

C₂

—

176

C₂

—

177

C₂

—

376

—

178

C₂

#20244818

yes

characteristic

	subgroup	G/H
center	C₄	#15689377
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1384	C₂
socle	C₂×C₄₇	#2076
prime radical	C₂×#10625350	D₄
hypercenter	#1384	D₄₇
hypocenter	C₄₇	#13499
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

subgroup

G/H

center

C₄

#15689377

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1384

C₂

C₂×C₄₇

D₄

D₄₇

C₄₇

G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1384

N	H ≅ G/N	type
1. #20245183	C₂	semidirect
4. #20241483	C₂	semidirect
5. C₄₇×#1384	C₂	semidirect
89. #1384	D₄₇	semidirect
104. C₄₇	#13499	semidirect

H ≅ G/N

type

1. #20245183

C₂

semidirect

4. #20241483

C₂

semidirect

5. C₄₇×#1384

C₂

semidirect

89. #1384

D₄₇

semidirect

104. C₄₇

#13499

semidirect

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

subgroup	index
#20245183	2
#20245003	2
#20243439	2
#20241483	2
C₄₇×#1384	2
#13499	47

subgroup

index

C₄₇×#1384

Subgroups — #26136175

Conjugacy classes of subgroups

Complements

Maximal subgroups