Group #26136157 — subgroups

Subgroups — #26136157

Order 12032.

284 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245001	C₂	1	1	yes	maximal, characteristic
2	#20243453	C₂	1	2	yes	maximal
3	#20241496	C₂	1	2	yes	maximal, complemented
4	#20241482	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄₇×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13094030	C₂²	1	1	yes	characteristic
7	C₂×#13093964	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₂²	1	2	yes	—
9	#15689122	—	2	2	no	—
10	(C₂×D₄₇)⋊C₁₆	—	2	2	no	—
11	#15688892	—	2	2	no	—
12	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#107×C₄₇	C₂²	1	2	yes	—
14	C₄₇×#367	—	2	2	no	—
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	#13093964	—	2	2	no	—
23	C₂×C₄₇⋊C₁₆	—	2	1	no	—
24	#13093960	—	2	2	no	—
25	#13093960	—	2	2	no	—
26	#13093947	—	4	2	no	—
27	#13093944	—	4	2	no	—
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂×C₁₆×C₄₇	—	2	2	no	—
31	C₂×C₁₆×C₄₇	D₄	1	2	yes	—
32	#107×C₄₇	—	2	2	no	—
33	#107×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	—	4	2	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₂²⋊D₄₇	—	2	1	no	—
37	#10625350	—	2	2	no	—
38	#10625350	—	2	2	no	—
39	C₂×D₁₈₈	D₈	1	1	yes	characteristic
40	C₂×C₄×D₄₇	—	2	1	no	—
41	C₂×Dic₉₄	D₈	1	1	yes	characteristic
42	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	#41	1	2	yes	—
44	C₂×C₈×C₄₇	#41	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	1	yes	characteristic
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	#10625329	—	2	2	no	—
49	#10625329	—	4	2	no	—
50	#10625329	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	2	1	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	#10625324	—	4	2	no	—
54	C₈×D₄₇	—	4	1	no	—
55	C₈×D₄₇	—	4	1	no	—
56	C₁₆×C₄₇	—	2	2	no	—
57	C₄₇⋊C₁₆	—	4	1	no	—
58	C₂³×C₄₇	#135	1	1	yes	characteristic
59	C₂²×D₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	8	2	no	—
61	C₂×C₄×C₄₇	—	16	2	no	—
62	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
63	C₂×C₄×C₄₇	#123	1	2	yes	—
64	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₄₇	#137	1	2	yes	—
66	C₂²⋊D₄₇	—	4	2	no	—
67	C₂×Dic₄₇	—	2	1	no	—
68	D₁₈₈	—	2	2	no	—
69	C₄×D₄₇	—	4	1	no	—
70	C₄×D₄₇	—	4	1	no	—
71	Dic₉₄	—	2	2	no	—
72	C₈×C₄₇	#123	1	1	yes	characteristic
73	C₈×C₄₇	#124	1	1	yes	characteristic
74	C₈×C₄₇	#126	1	2	yes	—
75	C₄₇⋊C₈	—	4	1	no	—
76	C₄₇⋊C₈	—	8	2	no	—
77	#13496	—	47	1	no	maximal, Sylow
78	C₂²×C₄₇	—	2	2	no	—
79	C₂²×C₄₇	#466	1	1	yes	characteristic
80	C₂²×C₄₇	#485	1	2	yes	—
81	C₂×D₄₇	—	4	1	no	—
82	C₂×D₄₇	—	4	2	no	—
83	C₄×C₄₇	—	16	2	no	—
84	C₄×C₄₇	#397	1	1	yes	characteristic
85	C₄×C₄₇	#470	1	1	yes	characteristic
86	C₄×C₄₇	#480	1	2	yes	—
87	Dic₄₇	—	4	1	no	—
88	C₂×#367	D₄₇	1	1	yes	characteristic, complemented
89	#2113	—	47	1	no	—
90	#1816	—	47	2	no	—
91	#1387	—	47	2	no	—
92	#1381	—	47	1	no	—
93	C₂×C₄₇	—	2	2	no	—
94	C₂×C₄₇	#2075	1	1	yes	characteristic
95	C₂×C₄₇	#2223	1	2	yes	—
96	D₄₇	—	8	1	no	—
97	C₂×#131	—	47	1	no	—
98	C₂×M₃₂	—	47	1	no	—
99	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
100	C₂×#108	—	47	2	no	—
101	C₂×#107	C₂×D₄₇	1	2	yes	—
102	#417	—	94	2	no	—
103	#367	—	2	2	no	—
104	C₂²⋊C₁₆	—	94	2	no	—
105	#343	—	94	2	no	—
106	C₄₇	#13496	1	1	yes	characteristic, complemented, Sylow
107	C₂×#41	—	47	1	no	—
108	#131	—	94	2	no	—
109	#131	—	94	2	no	—
110	C₂×M₁₆	—	47	1	no	—
111	C₂×M₁₆	—	94	1	no	—
112	C₂×M₁₆	—	94	2	no	—
113	C₂²×C₈	—	94	1	no	—
114	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×C₄⋊C₄	—	2	2	no	—
116	M₃₂	—	94	2	no	—
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	94	1	no	—
119	C₂×C₁₆	C₂²⋊D₄₇	1	2	yes	—
120	#108	—	94	2	no	—
121	#108	—	94	2	no	—
122	#107	—	2	2	no	—
123	#107	—	2	2	no	—
124	#103	—	188	2	no	—
125	#102	—	188	2	no	—
126	#41	—	94	2	no	—
127	#41	—	94	2	no	—
128	C₂×Q₈	—	47	1	no	—
129	C₂×D₄	—	47	1	no	—
130	C₂×D₄	—	94	1	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	94	1	no	—
133	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
134	M₁₆	—	94	2	no	—
135	M₁₆	—	188	2	no	—
136	M₁₆	—	188	2	no	—
137	M₁₆	—	188	2	no	—
138	C₂×C₈	—	94	1	no	—
139	C₂×C₈	—	188	2	no	—
140	C₂×C₈	—	188	1	no	—
141	C₂×C₈	—	188	1	no	—
142	C₂×C₈	#10625350	1	2	yes	—
143	C₂×C₈	#10625352	1	1	yes	characteristic
144	C₂×C₈	#10625354	1	2	yes	—
145	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₄⋊C₄	—	4	2	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	—	188	1	no	—
150	C₂³	—	94	1	no	—
151	C₂³	#13094043	1	1	yes	characteristic
152	Q₈	—	94	2	no	—
153	D₄	—	94	2	no	—
154	D₄	—	188	2	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	16	2	no	—
157	C₂×C₄	—	94	1	no	—
158	C₂×C₄	—	188	1	no	—
159	C₂×C₄	—	188	1	no	—
160	C₂×C₄	#13093995	1	1	yes	characteristic
161	C₂×C₄	#13094009	1	2	yes	—
162	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
163	C₂×C₄	#13094056	1	2	yes	—
164	C₈	—	376	2	no	—
165	C₈	—	188	1	no	—
166	C₈	#13094007	1	1	yes	characteristic
167	C₈	#13094013	1	1	yes	characteristic
168	C₈	#13094019	1	2	yes	—
169	C₂²	—	2	2	no	—
170	C₂²	—	188	2	no	—
171	C₂²	—	188	1	no	—
172	C₂²	#15689356	1	1	yes	characteristic
173	C₂²	#15689466	1	2	yes	—
174	C₄	—	16	2	no	—
175	C₄	—	188	1	no	—
176	C₄	#15689049	1	1	yes	characteristic
177	C₄	#15689378	1	1	yes	characteristic
178	C₄	#15689438	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	—	376	1	no	—
181	C₂	#20244812	1	1	yes	characteristic
182	C₂	#20245515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245001

C₂

yes

maximal, characteristic

#20243453

C₂

yes

maximal

#20241496

C₂

yes

maximal, complemented

#20241482

C₂

yes

maximal, characteristic, complemented

C₂×C₄₇×#367

C₂

yes

maximal, characteristic, complemented

C₂×#13094030

C₂²

yes

characteristic

C₂×#13093964

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

#15689122

—

(C₂×D₄₇)⋊C₁₆

—

#15688892

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#107×C₄₇

C₂²

yes

—

C₄₇×#367

—

C₂×#10625350

D₄

yes

—

—

—

D₄

yes

characteristic

C₂×#10625324

—

C₂×C₈×D₄₇

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

—

—

—

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

D₄

yes

—

#107×C₄₇

—

#107×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

D₈

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₈

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

—

C₄⋊C₄×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

—

C₂×C₄×C₄₇

#121

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#137

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#13496

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#466

yes

characteristic

C₂²×C₄₇

#485

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×#367

D₄₇

yes

characteristic, complemented

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#2075

yes

characteristic

C₂×C₄₇

#2223

yes

—

D₄₇

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

100

C₂×#108

—

101

C₂×#107

C₂×D₄₇

yes

—

102

#417

—

103

#367

—

104

C₂²⋊C₁₆

—

105

#343

—

106

C₄₇

#13496

yes

characteristic, complemented, Sylow

107

C₂×#41

—

108

#131

—

109

#131

—

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂×M₁₆

—

113

C₂²×C₈

—

114

C₂²×C₈

C₂²×D₄₇

yes

characteristic

115

C₂×C₄⋊C₄

—

116

M₃₂

—

117

C₂×C₁₆

—

118

C₂×C₁₆

—

119

C₂×C₁₆

C₂²⋊D₄₇

yes

—

120

#108

—

121

#108

—

122

#107

—

123

#107

—

124

#103

—

188

—

125

#102

—

188

—

126

#41

—

127

#41

—

128

C₂×Q₈

—

129

C₂×D₄

—

130

C₂×D₄

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

D₄×D₄₇

yes

characteristic

134

M₁₆

—

135

M₁₆

—

188

—

136

M₁₆

—

188

—

137

M₁₆

—

188

—

138

C₂×C₈

—

139

C₂×C₈

—

188

—

140

C₂×C₈

—

188

—

141

C₂×C₈

—

188

—

142

C₂×C₈

#10625350

yes

—

143

C₂×C₈

#10625352

yes

characteristic

144

C₂×C₈

#10625354

yes

—

145

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₁₆

—

149

C₁₆

—

188

—

150

C₂³

—

151

C₂³

#13094043

yes

characteristic

152

Q₈

—

153

D₄

—

154

D₄

—

188

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

188

—

159

C₂×C₄

—

188

—

160

C₂×C₄

#13093995

yes

characteristic

161

C₂×C₄

#13094009

yes

—

162

C₂×C₄

D₈×D₄₇

yes

characteristic

163

C₂×C₄

#13094056

yes

—

164

C₈

—

376

—

165

C₈

—

188

—

166

C₈

#13094007

yes

characteristic

167

C₈

#13094013

yes

characteristic

168

C₈

#13094019

yes

—

169

C₂²

—

170

C₂²

—

188

—

171

C₂²

—

188

—

172

C₂²

#15689356

yes

characteristic

173

C₂²

#15689466

yes

—

174

C₄

—

175

C₄

—

188

—

176

C₄

#15689049

yes

characteristic

177

C₄

#15689378

yes

characteristic

178

C₄

#15689438

yes

—

179

C₂

—

180

C₂

—

376

—

181

C₂

#20244812

yes

characteristic

182

C₂

#20245515

yes

—

	subgroup	G/H
center	C₂²	#15689356
commutator, cosocle	C₂²×C₈×C₄₇	C₂³
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#367	C₂
socle	C₂²×C₄₇	#466
prime radical	C₂×#10625350	D₄
hypercenter	C₂×#367	D₄₇
hypocenter	C₄₇	#13496
LCS	G ▷ C₂²×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

subgroup

G/H

center

C₂²

#15689356

commutator, cosocle

C₂²×C₈×C₄₇

C₂³

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#367

C₂

C₂²×C₄₇

D₄

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

N	H ≅ G/N	type
3. #20241496	C₂	semidirect
4. #20241482	C₂	semidirect
5. C₂×C₄₇×#367	C₂	semidirect
88. C₂×#367	D₄₇	semidirect
106. C₄₇	#13496	semidirect

H ≅ G/N

type

3. #20241496

C₂

semidirect

4. #20241482

C₂

semidirect

5. C₂×C₄₇×#367

C₂

semidirect

88. C₂×#367

D₄₇

semidirect

106. C₄₇

#13496

semidirect

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

subgroup	index
#20245001	2
#20243453	2
#20241496	2
#20241482	2
C₂×C₄₇×#367	2
#13496	47

subgroup

index

C₂×C₄₇×#367

Subgroups — #26136157

Conjugacy classes of subgroups

Complements

Maximal subgroups