Group #26121267 — subgroups

Subgroups — #26121267

Order 12032.

320 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 97 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	#20244328	C₂	1	1	yes	maximal, characteristic
2	#20242164	C₂	1	2	yes	maximal
3	#20241816	C₂	1	2	yes	maximal
4	#20241219	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1849	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₉₄	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
9	#15688877	—	2	2	no	—
10	#15688875	—	2	4	no	—
11	#15688853	—	2	4	no	—
12	C₂×#108×C₄₇	—	2	2	no	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₄	Q₈	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	—	2	1	no	—
17	C₄×Dic₉₄	—	2	2	no	—
18	C₄×Dic₉₄	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	Q₈	1	1	yes	characteristic
20	C₂×Dic₄₇⋊C₄	—	2	1	no	—
21	C₂×C₄×Dic₄₇	—	2	1	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
25	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	#108×C₄₇	—	4	4	no	—
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₄⋊C₈×C₄₇	—	2	2	no	—
32	#97×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	2	1	no	—
37	C₂²×Dic₄₇	—	2	1	no	—
38	C₂×Dic₉₄	—	2	2	no	—
39	C₂×Dic₉₄	—	4	2	no	—
40	M₁₆×C₄₇	—	4	4	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	4	2	no	—
44	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
45	C₄²×C₄₇	C₂×D₄	1	2	yes	—
46	C₄²×C₄₇	#41	1	2	yes	—
47	C₄⋊Dic₄₇	—	2	2	no	—
48	Dic₄₇⋊C₄	—	4	2	no	—
49	C₄×Dic₄₇	—	4	1	no	—
50	C₄×Dic₄₇	—	4	1	no	—
51	C₂×C₄₇⋊C₈	—	2	4	no	—
52	C₂×C₄₇⋊C₈	—	4	1	no	—
53	C₂×C₄₇⋊C₈	—	4	2	no	—
54	C₂³×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	4	1	no	—
56	C₂×C₄×C₄₇	—	4	1	no	—
57	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#123	1	2	yes	—
60	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
62	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
63	C₂×Dic₄₇	—	2	2	no	—
64	C₂×Dic₄₇	—	4	1	no	—
65	C₂×Dic₄₇	—	4	1	no	—
66	C₂×Dic₄₇	—	4	2	no	—
67	C₂×Dic₄₇	—	4	1	no	—
68	Dic₉₄	—	2	4	no	—
69	Dic₉₄	—	4	2	no	—
70	C₈×C₄₇	—	2	4	no	—
71	C₈×C₄₇	—	8	2	no	—
72	C₄₇⋊C₈	—	4	4	no	—
73	C₄₇⋊C₈	—	8	1	no	—
74	#10662	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	—	2	1	no	—
77	C₂²×C₄₇	#399	1	1	yes	characteristic
78	C₂²×C₄₇	#477	1	2	yes	—
79	C₄×C₄₇	—	2	2	no	—
80	C₄×C₄₇	—	2	2	no	—
81	C₄×C₄₇	#484	1	2	yes	—
82	C₄×C₄₇	#494	1	1	yes	characteristic
83	C₄×C₄₇	#496	1	1	yes	characteristic
84	Dic₄₇	—	4	2	no	—
85	Dic₄₇	—	8	1	no	—
86	#1985	—	47	1	no	—
87	#1849	D₄₇	1	1	yes	characteristic
88	#1571	—	47	2	no	—
89	#1487	—	47	2	no	—
90	#1275	—	47	1	no	—
91	C₂×C₄₇	—	2	2	no	—
92	C₂×C₄₇	#2091	1	1	yes	characteristic
93	C₂×C₄₇	#2095	1	1	yes	characteristic
94	C₂×C₄₇	#2098	1	1	yes	characteristic
95	C₂×C₄×Q₈	—	47	1	no	—
96	#431	—	47	2	no	—
97	C₂×#108	—	2	2	no	—
98	#423	C₂×D₄₇	1	2	yes	—
99	C₂×C₄⋊C₈	—	47	1	no	—
100	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
101	#336	—	94	2	no	—
102	#335	—	94	4	no	—
103	Q₈⋊C₈	—	94	4	no	—
104	C₄₇	#10662	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	47	1	no	—
106	C₂×M₁₆	—	2	2	no	—
107	C₂²×C₈	—	94	1	no	—
108	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
109	C₄×Q₈	—	94	2	no	—
110	C₄×Q₈	—	94	2	no	—
111	C₂×C₄⋊C₄	—	47	1	no	—
112	C₂×C₄⋊C₄	—	94	1	no	—
113	C₂×C₄²	—	94	1	no	—
114	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
115	#108	—	4	4	no	—
116	C₄⋊C₈	—	2	2	no	—
117	C₄⋊C₈	—	2	2	no	—
118	C₄⋊C₈	—	94	2	no	—
119	C₄⋊C₈	—	94	2	no	—
120	C₂²⋊C₈	—	94	2	no	—
121	#97	—	2	2	no	—
122	C₄×C₈	—	47	4	no	—
123	C₂×Q₈	—	94	2	no	—
124	C₂×Q₈	—	188	2	no	—
125	C₂²×C₄	—	94	1	no	—
126	C₂²×C₄	—	94	1	no	—
127	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
128	C₂²×C₄	#10625354	1	1	yes	characteristic
129	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
130	M₁₆	—	4	4	no	—
131	C₂×C₈	—	2	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	94	4	no	—
135	C₂×C₈	—	188	1	no	—
136	C₂×C₈	—	188	2	no	—
137	C₂×C₈	#10625337	1	4	yes	—
138	C₄⋊C₄	—	94	2	no	—
139	C₄⋊C₄	—	188	2	no	—
140	C₄²	—	188	1	no	—
141	C₄²	—	188	1	no	—
142	C₄²	#10625350	1	2	yes	—
143	C₄²	D₄×D₄₇	1	2	yes	—
144	C₂³	#13094000	1	1	yes	characteristic
145	Q₈	—	94	4	no	—
146	Q₈	—	188	2	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	—	94	2	no	—
150	C₂×C₄	—	188	2	no	—
151	C₂×C₄	—	188	1	no	—
152	C₂×C₄	—	188	1	no	—
153	C₂×C₄	—	188	1	no	—
154	C₂×C₄	#13094002	1	2	yes	—
155	C₂×C₄	#13094007	1	1	yes	characteristic
156	C₂×C₄	#13094009	1	2	yes	—
157	C₂×C₄	#13094020	1	1	yes	characteristic
158	C₂×C₄	C₂×#10625337	1	2	yes	—
159	C₂×C₄	#13094050	1	2	yes	—
160	C₈	—	2	4	no	—
161	C₈	—	8	2	no	—
162	C₈	—	376	1	no	—
163	C₈	—	188	4	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	#15689058	1	1	yes	characteristic
167	C₂²	#15689415	1	2	yes	—
168	C₄	—	2	2	no	—
169	C₄	—	2	2	no	—
170	C₄	—	376	1	no	—
171	C₄	—	188	2	no	—
172	C₄	#15689458	1	2	yes	—
173	C₄	#15689502	1	1	yes	characteristic
174	C₄	#15689510	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	#20244903	1	1	yes	characteristic
177	C₂	#20244927	1	1	yes	characteristic
178	C₂	#20244940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244328

C₂

yes

maximal, characteristic

#20242164

C₂

yes

maximal

#20241816

C₂

yes

maximal

#20241219

C₂

yes

maximal, characteristic

C₄₇×#1849

C₂

yes

maximal, characteristic

C₂×C₄×Dic₉₄

C₂²

yes

characteristic

#15689176

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

—

—

C₂×#108×C₄₇

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

Q₈

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₄×Dic₉₄

—

C₄×Dic₉₄

—

C₂×C₄⋊Dic₄₇

Q₈

yes

characteristic

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#108×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10662

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#399

yes

characteristic

C₂²×C₄₇

#477

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#484

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#1985

—

#1849

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

—

#431

—

C₂×#108

—

#423

C₂×D₄₇

yes

—

C₂×C₄⋊C₈

—

100

C₂×#97

C₂×D₄₇

yes

characteristic

101

#336

—

102

#335

—

103

Q₈⋊C₈

—

104

C₄₇

#10662

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂²×C₈

—

108

C₂²×C₈

C₂²⋊D₄₇

yes

—

109

C₄×Q₈

—

110

C₄×Q₈

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂²×D₄₇

yes

characteristic

115

#108

—

116

C₄⋊C₈

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₂²⋊C₈

—

121

#97

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂×Q₈

—

188

—

125

C₂²×C₄

—

126

C₂²×C₄

—

127

C₂²×C₄

Q₈×D₄₇

yes

characteristic

128

C₂²×C₄

#10625354

yes

characteristic

129

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

130

M₁₆

—

131

C₂×C₈

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

188

—

136

C₂×C₈

—

188

—

137

C₂×C₈

#10625337

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

188

—

140

C₄²

—

188

—

141

C₄²

—

188

—

142

C₄²

#10625350

yes

—

143

C₄²

D₄×D₄₇

yes

—

144

C₂³

#13094000

yes

characteristic

145

Q₈

—

146

Q₈

—

188

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

188

—

151

C₂×C₄

—

188

—

152

C₂×C₄

—

188

—

153

C₂×C₄

—

188

—

154

C₂×C₄

#13094002

yes

—

155

C₂×C₄

#13094007

yes

characteristic

156

C₂×C₄

#13094009

yes

—

157

C₂×C₄

#13094020

yes

characteristic

158

C₂×C₄

C₂×#10625337

yes

—

159

C₂×C₄

#13094050

yes

—

160

C₈

—

161

C₈

—

162

C₈

—

376

—

163

C₈

—

188

—

164

C₂²

—

165

C₂²

—

166

C₂²

#15689058

yes

characteristic

167

C₂²

#15689415

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

376

—

171

C₄

—

188

—

172

C₄

#15689458

yes

—

173

C₄

#15689502

yes

characteristic

174

C₄

#15689510

yes

characteristic

175

C₂

—

176

C₂

#20244903

yes

characteristic

177

C₂

#20244927

yes

characteristic

178

C₂

#20244940

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1849	C₂
socle	C₂²×C₄₇	#399
prime radical	C₂³×C₄₇	C₂²⋊Q₈
hypercenter	#1849	D₄₇
hypocenter	C₄₇	#10662
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

subgroup

G/H

center

C₂×C₄

#13094020

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1849

C₂

socle

C₂²×C₄₇

#399

prime radical

C₂³×C₄₇

C₂²⋊Q₈

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1849

N	H ≅ G/N	type
104. C₄₇	#10662	semidirect

H ≅ G/N

type

104. C₄₇

#10662

semidirect

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

subgroup	index
#20244328	2
#20242164	2
#20241816	2
#20241219	2
C₄₇×#1849	2
#10662	47

subgroup

index

C₄₇×#1849

Subgroups — #26121267

Conjugacy classes of subgroups

Complements

Maximal subgroups