Group #26121089 — subgroups

Subgroups — #26121089

Order 12032.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20244208	C₂	1	1	yes	maximal, characteristic
2	#20242061	C₂	1	2	yes	maximal
3	#20241248	C₂	1	2	yes	maximal
4	#20241233	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1838	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	1	yes	characteristic
7	#15689176	C₂²	1	2	yes	—
8	C₂×#13093949	—	2	2	no	—
9	#15688989	C₂²	1	1	yes	characteristic
10	#15688870	—	2	4	no	—
11	C₄₇×#423	C₂²	1	2	yes	—
12	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#341	—	2	2	no	—
14	C₂²×Dic₉₄	—	2	1	no	—
15	#13094024	—	2	2	no	—
16	#13094024	—	2	1	no	—
17	#13094024	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×C₄⋊Dic₄₇	—	2	1	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	#13093949	—	4	4	no	—
22	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄×C₄₇⋊C₈	—	2	2	no	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	#97×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	#41	1	2	yes	—
33	C₂²×Dic₄₇	—	4	1	no	—
34	C₂²×Dic₄₇	—	4	1	no	—
35	C₂×Dic₉₄	—	2	2	no	—
36	C₂×Dic₉₄	—	4	2	no	—
37	C₂×Dic₉₄	—	4	2	no	—
38	C₂×Dic₉₄	—	4	1	no	—
39	M₁₆×C₄₇	—	4	2	no	—
40	M₁₆×C₄₇	—	4	2	no	—
41	C₂×C₈×C₄₇	—	2	2	no	—
42	C₂×C₈×C₄₇	—	2	2	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₇	#41	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	2	yes	—
48	C₄⋊Dic₄₇	—	2	4	no	—
49	C₄⋊Dic₄₇	—	4	2	no	—
50	C₄⋊Dic₄₇	—	4	2	no	—
51	C₂×C₄₇⋊C₈	—	4	2	no	—
52	C₂³×C₄₇	#124	1	1	yes	characteristic
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	#121	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#123	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#123	1	2	yes	—
57	C₂×C₄×C₄₇	#126	1	2	yes	—
58	C₂×C₄×C₄₇	#135	1	2	yes	—
59	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	—	8	1	no	—
62	C₂×Dic₄₇	—	8	1	no	—
63	C₂×Dic₄₇	—	8	2	no	—
64	C₂×Dic₄₇	—	8	1	no	—
65	Dic₉₄	—	4	4	no	—
66	Dic₉₄	—	8	2	no	—
67	C₈×C₄₇	—	4	2	no	—
68	C₈×C₄₇	—	8	2	no	—
69	C₄₇⋊C₈	—	8	2	no	—
70	#10635	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	2	2	no	—
72	C₂²×C₄₇	#397	1	1	yes	characteristic
73	C₂²×C₄₇	#488	1	1	yes	characteristic
74	C₂²×C₄₇	#490	1	1	yes	characteristic
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	#463	1	2	yes	—
78	C₄×C₄₇	#482	1	2	yes	—
79	Dic₄₇	—	8	2	no	—
80	Dic₄₇	—	16	1	no	—
81	#1960	—	47	1	no	—
82	#1838	D₄₇	1	1	yes	characteristic
83	#1546	—	47	2	no	—
84	#1286	—	47	2	no	—
85	#1281	—	47	1	no	—
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	#2073	1	1	yes	characteristic
89	C₂×C₄₇	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	—	47	1	no	—
91	#431	—	47	2	no	—
92	#423	C₂×D₄₇	1	2	yes	—
93	C₂×#103	—	94	2	no	—
94	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
95	#382	—	47	1	no	—
96	#341	—	2	2	no	—
97	#332	—	94	4	no	—
98	C₄₇	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	94	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
102	C₄⋊Q₈	—	47	2	no	—
103	C₄⋊Q₈	—	94	2	no	—
104	C₄⋊Q₈	—	94	1	no	—
105	C₂×C₄⋊C₄	—	94	2	no	—
106	C₂×C₄⋊C₄	—	94	1	no	—
107	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	94	2	no	—
111	#103	—	188	4	no	—
112	C₂²⋊C₈	—	94	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	94	2	no	—
115	#95	—	94	2	no	—
116	C₂×Q₈	—	94	2	no	—
117	C₂×Q₈	—	188	2	no	—
118	C₂×Q₈	—	188	2	no	—
119	C₂×Q₈	—	188	1	no	—
120	C₂²×C₄	—	188	1	no	—
121	C₂²×C₄	—	188	1	no	—
122	C₂²×C₄	#10625354	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	188	2	no	—
131	C₄⋊C₄	—	94	4	no	—
132	C₄⋊C₄	—	188	2	no	—
133	C₄⋊C₄	—	188	2	no	—
134	C₄²	#10625350	1	2	yes	—
135	C₄²	D₄×D₄₇	1	1	yes	characteristic
136	C₄²	#10625352	1	1	yes	characteristic
137	C₂³	#13094013	1	1	yes	characteristic
138	Q₈	—	376	2	no	—
139	Q₈	—	188	4	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	376	1	no	—
142	C₂×C₄	—	376	1	no	—
143	C₂×C₄	—	376	2	no	—
144	C₂×C₄	—	376	1	no	—
145	C₂×C₄	—	188	2	no	—
146	C₂×C₄	#13093995	1	1	yes	characteristic
147	C₂×C₄	#13094007	1	1	yes	characteristic
148	C₂×C₄	#13094009	1	2	yes	—
149	C₂×C₄	#13094019	1	2	yes	—
150	C₂×C₄	#13094044	1	2	yes	—
151	C₂×C₄	#13094049	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	376	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#15689049	1	1	yes	characteristic
157	C₂²	#15689480	1	1	yes	characteristic
158	C₂²	#15689489	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	376	2	no	—
162	C₄	—	752	1	no	—
163	C₄	#15689335	1	2	yes	—
164	C₄	#15689447	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#20244802	1	1	yes	characteristic
168	C₂	#20244853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244208

C₂

yes

maximal, characteristic

#20242061

C₂

yes

maximal

#20241248

C₂

yes

maximal

#20241233

C₂

yes

maximal, characteristic

C₄₇×#1838

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

characteristic

#15689176

C₂²

yes

—

C₂×#13093949

—

#15688989

C₂²

yes

characteristic

#15688870

—

C₄₇×#423

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂²×Dic₉₄

—

#13094024

—

#13094024

—

#13094024

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

#13093949

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

#97×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#126

yes

—

C₂×C₄×C₄₇

#135

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

—

#10635

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#463

yes

—

C₄×C₄₇

#482

yes

—

Dic₄₇

—

Dic₄₇

—

#1960

—

#1838

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2073

yes

characteristic

C₂×C₄₇

#2082

yes

—

C₂×C₄⋊Q₈

—

#431

—

#423

C₂×D₄₇

yes

—

C₂×#103

—

C₂×#97

C₂×D₄₇

yes

—

—

—

C₄₇

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₇

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₄₇

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

188

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

188

—

118

C₂×Q₈

—

188

—

119

C₂×Q₈

—

188

—

120

C₂²×C₄

—

188

—

121

C₂²×C₄

—

188

—

122

C₂²×C₄

#10625354

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

188

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

188

—

133

C₄⋊C₄

—

188

—

134

C₄²

#10625350

yes

—

135

C₄²

D₄×D₄₇

yes

characteristic

136

C₄²

#10625352

yes

characteristic

137

C₂³

#13094013

yes

characteristic

138

Q₈

—

376

—

139

Q₈

—

188

—

140

C₂×C₄

—

141

C₂×C₄

—

376

—

142

C₂×C₄

—

376

—

143

C₂×C₄

—

376

—

144

C₂×C₄

—

376

—

145

C₂×C₄

—

188

—

146

C₂×C₄

#13093995

yes

characteristic

147

C₂×C₄

#13094007

yes

characteristic

148

C₂×C₄

#13094009

yes

—

149

C₂×C₄

#13094019

yes

—

150

C₂×C₄

#13094044

yes

—

151

C₂×C₄

#13094049

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

376

—

155

C₂²

—

156

C₂²

#15689049

yes

characteristic

157

C₂²

#15689480

yes

characteristic

158

C₂²

#15689489

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

376

—

162

C₄

—

752

—

163

C₄

#15689335

yes

—

164

C₄

#15689447

yes

—

165

C₂

—

166

C₂

—

167

C₂

#20244802

yes

characteristic

168

C₂

#20244853

yes

—

	subgroup	G/H
center	C₂²	#15689049
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1838	C₂
socle	C₂²×C₄₇	#397
prime radical	C₂³×C₄₇	#124
hypercenter	#1838	D₄₇
hypocenter	C₄₇	#10635
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#15689049

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1838

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
98. C₄₇	#10635	semidirect

H ≅ G/N

type

98. C₄₇

#10635

semidirect

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

subgroup	index
#20244208	2
#20242061	2
#20241248	2
#20241233	2
C₄₇×#1838	2
#10635	47

subgroup

index

C₄₇×#1838

Subgroups — #26121089

Conjugacy classes of subgroups

Complements

Maximal subgroups