Group #26105536 — subgroups

Subgroups — #26105536

Order 12032.

698 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20244921	C₂	1	3	yes	maximal, complemented
2	#20241209	C₂	1	3	yes	maximal, complemented
3	C₄₇×#2679	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093955	—	2	3	no	—
5	C₂×#13093945	—	2	3	no	—
6	#15688993	C₂²	1	1	yes	characteristic, complemented
7	#15688878	C₂²	1	3	yes	complemented
8	C₂×#120×C₄₇	—	2	3	no	—
9	C₄₇×#394	—	2	3	no	—
10	C₄₇×#390	C₂²	1	3	yes	—
11	C₂²×C₄₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₄₇	—	2	3	no	—
13	#13093955	—	4	6	no	—
14	#13093945	—	4	6	no	—
15	C₂⁵×C₄₇	—	2	1	no	—
16	C₂²×D₄×C₄₇	—	2	3	no	—
17	C₂²×D₄×C₄₇	D₄	1	3	yes	—
18	#120×C₄₇	—	4	6	no	—
19	#120×C₄₇	—	4	6	no	—
20	C₂×C₄⋊C₄×C₄₇	D₄	1	3	yes	—
21	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
22	C₂×C₂²⋊C₄×C₄₇	—	2	6	no	—
23	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	#95×C₄₇	—	2	1	no	—
25	C₂⁴×C₄₇	—	2	3	no	—
26	C₂⁴×C₄₇	—	4	6	no	—
27	C₂⁴×C₄₇	—	8	3	no	—
28	C₂⁴×C₄₇	—	8	1	no	—
29	C₂×D₄×C₄₇	—	2	3	no	—
30	C₂×D₄×C₄₇	—	4	3	no	—
31	C₂×D₄×C₄₇	—	4	3	no	—
32	C₂×D₄×C₄₇	—	4	12	no	—
33	C₂×D₄×C₄₇	—	8	3	no	—
34	C₂×D₄×C₄₇	SD₁₆	1	6	yes	—
35	C₂²×C₄×C₄₇	—	2	3	no	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
37	C₂²×Dic₄₇	—	4	1	no	—
38	C₄⋊C₄×C₄₇	—	4	3	no	—
39	C₂²⋊C₄×C₄₇	—	4	6	no	—
40	C₂²⋊C₄×C₄₇	—	4	6	no	—
41	C₂²⋊C₄×C₄₇	—	4	6	no	—
42	C₄²×C₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	3	no	—
44	C₄⋊Dic₄₇	—	4	3	no	—
45	C₂×C₄₇⋊C₈	—	2	6	no	—
46	C₂×C₄₇⋊C₈	—	4	3	no	—
47	C₂×C₄₇⋊C₈	—	4	3	no	—
48	C₂³×C₄₇	—	2	6	no	—
49	C₂³×C₄₇	—	4	6	no	—
50	C₂³×C₄₇	—	4	3	no	—
51	C₂³×C₄₇	—	4	6	no	—
52	C₂³×C₄₇	—	4	3	no	—
53	C₂³×C₄₇	—	8	12	no	—
54	C₂³×C₄₇	—	8	12	no	—
55	C₂³×C₄₇	—	8	4	no	—
56	C₂³×C₄₇	#120	1	1	yes	characteristic
57	D₄×C₄₇	—	4	6	no	—
58	D₄×C₄₇	—	8	3	no	—
59	D₄×C₄₇	—	8	12	no	—
60	C₂×C₄×C₄₇	—	2	3	no	—
61	C₂×C₄×C₄₇	—	2	3	no	—
62	C₂×C₄×C₄₇	—	4	6	no	—
63	C₂×C₄×C₄₇	—	4	3	no	—
64	C₂×C₄×C₄₇	C₂×SD₁₆	1	3	yes	—
65	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	3	yes	—
66	C₂×Dic₄₇	—	8	3	no	—
67	C₄₇⋊C₈	—	4	6	no	—
68	#7822	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	—	4	6	no	—
70	C₂²×C₄₇	—	4	12	no	—
71	C₂²×C₄₇	—	8	16	no	—
72	C₂²×C₄₇	—	8	6	no	—
73	C₂²×C₄₇	—	8	3	no	—
74	C₂²×C₄₇	—	8	3	no	—
75	C₂²×C₄₇	#450	1	3	yes	—
76	C₂²×C₄₇	#453	1	3	yes	—
77	C₂²×C₄₇	#458	1	1	yes	characteristic
78	C₄×C₄₇	—	2	3	no	—
79	C₄×C₄₇	—	2	3	no	—
80	C₄×C₄₇	—	8	3	no	—
81	Dic₄₇	—	16	1	no	—
82	#2679	D₄₇	1	1	yes	characteristic
83	#2094	—	47	3	no	—
84	#1270	—	47	3	no	—
85	C₂×C₄₇	—	8	6	no	—
86	C₂×C₄₇	#1613	1	3	yes	—
87	C₂×C₄₇	#1629	1	3	yes	—
88	C₂×C₄₇	#1655	1	1	yes	characteristic
89	C₂×#120	—	2	3	no	—
90	C₂×#106	—	94	3	no	—
91	C₂×#102	—	94	3	no	—
92	#394	—	2	3	no	—
93	#390	C₂×D₄₇	1	3	yes	—
94	#384	—	47	1	no	—
95	#336	—	47	3	no	—
96	C₄₇	#7822	1	1	yes	characteristic, complemented, Sylow
97	C₂⁵	—	2	1	no	—
98	C₂²×D₄	—	2	3	no	—
99	C₂²×D₄	C₂²⋊D₄₇	1	3	yes	—
100	C₂²×C₈	—	94	3	no	—
101	#120	—	4	6	no	—
102	#120	—	4	6	no	—
103	C₂×C₄⋊C₄	—	94	3	no	—
104	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	3	yes	—
105	C₂×C₂²⋊C₄	—	2	3	no	—
106	C₂×C₂²⋊C₄	—	2	6	no	—
107	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
108	#106	—	188	6	no	—
109	#102	—	188	6	no	—
110	#95	—	2	1	no	—
111	C₂⁴	—	2	3	no	—
112	C₂⁴	—	4	6	no	—
113	C₂⁴	—	8	3	no	—
114	C₂⁴	—	8	1	no	—
115	C₂×D₄	—	2	3	no	—
116	C₂×D₄	—	4	3	no	—
117	C₂×D₄	—	4	3	no	—
118	C₂×D₄	—	4	12	no	—
119	C₂×D₄	—	8	3	no	—
120	C₂×D₄	#10625335	1	6	yes	—
121	C₂²×C₄	—	2	3	no	—
122	C₂²×C₄	—	188	1	no	—
123	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
124	C₂×C₈	—	94	6	no	—
125	C₂×C₈	—	188	3	no	—
126	C₂×C₈	—	188	3	no	—
127	C₄⋊C₄	—	4	3	no	—
128	C₄⋊C₄	—	188	3	no	—
129	C₄⋊C₄	—	188	3	no	—
130	C₂²⋊C₄	—	4	6	no	—
131	C₂²⋊C₄	—	4	6	no	—
132	C₂²⋊C₄	—	4	6	no	—
133	C₄²	—	4	1	no	—
134	C₂³	—	2	6	no	—
135	C₂³	—	4	6	no	—
136	C₂³	—	4	3	no	—
137	C₂³	—	4	6	no	—
138	C₂³	—	4	3	no	—
139	C₂³	—	8	12	no	—
140	C₂³	—	8	12	no	—
141	C₂³	—	8	4	no	—
142	C₂³	#13093993	1	1	yes	characteristic
143	D₄	—	4	6	no	—
144	D₄	—	8	3	no	—
145	D₄	—	8	12	no	—
146	C₂×C₄	—	2	3	no	—
147	C₂×C₄	—	2	3	no	—
148	C₂×C₄	—	4	6	no	—
149	C₂×C₄	—	4	3	no	—
150	C₂×C₄	—	376	3	no	—
151	C₂×C₄	C₂×#10625335	1	3	yes	—
152	C₂×C₄	#13094050	1	3	yes	—
153	C₈	—	188	6	no	—
154	C₂²	—	4	12	no	—
155	C₂²	—	4	6	no	—
156	C₂²	—	8	16	no	—
157	C₂²	—	8	3	no	—
158	C₂²	—	8	6	no	—
159	C₂²	—	8	3	no	—
160	C₂²	#15689274	1	3	yes	—
161	C₂²	#15689291	1	3	yes	—
162	C₂²	#15689313	1	1	yes	characteristic
163	C₄	—	2	3	no	—
164	C₄	—	2	3	no	—
165	C₄	—	8	3	no	—
166	C₄	—	752	1	no	—
167	C₂	—	8	6	no	—
168	C₂	#20242408	1	3	yes	—
169	C₂	#20242504	1	3	yes	—
170	C₂	#20242677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244921

C₂

yes

maximal, complemented

#20241209

C₂

yes

maximal, complemented

C₄₇×#2679

C₂

yes

maximal, characteristic

C₂×#13093955

—

C₂×#13093945

—

#15688993

C₂²

yes

characteristic, complemented

#15688878

C₂²

yes

complemented

C₂×#120×C₄₇

—

C₄₇×#394

—

C₄₇×#390

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊Dic₄₇

—

#13093955

—

#13093945

—

C₂⁵×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

D₄

yes

—

#120×C₄₇

—

#120×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#95×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

SD₁₆

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#120

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

#7822

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#450

yes

—

C₂²×C₄₇

#453

yes

—

C₂²×C₄₇

#458

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2679

D₄₇

yes

characteristic

#2094

—

#1270

—

C₂×C₄₇

—

C₂×C₄₇

#1613

yes

—

C₂×C₄₇

#1629

yes

—

C₂×C₄₇

yes

—

—

—

—

C₂×D₄₇

yes

—

#384

—

#336

—

C₄₇

#7822

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×D₄

—

C₂²×D₄

C₂²⋊D₄₇

yes

—

100

C₂²×C₈

—

101

#120

—

102

#120

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

105

C₂×C₂²⋊C₄

—

106

C₂×C₂²⋊C₄

—

107

C₂×C₄²

C₂²×D₄₇

yes

characteristic

108

#106

—

188

—

109

#102

—

188

—

110

#95

—

111

C₂⁴

—

112

C₂⁴

—

113

C₂⁴

—

114

C₂⁴

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂×D₄

#10625335

yes

—

121

C₂²×C₄

—

122

C₂²×C₄

—

188

—

123

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

124

C₂×C₈

—

125

C₂×C₈

—

188

—

126

C₂×C₈

—

188

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

188

—

129

C₄⋊C₄

—

188

—

130

C₂²⋊C₄

—

131

C₂²⋊C₄

—

132

C₂²⋊C₄

—

133

C₄²

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

—

141

C₂³

—

142

C₂³

#13093993

yes

characteristic

143

D₄

—

144

D₄

—

145

D₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

376

—

151

C₂×C₄

C₂×#10625335

yes

—

152

C₂×C₄

#13094050

yes

—

153

C₈

—

188

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

#15689274

yes

—

161

C₂²

#15689291

yes

—

162

C₂²

#15689313

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

752

—

167

C₂

—

168

C₂

#20242408

yes

—

169

C₂

#20242504

yes

—

170

C₂

#20242677

yes

characteristic

	subgroup	G/H
center	C₂³	#13093993
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting, prime radical	C₄₇×#2679	C₂
socle	C₂³×C₄₇	#120
hypercenter	#2679	D₄₇
hypocenter	C₄₇	#7822
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2679

subgroup

G/H

center

C₂³

#13093993

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting, prime radical

C₄₇×#2679

C₂

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂³×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2679

N	H ≅ G/N	type
1. #20244921	C₂	semidirect
2. #20241209	C₂	semidirect
6. #15688993	C₂²	semidirect
7. #15688878	C₂²	semidirect
96. C₄₇	#7822	semidirect

H ≅ G/N

type

1. #20244921

C₂

semidirect

2. #20241209

C₂

semidirect

6. #15688993

C₂²

semidirect

7. #15688878

C₂²

semidirect

96. C₄₇

#7822

semidirect

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

subgroup	index
#20244921	2
#20241209	2
C₄₇×#2679	2
#7822	47

subgroup

index

#20244921

#20241209

C₄₇×#2679

#7822

Subgroups — #26105536

Conjugacy classes of subgroups

Complements

Maximal subgroups