Group #26103661 — subgroups

Subgroups — #26103661

Order 12032.

268 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20241334	C₂	1	2	yes	maximal, complemented
2	C₄₇×#1957	C₂	1	1	yes	maximal, characteristic
3	#15688849	—	2	2	no	—
4	C₂×#127×C₄₇	C₄	1	1	yes	characteristic
5	C₂×#102×C₄₇	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
7	C₄₇×#384	C₂²	1	1	yes	characteristic
8	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
9	C₂²×D₄×C₄₇	—	2	1	no	—
10	C₂²×D₄×C₄₇	—	4	1	no	—
11	C₂²×C₈×C₄₇	—	2	1	no	—
12	#127×C₄₇	—	2	1	no	—
13	#127×C₄₇	—	4	1	no	—
14	#127×C₄₇	C₈	1	2	yes	complemented
15	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
16	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	#102×C₄₇	—	4	4	no	—
19	C₄×C₈×C₄₇	—	4	1	no	—
20	C₂⁴×C₄₇	—	4	1	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	—	4	2	no	—
23	C₂×D₄×C₄₇	—	4	2	no	—
24	C₂×D₄×C₄₇	—	4	2	no	—
25	C₂×D₄×C₄₇	—	4	2	no	—
26	C₂×D₄×C₄₇	—	4	1	no	—
27	C₂×D₄×C₄₇	—	8	2	no	—
28	C₂×D₄×C₄₇	—	8	2	no	—
29	C₂²×C₄×C₄₇	—	2	1	no	—
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	—	4	1	no	—
34	C₂×C₈×C₄₇	—	4	1	no	—
35	C₄⋊C₄×C₄₇	—	4	2	no	—
36	C₄⋊C₄×C₄₇	—	8	2	no	—
37	C₄²×C₄₇	—	2	1	no	—
38	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
39	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	—	8	2	no	—
41	C₂³×C₄₇	—	4	2	no	—
42	C₂³×C₄₇	—	8	2	no	—
43	C₂³×C₄₇	—	8	2	no	—
44	C₂³×C₄₇	—	8	1	no	—
45	C₂³×C₄₇	—	8	1	no	—
46	C₂³×C₄₇	#99	1	1	yes	characteristic
47	D₄×C₄₇	—	4	4	no	—
48	D₄×C₄₇	—	8	2	no	—
49	D₄×C₄₇	—	8	4	no	—
50	D₄×C₄₇	—	8	4	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	1	no	—
54	C₂×C₄×C₄₇	—	2	1	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	1	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	8	2	no	—
59	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
61	C₈×C₄₇	—	4	2	no	—
62	C₄₇⋊C₈	—	16	2	no	—
63	#7354	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	8	4	no	—
67	C₂²×C₄₇	—	8	2	no	—
68	C₂²×C₄₇	—	16	2	no	—
69	C₂²×C₄₇	—	16	1	no	—
70	C₂²×C₄₇	—	16	1	no	—
71	C₂²×C₄₇	#323	1	1	yes	characteristic
72	C₂²×C₄₇	#353	1	1	yes	characteristic
73	C₂²×C₄₇	#355	1	1	yes	characteristic
74	C₄×C₄₇	—	2	2	no	—
75	C₄×C₄₇	—	2	2	no	—
76	C₄×C₄₇	—	2	1	no	—
77	C₄×C₄₇	—	2	1	no	—
78	C₄×C₄₇	—	8	2	no	—
79	#1957	D₄₇	1	1	yes	characteristic
80	#1325	—	47	2	no	—
81	C₂×C₄₇	—	2	1	no	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	—	16	2	no	—
84	C₂×C₄₇	#1318	1	1	yes	characteristic
85	C₂×C₄₇	#1408	1	1	yes	characteristic
86	C₂×C₄₇	#1410	1	1	yes	characteristic
87	C₂×#127	Dic₄₇	1	1	yes	characteristic
88	C₂×#102	—	2	2	no	—
89	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
90	#384	C₂×D₄₇	1	1	yes	characteristic
91	#324	—	94	2	no	—
92	C₄₇	#7354	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	2	1	no	—
94	C₂²×D₄	—	4	1	no	—
95	C₂²×C₈	—	2	1	no	—
96	#127	—	2	1	no	—
97	#127	—	4	1	no	—
98	#127	C₄₇⋊C₈	1	2	yes	complemented
99	C₂×C₄⋊C₄	—	4	1	no	—
100	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
101	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
102	#102	—	4	4	no	—
103	C₂²⋊C₈	—	188	2	no	—
104	C₄×C₈	—	4	1	no	—
105	C₂⁴	—	4	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	4	2	no	—
110	C₂×D₄	—	4	2	no	—
111	C₂×D₄	—	4	1	no	—
112	C₂×D₄	—	8	2	no	—
113	C₂×D₄	—	8	2	no	—
114	C₂²×C₄	—	2	1	no	—
115	C₂²×C₄	—	2	2	no	—
116	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	1	no	—
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	376	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	8	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
125	C₄²	#10625329	1	1	yes	characteristic
126	C₂³	—	4	2	no	—
127	C₂³	—	8	2	no	—
128	C₂³	—	8	2	no	—
129	C₂³	—	8	1	no	—
130	C₂³	—	8	1	no	—
131	C₂³	#13093939	1	1	yes	characteristic
132	D₄	—	4	4	no	—
133	D₄	—	8	2	no	—
134	D₄	—	8	4	no	—
135	D₄	—	8	4	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
145	C₂×C₄	#13093941	1	1	yes	characteristic
146	C₈	—	4	2	no	—
147	C₈	—	752	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	8	4	no	—
151	C₂²	—	8	2	no	—
152	C₂²	—	16	2	no	—
153	C₂²	—	16	1	no	—
154	C₂²	—	16	1	no	—
155	C₂²	#15688847	1	1	yes	characteristic
156	C₂²	#15688916	1	1	yes	characteristic
157	C₂²	#15688920	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	8	2	no	—
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	—	16	2	no	—
166	C₂	#20241320	1	1	yes	characteristic
167	C₂	#20241551	1	1	yes	characteristic
168	C₂	#20241555	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241334

C₂

yes

maximal, complemented

C₄₇×#1957

C₂

yes

maximal, characteristic

#15688849

—

C₂×#127×C₄₇

C₄

yes

characteristic

C₂×#102×C₄₇

—

C₂×C₄×C₈×C₄₇

C₄

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₂²×D₄×C₄₇

—

C₂²×D₄×C₄₇

—

C₂²×C₈×C₄₇

—

#127×C₄₇

—

#127×C₄₇

—

#127×C₄₇

C₈

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#102×C₄₇

—

C₄×C₈×C₄₇

—

C₂⁴×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

M₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#99

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₇

#100

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

#7354

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1957

D₄₇

yes

characteristic

#1325

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

Dic₄₇

yes

characteristic

C₂×#102

—

C₂×C₄×C₈

Dic₄₇

yes

characteristic

#384

C₂×D₄₇

yes

characteristic

#324

—

C₄₇

#7354

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂²×C₈

—

#127

—

#127

—

#127

C₄₇⋊C₈

yes

complemented

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

101

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

102

#102

—

103

C₂²⋊C₈

—

188

—

104

C₄×C₈

—

105

C₂⁴

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

376

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

125

C₄²

#10625329

yes

characteristic

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

#13093939

yes

characteristic

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

145

C₂×C₄

#13093941

yes

characteristic

146

C₈

—

147

C₈

—

752

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

#15688847

yes

characteristic

156

C₂²

#15688916

yes

characteristic

157

C₂²

#15688920

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

#20241320

yes

characteristic

167

C₂

#20241551

yes

characteristic

168

C₂

#20241555

yes

characteristic

	subgroup	G/H
center	C₂²	#15688916
commutator	C₄²×C₄₇	C₂×C₈
Frattini	#384	C₂×D₄₇
Fitting	C₄₇×#1957	C₂
socle	C₂²×C₄₇	#353
prime radical	C₂×#127×C₄₇	C₄
cosocle	C₄₇×#384	C₂²
hypercenter	#1957	D₄₇
hypocenter	C₄₇	#7354
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

subgroup

G/H

center

C₂²

#15688916

commutator

C₄²×C₄₇

C₂×C₈

Frattini

#384

C₂×D₄₇

Fitting

C₄₇×#1957

C₂

socle

C₂²×C₄₇

#353

prime radical

C₂×#127×C₄₇

C₄

cosocle

C₄₇×#384

C₂²

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1957

N	H ≅ G/N	type
1. #20241334	C₂	semidirect
14. #127×C₄₇	C₈	semidirect
92. C₄₇	#7354	semidirect
98. #127	C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. #20241334

C₂

semidirect

14. #127×C₄₇

C₈

semidirect

92. C₄₇

#7354

semidirect

98. #127

C₄₇⋊C₈

semidirect

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

subgroup	index
#20241334	2
C₄₇×#1957	2
#7354	47

subgroup

index

#20241334

C₄₇×#1957

#7354

Subgroups — #26103661

Conjugacy classes of subgroups

Complements

Maximal subgroups