Group #26103555 — subgroups

Subgroups — #26103555

Order 12032.

456 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20243316	C₂	1	2	yes	maximal
2	C₄₇×#1780	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093943	—	2	2	no	—
4	C₂×#13093941	C₄	1	4	yes	—
5	#15689087	—	2	2	no	—
6	C₂²×#41×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₄₇	—	2	2	no	—
8	C₂×C₂²⋊C₈×C₄₇	C₄	1	2	yes	—
9	C₂×#10625329	—	2	4	no	—
10	#13093943	—	4	4	no	—
11	#13093941	—	4	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
13	C₂×#41×C₄₇	—	4	2	no	—
14	C₂×#41×C₄₇	—	4	4	no	—
15	C₂²×Q₈×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×D₄×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×D₄×C₄₇	Q₈	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	D₄	1	2	yes	—
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂²⋊C₈×C₄₇	—	4	4	no	—
22	C₂²⋊C₈×C₄₇	—	4	2	no	—
23	C₂⁴×C₄₇	C₄²	1	1	yes	characteristic
24	C₂⁴×C₄₇	C₄⋊C₄	1	2	yes	—
25	#41×C₄₇	—	4	16	no	—
26	C₂×Q₈×C₄₇	—	2	4	no	—
27	C₂×Q₈×C₄₇	—	4	1	no	—
28	C₂×D₄×C₄₇	—	2	4	no	—
29	C₂×D₄×C₄₇	—	4	2	no	—
30	C₂×D₄×C₄₇	—	4	2	no	—
31	C₂×D₄×C₄₇	—	4	1	no	—
32	C₂×D₄×C₄₇	—	4	2	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	—	2	4	no	—
36	C₂²×C₄×C₄₇	—	4	2	no	—
37	C₂²×C₄×C₄₇	—	4	2	no	—
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	4	4	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	#10625329	—	4	4	no	—
44	#10625329	—	4	4	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂³×C₄₇	—	2	1	no	—
47	C₂³×C₄₇	—	2	2	no	—
48	C₂³×C₄₇	—	2	2	no	—
49	C₂³×C₄₇	—	2	2	no	—
50	C₂³×C₄₇	—	2	2	no	—
51	C₂³×C₄₇	—	4	4	no	—
52	C₂³×C₄₇	—	4	2	no	—
53	C₂³×C₄₇	#95	1	1	yes	characteristic
54	Q₈×C₄₇	—	4	4	no	—
55	D₄×C₄₇	—	4	8	no	—
56	D₄×C₄₇	—	4	4	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	2	1	no	—
61	C₂×C₄×C₄₇	—	2	2	no	—
62	C₂×C₄×C₄₇	—	4	8	no	—
63	C₂×C₄×C₄₇	—	4	4	no	—
64	C₂×C₄×C₄₇	#99	1	4	yes	—
65	C₂×C₄×C₄₇	#99	1	2	yes	—
66	C₈×C₄₇	—	8	2	no	—
67	C₄₇⋊C₈	—	8	4	no	—
68	#7306	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	4	4	no	—
71	C₂²×C₄₇	—	4	2	no	—
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	—	4	4	no	—
74	C₂²×C₄₇	—	4	2	no	—
75	C₂²×C₄₇	—	8	2	no	—
76	C₂²×C₄₇	—	8	2	no	—
77	C₂²×C₄₇	#342	1	2	yes	—
78	C₂²×C₄₇	#342	1	1	yes	characteristic
79	C₄×C₄₇	—	2	4	no	—
80	C₄×C₄₇	—	2	2	no	—
81	C₄×C₄₇	—	4	1	no	—
82	#1785	—	47	2	no	—
83	#1780	D₄₇	1	1	yes	characteristic
84	C₂×C₄₇	—	2	2	no	—
85	C₂×C₄₇	—	8	2	no	—
86	C₂×C₄₇	—	8	1	no	—
87	C₂×C₄₇	#1304	1	1	yes	characteristic
88	C₂×C₄₇	#1392	1	2	yes	—
89	C₂²×#41	C₂×D₄₇	1	1	yes	characteristic
90	C₂×#101	—	94	2	no	—
91	C₂×#100	—	47	4	no	—
92	#407	—	94	2	no	—
93	C₂×C₂²⋊C₈	—	2	2	no	—
94	C₂×C₂²⋊C₈	Dic₄₇	1	2	yes	—
95	C₄₇	#7306	1	1	yes	characteristic, complemented, Sylow
96	C₂×#41	—	4	2	no	—
97	C₂×#41	—	4	4	no	—
98	C₂²×Q₈	D₁₈₈	1	1	yes	characteristic
99	C₂²×D₄	Dic₉₄	1	1	yes	characteristic
100	C₂²×D₄	C₄×D₄₇	1	2	yes	—
101	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
102	C₂³×C₄	C₂²⋊D₄₇	1	2	yes	—
103	C₂×M₁₆	—	94	4	no	—
104	C₂²×C₈	—	2	2	no	—
105	#101	—	188	4	no	—
106	#100	—	188	4	no	—
107	C₂²⋊C₈	—	4	4	no	—
108	C₂²⋊C₈	—	4	2	no	—
109	C₂²⋊C₈	—	188	4	no	—
110	C₂⁴	C₄×Dic₄₇	1	1	yes	characteristic
111	C₂⁴	Dic₄₇⋊C₄	1	2	yes	—
112	#41	—	4	16	no	—
113	C₂×Q₈	—	2	4	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×D₄	—	2	4	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	1	no	—
118	C₂×D₄	—	4	2	no	—
119	C₂×D₄	—	4	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	2	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
126	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
127	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
128	M₁₆	—	188	4	no	—
129	M₁₆	—	188	4	no	—
130	C₂×C₈	—	4	4	no	—
131	C₂×C₈	—	4	2	no	—
132	C₂×C₈	—	188	4	no	—
133	C₂³	—	2	2	no	—
134	C₂³	—	2	1	no	—
135	C₂³	—	2	2	no	—
136	C₂³	—	2	2	no	—
137	C₂³	—	2	2	no	—
138	C₂³	—	4	4	no	—
139	C₂³	—	4	2	no	—
140	C₂³	C₄₇⋊#95	1	1	yes	characteristic
141	Q₈	—	4	4	no	—
142	D₄	—	4	8	no	—
143	D₄	—	4	4	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	2	2	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	2	1	no	—
149	C₂×C₄	—	4	8	no	—
150	C₂×C₄	—	4	4	no	—
151	C₂×C₄	#13093938	1	4	yes	—
152	C₂×C₄	#13093939	1	2	yes	—
153	C₈	—	8	2	no	—
154	C₈	—	376	4	no	—
155	C₂²	—	2	2	no	—
156	C₂²	—	4	4	no	—
157	C₂²	—	4	4	no	—
158	C₂²	—	4	2	no	—
159	C₂²	—	4	2	no	—
160	C₂²	—	4	1	no	—
161	C₂²	—	8	2	no	—
162	C₂²	—	8	2	no	—
163	C₂²	#15688889	1	2	yes	—
164	C₂²	#15688890	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	2	4	no	—
167	C₄	—	4	1	no	—
168	C₂	—	2	2	no	—
169	C₂	—	8	2	no	—
170	C₂	—	8	1	no	—
171	C₂	C₄₇⋊#1304	1	1	yes	characteristic
172	C₂	#20241511	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243316

C₂

yes

maximal

C₄₇×#1780

C₂

yes

maximal, characteristic

C₂×#13093943

—

C₂×#13093941

C₄

yes

—

#15689087

—

C₂²×#41×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₄₇

—

C₂×C₂²⋊C₈×C₄₇

C₄

yes

—

C₂×#10625329

—

#13093943

—

#13093941

—

C₂²⋊(C₄₇⋊C₈)

—

C₂×#41×C₄₇

—

C₂×#41×C₄₇

—

C₂²×Q₈×C₄₇

D₄

yes

characteristic

C₂²×D₄×C₄₇

C₂×C₄

yes

—

C₂²×D₄×C₄₇

Q₈

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₄²

yes

characteristic

C₂⁴×C₄₇

C₄⋊C₄

yes

—

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×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×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#95

yes

characteristic

Q₈×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

#99

yes

—

C₂×C₄×C₄₇

#99

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7306

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#342

yes

—

C₂²×C₄₇

#342

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1785

—

#1780

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1304

yes

characteristic

C₂×C₄₇

#1392

yes

—

C₂²×#41

C₂×D₄₇

yes

—

—

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

Dic₄₇

yes

—

C₄₇

#7306

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₂²×Q₈

D₁₈₈

yes

characteristic

C₂²×D₄

Dic₉₄

yes

characteristic

100

C₂²×D₄

C₄×D₄₇

yes

—

101

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

102

C₂³×C₄

C₂²⋊D₄₇

yes

—

103

C₂×M₁₆

—

104

C₂²×C₈

—

105

#101

—

188

—

106

#100

—

188

—

107

C₂²⋊C₈

—

108

C₂²⋊C₈

—

109

C₂²⋊C₈

—

188

—

110

C₂⁴

C₄×Dic₄₇

yes

characteristic

111

C₂⁴

Dic₄₇⋊C₄

yes

—

112

#41

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

126

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

127

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

128

M₁₆

—

188

—

129

M₁₆

—

188

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

—

188

—

133

C₂³

—

134

C₂³

—

135

C₂³

—

136

C₂³

—

137

C₂³

—

138

C₂³

—

139

C₂³

—

140

C₂³

C₄₇⋊#95

yes

characteristic

141

Q₈

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#13093938

yes

—

152

C₂×C₄

#13093939

yes

—

153

C₈

—

154

C₈

—

376

—

155

C₂²

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

—

163

C₂²

#15688889

yes

—

164

C₂²

#15688890

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₂

—

169

C₂

—

170

C₂

—

171

C₂

C₄₇⋊#1304

yes

characteristic

172

C₂

#20241511

yes

—

	subgroup	G/H
center	C₂²	#15688890
commutator	C₂⁴×C₄₇	C₄²
Frattini	C₂²×#41	C₂×D₄₇
Fitting	C₄₇×#1780	C₂
socle	C₂²×C₄₇	#342
prime radical, cosocle	C₂²×#41×C₄₇	C₂²
hypercenter	#1780	D₄₇
hypocenter	C₄₇	#7306
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #1780

subgroup

G/H

center

C₂²

#15688890

commutator

C₂⁴×C₄₇

C₄²

Frattini

C₂²×#41

C₂×D₄₇

Fitting

C₄₇×#1780

C₂

socle

C₂²×C₄₇

#342

prime radical, cosocle

C₂²×#41×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ #1780

N	H ≅ G/N	type
95. C₄₇	#7306	semidirect

H ≅ G/N

type

95. C₄₇

#7306

semidirect

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

subgroup	index
#20243316	2
C₄₇×#1780	2
#7306	47

subgroup

index

#20243316

C₄₇×#1780

#7306

Subgroups — #26103555

Conjugacy classes of subgroups

Complements

Maximal subgroups