Group #26103080 — subgroups

Subgroups — #26103080

Order 12032.

182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20244141	C₂	1	1	yes	maximal, characteristic
2	#20241468	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#1281	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093964	—	2	1	no	—
5	C₂×#13093960	C₄	1	1	yes	characteristic
6	#15689161	C₄	1	1	yes	characteristic
7	#15689149	—	2	1	no	—
8	#15688883	—	2	1	no	—
9	C₄₇×#431	—	2	1	no	—
10	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×#10625329	—	2	1	no	—
12	#13093983	—	2	1	no	—
13	#13093964	—	4	1	no	—
14	#13093964	—	4	1	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	#13093960	—	4	1	no	—
17	#13093958	—	2	1	no	—
18	#13093956	C₈	1	2	yes	complemented
19	Dic₄₇⋊C₈	—	4	1	no	—
20	Dic₄₇⋊C₈	—	4	1	no	—
21	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₄₇	—	4	1	no	—
25	C₂²⋊C₈×C₄₇	—	2	2	no	—
26	#97×C₄₇	—	4	1	no	—
27	C₄×C₈×C₄₇	—	4	1	no	—
28	C₂²×C₄×C₄₇	—	2	1	no	—
29	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	—	2	1	no	—
31	C₂×C₈×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₂×C₈×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₄₇	M₁₆	1	1	yes	characteristic
37	C₂×C₈×C₄₇	D₈	1	1	yes	characteristic
38	C₂×C₈×C₄₇	SD₁₆	1	1	yes	characteristic
39	C₄²×C₄₇	—	2	1	no	—
40	C₄²×C₄₇	SD₁₆	1	1	yes	characteristic
41	C₄²×C₄₇	Q₁₆	1	1	yes	characteristic
42	C₂²⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	2	2	no	—
44	C₄×Dic₄₇	—	4	1	no	—
45	#10625329	—	4	1	no	—
46	#10625329	—	4	1	no	—
47	C₂×C₄₇⋊C₈	—	4	1	no	—
48	C₄₇⋊C₁₆	—	4	2	no	—
49	C₂³×C₄₇	#104	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#101	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#102	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#104	1	1	yes	characteristic
59	C₂×Dic₄₇	—	4	2	no	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	—	2	1	no	—
63	C₈×C₄₇	—	2	1	no	—
64	C₈×C₄₇	—	8	2	no	—
65	C₄₇⋊C₈	—	8	1	no	—
66	#7125	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	4	1	no	—
68	C₂²×C₄₇	D₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄₇	#328	1	1	yes	characteristic
70	C₂²×C₄₇	#330	1	1	yes	characteristic
71	C₄×C₄₇	—	4	1	no	—
72	C₄×C₄₇	—	4	1	no	—
73	C₄×C₄₇	#324	1	1	yes	characteristic
74	C₄×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
75	C₄×C₄₇	#327	1	1	yes	characteristic
76	C₄×C₄₇	#332	1	1	yes	characteristic
77	Dic₄₇	—	8	2	no	—
78	#1946	—	47	1	no	—
79	#1375	—	47	1	no	—
80	#1281	D₄₇	1	1	yes	characteristic
81	C₂×C₄₇	—	2	1	no	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	#1271	1	1	yes	characteristic
85	C₂×M₃₂	—	94	1	no	—
86	#431	—	2	1	no	—
87	C₂×#108	—	47	1	no	—
88	#427	—	47	1	no	—
89	#424	—	94	1	no	—
90	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
91	#339	—	94	1	no	—
92	C₄₇	#7125	1	1	yes	characteristic, complemented, Sylow
93	C₂×M₁₆	—	94	1	no	—
94	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
95	C₂²×C₈	D₁₈₈	1	1	yes	characteristic
96	#117	—	94	1	no	—
97	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
98	M₃₂	—	188	1	no	—
99	M₃₂	—	188	1	no	—
100	C₂×C₁₆	—	94	2	no	—
101	#108	—	188	1	no	—
102	#107	—	94	1	no	—
103	#106	—	47	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	188	1	no	—
106	C₄⋊C₈	—	188	1	no	—
107	C₂²⋊C₈	—	2	2	no	—
108	#97	—	4	1	no	—
109	C₄×C₈	—	4	1	no	—
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
112	M₁₆	—	188	1	no	—
113	M₁₆	—	188	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	188	1	no	—
120	C₂×C₈	C₈×D₄₇	1	1	yes	characteristic
121	C₂×C₈	#10625324	1	1	yes	characteristic
122	C₂×C₈	#10625325	1	1	yes	characteristic
123	C₂×C₈	D₃₇₆	1	1	yes	characteristic
124	C₄⋊C₄	—	94	2	no	—
125	C₂²⋊C₄	—	188	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	188	1	no	—
128	C₄²	#10625335	1	1	yes	characteristic
129	C₄²	#10625337	1	1	yes	characteristic
130	C₁₆	—	188	2	no	—
131	C₂³	#13093952	1	1	yes	characteristic
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	8	1	no	—
135	C₂×C₄	—	188	2	no	—
136	C₂×C₄	(C₂×D₄₇)⋊C₈	1	1	yes	characteristic
137	C₂×C₄	#13093938	1	1	yes	characteristic
138	C₂×C₄	#13093942	1	1	yes	characteristic
139	C₂×C₄	#13093946	1	1	yes	characteristic
140	C₂×C₄	#13093947	1	1	yes	characteristic
141	C₂×C₄	#13093950	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	376	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#15688852	1	1	yes	characteristic
150	C₂²	#15688859	1	1	yes	characteristic
151	C₂²	#15688864	1	1	yes	characteristic
152	C₄	—	4	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	376	2	no	—
155	C₄	#15688848	1	1	yes	characteristic
156	C₄	#15688853	1	1	yes	characteristic
157	C₄	#15688858	1	1	yes	characteristic
158	C₄	#15688868	1	1	yes	characteristic
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#20241210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20244141

C₂

yes

maximal, characteristic

#20241468

C₂

yes

maximal, characteristic

C₄₇×#1281

C₂

yes

maximal, characteristic

C₂×#13093964

—

C₂×#13093960

C₄

yes

characteristic

#15689161

C₄

yes

characteristic

#15689149

—

#15688883

—

C₄₇×#431

—

C₂×#97×C₄₇

C₂²

yes

—

—

—

—

C₂×C₄₇⋊C₁₆

—

#13093960

—

#13093958

—

#13093956

C₈

yes

complemented

Dic₄₇⋊C₈

—

Dic₄₇⋊C₈

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₈

yes

characteristic

C₂×C₈×C₄₇

M₁₆

yes

characteristic

C₂×C₈×C₄₇

D₈

yes

characteristic

C₂×C₈×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

characteristic

C₄²×C₄₇

Q₁₆

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#104

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7125

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

D₄⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#324

yes

characteristic

C₄×C₄₇

Q₈⋊C₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1946

—

#1375

—

#1281

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1271

yes

characteristic

C₂×M₃₂

—

—

—

—

—

C₂×#97

C₂×D₄₇

yes

characteristic

#339

—

C₄₇

#7125

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

D₁₈₈

yes

characteristic

#117

—

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

M₃₂

—

188

—

M₃₂

—

188

—

100

C₂×C₁₆

—

101

#108

—

188

—

102

#107

—

103

#106

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

188

—

106

C₄⋊C₈

—

188

—

107

C₂²⋊C₈

—

108

#97

—

109

C₄×C₈

—

110

C₂²×C₄

—

111

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

112

M₁₆

—

188

—

113

M₁₆

—

188

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

188

—

120

C₂×C₈

C₈×D₄₇

yes

characteristic

121

C₂×C₈

#10625324

yes

characteristic

122

C₂×C₈

#10625325

yes

characteristic

123

C₂×C₈

D₃₇₆

yes

characteristic

124

C₄⋊C₄

—

125

C₂²⋊C₄

—

188

—

126

C₄²

—

127

C₄²

—

188

—

128

C₄²

#10625335

yes

characteristic

129

C₄²

#10625337

yes

characteristic

130

C₁₆

—

188

—

131

C₂³

#13093952

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

188

—

136

C₂×C₄

(C₂×D₄₇)⋊C₈

yes

characteristic

137

C₂×C₄

#13093938

yes

characteristic

138

C₂×C₄

#13093942

yes

characteristic

139

C₂×C₄

#13093946

yes

characteristic

140

C₂×C₄

#13093947

yes

characteristic

141

C₂×C₄

#13093950

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

376

—

148

C₂²

—

149

C₂²

#15688852

yes

characteristic

150

C₂²

#15688859

yes

characteristic

151

C₂²

#15688864

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

376

—

155

C₄

#15688848

yes

characteristic

156

C₄

#15688853

yes

characteristic

157

C₄

#15688858

yes

characteristic

158

C₄

#15688868

yes

characteristic

159

C₂

—

160

C₂

—

161

C₂

—

162

C₂

#20241210

yes

characteristic

	subgroup	G/H
center	C₄	#15688858
commutator	C₂×C₈×C₄₇	C₂×C₈
Frattini	C₂×#97	C₂×D₄₇
Fitting	C₄₇×#1281	C₂
socle	C₂×C₄₇	#1271
prime radical	C₂³×C₄₇	#104
cosocle	C₂×#97×C₄₇	C₂²
hypercenter	#1281	D₄₇
hypocenter	C₄₇	#7125
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1281

subgroup

G/H

center

C₄

#15688858

commutator

C₂×C₈×C₄₇

C₂×C₈

Frattini

C₂×#97

C₂×D₄₇

Fitting

C₄₇×#1281

C₂

C₂×C₄₇

C₂³×C₄₇

C₂×#97×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1281

N	H ≅ G/N	type
18. #13093956	C₈	semidirect
92. C₄₇	#7125	semidirect

H ≅ G/N

type

18. #13093956

C₈

semidirect

92. C₄₇

#7125

semidirect

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

subgroup	index
#20244141	2
#20241468	2
C₄₇×#1281	2
#7125	47

subgroup

index

#20244141

#20241468

C₄₇×#1281

#7125

Subgroups — #26103080

Conjugacy classes of subgroups

Complements

Maximal subgroups