Group #26116247 — subgroups

Subgroups — #26116247

Order 12032.

406 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 159 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	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20241843	C₂	1	4	yes	maximal
3	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#3160	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	—	2	2	no	—
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	Q₈⋊(C₄₇⋊C₈)	—	2	8	no	—
9	#15688844	—	2	2	no	—
10	C₄₇×#533	C₂²	1	2	yes	—
11	C₄₇×#500	C₄	1	2	yes	—
12	C₄₇×#499	—	2	2	no	—
13	C₄₇×#498	C₄	1	2	yes	—
14	C₄₇×#427	—	2	2	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	—	2	2	no	—
18	#13093960	—	4	4	no	—
19	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
22	C₂²×C₈×C₄₇	D₄	1	2	yes	—
23	C₄⋊Q₈×C₄₇	C₂×C₄	1	4	yes	—
24	#125×C₄₇	—	2	2	no	—
25	C₂²⋊Q₈×C₄₇	—	2	4	no	—
26	C₄×Q₈×C₄₇	—	2	4	no	—
27	#117×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	#107×C₄₇	—	2	4	no	—
30	#106×C₄₇	—	2	4	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	C₂×Q₈×C₄₇	—	2	4	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	#41	1	2	yes	—
37	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
40	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
41	C₄⋊C₄×C₄₇	—	2	4	no	—
42	C₄⋊C₄×C₄₇	—	2	4	no	—
43	C₄⋊C₄×C₄₇	—	4	4	no	—
44	C₂²⋊C₄×C₄₇	—	4	2	no	—
45	C₄²×C₄₇	—	4	2	no	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×D₄	1	2	yes	—
49	#10625329	—	4	4	no	—
50	C₂×C₄₇⋊C₈	—	2	4	no	—
51	C₂×C₄₇⋊C₈	—	4	4	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₂³×C₄₇	#124	1	1	yes	characteristic
54	Q₈×C₄₇	—	4	8	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	4	4	no	—
57	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
59	C₂×C₄×C₄₇	#121	1	2	yes	—
60	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
63	C₈×C₄₇	#103	1	4	yes	—
64	C₈×C₄₇	#103	1	4	yes	—
65	C₄₇⋊C₈	—	4	4	no	—
66	C₄₇⋊C₈	—	8	2	no	—
67	#9861	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	2	no	—
69	C₂²×C₄₇	#390	1	1	yes	characteristic
70	C₂²×C₄₇	#487	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	8	4	no	—
73	C₄×C₄₇	C₂×#103	1	2	yes	—
74	C₄×C₄₇	#444	1	4	yes	—
75	C₄×C₄₇	#492	1	1	yes	characteristic
76	C₄×C₄₇	#494	1	1	yes	characteristic
77	#3160	D₄₇	1	1	yes	characteristic
78	#1848	—	47	1	no	—
79	#1493	—	47	4	no	—
80	C₂×#322	—	47	1	no	—
81	C₂×C₄₇	—	2	2	no	—
82	C₂×C₄₇	#1956	1	1	yes	characteristic
83	C₂×C₄₇	#1975	1	2	yes	—
84	#533	C₂×D₄₇	1	2	yes	—
85	#500	Dic₄₇	1	2	yes	—
86	#499	—	2	2	no	—
87	#498	Dic₄₇	1	2	yes	—
88	C₂×#108	—	94	2	no	—
89	#427	—	2	2	no	—
90	#423	—	47	2	no	—
91	C₂×C₄×C₈	—	47	2	no	—
92	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
93	Q₈⋊C₈	—	94	8	no	—
94	#322	—	94	2	no	—
95	C₄₇	#9861	1	1	yes	characteristic, complemented, Sylow
96	C₂×M₁₆	—	94	2	no	—
97	C₂²×C₈	—	94	2	no	—
98	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
99	C₄⋊Q₈	C₂×Dic₄₇	1	4	yes	—
100	#125	—	2	2	no	—
101	C₂²⋊Q₈	—	2	4	no	—
102	C₄×Q₈	—	2	4	no	—
103	#117	—	2	2	no	—
104	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
105	#108	—	188	4	no	—
106	#107	—	2	4	no	—
107	#106	—	2	4	no	—
108	C₄⋊C₈	—	94	4	no	—
109	C₄×C₈	—	47	4	no	—
110	C₄×C₈	—	94	2	no	—
111	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
112	C₄×C₈	C₂×Dic₄₇	1	1	yes	characteristic
113	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
114	C₂×Q₈	—	2	4	no	—
115	C₂²×C₄	#10625352	1	2	yes	—
116	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	M₁₆	—	188	4	no	—
118	C₂×C₈	—	94	4	no	—
119	C₂×C₈	—	188	2	no	—
120	C₂×C₈	—	188	4	no	—
121	C₂×C₈	#10625336	1	4	yes	—
122	C₂×C₈	#10625337	1	4	yes	—
123	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
124	C₂×C₈	C₂²⋊Dic₄₇	1	2	yes	—
125	C₄⋊C₄	—	2	4	no	—
126	C₄⋊C₄	—	2	4	no	—
127	C₄⋊C₄	—	4	4	no	—
128	C₂²⋊C₄	—	4	2	no	—
129	C₄²	—	4	2	no	—
130	C₄²	D₄×D₄₇	1	2	yes	—
131	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
132	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
133	C₂³	#13094011	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
138	C₂×C₄	D₄×Dic₄₇	1	2	yes	—
139	C₂×C₄	#13093995	1	2	yes	—
140	C₂×C₄	#13094020	1	1	yes	characteristic
141	C₂×C₄	C₂×#10625336	1	2	yes	—
142	C₂×C₄	C₂×#10625337	1	2	yes	—
143	C₈	—	376	2	no	—
144	C₈	—	188	4	no	—
145	C₈	#13093948	1	4	yes	—
146	C₈	#13093948	1	4	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	#15689026	1	1	yes	characteristic
149	C₂²	#15689473	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	8	4	no	—
152	C₄	C₂×#13093948	1	2	yes	—
153	C₄	#15689245	1	4	yes	—
154	C₄	#15689497	1	1	yes	characteristic
155	C₄	#15689502	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20244191	1	1	yes	characteristic
158	C₂	#20244282	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal, characteristic

#20241843

C₂

yes

maximal

C₂×#15688844

C₂

yes

maximal, characteristic

C₄₇×#3160

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

Q₈⋊(C₄₇⋊C₈)

—

#15688844

—

C₄₇×#533

C₂²

yes

—

C₄₇×#500

C₄

yes

—

C₄₇×#499

—

C₄₇×#498

C₄

yes

—

C₄₇×#427

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

C₂²×C₄₇⋊C₈

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₄⋊Q₈×C₄₇

C₂×C₄

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

#103

yes

—

C₈×C₄₇

#103

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9861

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#487

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#103

yes

—

C₄×C₄₇

#444

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#1956

yes

characteristic

C₂×C₄₇

#1975

yes

—

#533

C₂×D₄₇

yes

—

#500

Dic₄₇

yes

—

#499

—

#498

Dic₄₇

yes

—

C₂×#108

—

#427

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

Q₈⋊C₈

—

#322

—

C₄₇

#9861

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₄⋊Q₈

C₂×Dic₄₇

yes

—

100

#125

—

101

C₂²⋊Q₈

—

102

C₄×Q₈

—

103

#117

—

104

C₂×C₄²

C₂²×D₄₇

yes

characteristic

105

#108

—

188

—

106

#107

—

107

#106

—

108

C₄⋊C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

C₂×Dic₄₇

yes

characteristic

112

C₄×C₈

C₂×Dic₄₇

yes

characteristic

113

C₄×C₈

C₂²⋊D₄₇

yes

—

114

C₂×Q₈

—

115

C₂²×C₄

#10625352

yes

—

116

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

117

M₁₆

—

188

—

118

C₂×C₈

—

119

C₂×C₈

—

188

—

120

C₂×C₈

—

188

—

121

C₂×C₈

#10625336

yes

—

122

C₂×C₈

#10625337

yes

—

123

C₂×C₈

C₂²⋊Dic₄₇

yes

—

124

C₂×C₈

C₂²⋊Dic₄₇

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₂²⋊C₄

—

129

C₄²

—

130

C₄²

D₄×D₄₇

yes

—

131

C₄²

C₂²×Dic₄₇

yes

characteristic

132

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

133

C₂³

#13094011

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

138

C₂×C₄

D₄×Dic₄₇

yes

—

139

C₂×C₄

#13093995

yes

—

140

C₂×C₄

#13094020

yes

characteristic

141

C₂×C₄

C₂×#10625336

yes

—

142

C₂×C₄

C₂×#10625337

yes

—

143

C₈

—

376

—

144

C₈

—

188

—

145

C₈

#13093948

yes

—

146

C₈

#13093948

yes

—

147

C₂²

—

148

C₂²

#15689026

yes

characteristic

149

C₂²

#15689473

yes

—

150

C₄

—

151

C₄

—

152

C₄

C₂×#13093948

yes

—

153

C₄

#15689245

yes

—

154

C₄

#15689497

yes

characteristic

155

C₄

#15689502

yes

characteristic

156

C₂

—

157

C₂

#20244191

yes

characteristic

158

C₂

#20244282

yes

—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#3160	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#3160	D₄₇
hypocenter	C₄₇	#9861
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#3160

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3160

N	H ≅ G/N	type
95. C₄₇	#9861	semidirect

H ≅ G/N

type

95. C₄₇

#9861

semidirect

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

subgroup	index
#20243560	2
#20241843	2
C₂×#15688844	2
C₄₇×#3160	2
#9861	47

subgroup

index

C₄₇×#3160

Subgroups — #26116247

Conjugacy classes of subgroups

Complements

Maximal subgroups