Group #26109929 — subgroups

Subgroups — #26109929

Order 12032.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20243141	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241233	C₂	1	1	yes	maximal, characteristic
4	#20241216	C₂	1	2	yes	maximal
5	C₄₇×#1281	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	#15688969	C₂²	1	1	yes	characteristic
9	#15688877	C₄	1	2	yes	—
10	#15688877	C₄	1	2	yes	—
11	#15688875	—	2	2	no	—
12	#15688873	—	2	2	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₂×#97×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄×C₄₇⋊C₈	—	2	2	no	—
21	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
22	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
24	C₄₇⋊#95	—	2	2	no	—
25	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₂²⋊C₈×C₄₇	—	2	2	no	—
29	#97×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	#41	1	2	yes	—
33	C₂²×Dic₄₇	—	2	1	no	—
34	C₂²×Dic₄₇	—	2	1	no	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	—	2	2	no	—
37	C₂×C₈×C₄₇	—	2	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₄₇	#41	1	2	yes	—
42	C₄×Dic₄₇	—	2	2	no	—
43	C₄×Dic₄₇	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂³×C₄₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#124	1	2	yes	—
54	C₂×C₄×C₄₇	#125	1	2	yes	—
55	C₂×C₄×C₄₇	#131	1	2	yes	—
56	C₂×C₄×C₄₇	#131	1	2	yes	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₂×Dic₄₇	—	4	2	no	—
59	C₂×Dic₄₇	—	4	1	no	—
60	C₂×Dic₄₇	—	4	1	no	—
61	C₈×C₄₇	—	4	2	no	—
62	C₈×C₄₇	—	8	2	no	—
63	C₄₇⋊C₈	—	2	2	no	—
64	C₄₇⋊C₈	—	2	2	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	#8782	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	#382	1	1	yes	characteristic
69	C₂²×C₄₇	#433	1	1	yes	characteristic
70	C₂²×C₄₇	#433	1	1	yes	characteristic
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	#424	1	2	yes	—
74	C₄×C₄₇	#433	1	2	yes	—
75	Dic₄₇	—	8	1	no	—
76	Dic₄₇	—	8	1	no	—
77	#1746	—	47	1	no	—
78	#1564	—	47	2	no	—
79	#1281	—	47	1	no	—
80	#1281	D₄₇	1	1	yes	characteristic
81	#1274	—	47	2	no	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	#1832	1	1	yes	characteristic
85	C₂×C₄₇	#1833	1	2	yes	—
86	#431	—	47	2	no	—
87	#431	C₂×D₄₇	1	2	yes	—
88	C₂×#97	C₂×D₄₇	1	1	yes	characteristic
89	C₂×C₄×C₈	—	47	1	no	—
90	#376	—	47	1	no	—
91	#336	—	47	2	no	—
92	#336	—	47	2	no	—
93	#335	—	94	2	no	—
94	#334	—	94	2	no	—
95	C₄₇	#8782	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	47	2	no	—
97	C₂²×C₈	C₄×D₄₇	1	2	yes	—
98	C₂×C₄²	—	47	1	no	—
99	C₂×C₄²	—	47	1	no	—
100	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	94	2	no	—
103	C₂²⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	—	94	2	no	—
105	#97	—	2	2	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	—	47	1	no	—
108	C₄×C₈	—	47	1	no	—
109	C₄×C₈	—	47	2	no	—
110	C₄×C₈	—	94	2	no	—
111	#95	—	94	2	no	—
112	C₂²×C₄	—	94	1	no	—
113	C₂²×C₄	—	94	1	no	—
114	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	#10625350	1	2	yes	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	2	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	47	2	no	—
121	C₂×C₈	—	47	2	no	—
122	C₂×C₈	—	94	2	no	—
123	C₂×C₈	—	94	2	no	—
124	C₂×C₈	—	188	2	no	—
125	C₄²	—	94	2	no	—
126	C₄²	—	94	2	no	—
127	C₄²	D₄×D₄₇	1	1	yes	characteristic
128	C₄²	#10625352	1	2	yes	—
129	C₄²	Q₈×D₄₇	1	1	yes	characteristic
130	C₂³	#13093981	1	1	yes	characteristic
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	188	2	no	—
133	C₂×C₄	—	188	2	no	—
134	C₂×C₄	—	188	1	no	—
135	C₂×C₄	—	188	1	no	—
136	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄	#13093980	1	1	yes	characteristic
138	C₂×C₄	#13094010	1	2	yes	—
139	C₂×C₄	#13094014	1	2	yes	—
140	C₂×C₄	#13094030	1	2	yes	—
141	C₂×C₄	#13094032	1	2	yes	—
142	C₈	—	4	2	no	—
143	C₈	—	8	2	no	—
144	C₈	—	94	2	no	—
145	C₈	—	94	2	no	—
146	C₈	—	376	2	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#15688987	1	1	yes	characteristic
149	C₂²	#15689182	1	1	yes	characteristic
150	C₂²	#15689185	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	376	1	no	—
154	C₄	—	376	1	no	—
155	C₄	#15689148	1	2	yes	—
156	C₄	#15689181	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	#20243506	1	1	yes	characteristic
160	C₂	#20243509	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243141

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241233

C₂

yes

maximal, characteristic

#20241216

C₂

yes

maximal

C₄₇×#1281

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688969

C₂²

yes

characteristic

#15688877

C₄

yes

—

#15688877

C₄

yes

—

#15688875

—

#15688873

—

C₄₇×#431

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

Q₈

yes

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8782

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#424

yes

—

C₄×C₄₇

#433

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1274

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1832

yes

characteristic

C₂×C₄₇

#1833

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×#97

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

—

—

—

—

—

—

C₄₇

#8782

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

C₂²×D₄₇

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₂²⋊C₈

—

105

#97

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

#95

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

115

C₂²×C₄

#10625350

yes

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

188

—

125

C₄²

—

126

C₄²

—

127

C₄²

D₄×D₄₇

yes

characteristic

128

C₄²

#10625352

yes

—

129

C₄²

Q₈×D₄₇

yes

characteristic

130

C₂³

#13093981

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

188

—

133

C₂×C₄

—

188

—

134

C₂×C₄

—

188

—

135

C₂×C₄

—

188

—

136

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

137

C₂×C₄

#13093980

yes

characteristic

138

C₂×C₄

#13094010

yes

—

139

C₂×C₄

#13094014

yes

—

140

C₂×C₄

#13094030

yes

—

141

C₂×C₄

#13094032

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

376

—

147

C₂²

—

148

C₂²

#15688987

yes

characteristic

149

C₂²

#15689182

yes

characteristic

150

C₂²

#15689185

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

376

—

154

C₄

—

376

—

155

C₄

#15689148

yes

—

156

C₄

#15689181

yes

—

157

C₂

—

158

C₂

—

159

C₂

#20243506

yes

characteristic

160

C₂

#20243509

yes

—

	subgroup	G/H
center	C₂×C₄	#13093980
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1281	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1281	D₄₇
hypocenter	C₄₇	#8782
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

subgroup

G/H

center

C₂×C₄

#13093980

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1281

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1281

N	H ≅ G/N	type
95. C₄₇	#8782	semidirect

H ≅ G/N

type

95. C₄₇

#8782

semidirect

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

subgroup	index
#20243141	2
#20242134	2
#20241233	2
#20241216	2
C₄₇×#1281	2
#8782	47

subgroup

index

C₄₇×#1281

Subgroups — #26109929

Conjugacy classes of subgroups

Complements

Maximal subgroups