Group #26132627 — subgroups

Subgroups — #26132627

Order 12032.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₂×C₄⋊(C₄₇⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#20243074	C₂	1	1	yes	maximal, characteristic
3	#20241465	C₂	1	4	yes	maximal
4	C₂×C₄₇×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093935	C₄	1	1	yes	characteristic
7	C₂×#13093934	C₄	1	2	yes	—
8	C₂×C₈×Dic₄₇	C₂²	1	2	yes	—
9	#15688969	C₄	1	1	yes	characteristic
10	C₄⋊(C₄₇⋊C₁₆)	—	2	2	no	—
11	C₂²×C₁₆×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₄₇	C₂×C₄	1	2	yes	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	—	2	4	no	—
18	#13093935	—	2	2	no	—
19	#13093934	—	2	4	no	—
20	C₈×Dic₄₇	—	2	2	no	—
21	C₈×Dic₄₇	—	2	2	no	—
22	C₄₇⋊#95	C₈	1	2	yes	—
23	C₄₇⋊#95	C₈	1	2	yes	—
24	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₄₇	—	2	2	no	—
28	C₂×C₁₆×C₄₇	—	2	4	no	—
29	C₄×C₈×C₄₇	D₄	1	2	yes	—
30	C₄×C₈×C₄₇	Q₈	1	2	yes	—
31	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₄₇	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₄₇	—	2	2	no	—
36	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₇	M₁₆	1	2	yes	—
43	C₄×Dic₄₇	—	2	2	no	—
44	C₄×Dic₄₇	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	2	2	no	—
47	C₂×C₄₇⋊C₈	M₁₆	1	4	yes	—
48	C₁₆×C₄₇	—	4	2	no	—
49	C₄₇⋊C₁₆	—	4	2	no	—
50	C₂³×C₄₇	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#117	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	—	2	4	no	—
61	C₂×Dic₄₇	—	2	4	no	—
62	C₈×C₄₇	—	2	2	no	—
63	C₈×C₄₇	#124	1	1	yes	characteristic
64	C₈×C₄₇	#124	1	1	yes	characteristic
65	C₈×C₄₇	#125	1	2	yes	—
66	C₄₇⋊C₈	—	2	4	no	—
67	#12880	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₄₇	#431	1	2	yes	—
70	C₂²×C₄₇	#504	1	2	yes	—
71	C₂²×C₄₇	#504	1	2	yes	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	#382	1	1	yes	characteristic
75	C₄×C₄₇	#423	1	1	yes	characteristic
76	C₄×C₄₇	#432	1	2	yes	—
77	Dic₄₇	—	4	4	no	—
78	C₂×C₄⋊C₁₆	—	47	1	no	—
79	C₂×C₄⋊C₁₆	D₄₇	1	1	yes	characteristic
80	#1726	—	47	1	no	—
81	#1374	—	47	4	no	—
82	C₂×C₄₇	#1831	1	1	yes	characteristic
83	C₂×C₄₇	#2151	1	1	yes	characteristic
84	C₂×C₄₇	#2151	1	1	yes	characteristic
85	C₂×C₄₇	#2164	1	4	yes	—
86	C₂²×C₁₆	—	47	2	no	—
87	C₂²×C₁₆	C₂×D₄₇	1	2	yes	—
88	C₂×#97	—	47	2	no	—
89	C₂×#97	—	47	1	no	—
90	C₂×C₄×C₈	—	47	2	no	—
91	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
92	#376	—	47	1	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	94	2	no	—
95	C₄₇	#12880	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	47	2	no	—
97	C₂²×C₈	C₄×D₄₇	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×C₄²	—	47	2	no	—
100	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	94	4	no	—
104	C₂×C₁₆	—	94	2	no	—
105	#97	—	94	4	no	—
106	#97	—	94	2	no	—
107	C₄×C₈	—	94	2	no	—
108	C₄×C₈	—	94	2	no	—
109	C₄×C₈	Dic₉₄	1	2	yes	—
110	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
111	#95	—	47	2	no	—
112	#95	—	47	2	no	—
113	C₂²×C₄	—	47	4	no	—
114	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₄₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	47	4	no	—
119	C₂×C₈	—	94	2	no	—
120	C₂×C₈	—	94	2	no	—
121	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
122	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
123	C₂×C₈	#10625352	1	2	yes	—
124	C₂×C₈	#10625354	1	2	yes	—
125	C₂×C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	C₄²	—	94	2	no	—
127	C₄²	—	94	2	no	—
128	C₄²	#10625324	1	2	yes	—
129	C₄²	Dic₄₇⋊C₄	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	188	2	no	—
132	C₂³	C₂×C₈×D₄₇	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	—	94	4	no	—
137	C₂×C₄	—	94	4	no	—
138	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
139	C₂×C₄	Dic₄₇⋊C₈	1	2	yes	—
140	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
141	C₂×C₄	#13093982	1	2	yes	—
142	C₂×C₄	M₁₆×D₄₇	1	2	yes	—
143	C₂×C₄	C₂×#10625324	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	94	4	no	—
146	C₈	#13094011	1	1	yes	characteristic
147	C₈	#13094013	1	1	yes	characteristic
148	C₈	#13094015	1	2	yes	—
149	C₂²	C₂×Dic₄₇⋊C₈	1	1	yes	characteristic
150	C₂²	#15689175	1	2	yes	—
151	C₂²	#15689539	1	2	yes	—
152	C₂²	#15689541	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	188	4	no	—
156	C₄	#15688988	1	1	yes	characteristic
157	C₄	#15689146	1	1	yes	characteristic
158	C₄	#15689179	1	2	yes	—
159	C₂	#20243504	1	1	yes	characteristic
160	C₂	#20245146	1	1	yes	characteristic
161	C₂	#20245148	1	1	yes	characteristic
162	C₂	#20245194	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₁₆)

C₂

yes

maximal, characteristic

#20243074

C₂

yes

maximal, characteristic

#20241465

C₂

yes

maximal

C₂×C₄₇×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₂×#13093935

C₄

yes

characteristic

C₂×#13093934

C₄

yes

—

C₂×C₈×Dic₄₇

C₂²

yes

—

#15688969

C₄

yes

characteristic

C₄⋊(C₄₇⋊C₁₆)

—

C₂²×C₁₆×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×C₄⋊C₁₆

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₄₇

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

#13093935

—

#13093934

—

C₈×Dic₄₇

—

C₈×Dic₄₇

—

C₄₇⋊#95

C₈

yes

—

C₄₇⋊#95

C₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

M₁₆

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₈×C₄₇

yes

—

C₄₇⋊C₈

—

#12880

—

maximal, Sylow

C₂²×C₄₇

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₄₇

#431

yes

—

C₂²×C₄₇

#504

yes

—

C₂²×C₄₇

#504

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₄₇

yes

characteristic

#1726

—

#1374

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄₇

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

characteristic

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₄₇

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

#97

—

106

#97

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

Dic₉₄

yes

—

110

C₄×C₈

C₂²⋊D₄₇

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₄₇

yes

characteristic

115

C₂²×C₄

C₈×D₄₇

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

Dic₄₇⋊C₄

yes

—

122

C₂×C₈

C₂×Dic₉₄

yes

characteristic

123

C₂×C₈

#10625352

yes

—

124

C₂×C₈

#10625354

yes

—

125

C₂×C₈

C₂×C₂²⋊D₄₇

yes

characteristic

126

C₄²

—

127

C₄²

—

128

C₄²

#10625324

yes

—

129

C₄²

Dic₄₇⋊C₄

yes

—

130

C₁₆

—

131

C₁₆

—

188

—

132

C₂³

C₂×C₈×D₄₇

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

Dic₄₇⋊C₈

yes

—

139

C₂×C₄

Dic₄₇⋊C₈

yes

—

140

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

141

C₂×C₄

#13093982

yes

—

142

C₂×C₄

M₁₆×D₄₇

yes

—

143

C₂×C₄

C₂×#10625324

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

#13094011

yes

characteristic

147

C₈

#13094013

yes

characteristic

148

C₈

#13094015

yes

—

149

C₂²

C₂×Dic₄₇⋊C₈

yes

characteristic

150

C₂²

#15689175

yes

—

151

C₂²

#15689539

yes

—

152

C₂²

#15689541

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

188

—

156

C₄

#15688988

yes

characteristic

157

C₄

#15689146

yes

characteristic

158

C₄

#15689179

yes

—

159

C₂

#20243504

yes

characteristic

160

C₂

#20245146

yes

characteristic

161

C₂

#20245148

yes

characteristic

162

C₂

#20245194

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₄₇
commutator, socle, prime radical	C₂³×C₄₇	C₂²×C₈
Fitting	C₂×C₄₇×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₄₇
hypocenter	C₄₇	#12880
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₄₇

commutator, socle, prime radical

C₂³×C₄₇

C₂²×C₈

Fitting

C₂×C₄₇×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₄₇

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
95. C₄₇	#12880	semidirect

H ≅ G/N

type

95. C₄₇

#12880

semidirect

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

subgroup	index
C₂×C₄⋊(C₄₇⋊C₁₆)	2
#20243074	2
#20241465	2
C₂×C₄₇×C₄⋊C₁₆	2
#12880	47

subgroup

index

C₂×C₄⋊(C₄₇⋊C₁₆)

#20243074

#20241465

C₂×C₄₇×C₄⋊C₁₆

#12880

Subgroups — #26132627

Conjugacy classes of subgroups

Complements

Maximal subgroups