Group #26106708 — subgroups

Subgroups — #26106708

Order 12032.

446 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20243134	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄₇⋊#1273	C₂	1	4	yes	maximal
4	C₄₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
6	C₂×C₄×C₄₇⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
11	C₄³×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#336	C₄	1	4	yes	—
13	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×C₄₇⋊C₈	—	2	4	no	—
16	C₄×C₄₇⋊C₈	—	2	4	no	—
17	C₄×C₄₇⋊C₈	—	2	4	no	—
18	C₄×C₄₇⋊C₈	C₈	1	8	yes	complemented
19	C₂²×C₈×C₄₇	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₄₇	—	2	4	no	—
26	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
27	C₂²×C₄×C₄₇	C₄²	1	2	yes	—
28	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	4	no	—
32	C₂×C₈×C₄₇	—	2	8	no	—
33	C₄²×C₄₇	—	2	4	no	—
34	C₄²×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	C₂×C₈	1	4	yes	—
36	C₄²×C₄₇	M₁₆	1	4	yes	—
37	C₄²×C₄₇	#41	1	4	yes	—
38	C₂×C₄₇⋊C₈	—	2	4	no	—
39	C₂×C₄₇⋊C₈	—	2	4	no	—
40	C₂×C₄₇⋊C₈	—	2	4	no	—
41	C₂×C₄₇⋊C₈	—	2	8	no	—
42	C₂×C₄₇⋊C₈	C₂×C₈	1	8	yes	—
43	C₂³×C₄₇	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₄₇	#97	1	4	yes	—
50	C₂×C₄×C₄₇	#117	1	2	yes	—
51	C₂×C₄×C₄₇	#117	1	2	yes	—
52	C₂×C₄×C₄₇	#117	1	2	yes	—
53	C₂×C₄×C₄₇	C₂²×C₈	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×M₁₆	1	2	yes	—
55	C₂×C₄×C₄₇	#131	1	4	yes	—
56	C₂×C₄×C₄₇	#131	1	4	yes	—
57	C₈×C₄₇	—	4	4	no	—
58	C₄₇⋊C₈	—	2	8	no	—
59	C₄₇⋊C₈	—	4	4	no	—
60	#8133	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	#376	1	1	yes	characteristic
62	C₂²×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₄₇	C₂×#97	1	1	yes	characteristic
64	C₂²×C₄₇	#405	1	2	yes	—
65	C₂²×C₄₇	#405	1	2	yes	—
66	C₄×C₄₇	—	2	4	no	—
67	C₄×C₄₇	—	2	4	no	—
68	C₄×C₄₇	#431	1	4	yes	—
69	C₄×C₄₇	#433	1	4	yes	—
70	C₄×C₄₇	#433	1	4	yes	—
71	#1744	—	47	1	no	—
72	#1744	D₄₇	1	1	yes	characteristic
73	C₄²×C₈	—	47	1	no	—
74	#1273	—	47	4	no	—
75	C₂×C₄₇	#1450	1	2	yes	—
76	C₂×C₄₇	#1452	1	2	yes	—
77	C₂×C₄₇	#1744	1	1	yes	characteristic
78	C₂×C₄₇	#1746	1	1	yes	characteristic
79	C₂×C₄₇	#1746	1	1	yes	characteristic
80	C₂×C₄×C₈	—	47	2	no	—
81	C₂×C₄×C₈	—	47	2	no	—
82	C₂×C₄×C₈	—	47	2	no	—
83	C₂×C₄×C₈	—	47	2	no	—
84	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
85	C₄³	C₂×D₄₇	1	1	yes	characteristic
86	#336	—	47	4	no	—
87	#336	Dic₄₇	1	4	yes	—
88	C₄₇	#8133	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	47	4	no	—
90	C₂²×C₈	—	47	4	no	—
91	C₂²×C₈	C₂×Dic₄₇	1	4	yes	—
92	C₂×C₄²	C₄×D₄₇	1	2	yes	—
93	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
96	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	47	8	no	—
99	C₄×C₈	—	94	4	no	—
100	C₄×C₈	—	94	4	no	—
101	C₄×C₈	—	94	4	no	—
102	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
103	C₂²×C₄	C₄×Dic₄₇	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	47	8	no	—
110	C₂×C₈	—	94	4	no	—
111	C₂×C₈	—	94	4	no	—
112	C₂×C₈	—	94	4	no	—
113	C₂×C₈	—	94	8	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	C₈×D₄₇	1	4	yes	—
117	C₄²	#10625324	1	4	yes	—
118	C₄²	#10625350	1	4	yes	—
119	C₂³	C₂×C₄×Dic₄₇	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₈×Dic₄₇	1	4	yes	—
125	C₂×C₄	#13093934	1	4	yes	—
126	C₂×C₄	#13093981	1	2	yes	—
127	C₂×C₄	#13093981	1	2	yes	—
128	C₂×C₄	#13093983	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₄₇	1	2	yes	—
130	C₂×C₄	C₂×#10625324	1	2	yes	—
131	C₂×C₄	#13094031	1	4	yes	—
132	C₂×C₄	#13094032	1	4	yes	—
133	C₈	—	4	4	no	—
134	C₈	—	94	8	no	—
135	C₈	—	188	4	no	—
136	C₂²	#15688969	1	1	yes	characteristic
137	C₂²	C₂×C₈×Dic₄₇	1	1	yes	characteristic
138	C₂²	C₂×#13093934	1	1	yes	characteristic
139	C₂²	#15689079	1	2	yes	—
140	C₂²	#15689081	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	#15689174	1	4	yes	—
144	C₄	#15689184	1	4	yes	—
145	C₄	#15689185	1	4	yes	—
146	C₂	#20241672	1	2	yes	—
147	C₂	#20241681	1	2	yes	—
148	C₂	#20243133	1	1	yes	characteristic
149	C₂	#20243140	1	1	yes	characteristic
150	C₂	#20243141	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243134

C₂

yes

maximal, characteristic

C₄²×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇⋊#1273

C₂

yes

maximal

C₄₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₄

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

C₈

yes

complemented

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₄²

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

—

C₄²×C₄₇

M₁₆

yes

—

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₈

yes

—

C₂³×C₄₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

C₂²×C₈

yes

—

C₂×C₄×C₄₇

C₂×M₁₆

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂²×C₄₇

C₂×#97

yes

characteristic

C₂²×C₄₇

#405

yes

—

C₂²×C₄₇

#405

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#431

yes

—

C₄×C₄₇

#433

yes

—

C₄×C₄₇

#433

yes

—

#1744

—

#1744

D₄₇

yes

characteristic

C₄²×C₈

—

#1273

—

C₂×C₄₇

#1450

yes

—

C₂×C₄₇

#1452

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₄³

C₂×D₄₇

yes

characteristic

#336

—

#336

Dic₄₇

yes

—

C₄₇

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₄×D₄₇

yes

characteristic

C₂×C₄²

C₂×Dic₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂²×C₄

C₄×Dic₄₇

yes

—

103

C₂²×C₄

C₄×Dic₄₇

yes

—

104

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

106

C₂²×C₄

C₂²×Dic₄₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₈×D₄₇

yes

—

117

C₄²

#10625324

yes

—

118

C₄²

#10625350

yes

—

119

C₂³

C₂×C₄×Dic₄₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₈×Dic₄₇

yes

—

125

C₂×C₄

#13093934

yes

—

126

C₂×C₄

#13093981

yes

—

127

C₂×C₄

#13093981

yes

—

128

C₂×C₄

#13093983

yes

—

129

C₂×C₄

C₂×C₈×D₄₇

yes

—

130

C₂×C₄

C₂×#10625324

yes

—

131

C₂×C₄

#13094031

yes

—

132

C₂×C₄

#13094032

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

188

—

136

C₂²

#15688969

yes

characteristic

137

C₂²

C₂×C₈×Dic₄₇

yes

characteristic

138

C₂²

C₂×#13093934

yes

characteristic

139

C₂²

#15689079

yes

—

140

C₂²

#15689081

yes

—

141

C₄

—

142

C₄

—

143

C₄

#15689174

yes

—

144

C₄

#15689184

yes

—

145

C₄

#15689185

yes

—

146

C₂

#20241672

yes

—

147

C₂

#20241681

yes

—

148

C₂

#20243133

yes

characteristic

149

C₂

#20243140

yes

characteristic

150

C₂

#20243141

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄×C₈
Fitting	C₄₇×#1744	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄²
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1744	D₄₇
hypocenter	C₄₇	#8133
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄×C₈

Fitting

C₄₇×#1744

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄²

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
18. C₄×C₄₇⋊C₈	C₈	semidirect
88. C₄₇	#8133	semidirect

H ≅ G/N

type

18. C₄×C₄₇⋊C₈

C₈

semidirect

88. C₄₇

#8133

semidirect

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

subgroup	index
#20243134	2
C₄²×C₄₇⋊C₈	2
C₄₇⋊#1273	2
C₄₇×#1744	2
#8133	47

subgroup

index

#20243134

C₄²×C₄₇⋊C₈

C₄₇⋊#1273

C₄₇×#1744

#8133

Subgroups — #26106708

Conjugacy classes of subgroups

Complements

Maximal subgroups