Group #26109746 — subgroups

Subgroups — #26109746

Order 12032.

418 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 115 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×#15688849	C₂	1	2	yes	maximal
2	#20241236	C₂	1	4	yes	maximal
3	C₄₇×#2313	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
5	#15688849	—	2	4	no	—
6	C₂²×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#387	—	2	4	no	—
8	C₄₇×#381	C₄	1	2	yes	—
9	C₄₇×#380	C₄	1	2	yes	—
10	C₄×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄₇⋊C₈	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	4	no	—
14	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
18	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
19	C₂×C₄⋊C₄×C₄₇	D₄	1	4	yes	—
20	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
21	C₂×C₄²×C₄₇	—	2	2	no	—
22	#95×C₄₇	—	2	2	no	—
23	#95×C₄₇	—	2	4	no	—
24	#95×C₄₇	—	2	2	no	—
25	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	—	2	4	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	—	2	4	no	—
30	C₂²×C₄×C₄₇	—	8	1	no	—
31	C₂²×C₄×C₄₇	—	8	1	no	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₄₇	#41	1	4	yes	—
36	C₄⋊C₄×C₄₇	—	2	8	no	—
37	C₄⋊C₄×C₄₇	—	8	2	no	—
38	C₂²⋊C₄×C₄₇	—	2	8	no	—
39	C₄²×C₄₇	—	8	2	no	—
40	C₂×C₄₇⋊C₈	—	4	4	no	—
41	C₂×C₄₇⋊C₈	—	8	4	no	—
42	C₂³×C₄₇	—	4	1	no	—
43	C₂³×C₄₇	—	4	1	no	—
44	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂³×C₄₇	#117	1	2	yes	—
46	C₂³×C₄₇	#131	1	4	yes	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	—	4	4	no	—
51	C₂×C₄×C₄₇	—	4	2	no	—
52	C₂×C₄×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	—	4	4	no	—
57	C₂×C₄×C₄₇	—	8	2	no	—
58	C₂×C₄×C₄₇	#123	1	4	yes	—
59	C₄₇⋊C₈	—	8	4	no	—
60	#8737	—	47	1	no	maximal, Sylow
61	C₂²×C₄₇	—	2	1	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	2	no	—
65	C₂²×C₄₇	—	4	4	no	—
66	C₂²×C₄₇	#381	1	1	yes	characteristic
67	C₂²×C₄₇	#408	1	2	yes	—
68	C₂²×C₄₇	#433	1	4	yes	—
69	C₄×C₄₇	—	2	4	no	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	—	8	4	no	—
73	#2313	D₄₇	1	1	yes	characteristic
74	C₂×#324	—	47	2	no	—
75	#1282	—	47	4	no	—
76	C₂×C₄₇	—	2	2	no	—
77	C₂×C₄₇	—	2	4	no	—
78	C₂×C₄₇	#1818	1	1	yes	characteristic
79	C₂×C₄₇	#1828	1	1	yes	characteristic
80	C₂×C₄₇	#1830	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
82	C₂×C₂²⋊C₈	—	47	4	no	—
83	#387	—	2	4	no	—
84	#381	Dic₄₇	1	2	yes	—
85	#380	Dic₄₇	1	2	yes	—
86	C₄×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
87	#324	—	94	4	no	—
88	C₄₇	#8737	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
91	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
92	C₂²×C₈	—	94	4	no	—
93	C₂×C₄⋊C₄	—	2	4	no	—
94	C₂×C₄⋊C₄	—	2	2	no	—
95	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	4	yes	—
96	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	4	yes	—
97	C₂×C₄²	—	2	2	no	—
98	C₂²⋊C₈	—	94	4	no	—
99	C₂²⋊C₈	—	94	4	no	—
100	#95	—	2	2	no	—
101	#95	—	2	4	no	—
102	#95	—	2	2	no	—
103	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	4	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	8	1	no	—
109	C₂²×C₄	—	8	1	no	—
110	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
111	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
112	C₂²×C₄	#10625350	1	4	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
114	C₂×C₈	—	376	4	no	—
115	C₂×C₈	—	188	4	no	—
116	C₄⋊C₄	—	2	8	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₂²⋊C₄	—	2	8	no	—
119	C₄²	—	8	2	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	1	no	—
122	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
123	C₂³	#13093983	1	2	yes	—
124	C₂³	#13094031	1	4	yes	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	4	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	#13094007	1	4	yes	—
137	C₈	—	376	4	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	4	no	—
143	C₂²	#15688986	1	1	yes	characteristic
144	C₂²	#15689090	1	2	yes	—
145	C₂²	#15689184	1	4	yes	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	4	no	—
150	C₂	—	2	2	no	—
151	C₂	—	2	4	no	—
152	C₂	#20243459	1	1	yes	characteristic
153	C₂	#20243496	1	1	yes	characteristic
154	C₂	#20243502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688849

C₂

yes

maximal

#20241236

C₂

yes

maximal

C₄₇×#2313

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688849

—

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₄₇×#387

—

C₄₇×#381

C₄

yes

—

C₄₇×#380

C₄

yes

—

C₄×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

—

C₂³×C₄₇

#131

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#8737

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#381

yes

characteristic

C₂²×C₄₇

#408

yes

—

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#2313

D₄₇

yes

characteristic

C₂×#324

—

#1282

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

#387

—

#381

Dic₄₇

yes

—

#380

Dic₄₇

yes

—

C₄×C₂²⋊C₄

C₂×D₄₇

yes

—

#324

—

C₄₇

#8737

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₂²⋊C₄

C₂×Dic₄₇

yes

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#95

—

101

#95

—

102

#95

—

103

C₂⁴

C₂²×Dic₄₇

yes

characteristic

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

111

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

112

C₂²×C₄

#10625350

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

114

C₂×C₈

—

376

—

115

C₂×C₈

—

188

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

123

C₂³

#13093983

yes

—

124

C₂³

#13094031

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#13094007

yes

—

137

C₈

—

376

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#15688986

yes

characteristic

144

C₂²

#15689090

yes

—

145

C₂²

#15689184

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₂

—

151

C₂

—

152

C₂

#20243459

yes

characteristic

153

C₂

#20243496

yes

characteristic

154

C₂

#20243502

yes

characteristic

	subgroup	G/H
center	C₂²	#15688986
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#2313	C₂
socle	C₂²×C₄₇	#381
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#2313	D₄₇
hypocenter	C₄₇	#8737
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2313

subgroup

G/H

center

C₂²

#15688986

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₄₇×#2313

C₂

socle

C₂²×C₄₇

#381

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2313

N	H ≅ G/N	type
88. C₄₇	#8737	semidirect

H ≅ G/N

type

88. C₄₇

#8737

semidirect

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

subgroup	index
C₂×#15688849	2
#20241236	2
C₄₇×#2313	2
#8737	47

subgroup

index

C₂×#15688849

#20241236

C₄₇×#2313

#8737

Subgroups — #26109746

Conjugacy classes of subgroups

Complements

Maximal subgroups