Group #26110030 — subgroups

Subgroups — #26110030

Order 12032.

402 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20243540	C₂	1	1	yes	maximal, characteristic
2	#20243134	C₂	1	1	yes	maximal, characteristic
3	C₂×#15688873	C₂	1	4	yes	maximal
4	C₄₇×#1842	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
8	#15688878	C₄	1	2	yes	—
9	#15688878	C₄	1	2	yes	—
10	#15688873	—	2	8	no	—
11	C₂×C₄⋊C₈×C₄₇	—	2	2	no	—
12	C₂×C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄₇⋊C₈	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
18	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
19	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
20	C₂²×C₈×C₄₇	—	2	2	no	—
21	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₇	D₄	1	2	yes	—
24	C₂×C₄²×C₄₇	Q₈	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	—	2	4	no	—
27	C₄⋊C₈×C₄₇	—	4	4	no	—
28	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₈×C₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	—	4	4	no	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	—	4	2	no	—
38	C₄²×C₄₇	SD₁₆	1	4	yes	—
39	C₄²×C₄₇	SD₁₆	1	4	yes	—
40	C₄²×C₄₇	C₂×D₄	1	2	yes	—
41	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
42	C₂×C₄₇⋊C₈	—	2	4	no	—
43	C₂×C₄₇⋊C₈	—	2	4	no	—
44	C₂×C₄₇⋊C₈	—	4	2	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
47	C₂³×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	#106	1	4	yes	—
54	C₂×C₄×C₄₇	#106	1	4	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₄₇	#131	1	2	yes	—
60	C₂×C₄×C₄₇	#131	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
63	C₈×C₄₇	—	8	2	no	—
64	C₄₇⋊C₈	—	2	8	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	#8809	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	#384	1	1	yes	characteristic
68	C₂²×C₄₇	#424	1	1	yes	characteristic
69	C₂²×C₄₇	#424	1	1	yes	characteristic
70	C₂²×C₄₇	C₂×#106	1	2	yes	—
71	C₂²×C₄₇	#430	1	2	yes	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	—	2	2	no	—
75	C₄×C₄₇	#424	1	4	yes	—
76	C₄×C₄₇	#492	1	1	yes	characteristic
77	C₄×C₄₇	#492	1	1	yes	characteristic
78	C₄×C₄₇	#498	1	2	yes	—
79	#1842	—	47	1	no	—
80	#1842	D₄₇	1	1	yes	characteristic
81	#1744	—	47	1	no	—
82	C₂×#334	—	47	4	no	—
83	C₂×C₄₇	#1570	1	2	yes	—
84	C₂×C₄₇	#1570	1	2	yes	—
85	C₂×C₄₇	#1843	1	1	yes	characteristic
86	C₂×C₄₇	#1844	1	1	yes	characteristic
87	C₂×C₄₇	#1849	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	2	2	no	—
89	C₂×C₄⋊C₈	—	47	2	no	—
90	C₂×C₄⋊C₈	—	94	2	no	—
91	C₂×C₄⋊C₈	C₂×D₄₇	1	2	yes	—
92	C₂×C₄×C₈	—	47	2	no	—
93	C₄³	C₂×D₄₇	1	1	yes	characteristic
94	#336	—	47	2	no	—
95	#336	—	47	2	no	—
96	#334	—	94	8	no	—
97	C₄₇	#8809	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	2	2	no	—
99	C₂²×C₈	—	47	4	no	—
100	C₂²×C₈	—	94	2	no	—
101	C₂×C₄²	Dic₉₄	1	2	yes	—
102	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
103	C₂×C₄²	C₄×D₄₇	1	1	yes	characteristic
104	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
105	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₄⋊C₈	—	4	4	no	—
108	C₄⋊C₈	—	94	4	no	—
109	C₄⋊C₈	—	188	4	no	—
110	C₄×C₈	—	47	4	no	—
111	C₄×C₈	—	47	4	no	—
112	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
113	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
114	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
115	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	47	8	no	—
120	C₂×C₈	—	94	4	no	—
121	C₂×C₈	—	94	4	no	—
122	C₂×C₈	—	188	2	no	—
123	C₂×C₈	—	188	4	no	—
124	C₄²	—	2	2	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#10625335	1	4	yes	—
128	C₄²	#10625336	1	4	yes	—
129	C₄²	D₄×D₄₇	1	2	yes	—
130	C₄²	Q₈×D₄₇	1	2	yes	—
131	C₂³	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	#13093955	1	4	yes	—
138	C₂×C₄	#13093955	1	4	yes	—
139	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
140	C₂×C₄	#13094020	1	1	yes	characteristic
141	C₂×C₄	#13094022	1	1	yes	characteristic
142	C₂×C₄	#13094023	1	2	yes	—
143	C₂×C₄	#13094030	1	2	yes	—
144	C₂×C₄	#13094032	1	2	yes	—
145	C₂×C₄	C₂×#10625336	1	2	yes	—
146	C₂×C₄	C₂×#10625335	1	2	yes	—
147	C₈	—	8	2	no	—
148	C₈	—	94	8	no	—
149	C₈	—	376	2	no	—
150	C₂²	#15688992	1	1	yes	characteristic
151	C₂²	#15689149	1	1	yes	characteristic
152	C₂²	#15689151	1	1	yes	characteristic
153	C₂²	C₂×#13093955	1	2	yes	—
154	C₂²	#15689171	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	2	4	no	—
157	C₄	—	2	4	no	—
158	C₄	#15689148	1	4	yes	—
159	C₄	#15689495	1	1	yes	characteristic
160	C₄	#15689497	1	1	yes	characteristic
161	C₄	#15689519	1	2	yes	—
162	C₂	#20242158	1	2	yes	—
163	C₂	#20242160	1	2	yes	—
164	C₂	#20243541	1	1	yes	characteristic
165	C₂	#20243544	1	1	yes	characteristic
166	C₂	#20243562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243540

C₂

yes

maximal, characteristic

#20243134

C₂

yes

maximal, characteristic

C₂×#15688873

C₂

yes

maximal

C₄₇×#1842

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688878

C₄

yes

—

#15688878

C₄

yes

—

#15688873

—

C₂×C₄⋊C₈×C₄₇

—

C₂×C₄⋊C₈×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#106

yes

—

C₂×C₄×C₄₇

#106

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#430

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#424

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

#1842

—

#1842

D₄₇

yes

characteristic

#1744

—

C₂×#334

—

C₂×C₄₇

#1570

yes

—

C₂×C₄₇

#1570

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₄³

C₂×D₄₇

yes

—

—

—

C₄₇

#8809

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

Dic₉₄

yes

—

102

C₂×C₄²

C₄×D₄₇

yes

characteristic

103

C₂×C₄²

C₄×D₄₇

yes

characteristic

104

C₂×C₄²

C₂²⋊D₄₇

yes

—

105

C₂×C₄²

C₂²×D₄₇

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

109

C₄⋊C₈

—

188

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₂²×C₄

Dic₄₇⋊C₄

yes

—

113

C₂²×C₄

Dic₄₇⋊C₄

yes

—

114

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

115

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

116

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

188

—

123

C₂×C₈

—

188

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#10625335

yes

—

128

C₄²

#10625336

yes

—

129

C₄²

D₄×D₄₇

yes

—

130

C₄²

Q₈×D₄₇

yes

—

131

C₂³

C₂×Dic₄₇⋊C₄

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13093955

yes

—

138

C₂×C₄

#13093955

yes

—

139

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

140

C₂×C₄

#13094020

yes

characteristic

141

C₂×C₄

#13094022

yes

characteristic

142

C₂×C₄

#13094023

yes

—

143

C₂×C₄

#13094030

yes

—

144

C₂×C₄

#13094032

yes

—

145

C₂×C₄

C₂×#10625336

yes

—

146

C₂×C₄

C₂×#10625335

yes

—

147

C₈

—

148

C₈

—

149

C₈

—

376

—

150

C₂²

#15688992

yes

characteristic

151

C₂²

#15689149

yes

characteristic

152

C₂²

#15689151

yes

characteristic

153

C₂²

C₂×#13093955

yes

—

154

C₂²

#15689171

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#15689148

yes

—

159

C₄

#15689495

yes

characteristic

160

C₄

#15689497

yes

characteristic

161

C₄

#15689519

yes

—

162

C₂

#20242158

yes

—

163

C₂

#20242160

yes

—

164

C₂

#20243541

yes

characteristic

165

C₂

#20243544

yes

characteristic

166

C₂

#20243562

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1842	C₂
socle, prime radical	C₂³×C₄₇	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1842	D₄₇
hypocenter	C₄₇	#8809
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

subgroup

G/H

center

C₂²×C₄

C₂×C₂²⋊D₄₇

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1842

C₂

socle, prime radical

C₂³×C₄₇

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1842

N	H ≅ G/N	type
97. C₄₇	#8809	semidirect

H ≅ G/N

type

97. C₄₇

#8809

semidirect

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

subgroup	index
#20243540	2
#20243134	2
C₂×#15688873	2
C₄₇×#1842	2
#8809	47

subgroup

index

C₄₇×#1842

Subgroups — #26110030

Conjugacy classes of subgroups

Complements

Maximal subgroups