Group #26134874 — subgroups

Subgroups — #26134874

Order 12032.

366 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 159 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×#15689525	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal
4	C₄₇×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×#13094016	—	2	1	no	—
6	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
7	#15689525	—	2	2	no	—
8	#15689525	C₄	1	4	yes	complemented
9	C₂×#13093958	C₂²	1	2	yes	—
10	C₂×#13093956	—	2	1	no	—
11	C₄×C₄₇⋊C₁₆	—	2	2	no	—
12	C₂×#107×C₄₇	C₂²	1	2	yes	—
13	C₂×#106×C₄₇	—	2	1	no	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#382	—	2	1	no	—
16	#13094016	—	2	4	no	—
17	#13094016	—	4	2	no	—
18	C₂×C₄⋊Dic₄₇	—	2	2	no	—
19	C₂×Dic₄₇⋊C₄	—	4	2	no	—
20	C₂×C₄₇⋊C₁₆	—	2	2	no	—
21	C₂×C₄₇⋊C₁₆	—	2	2	no	—
22	C₂×C₄₇⋊C₁₆	Q₈	1	4	yes	—
23	#13093958	—	2	2	no	—
24	#13093958	C₂×C₄	1	4	yes	—
25	#13093956	—	2	4	no	—
26	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	#107×C₄₇	—	2	4	no	—
31	#106×C₄₇	—	4	2	no	—
32	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₇	D₄	1	2	yes	—
34	#95×C₄₇	—	4	2	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	D₈	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
41	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
42	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₄₇	#41	1	2	yes	—
46	C₄⋊C₄×C₄₇	—	4	2	no	—
47	C₄⋊C₄×C₄₇	—	4	2	no	—
48	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₄²×C₄₇	SD₁₆	1	2	yes	—
50	C₄⋊Dic₄₇	—	2	4	no	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	Dic₄₇⋊C₄	—	4	8	no	—
53	C₄₇⋊C₁₆	—	2	4	no	—
54	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	8	2	no	—
57	C₂×C₄×C₄₇	—	8	4	no	—
58	C₂×C₄×C₄₇	#102	1	4	yes	—
59	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₄₇	—	4	4	no	—
66	C₂×Dic₄₇	—	8	4	no	—
67	C₈×C₄₇	#103	1	4	yes	—
68	C₈×C₄₇	C₄×Q₈	1	2	yes	—
69	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
70	#13281	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
72	C₂²×C₄₇	#486	1	1	yes	characteristic
73	C₂²×C₄₇	#500	1	1	yes	characteristic
74	C₂²×C₄₇	#508	1	4	yes	—
75	C₄×C₄₇	—	2	4	no	—
76	C₄×C₄₇	—	16	2	no	—
77	C₄×C₄₇	#391	1	1	yes	characteristic
78	C₄×C₄₇	C₂×#103	1	1	yes	characteristic
79	C₄×C₄₇	#487	1	1	yes	characteristic
80	C₄×C₄₇	#498	1	1	yes	characteristic
81	Dic₄₇	—	8	4	no	—
82	C₂×#499	—	47	1	no	—
83	C₂×C₄×C₁₆	—	47	1	no	—
84	#1846	D₄₇	1	1	yes	characteristic
85	#1380	—	47	4	no	—
86	C₂×C₄₇	#1990	1	1	yes	characteristic
87	C₂×C₄₇	#2138	1	2	yes	—
88	C₂×C₄₇	#2242	1	2	yes	—
89	C₂×C₄₇	#2253	1	2	yes	—
90	C₂×#125	—	94	1	no	—
91	C₂²×C₁₆	—	47	2	no	—
92	#499	—	47	4	no	—
93	#499	—	94	2	no	—
94	C₂×#107	—	47	2	no	—
95	C₂×#107	C₂×D₄₇	1	2	yes	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	—	94	1	no	—
98	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
99	#382	—	2	1	no	—
100	C₄×C₁₆	—	94	2	no	—
101	C₄₇	#13281	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
104	#125	—	94	4	no	—
105	#125	—	188	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	94	2	no	—
108	C₂×C₄⋊C₄	—	188	2	no	—
109	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂×C₁₆	—	47	4	no	—
111	C₂×C₁₆	—	94	2	no	—
112	C₂×C₁₆	—	94	2	no	—
113	#107	—	2	4	no	—
114	#107	—	47	4	no	—
115	#107	—	94	2	no	—
116	#106	—	4	2	no	—
117	#106	—	94	4	no	—
118	C₄×C₈	C₄×D₄₇	1	2	yes	—
119	C₄×C₈	D₁₈₈	1	2	yes	—
120	#95	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	188	2	no	—
123	C₂²×C₄	#10625334	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
126	C₂×C₈	#10625335	1	2	yes	—
127	C₂×C₈	#10625337	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₂×C₈	C₂×D₁₈₈	1	1	yes	characteristic
130	C₂×C₈	#10625352	1	2	yes	—
131	C₂×C₈	Q₈×D₄₇	1	2	yes	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	94	4	no	—
135	C₄⋊C₄	—	188	8	no	—
136	C₄⋊C₄	—	188	2	no	—
137	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
138	C₄²	#10625336	1	2	yes	—
139	C₁₆	—	94	4	no	—
140	C₂³	C₂×#10625334	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	376	4	no	—
145	C₂×C₄	—	188	4	no	—
146	C₂×C₄	#13093944	1	4	yes	—
147	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#13094011	1	1	yes	characteristic
149	C₂×C₄	#13094022	1	1	yes	characteristic
150	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
151	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
152	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
153	C₈	#13093947	1	4	yes	—
154	C₈	#13093989	1	2	yes	—
155	C₈	#13094004	1	2	yes	—
156	C₂²	C₂×#13093944	1	1	yes	characteristic
157	C₂²	#15689469	1	1	yes	characteristic
158	C₂²	#15689526	1	1	yes	characteristic
159	C₂²	#15689553	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	16	2	no	—
162	C₄	—	376	4	no	—
163	C₄	#15689028	1	1	yes	characteristic
164	C₄	C₂×#13093947	1	1	yes	characteristic
165	C₄	#15689473	1	1	yes	characteristic
166	C₄	#15689519	1	1	yes	characteristic
167	C₂	#20244348	1	1	yes	characteristic
168	C₂	#20245093	1	2	yes	—
169	C₂	#20245619	1	2	yes	—
170	C₂	#20245664	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689525

C₂

yes

maximal, characteristic

C₂×C₄×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

#20241479

C₂

yes

maximal

C₄₇×#1846

C₂

yes

maximal, characteristic

C₂×#13094016

—

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

#15689525

—

#15689525

C₄

yes

complemented

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

C₄×C₄₇⋊C₁₆

—

C₂×#107×C₄₇

C₂²

yes

—

C₂×#106×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#382

—

#13094016

—

#13094016

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

Q₈

yes

—

#13093958

—

#13093958

C₂×C₄

yes

—

#13093956

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#106×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#103

yes

—

C₈×C₄₇

C₄×Q₈

yes

—

C₈×C₄₇

C₂²⋊Q₈

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×#499

—

C₂×C₄×C₁₆

—

#1846

D₄₇

yes

characteristic

#1380

—

C₂×C₄₇

#1990

yes

characteristic

C₂×C₄₇

#2138

yes

—

C₂×C₄₇

#2242

yes

—

C₂×C₄₇

#2253

yes

—

C₂×#125

—

C₂²×C₁₆

—

—

—

—

C₂×D₄₇

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#382

—

100

C₄×C₁₆

—

101

C₄₇

#13281

yes

characteristic, complemented, Sylow

102

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

103

C₂²×C₈

C₂²×D₄₇

yes

characteristic

104

#125

—

105

#125

—

188

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

188

—

109

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

#107

—

114

#107

—

115

#107

—

116

#106

—

117

#106

—

118

C₄×C₈

C₄×D₄₇

yes

—

119

C₄×C₈

D₁₈₈

yes

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

188

—

123

C₂²×C₄

#10625334

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

125

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

126

C₂×C₈

#10625335

yes

—

127

C₂×C₈

#10625337

yes

—

128

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

129

C₂×C₈

C₂×D₁₈₈

yes

characteristic

130

C₂×C₈

#10625352

yes

—

131

C₂×C₈

Q₈×D₄₇

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

188

—

136

C₄⋊C₄

—

188

—

137

C₄²

(C₂×D₄₇)⋊C₄

yes

—

138

C₄²

#10625336

yes

—

139

C₁₆

—

140

C₂³

C₂×#10625334

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

376

—

145

C₂×C₄

—

188

—

146

C₂×C₄

#13093944

yes

—

147

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

148

C₂×C₄

#13094011

yes

characteristic

149

C₂×C₄

#13094022

yes

characteristic

150

C₂×C₄

C₂×#10625336

yes

characteristic

151

C₂×C₄

C₂×#10625335

yes

characteristic

152

C₂×C₄

C₂×#10625337

yes

characteristic

153

C₈

#13093947

yes

—

154

C₈

#13093989

yes

—

155

C₈

#13094004

yes

—

156

C₂²

C₂×#13093944

yes

characteristic

157

C₂²

#15689469

yes

characteristic

158

C₂²

#15689526

yes

characteristic

159

C₂²

#15689553

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

376

—

163

C₄

#15689028

yes

characteristic

164

C₄

C₂×#13093947

yes

characteristic

165

C₄

#15689473

yes

characteristic

166

C₄

#15689519

yes

characteristic

167

C₂

#20244348

yes

characteristic

168

C₂

#20245093

yes

—

169

C₂

#20245619

yes

—

170

C₂

#20245664

yes

—

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1846	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1846	D₄₇
hypocenter	C₄₇	#13281
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1846

C₂

socle, prime radical

C₂³×C₄₇

C₂×D₈

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
8. #15689525	C₄	semidirect
101. C₄₇	#13281	semidirect

H ≅ G/N

type

8. #15689525

C₄

semidirect

101. C₄₇

#13281

semidirect

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

subgroup	index
C₂×#15689525	2
C₂×C₄×C₄₇⋊C₁₆	2
#20241479	2
C₄₇×#1846	2
#13281	47

subgroup

index

C₂×#15689525

C₂×C₄×C₄₇⋊C₁₆

#20241479

C₄₇×#1846

#13281

Subgroups — #26134874

Conjugacy classes of subgroups

Complements

Maximal subgroups