Group #26125194 — subgroups

Subgroups — #26125194

Order 12032.

926 conjugacy classes of subgroups; only the 251 normal subgroups are shown.

normal subgroups only

This group has 926 conjugacy classes of subgroups — too many to draw. Only its 251 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×#15688892	C₂	1	4	yes	maximal
2	#20243082	C₂	1	2	yes	maximal
3	C₂²×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	4	yes	—
5	#15688892	—	2	8	no	—
6	#15688878	—	2	4	no	—
7	C₂×#125×C₄₇	—	2	8	no	—
8	C₂×#125×C₄₇	—	2	4	no	—
9	C₂²×C₄⋊C₄×C₄₇	C₄	1	4	yes	—
10	C₂²×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×#10625329	—	2	4	no	—
13	C₂×#10625329	—	2	8	no	—
14	C₂×#10625329	—	2	8	no	—
15	C₂×#10625329	Q₈	1	8	yes	—
16	C₂²×C₄₇⋊C₈	—	2	4	no	—
17	C₂³×C₄×C₄₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
19	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	#125×C₄₇	—	2	32	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
22	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
23	C₂×C₄⋊C₄×C₄₇	—	2	8	no	—
24	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
25	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
26	C₂×C₄⋊C₄×C₄₇	D₄	1	8	yes	—
27	C₂×C₄²×C₄₇	—	2	4	no	—
28	C₂×C₄²×C₄₇	D₄	1	4	yes	—
29	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	2	4	no	—
34	C₂²×C₄×C₄₇	—	4	4	no	—
35	C₂²×C₄×C₄₇	—	4	8	no	—
36	C₂²×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₄₇	C₂²⋊C₄	1	8	yes	—
38	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
39	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
40	C₂²×C₄×C₄₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₄×C₄₇	#41	1	4	yes	—
42	C₄⋊C₄×C₄₇	—	2	32	no	—
43	C₄⋊C₄×C₄₇	—	2	16	no	—
44	C₄²×C₄₇	—	2	8	no	—
45	#10625329	—	2	16	no	—
46	#10625329	—	4	8	no	—
47	C₂×C₄₇⋊C₈	—	2	8	no	—
48	C₂×C₄₇⋊C₈	—	4	4	no	—
49	C₂×C₄₇⋊C₈	—	4	4	no	—
50	C₂³×C₄₇	—	2	4	no	—
51	C₂³×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
52	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂³×C₄₇	C₄×Q₈	1	4	yes	—
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	2	8	no	—
56	C₂×C₄×C₄₇	—	2	8	no	—
57	C₂×C₄×C₄₇	—	4	4	no	—
58	C₂×C₄×C₄₇	—	4	4	no	—
59	C₂×C₄×C₄₇	—	4	8	no	—
60	C₂×C₄×C₄₇	—	4	8	no	—
61	C₂×C₄×C₄₇	—	4	8	no	—
62	C₂×C₄×C₄₇	#120	1	4	yes	—
63	C₂×C₄×C₄₇	C₂²⋊Q₈	1	4	yes	—
64	C₂×C₄×C₄₇	C₂²⋊Q₈	1	8	yes	—
65	C₂×C₄×C₄₇	#124	1	2	yes	—
66	C₂×C₄×C₄₇	#124	1	4	yes	—
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
68	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
69	C₄₇⋊C₈	—	4	8	no	—
70	#11383	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	2	8	no	—
72	C₂²×C₄₇	—	2	2	no	—
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	#379	1	1	yes	characteristic
75	C₂²×C₄₇	#391	1	4	yes	—
76	C₂²×C₄₇	#391	1	2	yes	—
77	C₂²×C₄₇	#413	1	4	yes	—
78	C₄×C₄₇	—	4	4	no	—
79	C₄×C₄₇	—	4	8	no	—
80	C₄×C₄₇	#393	1	4	yes	—
81	C₄×C₄₇	#395	1	4	yes	—
82	C₂²×#125	D₄₇	1	1	yes	characteristic
83	C₂×#343	—	47	4	no	—
84	#1728	—	47	2	no	—
85	C₂×C₄₇	—	2	4	no	—
86	C₂×C₄₇	#1446	1	1	yes	characteristic
87	C₂×C₄₇	#1785	1	2	yes	—
88	C₂×C₄₇	#1998	1	4	yes	—
89	C₂²×M₁₆	—	47	4	no	—
90	C₂×#125	—	2	8	no	—
91	C₂×#125	—	2	4	no	—
92	C₂²×C₄⋊C₄	Dic₄₇	1	4	yes	—
93	C₂²×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
94	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
95	#343	—	94	8	no	—
96	#336	—	94	4	no	—
97	C₄₇	#11383	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
99	C₂³×C₄	C₂×Dic₄₇	1	2	yes	—
100	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×M₁₆	—	47	8	no	—
102	C₂×M₁₆	—	94	4	no	—
103	C₂×M₁₆	—	94	8	no	—
104	C₂×M₁₆	—	94	8	no	—
105	C₂²×C₈	—	94	4	no	—
106	#125	—	2	32	no	—
107	C₂×C₄⋊C₄	—	2	8	no	—
108	C₂×C₄⋊C₄	—	2	8	no	—
109	C₂×C₄⋊C₄	—	2	8	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	4	no	—
112	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	8	yes	—
113	C₂×C₄²	—	2	4	no	—
114	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
115	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	2	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	—	4	4	no	—
121	C₂²×C₄	—	4	8	no	—
122	C₂²×C₄	C₂²⋊Dic₄₇	1	4	yes	—
123	C₂²×C₄	C₂²⋊Dic₄₇	1	8	yes	—
124	C₂²×C₄	Q₈×D₄₇	1	4	yes	—
125	C₂²×C₄	#10625354	1	4	yes	—
126	C₂²×C₄	C₂×C₂²⋊D₄₇	1	4	yes	—
127	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
128	M₁₆	—	94	16	no	—
129	M₁₆	—	188	8	no	—
130	C₂×C₈	—	94	8	no	—
131	C₂×C₈	—	188	4	no	—
132	C₂×C₈	—	188	4	no	—
133	C₄⋊C₄	—	2	32	no	—
134	C₄⋊C₄	—	2	16	no	—
135	C₄²	—	2	8	no	—
136	C₂³	—	2	4	no	—
137	C₂³	C₂×C₂²⋊Dic₄₇	1	2	yes	—
138	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
139	C₂³	Q₈×Dic₄₇	1	4	yes	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	8	no	—
146	C₂×C₄	—	4	8	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	#13093993	1	4	yes	—
149	C₂×C₄	#13094000	1	4	yes	—
150	C₂×C₄	#13094000	1	8	yes	—
151	C₂×C₄	#13094013	1	2	yes	—
152	C₂×C₄	#13094013	1	4	yes	—
153	C₂×C₄	#13094022	1	4	yes	—
154	C₂×C₄	#13094022	1	2	yes	—
155	C₈	—	188	8	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	2	2	no	—
158	C₂²	—	2	8	no	—
159	C₂²	#15688979	1	1	yes	characteristic
160	C₂²	#15689030	1	4	yes	—
161	C₂²	#15689030	1	2	yes	—
162	C₂²	#15689107	1	4	yes	—
163	C₄	—	4	4	no	—
164	C₄	—	4	8	no	—
165	C₄	#15689034	1	4	yes	—
166	C₄	#15689042	1	4	yes	—
167	C₂	—	2	4	no	—
168	C₂	#20241659	1	1	yes	characteristic
169	C₂	#20243316	1	2	yes	—
170	C₂	#20244382	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688892

C₂

yes

maximal

#20243082

C₂

yes

maximal

C₂²×#125×C₄₇

C₂

yes

maximal, characteristic

C₂²×#10625329

C₂²

yes

—

#15688892

—

#15688878

—

C₂×#125×C₄₇

—

C₂×#125×C₄₇

—

C₂²×C₄⋊C₄×C₄₇

C₄

yes

—

C₂²×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂²×C₄²×C₄₇

C₂²

yes

—

—

—

Q₈

yes

—

C₂²×C₄₇⋊C₈

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂³

yes

characteristic

#125×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

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

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄×Q₈

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

#120

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₄₇⋊C₈

—

#11383

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

characteristic

C₂²×C₄₇

#391

yes

—

C₂²×C₄₇

#391

yes

—

C₂²×C₄₇

#413

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#393

yes

—

C₄×C₄₇

#395

yes

—

C₂²×#125

D₄₇

yes

characteristic

C₂×#343

—

#1728

—

C₂×C₄₇

—

C₂×C₄₇

#1446

yes

characteristic

C₂×C₄₇

#1785

yes

—

C₂×C₄₇

#1998

yes

—

C₂²×M₁₆

—

C₂×#125

—

C₂×#125

—

C₂²×C₄⋊C₄

Dic₄₇

yes

—

C₂²×C₄⋊C₄

C₂×D₄₇

yes

—

C₂²×C₄²

C₂×D₄₇

yes

characteristic

#343

—

#336

—

C₄₇

#11383

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

—

100

C₂³×C₄

C₂²×D₄₇

yes

characteristic

101

C₂×M₁₆

—

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂×M₁₆

—

105

C₂²×C₈

—

106

#125

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂²⋊D₄₇

yes

—

115

C₂⁴

C₂²×Dic₄₇

yes

characteristic

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

—

122

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

123

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

124

C₂²×C₄

Q₈×D₄₇

yes

—

125

C₂²×C₄

#10625354

yes

—

126

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

127

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

128

M₁₆

—

129

M₁₆

—

188

—

130

C₂×C₈

—

131

C₂×C₈

—

188

—

132

C₂×C₈

—

188

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₂³

—

137

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

138

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

139

C₂³

Q₈×Dic₄₇

yes

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

#13093993

yes

—

149

C₂×C₄

#13094000

yes

—

150

C₂×C₄

#13094000

yes

—

151

C₂×C₄

#13094013

yes

—

152

C₂×C₄

#13094013

yes

—

153

C₂×C₄

#13094022

yes

—

154

C₂×C₄

#13094022

yes

—

155

C₈

—

188

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#15688979

yes

characteristic

160

C₂²

#15689030

yes

—

161

C₂²

#15689030

yes

—

162

C₂²

#15689107

yes

—

163

C₄

—

164

C₄

—

165

C₄

#15689034

yes

—

166

C₄

#15689042

yes

—

167

C₂

—

168

C₂

#20241659

yes

characteristic

169

C₂

#20243316

yes

—

170

C₂

#20244382

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂²×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂²×#125	D₄₇
hypocenter	C₄₇	#11383
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₂²×#125×C₄₇

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂³×C₄×C₄₇

C₂³

hypercenter

C₂²×#125

D₄₇

hypocenter

C₄₇

#11383

LCS

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×#125

N	H ≅ G/N	type
97. C₄₇	#11383	semidirect

H ≅ G/N

type

97. C₄₇

#11383

semidirect

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

subgroup	index
C₂×#15688892	2
#20243082	2
C₂²×#125×C₄₇	2
#11383	47

subgroup

index

C₂×#15688892

#20243082

C₂²×#125×C₄₇

#11383

Subgroups — #26125194

Conjugacy classes of subgroups

Complements

Maximal subgroups