Group #26125231 — subgroups

Subgroups — #26125231

Order 12032.

842 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 235 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	#20243127	C₂	1	3	yes	maximal
2	#20243085	C₂	1	3	yes	maximal
3	C₄₇×#3458	C₂	1	1	yes	maximal, characteristic
4	C₂²×#10625329	C₂²	1	3	yes	—
5	C₂³×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
6	#15688892	—	2	6	no	—
7	#15688878	—	2	6	no	—
8	C₄₇×#544	—	2	6	no	—
9	C₄₇×#543	C₄	1	4	yes	—
10	C₄₇×#533	—	2	6	no	—
11	C₂×#117×C₄₇	C₂²	1	3	yes	—
12	C₂×#10625329	—	2	3	no	—
13	C₂×#10625329	—	2	6	no	—
14	C₂×#10625329	—	2	6	no	—
15	C₂×#10625329	Q₈	1	6	yes	complemented
16	C₂²×C₄₇⋊C₈	—	2	3	no	—
17	C₂²×C₄₇⋊C₈	—	2	3	no	—
18	C₂²×C₄₇⋊C₈	—	2	3	no	—
19	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	complemented
20	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₄⋊Q₈×C₄₇	—	2	12	no	—
22	#125×C₄₇	—	2	12	no	—
23	C₂²⋊Q₈×C₄₇	—	2	24	no	—
24	C₄×Q₈×C₄₇	—	2	8	no	—
25	#117×C₄₇	—	2	6	no	—
26	#117×C₄₇	—	2	6	no	—
27	C₂×C₄⋊C₄×C₄₇	D₄	1	6	yes	—
28	C₂×C₂²⋊C₄×C₄₇	C₂×C₄	1	6	yes	—
29	C₂×C₄²×C₄₇	D₄	1	6	yes	—
30	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×Q₈×C₄₇	—	2	8	no	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	12	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
35	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₄×C₄₇	C₂×Q₈	1	3	yes	—
37	C₂²×C₄×C₄₇	#41	1	3	yes	—
38	C₂²×C₄×C₄₇	#41	1	1	yes	characteristic
39	C₄⋊C₄×C₄₇	—	2	24	no	—
40	C₄⋊C₄×C₄₇	—	2	12	no	—
41	C₂²⋊C₄×C₄₇	—	2	12	no	—
42	C₄²×C₄₇	—	2	6	no	—
43	C₄²×C₄₇	—	2	6	no	—
44	#10625329	—	2	12	no	—
45	#10625329	—	4	6	no	—
46	C₂×C₄₇⋊C₈	—	2	6	no	—
47	C₂×C₄₇⋊C₈	—	2	6	no	—
48	C₂×C₄₇⋊C₈	—	4	1	no	—
49	C₂×C₄₇⋊C₈	—	4	3	no	—
50	C₂×C₄₇⋊C₈	—	4	3	no	—
51	C₂×C₄₇⋊C₈	—	4	3	no	—
52	C₂³×C₄₇	—	4	2	no	—
53	C₂³×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
54	C₂³×C₄₇	C₄×Q₈	1	1	yes	characteristic
55	C₂³×C₄₇	C₄×Q₈	1	3	yes	—
56	Q₈×C₄₇	—	8	8	no	—
57	C₂×C₄×C₄₇	—	2	12	no	—
58	C₂×C₄×C₄₇	—	4	12	no	—
59	C₂×C₄×C₄₇	#120	1	3	yes	—
60	C₂×C₄×C₄₇	#120	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
63	C₂×C₄×C₄₇	C₂²⋊Q₈	1	6	yes	—
64	C₂×C₄×C₄₇	#124	1	3	yes	—
65	C₂×C₄×C₄₇	#124	1	3	yes	—
66	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
67	C₂×C₄×C₄₇	C₄⋊Q₈	1	3	yes	—
68	C₄₇⋊C₈	—	4	2	no	—
69	C₄₇⋊C₈	—	4	6	no	—
70	#11391	—	47	1	no	maximal, Sylow
71	C₂²×C₄₇	—	2	6	no	—
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	—	4	3	no	—
74	C₂²×C₄₇	#379	1	1	yes	characteristic
75	C₂²×C₄₇	#391	1	3	yes	—
76	C₂²×C₄₇	#391	1	3	yes	—
77	C₄×C₄₇	—	4	12	no	—
78	C₄×C₄₇	#393	1	3	yes	—
79	C₄×C₄₇	#393	1	1	yes	characteristic
80	C₄×C₄₇	#395	1	3	yes	—
81	C₄×C₄₇	#395	1	1	yes	characteristic
82	#3458	D₄₇	1	1	yes	characteristic
83	#1742	—	47	3	no	—
84	#1729	—	47	3	no	—
85	C₂×C₄₇	—	2	3	no	—
86	C₂×C₄₇	—	2	3	no	—
87	C₂×C₄₇	#1446	1	1	yes	characteristic
88	C₂×C₄₇	#1791	1	2	yes	—
89	C₂²×M₁₆	—	47	3	no	—
90	C₂³×C₈	—	47	1	no	—
91	#544	—	2	6	no	—
92	#543	Dic₄₇	1	4	yes	—
93	#533	—	2	6	no	—
94	C₂×#117	C₂×D₄₇	1	3	yes	—
95	#343	—	94	6	no	—
96	#336	—	94	6	no	—
97	C₄₇	#11391	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×M₁₆	—	47	6	no	—
100	C₂×M₁₆	—	94	3	no	—
101	C₂×M₁₆	—	94	6	no	—
102	C₂×M₁₆	—	94	6	no	—
103	C₂²×C₈	—	47	2	no	—
104	C₂²×C₈	—	94	3	no	—
105	C₂²×C₈	—	94	3	no	—
106	C₂²×C₈	—	94	3	no	—
107	C₄⋊Q₈	—	2	12	no	—
108	#125	—	2	12	no	—
109	C₂²⋊Q₈	—	2	24	no	—
110	C₄×Q₈	—	2	8	no	—
111	#117	—	2	6	no	—
112	#117	—	2	6	no	—
113	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	6	yes	—
114	C₂×C₂²⋊C₄	C₂×Dic₄₇	1	6	yes	—
115	C₂×C₄²	C₂²⋊D₄₇	1	6	yes	—
116	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
117	C₂×Q₈	—	2	8	no	—
118	C₂²×C₄	C₂²⋊Dic₄₇	1	12	yes	—
119	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
120	C₂²×C₄	Q₈×D₄₇	1	3	yes	—
121	C₂²×C₄	#10625354	1	3	yes	—
122	C₂²×C₄	#10625354	1	1	yes	characteristic
123	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
125	M₁₆	—	94	12	no	—
126	M₁₆	—	188	6	no	—
127	C₂×C₈	—	94	6	no	—
128	C₂×C₈	—	94	6	no	—
129	C₂×C₈	—	188	3	no	—
130	C₂×C₈	—	188	1	no	—
131	C₂×C₈	—	188	3	no	—
132	C₂×C₈	—	188	3	no	—
133	C₄⋊C₄	—	2	24	no	—
134	C₄⋊C₄	—	2	12	no	—
135	C₂²⋊C₄	—	2	12	no	—
136	C₄²	—	2	6	no	—
137	C₄²	—	2	6	no	—
138	C₂³	—	4	2	no	—
139	C₂³	C₂×C₂²⋊Dic₄₇	1	3	yes	—
140	C₂³	Q₈×Dic₄₇	1	1	yes	characteristic
141	C₂³	Q₈×Dic₄₇	1	3	yes	—
142	Q₈	—	8	8	no	—
143	C₂×C₄	—	2	12	no	—
144	C₂×C₄	—	4	12	no	—
145	C₂×C₄	#13093993	1	3	yes	—
146	C₂×C₄	#13093993	1	1	yes	characteristic
147	C₂×C₄	#13094000	1	3	yes	—
148	C₂×C₄	#13094000	1	3	yes	—
149	C₂×C₄	#13094000	1	6	yes	—
150	C₂×C₄	#13094013	1	3	yes	—
151	C₂×C₄	#13094013	1	3	yes	—
152	C₂×C₄	#13094022	1	3	yes	—
153	C₂×C₄	#13094022	1	3	yes	—
154	C₈	—	188	2	no	—
155	C₈	—	188	6	no	—
156	C₂²	—	2	6	no	—
157	C₂²	—	4	1	no	—
158	C₂²	—	4	3	no	—
159	C₂²	#15688979	1	1	yes	characteristic
160	C₂²	#15689030	1	3	yes	—
161	C₂²	#15689030	1	3	yes	—
162	C₄	—	4	12	no	—
163	C₄	#15689034	1	3	yes	—
164	C₄	#15689034	1	1	yes	characteristic
165	C₄	#15689042	1	3	yes	—
166	C₄	#15689042	1	1	yes	characteristic
167	C₂	—	2	3	no	—
168	C₂	—	2	3	no	—
169	C₂	#20241659	1	1	yes	characteristic
170	C₂	#20243339	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243127

C₂

yes

maximal

#20243085

C₂

yes

maximal

C₄₇×#3458

C₂

yes

maximal, characteristic

C₂²×#10625329

C₂²

yes

—

C₂³×C₄₇⋊C₈

C₂²

yes

characteristic

#15688892

—

#15688878

—

C₄₇×#544

—

C₄₇×#543

C₄

yes

—

C₄₇×#533

—

C₂×#117×C₄₇

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

Q₈

yes

complemented

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊Q₈×C₄₇

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₂²×C₄×C₄₇

#41

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

—

C₂³×C₄₇

C₄×Q₈

yes

characteristic

C₂³×C₄₇

C₄×Q₈

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊Q₈

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

—

C₄₇⋊C₈

—

#11391

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#379

yes

characteristic

C₂²×C₄₇

#391

yes

—

C₂²×C₄₇

#391

yes

—

C₄×C₄₇

—

C₄×C₄₇

#393

yes

—

C₄×C₄₇

#393

yes

characteristic

C₄×C₄₇

#395

yes

—

C₄×C₄₇

#395

yes

characteristic

#3458

D₄₇

yes

characteristic

#1742

—

#1729

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1446

yes

characteristic

C₂×C₄₇

#1791

yes

—

C₂²×M₁₆

—

C₂³×C₈

—

#544

—

#543

Dic₄₇

yes

—

#533

—

C₂×#117

C₂×D₄₇

yes

—

#343

—

#336

—

C₄₇

#11391

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂×M₁₆

—

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

C₂×M₁₆

—

103

C₂²×C₈

—

104

C₂²×C₈

—

105

C₂²×C₈

—

106

C₂²×C₈

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

#117

—

113

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

114

C₂×C₂²⋊C₄

C₂×Dic₄₇

yes

—

115

C₂×C₄²

C₂²⋊D₄₇

yes

—

116

C₂⁴

C₂²×Dic₄₇

yes

characteristic

117

C₂×Q₈

—

118

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

119

C₂²×C₄

Q₈×D₄₇

yes

characteristic

120

C₂²×C₄

Q₈×D₄₇

yes

—

121

C₂²×C₄

#10625354

yes

—

122

C₂²×C₄

#10625354

yes

characteristic

123

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

125

M₁₆

—

126

M₁₆

—

188

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

188

—

130

C₂×C₈

—

188

—

131

C₂×C₈

—

188

—

132

C₂×C₈

—

188

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₂²⋊C₄

—

136

C₄²

—

137

C₄²

—

138

C₂³

—

139

C₂³

C₂×C₂²⋊Dic₄₇

yes

—

140

C₂³

Q₈×Dic₄₇

yes

characteristic

141

C₂³

Q₈×Dic₄₇

yes

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#13093993

yes

—

146

C₂×C₄

#13093993

yes

characteristic

147

C₂×C₄

#13094000

yes

—

148

C₂×C₄

#13094000

yes

—

149

C₂×C₄

#13094000

yes

—

150

C₂×C₄

#13094013

yes

—

151

C₂×C₄

#13094013

yes

—

152

C₂×C₄

#13094022

yes

—

153

C₂×C₄

#13094022

yes

—

154

C₈

—

188

—

155

C₈

—

188

—

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

#15688979

yes

characteristic

160

C₂²

#15689030

yes

—

161

C₂²

#15689030

yes

—

162

C₄

—

163

C₄

#15689034

yes

—

164

C₄

#15689034

yes

characteristic

165

C₄

#15689042

yes

—

166

C₄

#15689042

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#20241659

yes

characteristic

170

C₂

#20243339

yes

—

	subgroup	G/H
center	C₂×C₄	#13093993
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₄₇×#3458	C₂
socle	C₂²×C₄₇	#379
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	#3458	D₄₇
hypocenter	C₄₇	#11391
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

subgroup

G/H

center

C₂×C₄

#13093993

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₄₇×#3458

C₂

socle

C₂²×C₄₇

#379

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3458

N	H ≅ G/N	type
15. C₂×#10625329	Q₈	semidirect
19. C₂²×C₄₇⋊C₈	Q₈	semidirect
97. C₄₇	#11391	semidirect

H ≅ G/N

type

15. C₂×#10625329

Q₈

semidirect

19. C₂²×C₄₇⋊C₈

Q₈

semidirect

97. C₄₇

#11391

semidirect

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

subgroup	index
#20243127	2
#20243085	2
C₄₇×#3458	2
#11391	47

subgroup

index

#20243127

#20243085

C₄₇×#3458

#11391

Subgroups — #26125231

Conjugacy classes of subgroups

Complements

Maximal subgroups