Group #26116127 — subgroups

Subgroups — #26116127

Order 12032.

542 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 195 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×#15689503	C₂	1	1	yes	maximal, characteristic
2	#20244349	C₂	1	4	yes	maximal
3	#20243545	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₈₈	—	2	2	no	—
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689503	—	2	2	no	—
8	#15689503	C₄	1	4	yes	complemented
9	C₂×#13093956	—	2	2	no	—
10	#15688993	C₂²	1	2	yes	—
11	#15688886	—	2	8	no	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#322	—	2	2	no	—
15	C₂²×Dic₉₄	—	2	2	no	—
16	C₂×Dic₁₈₈	—	2	8	no	—
17	C₂×Dic₁₈₈	—	4	4	no	—
18	C₂×Dic₁₈₈	—	4	4	no	—
19	#13094024	—	2	4	no	—
20	#13094024	—	2	2	no	—
21	#13094024	C₂×C₄	1	4	yes	—
22	C₂×C₄⋊Dic₄₇	—	2	4	no	—
23	C₂×C₄⋊Dic₄₇	—	2	4	no	—
24	C₂×C₄⋊Dic₄₇	—	2	2	no	—
25	#13093956	—	4	4	no	—
26	C₂²×C₈×C₄₇	D₄	1	4	yes	—
27	C₂²×C₈×C₄₇	D₄	1	2	yes	—
28	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	—	2	2	no	—
31	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₄₇	D₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
35	C₂²×Dic₄₇	—	4	2	no	—
36	C₂²×Dic₄₇	—	4	2	no	—
37	C₂×Dic₉₄	—	2	4	no	—
38	C₂×Dic₉₄	—	4	4	no	—
39	C₂×Dic₉₄	—	4	4	no	—
40	C₂×Dic₉₄	—	4	2	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	—	2	8	no	—
43	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
44	C₂×C₈×C₄₇	D₈	1	4	yes	—
45	C₂×C₈×C₄₇	SD₁₆	1	4	yes	—
46	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₇	#41	1	2	yes	—
49	C₄⋊Dic₄₇	—	2	8	no	—
50	C₄⋊Dic₄₇	—	4	4	no	—
51	C₄⋊Dic₄₇	—	4	4	no	—
52	C₄⋊Dic₄₇	—	4	8	no	—
53	Dic₁₈₈	—	4	16	no	—
54	C₂³×C₄₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	#121	1	2	yes	—
58	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
61	C₂×C₄×C₄₇	#137	1	8	yes	—
62	C₂×Dic₄₇	—	4	4	no	—
63	C₂×Dic₄₇	—	8	2	no	—
64	C₂×Dic₄₇	—	8	2	no	—
65	C₂×Dic₄₇	—	8	2	no	—
66	C₂×Dic₄₇	—	8	4	no	—
67	Dic₉₄	—	4	8	no	—
68	Dic₉₄	—	8	4	no	—
69	C₈×C₄₇	—	4	4	no	—
70	C₈×C₄₇	#102	1	8	yes	—
71	#9839	—	47	1	no	maximal, Sylow
72	C₂²×C₄₇	#390	1	1	yes	characteristic
73	C₂²×C₄₇	#492	1	1	yes	characteristic
74	C₂²×C₄₇	#493	1	1	yes	characteristic
75	C₂²×C₄₇	#497	1	4	yes	—
76	C₄×C₄₇	C₂×#102	1	2	yes	—
77	C₄×C₄₇	#441	1	4	yes	—
78	C₄×C₄₇	#470	1	4	yes	—
79	C₄×C₄₇	#486	1	2	yes	—
80	Dic₄₇	—	8	4	no	—
81	Dic₄₇	—	16	2	no	—
82	C₂×#494	—	47	1	no	—
83	#1990	—	47	4	no	—
84	#1844	—	47	1	no	—
85	C₂×#322	D₄₇	1	1	yes	characteristic
86	C₂×C₄₇	#1716	1	2	yes	—
87	C₂×C₄₇	#1722	1	2	yes	—
88	C₂×C₄₇	#1957	1	1	yes	characteristic
89	C₂×C₄₇	#1971	1	2	yes	—
90	C₂²×Q₁₆	—	94	2	no	—
91	C₂×C₄⋊Q₈	—	47	2	no	—
92	#494	—	47	4	no	—
93	#494	—	94	2	no	—
94	C₂×#106	—	94	2	no	—
95	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
97	#384	—	47	2	no	—
98	#340	—	94	8	no	—
99	#322	—	2	2	no	—
100	C₄₇	#9839	1	1	yes	characteristic, complemented, Sylow
101	C₂²×Q₈	—	94	2	no	—
102	C₂×Q₁₆	—	94	8	no	—
103	C₂×Q₁₆	—	188	4	no	—
104	C₂×Q₁₆	—	188	4	no	—
105	C₂²×C₈	D₁₈₈	1	4	yes	—
106	C₂²×C₈	D₁₈₈	1	2	yes	—
107	C₄⋊Q₈	—	47	4	no	—
108	C₄⋊Q₈	—	94	4	no	—
109	C₄⋊Q₈	—	94	2	no	—
110	C₂×C₄⋊C₄	—	94	4	no	—
111	C₂×C₄⋊C₄	—	94	4	no	—
112	C₂×C₄⋊C₄	—	94	2	no	—
113	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
114	#106	—	188	4	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	C₄×C₈	C₄×D₄₇	1	2	yes	—
118	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
119	C₂×Q₈	—	94	4	no	—
120	C₂×Q₈	—	188	4	no	—
121	C₂×Q₈	—	188	4	no	—
122	C₂×Q₈	—	188	2	no	—
123	C₂²×C₄	—	188	2	no	—
124	C₂²×C₄	—	188	2	no	—
125	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
127	Q₁₆	—	188	16	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	#10625325	1	4	yes	—
131	C₂×C₈	D₃₇₆	1	4	yes	—
132	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
133	C₄⋊C₄	—	94	8	no	—
134	C₄⋊C₄	—	188	4	no	—
135	C₄⋊C₄	—	188	8	no	—
136	C₄⋊C₄	—	188	4	no	—
137	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
138	C₄²	#10625350	1	2	yes	—
139	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂³	#13094021	1	1	yes	characteristic
141	Q₈	—	376	4	no	—
142	Q₈	—	188	8	no	—
143	C₂×C₄	—	376	2	no	—
144	C₂×C₄	—	376	2	no	—
145	C₂×C₄	—	376	2	no	—
146	C₂×C₄	—	376	4	no	—
147	C₂×C₄	—	188	4	no	—
148	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
149	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
150	C₂×C₄	#13093998	1	2	yes	—
151	C₂×C₄	#13094012	1	1	yes	characteristic
152	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
153	C₂×C₄	C₂×#10625325	1	2	yes	—
154	C₂×C₄	#13094057	1	8	yes	—
155	C₈	—	4	4	no	—
156	C₈	#13093946	1	8	yes	—
157	C₂²	#15689025	1	1	yes	characteristic
158	C₂²	#15689496	1	1	yes	characteristic
159	C₂²	#15689500	1	1	yes	characteristic
160	C₂²	#15689517	1	4	yes	—
161	C₄	—	376	4	no	—
162	C₄	—	752	2	no	—
163	C₄	C₂×#13093946	1	2	yes	—
164	C₄	#15689231	1	4	yes	—
165	C₄	#15689381	1	4	yes	—
166	C₄	#15689470	1	2	yes	—
167	C₂	#20243021	1	2	yes	—
168	C₂	#20243059	1	2	yes	—
169	C₂	#20244194	1	1	yes	characteristic
170	C₂	#20244259	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689503

C₂

yes

maximal, characteristic

#20244349

C₂

yes

maximal

#20243545

C₂

yes

maximal, characteristic

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

C₂²×Dic₁₈₈

—

C₂×#13094024

C₂²

yes

—

#15689503

—

#15689503

C₄

yes

complemented

C₂×#13093956

—

#15688993

C₂²

yes

—

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₂²×Dic₉₄

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13094024

—

#13094024

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#102

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

C₂×#102

yes

—

C₄×C₄₇

#441

yes

—

C₄×C₄₇

#470

yes

—

C₄×C₄₇

#486

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

#1716

yes

—

C₂×C₄₇

#1722

yes

—

C₂×C₄₇

#1957

yes

characteristic

C₂×C₄₇

#1971

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

#494

—

#494

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

100

C₄₇

#9839

yes

characteristic, complemented, Sylow

101

C₂²×Q₈

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

188

—

104

C₂×Q₁₆

—

188

—

105

C₂²×C₈

D₁₈₈

yes

—

106

C₂²×C₈

D₁₈₈

yes

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

—

109

C₄⋊Q₈

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄⋊C₄

—

113

C₂×C₄²

C₂²×D₄₇

yes

characteristic

114

#106

—

188

—

115

C₄×C₈

—

116

C₄×C₈

—

117

C₄×C₈

C₄×D₄₇

yes

—

118

C₄×C₈

C₂²⋊D₄₇

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

188

—

121

C₂×Q₈

—

188

—

122

C₂×Q₈

—

188

—

123

C₂²×C₄

—

188

—

124

C₂²×C₄

—

188

—

125

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

126

C₂²×C₄

C₂×D₁₈₈

yes

—

127

Q₁₆

—

188

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

#10625325

yes

—

131

C₂×C₈

D₃₇₆

yes

—

132

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

188

—

135

C₄⋊C₄

—

188

—

136

C₄⋊C₄

—

188

—

137

C₄²

C₂×C₄×D₄₇

yes

characteristic

138

C₄²

#10625350

yes

—

139

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

140

C₂³

#13094021

yes

characteristic

141

Q₈

—

376

—

142

Q₈

—

188

—

143

C₂×C₄

—

376

—

144

C₂×C₄

—

376

—

145

C₂×C₄

—

376

—

146

C₂×C₄

—

376

—

147

C₂×C₄

—

188

—

148

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

149

C₂×C₄

C₄×D₁₈₈

yes

—

150

C₂×C₄

#13093998

yes

—

151

C₂×C₄

#13094012

yes

characteristic

152

C₂×C₄

C₂×D₃₇₆

yes

—

153

C₂×C₄

C₂×#10625325

yes

—

154

C₂×C₄

#13094057

yes

—

155

C₈

—

156

C₈

#13093946

yes

—

157

C₂²

#15689025

yes

characteristic

158

C₂²

#15689496

yes

characteristic

159

C₂²

#15689500

yes

characteristic

160

C₂²

#15689517

yes

—

161

C₄

—

376

—

162

C₄

—

752

—

163

C₄

C₂×#13093946

yes

—

164

C₄

#15689231

yes

—

165

C₄

#15689381

yes

—

166

C₄

#15689470

yes

—

167

C₂

#20243021

yes

—

168

C₂

#20243059

yes

—

169

C₂

#20244194

yes

characteristic

170

C₂

#20244259

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#9839
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂³

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#322

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
8. #15689503	C₄	semidirect
100. C₄₇	#9839	semidirect

H ≅ G/N

type

8. #15689503

C₄

semidirect

100. C₄₇

#9839

semidirect

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

subgroup	index
C₂×#15689503	2
#20244349	2
#20243545	2
C₂×C₄₇×#322	2
#9839	47

subgroup

index

C₂×C₄₇×#322

Subgroups — #26116127

Conjugacy classes of subgroups

Complements

Maximal subgroups