Group #26119402 — subgroups

Subgroups — #26119402

Order 12032.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20251775	C₂	1	1	yes	maximal, characteristic
2	#20244365	C₂	1	1	yes	maximal, characteristic
3	#20241953	C₂	1	2	yes	maximal
4	#20241247	C₂	1	2	yes	maximal
5	C₄₇×#1276	C₂	1	1	yes	maximal, characteristic
6	#15689680	C₂²	1	1	yes	characteristic
7	#15689147	C₂²	1	1	yes	characteristic
8	#15689021	—	2	2	no	—
9	#15689007	—	2	4	no	—
10	#15688989	C₂²	1	2	yes	—
11	#15688883	—	2	2	no	—
12	#15688861	—	2	4	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₄³×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×#10625329	—	2	1	no	—
16	#13094024	Q₈	1	2	yes	—
17	#13094016	—	2	1	no	—
18	#13094003	—	2	2	no	—
19	C₄×Dic₉₄	—	2	2	no	—
20	#13093983	—	2	1	no	—
21	C₂×C₄⋊Dic₄₇	—	2	2	no	—
22	C₂×Dic₄₇⋊C₄	—	2	4	no	—
23	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
24	C₄₇⋊#95	—	2	4	no	—
25	C₄₇⋊#95	—	2	2	no	—
26	C₄₇⋊#95	D₄	1	4	yes	—
27	C₂×C₄²×C₄₇	—	2	2	no	—
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	C₂²⋊C₈×C₄₇	—	2	2	no	—
32	C₄×C₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×Dic₄₇	—	2	4	no	—
37	C₂×Dic₉₄	—	2	2	no	—
38	C₂×C₈×C₄₇	—	4	2	no	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	—	2	2	no	—
41	C₄²×C₄₇	—	4	4	no	—
42	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₄₇	#41	1	2	yes	—
44	C₄²×C₄₇	#41	1	1	yes	characteristic
45	C₂²⋊Dic₄₇	—	4	1	no	—
46	C₄⋊Dic₄₇	—	2	2	no	—
47	C₄⋊Dic₄₇	—	2	2	no	—
48	C₄⋊Dic₄₇	—	4	2	no	—
49	C₄⋊Dic₄₇	—	4	2	no	—
50	Dic₄₇⋊C₄	—	4	8	no	—
51	Dic₄₇⋊C₄	—	4	2	no	—
52	C₄×Dic₄₇	—	4	1	no	—
53	#10625329	—	4	2	no	—
54	C₂×C₄₇⋊C₈	—	4	1	no	—
55	C₂³×C₄₇	#120	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	4	2	no	—
58	C₂×C₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#124	1	2	yes	—
63	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#135	1	2	yes	—
65	C₂×C₄×C₄₇	#137	1	2	yes	—
66	C₂×Dic₄₇	—	4	8	no	—
67	C₂×Dic₄₇	—	4	2	no	—
68	C₂×Dic₄₇	—	4	4	no	—
69	Dic₉₄	—	8	2	no	—
70	C₈×C₄₇	—	8	2	no	—
71	C₄₇⋊C₈	—	8	1	no	—
72	#10369	—	47	1	no	maximal, Sylow
73	C₂²×C₄₇	—	2	2	no	—
74	C₂²×C₄₇	#393	1	1	yes	characteristic
75	C₂²×C₄₇	#452	1	2	yes	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	—	4	4	no	—
79	C₄×C₄₇	#479	1	2	yes	—
80	C₄×C₄₇	#488	1	1	yes	characteristic
81	C₄×C₄₇	#491	1	1	yes	characteristic
82	Dic₄₇	—	8	4	no	—
83	Dic₄₇	—	8	2	no	—
84	#2864	—	47	1	no	—
85	#1994	—	47	1	no	—
86	#1518	—	47	2	no	—
87	#1286	—	47	2	no	—
88	#1276	D₄₇	1	1	yes	characteristic
89	C₂×C₄₇	—	2	2	no	—
90	C₂×C₄₇	#2026	1	1	yes	characteristic
91	C₂×C₄₇	#2031	1	2	yes	—
92	#533	—	47	1	no	—
93	#431	C₂×D₄₇	1	2	yes	—
94	#423	—	47	1	no	—
95	#389	—	94	2	no	—
96	#387	—	94	4	no	—
97	#382	—	47	2	no	—
98	C₄³	C₂×D₄₇	1	1	yes	characteristic
99	#339	—	94	2	no	—
100	#328	—	94	4	no	—
101	C₄₇	#10369	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	94	1	no	—
103	C₄⋊Q₈	—	47	2	no	—
104	#125	—	94	1	no	—
105	C₂²⋊Q₈	—	94	2	no	—
106	C₄×Q₈	—	94	2	no	—
107	#117	—	94	1	no	—
108	C₂×C₄⋊C₄	—	94	4	no	—
109	C₂×C₄⋊C₄	—	94	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₁₈₈	1	2	yes	—
112	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	94	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	47	4	no	—
118	#95	—	94	4	no	—
119	#95	—	94	2	no	—
120	C₂×Q₈	—	94	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	94	4	no	—
123	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₄₇	1	2	yes	—
125	M₁₆	—	188	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	188	1	no	—
128	C₄⋊C₄	—	94	2	no	—
129	C₄⋊C₄	—	94	2	no	—
130	C₄⋊C₄	—	188	8	no	—
131	C₄⋊C₄	—	188	2	no	—
132	C₄⋊C₄	—	188	2	no	—
133	C₄⋊C₄	—	188	2	no	—
134	C₂²⋊C₄	—	188	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	188	1	no	—
139	C₄²	#10625350	1	2	yes	—
140	C₄²	#10625352	1	1	yes	characteristic
141	C₄²	Q₈×D₄₇	1	1	yes	characteristic
142	C₂³	#13093994	1	1	yes	characteristic
143	Q₈	—	376	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	188	8	no	—
149	C₂×C₄	—	188	4	no	—
150	C₂×C₄	—	188	2	no	—
151	C₂×C₄	#13094002	1	2	yes	—
152	C₂×C₄	#13094004	1	1	yes	characteristic
153	C₂×C₄	#13094010	1	2	yes	—
154	C₂×C₄	#13094012	1	1	yes	characteristic
155	C₂×C₄	#13094046	1	2	yes	—
156	C₂×C₄	#13094057	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	376	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#15689036	1	1	yes	characteristic
161	C₂²	#15689287	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	376	4	no	—
166	C₄	—	376	2	no	—
167	C₄	#15689431	1	2	yes	—
168	C₄	#15689478	1	1	yes	characteristic
169	C₄	#15689492	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#20244540	1	1	yes	characteristic
172	C₂	#20244561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20251775

C₂

yes

maximal, characteristic

#20244365

C₂

yes

maximal, characteristic

#20241953

C₂

yes

maximal

#20241247

C₂

yes

maximal

C₄₇×#1276

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

characteristic

#15689147

C₂²

yes

—

—

C₂²

yes

—

#15688883

—

#15688861

—

C₄₇×#431

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₂×#10625329

—

#13094024

Q₈

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

#13093983

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

D₄

yes

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#124

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#10369

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#393

yes

characteristic

C₂²×C₄₇

#452

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#479

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#2026

yes

characteristic

C₂×C₄₇

#2031

yes

—

#533

—

#431

C₂×D₄₇

yes

—

—

—

—

—

C₄³

C₂×D₄₇

yes

characteristic

#339

—

100

#328

—

101

C₄₇

#10369

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₄⋊Q₈

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

D₁₈₈

yes

—

112

C₂×C₄²

C₂²×D₄₇

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

119

#95

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

124

C₂²×C₄

D₄×D₄₇

yes

—

125

M₁₆

—

188

—

126

C₂×C₈

—

127

C₂×C₈

—

188

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

188

—

131

C₄⋊C₄

—

188

—

132

C₄⋊C₄

—

188

—

133

C₄⋊C₄

—

188

—

134

C₂²⋊C₄

—

188

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

188

—

139

C₄²

#10625350

yes

—

140

C₄²

#10625352

yes

characteristic

141

C₄²

Q₈×D₄₇

yes

characteristic

142

C₂³

#13093994

yes

characteristic

143

Q₈

—

376

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

188

—

149

C₂×C₄

—

188

—

150

C₂×C₄

—

188

—

151

C₂×C₄

#13094002

yes

—

152

C₂×C₄

#13094004

yes

characteristic

153

C₂×C₄

#13094010

yes

—

154

C₂×C₄

#13094012

yes

characteristic

155

C₂×C₄

#13094046

yes

—

156

C₂×C₄

#13094057

yes

—

157

C₈

—

158

C₈

—

376

—

159

C₂²

—

160

C₂²

#15689036

yes

characteristic

161

C₂²

#15689287

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

376

—

166

C₄

—

376

—

167

C₄

#15689431

yes

—

168

C₄

#15689478

yes

characteristic

169

C₄

#15689492

yes

characteristic

170

C₂

—

171

C₂

#20244540

yes

characteristic

172

C₂

#20244561

yes

—

	subgroup	G/H
center	C₂²	#15689036
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1276	C₂
socle	C₂²×C₄₇	#393
prime radical	C₂³×C₄₇	#120
hypercenter	#1276	D₄₇
hypocenter	C₄₇	#10369
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#15689036

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1276

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
101. C₄₇	#10369	semidirect

H ≅ G/N

type

101. C₄₇

#10369

semidirect

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

subgroup	index
#20251775	2
#20244365	2
#20241953	2
#20241247	2
C₄₇×#1276	2
#10369	47

subgroup

index

C₄₇×#1276

Subgroups — #26119402

Conjugacy classes of subgroups

Complements

Maximal subgroups