Group #26119401 — subgroups

Subgroups — #26119401

Order 12032.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20241952	C₂	1	2	yes	maximal
3	#20241248	C₂	1	2	yes	maximal
4	#20241221	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1994	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	#15689021	—	2	2	no	—
8	#15689007	—	2	4	no	—
9	#15688989	C₂²	1	2	yes	—
10	#15688860	—	2	4	no	—
11	C₄₇×#533	C₂²	1	1	yes	characteristic
12	C₄₇×#423	C₂²	1	1	yes	characteristic
13	C₄³×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#339	—	2	2	no	—
15	C₂×C₄⋊Dic₄₇	—	2	2	no	—
16	C₂×Dic₄₇⋊C₄	—	2	4	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄×C₄₇⋊C₈	—	2	2	no	—
20	C₄₇⋊#95	—	2	4	no	—
21	C₄₇⋊#95	—	2	2	no	—
22	C₄₇⋊#95	D₄	1	4	yes	—
23	C₂×M₁₆×C₄₇	—	2	1	no	—
24	C₄⋊Q₈×C₄₇	Q₈	1	2	yes	—
25	#125×C₄₇	—	2	1	no	—
26	C₂²⋊Q₈×C₄₇	—	2	2	no	—
27	C₄×Q₈×C₄₇	—	2	2	no	—
28	#117×C₄₇	—	2	1	no	—
29	C₂×C₄²×C₄₇	—	2	2	no	—
30	C₂×C₄²×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	—	2	2	no	—
33	C₂×Q₈×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	—	2	2	no	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₄₇	—	2	4	no	—
38	M₁₆×C₄₇	—	4	2	no	—
39	C₂×C₈×C₄₇	—	4	1	no	—
40	C₄⋊C₄×C₄₇	—	2	2	no	—
41	C₄⋊C₄×C₄₇	—	2	2	no	—
42	C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₂²⋊C₄×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	—	2	2	no	—
45	C₄²×C₄₇	—	2	2	no	—
46	C₄²×C₄₇	—	4	4	no	—
47	C₄²×C₄₇	—	4	1	no	—
48	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₇	#41	1	1	yes	characteristic
50	C₄²×C₄₇	#41	1	2	yes	—
51	C₄⋊Dic₄₇	—	4	2	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	Dic₄₇⋊C₄	—	4	8	no	—
54	C₂×C₄₇⋊C₈	—	4	2	no	—
55	C₂³×C₄₇	#120	1	1	yes	characteristic
56	Q₈×C₄₇	—	8	2	no	—
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	—	4	2	no	—
59	C₂×C₄×C₄₇	—	4	2	no	—
60	C₂×C₄×C₄₇	—	4	2	no	—
61	C₂×C₄×C₄₇	—	4	2	no	—
62	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
63	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#124	1	2	yes	—
65	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
66	C₂×C₄×C₄₇	#135	1	2	yes	—
67	C₂×C₄×C₄₇	#137	1	2	yes	—
68	C₂×Dic₄₇	—	4	8	no	—
69	C₂×Dic₄₇	—	4	4	no	—
70	C₈×C₄₇	—	8	1	no	—
71	C₄₇⋊C₈	—	8	2	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	—	8	2	no	—
80	C₄×C₄₇	#479	1	2	yes	—
81	C₄×C₄₇	#488	1	1	yes	characteristic
82	C₄×C₄₇	#491	1	1	yes	characteristic
83	Dic₄₇	—	8	4	no	—
84	#2864	—	47	1	no	—
85	#1994	D₄₇	1	1	yes	characteristic
86	#1518	—	47	2	no	—
87	#1286	—	47	2	no	—
88	#1276	—	47	1	no	—
89	C₂×C₄₇	—	2	2	no	—
90	C₂×C₄₇	#2026	1	1	yes	characteristic
91	C₂×C₄₇	#2031	1	2	yes	—
92	#533	C₂×D₄₇	1	1	yes	characteristic
93	#431	—	47	2	no	—
94	#423	C₂×D₄₇	1	1	yes	characteristic
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	—	2	2	no	—
100	#328	—	94	4	no	—
101	C₄₇	#10369	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	2	1	no	—
103	C₄⋊Q₈	Dic₉₄	1	2	yes	—
104	#125	—	2	1	no	—
105	C₂²⋊Q₈	—	2	2	no	—
106	C₄×Q₈	—	2	2	no	—
107	#117	—	2	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	C₂²⋊D₄₇	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	—	94	2	no	—
116	C₄×C₈	—	94	2	no	—
117	#95	—	47	4	no	—
118	#95	—	94	4	no	—
119	#95	—	94	2	no	—
120	C₂×Q₈	—	2	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	94	4	no	—
123	C₂²×C₄	D₄×D₄₇	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	188	2	no	—
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	2	2	no	—
130	C₄⋊C₄	—	4	2	no	—
131	C₄⋊C₄	—	188	8	no	—
132	C₄⋊C₄	—	188	2	no	—
133	C₄⋊C₄	—	188	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	4	1	no	—
139	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
140	C₄²	#10625350	1	1	yes	characteristic
141	C₄²	#10625352	1	2	yes	—
142	C₂³	#13093992	1	1	yes	characteristic
143	Q₈	—	8	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	—	4	2	no	—
149	C₂×C₄	—	188	8	no	—
150	C₂×C₄	—	188	4	no	—
151	C₂×C₄	#13094001	1	2	yes	—
152	C₂×C₄	#13094003	1	1	yes	characteristic
153	C₂×C₄	#13094010	1	2	yes	—
154	C₂×C₄	#13094011	1	1	yes	characteristic
155	C₂×C₄	#13094044	1	2	yes	—
156	C₂×C₄	#13094055	1	2	yes	—
157	C₈	—	8	1	no	—
158	C₈	—	376	2	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#15689035	1	1	yes	characteristic
161	C₂²	#15689283	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	8	2	no	—
166	C₄	—	376	4	no	—
167	C₄	#15689430	1	2	yes	—
168	C₄	#15689477	1	1	yes	characteristic
169	C₄	#15689491	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#20244539	1	1	yes	characteristic
172	C₂	#20244560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20251775

C₂

yes

maximal, characteristic

#20241952

C₂

yes

maximal

#20241248

C₂

yes

maximal

#20241221

C₂

yes

maximal, characteristic

C₄₇×#1994

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

#15689021

—

#15689007

—

#15688989

C₂²

yes

—

#15688860

—

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

characteristic

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#339

—

C₂×C₄⋊Dic₄₇

—

C₂×Dic₄₇⋊C₄

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

Q₈

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

C₄²×C₄₇

yes

C₄²×C₄₇

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#120

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

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

—

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

—

C₄×C₄₇

#479

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#2864

—

#1994

D₄₇

yes

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#2026

yes

characteristic

C₂×C₄₇

#2031

yes

—

#533

C₂×D₄₇

yes

characteristic

#431

—

#423

C₂×D₄₇

yes

—

—

—

C₄³

C₂×D₄₇

yes

characteristic

#339

—

100

#328

—

101

C₄₇

#10369

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₄⋊Q₈

Dic₉₄

yes

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

C₂²⋊D₄₇

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

D₄×D₄₇

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

188

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

188

—

132

C₄⋊C₄

—

188

—

133

C₄⋊C₄

—

188

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

C₂×Dic₉₄

yes

characteristic

140

C₄²

#10625350

yes

characteristic

141

C₄²

#10625352

yes

—

142

C₂³

#13093992

yes

characteristic

143

Q₈

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

188

—

150

C₂×C₄

—

188

—

151

C₂×C₄

#13094001

yes

—

152

C₂×C₄

#13094003

yes

characteristic

153

C₂×C₄

#13094010

yes

—

154

C₂×C₄

#13094011

yes

characteristic

155

C₂×C₄

#13094044

yes

—

156

C₂×C₄

#13094055

yes

—

157

C₈

—

158

C₈

—

376

—

159

C₂²

—

160

C₂²

#15689035

yes

characteristic

161

C₂²

#15689283

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

376

—

167

C₄

#15689430

yes

—

168

C₄

#15689477

yes

characteristic

169

C₄

#15689491

yes

characteristic

170

C₂

—

171

C₂

#20244539

yes

characteristic

172

C₂

#20244560

yes

—

	subgroup	G/H
center	C₂²	#15689035
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1994	C₂
socle	C₂²×C₄₇	#393
prime radical	C₂³×C₄₇	#120
hypercenter	#1994	D₄₇
hypocenter	C₄₇	#10369
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

subgroup

G/H

center

C₂²

#15689035

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1994

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1994

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
#20241952	2
#20241248	2
#20241221	2
C₄₇×#1994	2
#10369	47

subgroup

index

C₄₇×#1994

Subgroups — #26119401

Conjugacy classes of subgroups

Complements

Maximal subgroups