Group #26115763 — subgroups

Subgroups — #26115763

Order 12032.

390 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 123 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×#15689521	C₂	1	1	yes	maximal, characteristic
2	#20243526	C₂	1	1	yes	maximal, characteristic
3	#20242072	C₂	1	2	yes	maximal
4	#20242068	C₂	1	2	yes	maximal
5	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094024	C₂²	1	2	yes	—
7	#15689521	—	2	2	no	—
8	#15689521	C₄	1	4	yes	—
9	C₂×#13093956	—	2	2	no	—
10	#15689147	C₂²	1	1	yes	characteristic
11	#15689147	C₂²	1	1	yes	characteristic
12	#15688888	—	2	2	no	—
13	#15688871	—	2	4	no	—
14	#15688870	—	2	4	no	—
15	C₄₇×#431	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₄₇×#322	—	2	2	no	—
18	C₂²×Dic₉₄	—	2	2	no	—
19	C₂×#10625329	—	2	1	no	—
20	C₂×#10625329	—	2	1	no	—
21	#13094024	—	2	2	no	—
22	#13094024	—	2	2	no	—
23	#13094024	—	2	2	no	—
24	#13094024	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊Dic₄₇	—	2	2	no	—
26	C₂×C₄⋊Dic₄₇	—	2	2	no	—
27	#13093956	—	2	8	no	—
28	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
30	C₂²×C₈×C₄₇	D₄	1	2	yes	—
31	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₈×C₄₇	D₄	1	4	yes	—
33	C₂²⋊C₈×C₄₇	—	2	2	no	—
34	C₄×C₈×C₄₇	—	2	2	no	—
35	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₄₇	D₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	#41	1	2	yes	—
39	C₂²×Dic₄₇	—	4	2	no	—
40	C₂×Dic₉₄	—	2	4	no	—
41	C₂×Dic₉₄	—	4	2	no	—
42	C₂×Dic₉₄	—	8	4	no	—
43	C₂×C₈×C₄₇	—	2	2	no	—
44	C₂×C₈×C₄₇	—	2	2	no	—
45	C₂×C₈×C₄₇	—	2	4	no	—
46	C₂×C₈×C₄₇	C₂²⋊C₄	1	4	yes	—
47	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₄₇	C₂×D₄	1	2	yes	—
50	C₄⋊Dic₄₇	—	2	4	no	—
51	C₄⋊Dic₄₇	—	2	4	no	—
52	C₄⋊Dic₄₇	—	4	2	no	—
53	C₄⋊Dic₄₇	—	4	2	no	—
54	#10625329	—	4	2	no	—
55	#10625329	—	4	2	no	—
56	C₂×C₄₇⋊C₈	—	4	1	no	—
57	C₂×C₄₇⋊C₈	—	4	1	no	—
58	C₂³×C₄₇	#124	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	4	no	—
60	C₂×C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
62	C₂×C₄×C₄₇	#121	1	2	yes	—
63	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
64	C₂×C₄×C₄₇	#137	1	2	yes	—
65	C₂×C₄×C₄₇	#137	1	2	yes	—
66	C₂×Dic₄₇	—	4	4	no	—
67	C₂×Dic₄₇	—	8	4	no	—
68	Dic₉₄	—	8	4	no	—
69	Dic₉₄	—	8	4	no	—
70	C₈×C₄₇	—	2	4	no	—
71	C₈×C₄₇	—	4	4	no	—
72	C₄₇⋊C₈	—	8	1	no	—
73	C₄₇⋊C₈	—	8	1	no	—
74	#9776	—	47	1	no	maximal, Sylow
75	C₂²×C₄₇	—	2	2	no	—
76	C₂²×C₄₇	#390	1	1	yes	characteristic
77	C₂²×C₄₇	#491	1	2	yes	—
78	C₄×C₄₇	—	2	2	no	—
79	C₄×C₄₇	#419	1	2	yes	—
80	C₄×C₄₇	#471	1	4	yes	—
81	C₄×C₄₇	#497	1	1	yes	characteristic
82	C₄×C₄₇	#497	1	1	yes	characteristic
83	Dic₄₇	—	8	4	no	—
84	C₂×#498	—	47	1	no	—
85	#1837	—	47	1	no	—
86	#1549	—	47	2	no	—
87	#1548	—	47	2	no	—
88	#1452	D₄₇	1	1	yes	characteristic
89	C₂×C₄₇	—	2	2	no	—
90	C₂×C₄₇	#1960	1	1	yes	characteristic
91	C₂×C₄₇	#1977	1	2	yes	—
92	C₂×C₄⋊Q₈	—	47	2	no	—
93	#498	—	47	4	no	—
94	#498	—	94	2	no	—
95	#431	C₂×D₄₇	1	2	yes	—
96	C₂×#106	—	94	2	no	—
97	#423	—	47	1	no	—
98	#423	—	47	1	no	—
99	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
100	#341	—	94	2	no	—
101	#333	—	94	4	no	—
102	#332	—	94	4	no	—
103	#322	—	2	2	no	—
104	C₄₇	#9776	1	1	yes	characteristic, complemented, Sylow
105	C₂²×Q₈	—	94	2	no	—
106	C₂×M₁₆	—	94	1	no	—
107	C₂×M₁₆	—	94	1	no	—
108	C₂²×C₈	D₁₈₈	1	2	yes	—
109	C₄⋊Q₈	—	47	4	no	—
110	C₄⋊Q₈	—	94	2	no	—
111	C₄⋊Q₈	—	94	2	no	—
112	C₄⋊Q₈	—	94	2	no	—
113	C₂×C₄⋊C₄	—	94	2	no	—
114	C₂×C₄⋊C₄	—	94	2	no	—
115	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
116	#106	—	94	8	no	—
117	C₄⋊C₈	—	94	2	no	—
118	C₄⋊C₈	—	94	2	no	—
119	C₄⋊C₈	D₁₈₈	1	4	yes	—
120	C₂²⋊C₈	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	C₄×D₄₇	1	2	yes	—
123	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
124	C₂×Q₈	—	94	4	no	—
125	C₂×Q₈	—	376	4	no	—
126	C₂×Q₈	—	188	2	no	—
127	C₂²×C₄	—	188	2	no	—
128	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
129	C₂²×C₄	#10625350	1	2	yes	—
130	M₁₆	—	188	2	no	—
131	M₁₆	—	188	2	no	—
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	2	no	—
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	188	1	no	—
136	C₂×C₈	—	188	1	no	—
137	C₂×C₈	(C₂×D₄₇)⋊C₄	1	4	yes	—
138	C₄⋊C₄	—	94	4	no	—
139	C₄⋊C₄	—	94	4	no	—
140	C₄⋊C₄	—	188	2	no	—
141	C₄⋊C₄	—	188	2	no	—
142	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
143	C₄²	C₂×D₁₈₈	1	2	yes	—
144	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
145	C₂³	#13094012	1	1	yes	characteristic
146	Q₈	—	376	4	no	—
147	Q₈	—	376	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	—	376	4	no	—
150	C₂×C₄	—	188	4	no	—
151	C₂×C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
152	C₂×C₄	C₄×D₁₈₈	1	2	yes	—
153	C₂×C₄	#13093998	1	2	yes	—
154	C₂×C₄	#13094021	1	1	yes	characteristic
155	C₂×C₄	#13094057	1	2	yes	—
156	C₂×C₄	#13094057	1	2	yes	—
157	C₈	—	2	4	no	—
158	C₈	—	4	4	no	—
159	C₈	—	376	1	no	—
160	C₈	—	376	1	no	—
161	C₂²	—	2	2	no	—
162	C₂²	#15689025	1	1	yes	characteristic
163	C₂²	#15689492	1	2	yes	—
164	C₄	—	2	2	no	—
165	C₄	—	376	4	no	—
166	C₄	#15689133	1	2	yes	—
167	C₄	#15689386	1	4	yes	—
168	C₄	#15689517	1	1	yes	characteristic
169	C₄	#15689517	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#20244208	1	1	yes	characteristic
172	C₂	#20244291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15689521

C₂

yes

maximal, characteristic

#20243526

C₂

yes

maximal, characteristic

#20242072

C₂

yes

maximal

#20242068

C₂

yes

maximal

C₄₇×#1452

C₂

yes

maximal, characteristic

C₂×#13094024

C₂²

yes

—

#15689521

—

#15689521

C₄

yes

—

C₂×#13093956

—

#15689147

C₂²

yes

characteristic

#15689147

C₂²

yes

—

—

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₂²×Dic₉₄

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093956

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

D₄

yes

—

C₂²⋊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

#41

yes

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

#137

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#9776

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#390

yes

characteristic

C₂²×C₄₇

#491

yes

—

C₄×C₄₇

—

C₄×C₄₇

#419

yes

—

C₄×C₄₇

#471

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1960

yes

characteristic

C₂×C₄₇

#1977

yes

—

C₂×C₄⋊Q₈

—

#498

—

#498

—

#431

C₂×D₄₇

yes

—

C₂×#106

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

100

#341

—

101

#333

—

102

#332

—

103

#322

—

104

C₄₇

#9776

yes

characteristic, complemented, Sylow

105

C₂²×Q₈

—

106

C₂×M₁₆

—

107

C₂×M₁₆

—

108

C₂²×C₈

D₁₈₈

yes

—

109

C₄⋊Q₈

—

110

C₄⋊Q₈

—

111

C₄⋊Q₈

—

112

C₄⋊Q₈

—

113

C₂×C₄⋊C₄

—

114

C₂×C₄⋊C₄

—

115

C₂×C₄²

C₂²×D₄₇

yes

characteristic

116

#106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

D₁₈₈

yes

—

120

C₂²⋊C₈

—

121

C₄×C₈

—

122

C₄×C₈

C₄×D₄₇

yes

—

123

C₄×C₈

C₂²⋊D₄₇

yes

—

124

C₂×Q₈

—

125

C₂×Q₈

—

376

—

126

C₂×Q₈

—

188

—

127

C₂²×C₄

—

188

—

128

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

129

C₂²×C₄

#10625350

yes

—

130

M₁₆

—

188

—

131

M₁₆

—

188

—

132

C₂×C₈

—

133

C₂×C₈

—

134

C₂×C₈

—

135

C₂×C₈

—

188

—

136

C₂×C₈

—

188

—

137

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

138

C₄⋊C₄

—

139

C₄⋊C₄

—

140

C₄⋊C₄

—

188

—

141

C₄⋊C₄

—

188

—

142

C₄²

C₂×C₄×D₄₇

yes

characteristic

143

C₄²

C₂×D₁₈₈

yes

—

144

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

145

C₂³

#13094012

yes

characteristic

146

Q₈

—

376

—

147

Q₈

—

376

—

148

C₂×C₄

—

149

C₂×C₄

—

376

—

150

C₂×C₄

—

188

—

151

C₂×C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

152

C₂×C₄

C₄×D₁₈₈

yes

—

153

C₂×C₄

#13093998

yes

—

154

C₂×C₄

#13094021

yes

characteristic

155

C₂×C₄

#13094057

yes

—

156

C₂×C₄

#13094057

yes

—

157

C₈

—

158

C₈

—

159

C₈

—

376

—

160

C₈

—

376

—

161

C₂²

—

162

C₂²

#15689025

yes

characteristic

163

C₂²

#15689492

yes

—

164

C₄

—

165

C₄

—

376

—

166

C₄

#15689133

yes

—

167

C₄

#15689386

yes

—

168

C₄

#15689517

yes

characteristic

169

C₄

#15689517

yes

characteristic

170

C₂

—

171

C₂

#20244208

yes

characteristic

172

C₂

#20244291

yes

—

	subgroup	G/H
center	C₂²	#15689025
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1452	C₂
socle	C₂²×C₄₇	#390
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#9776
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#15689025

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1452

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
104. C₄₇	#9776	semidirect

H ≅ G/N

type

104. C₄₇

#9776

semidirect

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

subgroup	index
C₂×#15689521	2
#20243526	2
#20242072	2
#20242068	2
C₄₇×#1452	2
#9776	47

subgroup

index

C₄₇×#1452

Subgroups — #26115763

Conjugacy classes of subgroups

Complements

Maximal subgroups