Group #26132665 — subgroups

Subgroups — #26132665

Order 12032.

318 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 155 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×#15688899	C₂	1	1	yes	maximal, characteristic
2	#20243552	C₂	1	1	yes	maximal, characteristic
3	#20241479	C₂	1	4	yes	maximal
4	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13093958	C₂²	1	2	yes	—
7	C₂×#13093956	—	2	1	no	—
8	#15688989	—	2	1	no	—
9	#15688899	C₄	1	4	yes	complemented
10	C₂×#107×C₄₇	—	2	1	no	—
11	C₂×#107×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄₇×#384	—	2	1	no	—
14	C₂×C₄⋊Dic₄₇	—	2	2	no	—
15	C₂×C₄₇⋊C₁₆	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	—	2	2	no	—
17	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
18	#13093958	—	2	4	no	—
19	#13093956	—	4	2	no	—
20	C₄₇⋊#95	—	4	2	no	—
21	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
24	C₂×C₄⋊C₄×C₄₇	—	4	2	no	—
25	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
26	#107×C₄₇	—	2	4	no	—
27	#107×C₄₇	—	4	2	no	—
28	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₄₇	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₇	—	4	2	no	—
31	C₂²×C₄×C₄₇	D₈	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₄₇	—	4	2	no	—
34	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₇	#41	1	2	yes	—
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄⋊C₄×C₄₇	—	4	2	no	—
42	C₄⋊C₄×C₄₇	—	4	2	no	—
43	C₄⋊C₄×C₄₇	—	8	4	no	—
44	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
45	C₄²×C₄₇	Q₁₆	1	2	yes	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	C₄⋊Dic₄₇	—	4	2	no	—
48	C₄₇⋊C₁₆	—	2	4	no	—
49	C₂³×C₄₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	8	2	no	—
53	C₂×C₄×C₄₇	—	8	4	no	—
54	C₂×C₄×C₄₇	#107	1	4	yes	—
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
57	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×Dic₄₇	—	8	2	no	—
62	C₂×Dic₄₇	—	8	2	no	—
63	C₂×Dic₄₇	—	8	2	no	—
64	C₈×C₄₇	#106	1	4	yes	—
65	C₈×C₄₇	#117	1	2	yes	—
66	C₈×C₄₇	#125	1	2	yes	—
67	#12890	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
69	C₂²×C₄₇	#486	1	1	yes	characteristic
70	C₂²×C₄₇	#486	1	1	yes	characteristic
71	C₂²×C₄₇	#509	1	2	yes	—
72	C₂²×C₄₇	#510	1	2	yes	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	—	16	2	no	—
75	C₄×C₄₇	#382	1	1	yes	characteristic
76	C₄×C₄₇	C₂×#106	1	1	yes	characteristic
77	C₄×C₄₇	#487	1	1	yes	characteristic
78	C₄×C₄₇	#487	1	1	yes	characteristic
79	Dic₄₇	—	16	2	no	—
80	C₂×#346	—	47	1	no	—
81	#1846	—	47	1	no	—
82	#1845	D₄₇	1	1	yes	characteristic
83	#1380	—	47	4	no	—
84	C₂×C₄₇	#1834	1	1	yes	characteristic
85	C₂×C₄₇	#2159	1	2	yes	—
86	C₂×C₄₇	#2243	1	2	yes	—
87	C₂×C₄₇	#2244	1	1	yes	characteristic
88	C₂×C₄₇	#2245	1	1	yes	characteristic
89	C₂²×C₁₆	—	47	2	no	—
90	C₂×#107	—	2	1	no	—
91	C₂×#107	—	47	2	no	—
92	C₂×#107	C₂×D₄₇	1	2	yes	—
93	C₂×#106	—	94	1	no	—
94	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
95	#384	—	2	1	no	—
96	#382	—	94	1	no	—
97	#346	—	47	4	no	—
98	C₄₇	#12890	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	4	2	no	—
103	C₂×C₄⋊C₄	—	94	2	no	—
104	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×C₁₆	—	47	4	no	—
106	C₂×C₁₆	—	94	2	no	—
107	C₂×C₁₆	—	94	2	no	—
108	#107	—	2	4	no	—
109	#107	—	4	2	no	—
110	#107	—	94	4	no	—
111	#106	—	188	2	no	—
112	C₄×C₈	Dic₉₄	1	2	yes	—
113	C₄×C₈	C₄×D₄₇	1	2	yes	—
114	#95	—	188	2	no	—
115	C₂²×C₄	—	4	2	no	—
116	C₂²×C₄	—	188	2	no	—
117	C₂²×C₄	#10625334	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
120	C₂×C₈	#10625335	1	2	yes	—
121	C₂×C₈	#10625336	1	2	yes	—
122	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
123	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
124	C₂×C₈	#10625352	1	2	yes	—
125	C₂×C₈	#10625354	1	2	yes	—
126	C₄⋊C₄	—	4	2	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	8	4	no	—
129	C₄⋊C₄	—	188	2	no	—
130	C₄⋊C₄	—	188	2	no	—
131	C₄²	Dic₄₇⋊C₄	1	2	yes	—
132	C₄²	#10625337	1	2	yes	—
133	C₁₆	—	94	4	no	—
134	C₂³	C₂×#10625334	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	376	2	no	—
140	C₂×C₄	—	376	2	no	—
141	C₂×C₄	—	376	2	no	—
142	C₂×C₄	#13093957	1	4	yes	—
143	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
144	C₂×C₄	#13094011	1	1	yes	characteristic
145	C₂×C₄	#13094013	1	1	yes	characteristic
146	C₂×C₄	C₂×#10625336	1	1	yes	characteristic
147	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
148	C₂×C₄	C₂×#10625337	1	1	yes	characteristic
149	C₈	#13093955	1	4	yes	—
150	C₈	#13093982	1	2	yes	—
151	C₈	#13094015	1	2	yes	—
152	C₂²	C₂×#13093957	1	1	yes	characteristic
153	C₂²	#15689469	1	1	yes	characteristic
154	C₂²	#15689471	1	1	yes	characteristic
155	C₂²	#15689559	1	2	yes	—
156	C₂²	#15689564	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	16	2	no	—
159	C₄	—	752	2	no	—
160	C₄	#15688988	1	1	yes	characteristic
161	C₄	C₂×#13093955	1	1	yes	characteristic
162	C₄	#15689473	1	1	yes	characteristic
163	C₄	#15689475	1	1	yes	characteristic
164	C₂	#20243515	1	1	yes	characteristic
165	C₂	#20245174	1	2	yes	—
166	C₂	#20245627	1	2	yes	—
167	C₂	#20245629	1	1	yes	characteristic
168	C₂	#20245633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688899

C₂

yes

maximal, characteristic

#20243552

C₂

yes

maximal, characteristic

#20241479

C₂

yes

maximal

C₄₇×#1845

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

—

C₂×#13093958

C₂²

yes

—

C₂×#13093956

—

#15688989

—

#15688899

C₄

yes

complemented

C₂×#107×C₄₇

—

C₂×#107×C₄₇

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093958

—

#13093956

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

characteristic

#107×C₄₇

—

#107×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

SD₁₆

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂×C₈×C₄₇

#41

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

Q₁₆

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#106

yes

—

C₈×C₄₇

#117

yes

—

C₈×C₄₇

#125

yes

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#510

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

C₂×#346

—

#1846

—

#1845

D₄₇

yes

characteristic

#1380

—

C₂×C₄₇

#1834

yes

characteristic

C₂×C₄₇

#2159

yes

—

C₂×C₄₇

#2243

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

—

C₂×#107

C₂×D₄₇

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#12890

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

100

C₂²×C₈

C₂²×D₄₇

yes

characteristic

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

C₂×C₁₆

—

108

#107

—

109

#107

—

110

#107

—

111

#106

—

188

—

112

C₄×C₈

Dic₉₄

yes

—

113

C₄×C₈

C₄×D₄₇

yes

—

114

#95

—

188

—

115

C₂²×C₄

—

116

C₂²×C₄

—

188

—

117

C₂²×C₄

#10625334

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

119

C₂×C₈

Dic₄₇⋊C₄

yes

—

120

C₂×C₈

#10625335

yes

—

121

C₂×C₈

#10625336

yes

—

122

C₂×C₈

C₂×Dic₉₄

yes

characteristic

123

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

124

C₂×C₈

#10625352

yes

—

125

C₂×C₈

#10625354

yes

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

188

—

130

C₄⋊C₄

—

188

—

131

C₄²

Dic₄₇⋊C₄

yes

—

132

C₄²

#10625337

yes

—

133

C₁₆

—

134

C₂³

C₂×#10625334

yes

characteristic

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

376

—

140

C₂×C₄

—

376

—

141

C₂×C₄

—

376

—

142

C₂×C₄

#13093957

yes

—

143

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

144

C₂×C₄

#13094011

yes

characteristic

145

C₂×C₄

#13094013

yes

characteristic

146

C₂×C₄

C₂×#10625336

yes

characteristic

147

C₂×C₄

C₂×#10625335

yes

characteristic

148

C₂×C₄

C₂×#10625337

yes

characteristic

149

C₈

#13093955

yes

—

150

C₈

#13093982

yes

—

151

C₈

#13094015

yes

—

152

C₂²

C₂×#13093957

yes

characteristic

153

C₂²

#15689469

yes

characteristic

154

C₂²

#15689471

yes

characteristic

155

C₂²

#15689559

yes

—

156

C₂²

#15689564

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

752

—

160

C₄

#15688988

yes

characteristic

161

C₄

C₂×#13093955

yes

characteristic

162

C₄

#15689473

yes

characteristic

163

C₄

#15689475

yes

characteristic

164

C₂

#20243515

yes

characteristic

165

C₂

#20245174

yes

—

166

C₂

#20245627

yes

—

167

C₂

#20245629

yes

characteristic

168

C₂

#20245633

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#10625334
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	C₂×D₈
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#12890
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂³

C₂×#10625334

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1845

C₂

socle, prime radical

C₂³×C₄₇

C₂×D₈

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
9. #15688899	C₄	semidirect
98. C₄₇	#12890	semidirect

H ≅ G/N

type

9. #15688899

C₄

semidirect

98. C₄₇

#12890

semidirect

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

subgroup	index
C₂×#15688899	2
#20243552	2
#20241479	2
C₄₇×#1845	2
#12890	47

subgroup

index

C₄₇×#1845

Subgroups — #26132665

Conjugacy classes of subgroups

Complements

Maximal subgroups