Group #25015193 — subgroups

Subgroups — #25015193

Order 11008.

366 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 159 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×#14583301	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#19132686	C₂	1	4	yes	maximal
4	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063317	—	2	1	no	—
6	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
7	#14583301	—	2	2	no	—
8	#14583301	C₄	1	4	yes	complemented
9	C₂×#13063259	C₂²	1	2	yes	—
10	C₂×#13063257	—	2	1	no	—
11	C₄×C₄₃⋊C₁₆	—	2	2	no	—
12	C₂×#107×C₄₃	C₂²	1	2	yes	—
13	C₂×#106×C₄₃	—	2	1	no	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#382	—	2	1	no	—
16	#13063317	—	2	4	no	—
17	#13063317	—	4	2	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×Dic₄₃⋊C₄	—	4	2	no	—
20	C₂×C₄₃⋊C₁₆	—	2	2	no	—
21	C₂×C₄₃⋊C₁₆	—	2	2	no	—
22	C₂×C₄₃⋊C₁₆	Q₈	1	4	yes	—
23	#13063259	—	2	2	no	—
24	#13063259	C₂×C₄	1	4	yes	—
25	#13063257	—	2	4	no	—
26	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
29	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
30	#107×C₄₃	—	2	4	no	—
31	#106×C₄₃	—	4	2	no	—
32	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₄₃	D₄	1	2	yes	—
34	#95×C₄₃	—	4	2	no	—
35	C₂²×C₄×C₄₃	—	4	2	no	—
36	C₂²×C₄×C₄₃	D₈	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₄₃	—	4	2	no	—
39	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
40	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₄₃	Q₁₆	1	2	yes	—
42	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
45	C₂×C₈×C₄₃	#41	1	2	yes	—
46	C₄⋊C₄×C₄₃	—	4	2	no	—
47	C₄⋊C₄×C₄₃	—	4	2	no	—
48	C₄²×C₄₃	C₂²⋊C₄	1	2	yes	—
49	C₄²×C₄₃	SD₁₆	1	2	yes	—
50	C₄⋊Dic₄₃	—	2	4	no	—
51	C₄⋊Dic₄₃	—	4	2	no	—
52	Dic₄₃⋊C₄	—	4	8	no	—
53	C₄₃⋊C₁₆	—	2	4	no	—
54	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₃	—	2	4	no	—
56	C₂×C₄×C₄₃	—	8	2	no	—
57	C₂×C₄×C₄₃	—	8	4	no	—
58	C₂×C₄×C₄₃	#102	1	4	yes	—
59	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
61	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
65	C₂×Dic₄₃	—	4	4	no	—
66	C₂×Dic₄₃	—	8	4	no	—
67	C₈×C₄₃	#103	1	4	yes	—
68	C₈×C₄₃	C₄×Q₈	1	2	yes	—
69	C₈×C₄₃	C₂²⋊Q₈	1	2	yes	—
70	#13281	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
72	C₂²×C₄₃	#486	1	1	yes	characteristic
73	C₂²×C₄₃	#500	1	1	yes	characteristic
74	C₂²×C₄₃	#508	1	4	yes	—
75	C₄×C₄₃	—	2	4	no	—
76	C₄×C₄₃	—	16	2	no	—
77	C₄×C₄₃	#391	1	1	yes	characteristic
78	C₄×C₄₃	C₂×#103	1	1	yes	characteristic
79	C₄×C₄₃	#487	1	1	yes	characteristic
80	C₄×C₄₃	#498	1	1	yes	characteristic
81	Dic₄₃	—	8	4	no	—
82	C₂×#499	—	43	1	no	—
83	C₂×C₄×C₁₆	—	43	1	no	—
84	#1846	D₄₃	1	1	yes	characteristic
85	#1380	—	43	4	no	—
86	C₂×C₄₃	#1990	1	1	yes	characteristic
87	C₂×C₄₃	#2138	1	2	yes	—
88	C₂×C₄₃	#2242	1	2	yes	—
89	C₂×C₄₃	#2253	1	2	yes	—
90	C₂×#125	—	86	1	no	—
91	C₂²×C₁₆	—	43	2	no	—
92	#499	—	43	4	no	—
93	#499	—	86	2	no	—
94	C₂×#107	—	43	2	no	—
95	C₂×#107	C₂×D₄₃	1	2	yes	—
96	C₂×#106	—	2	1	no	—
97	C₂×#106	—	86	1	no	—
98	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
99	#382	—	2	1	no	—
100	C₄×C₁₆	—	86	2	no	—
101	C₄₃	#13281	1	1	yes	characteristic, complemented, Sylow
102	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
103	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
104	#125	—	86	4	no	—
105	#125	—	172	2	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₄⋊C₄	—	86	2	no	—
108	C₂×C₄⋊C₄	—	172	2	no	—
109	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
110	C₂×C₁₆	—	43	4	no	—
111	C₂×C₁₆	—	86	2	no	—
112	C₂×C₁₆	—	86	2	no	—
113	#107	—	2	4	no	—
114	#107	—	43	4	no	—
115	#107	—	86	2	no	—
116	#106	—	4	2	no	—
117	#106	—	86	4	no	—
118	C₄×C₈	C₄×D₄₃	1	2	yes	—
119	C₄×C₈	D₁₇₂	1	2	yes	—
120	#95	—	4	2	no	—
121	C₂²×C₄	—	4	2	no	—
122	C₂²×C₄	—	172	2	no	—
123	C₂²×C₄	#10621882	1	2	yes	—
124	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
125	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
126	C₂×C₈	#10621883	1	2	yes	—
127	C₂×C₈	#10621885	1	2	yes	—
128	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
129	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
130	C₂×C₈	#10621900	1	2	yes	—
131	C₂×C₈	Q₈×D₄₃	1	2	yes	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	86	4	no	—
135	C₄⋊C₄	—	172	8	no	—
136	C₄⋊C₄	—	172	2	no	—
137	C₄²	(C₂×D₄₃)⋊C₄	1	2	yes	—
138	C₄²	#10621884	1	2	yes	—
139	C₁₆	—	86	4	no	—
140	C₂³	C₂×#10621882	1	1	yes	characteristic
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	344	4	no	—
145	C₂×C₄	—	172	4	no	—
146	C₂×C₄	#13063245	1	4	yes	—
147	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
148	C₂×C₄	#13063312	1	1	yes	characteristic
149	C₂×C₄	#13063323	1	1	yes	characteristic
150	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
151	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
152	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
153	C₈	#13063248	1	4	yes	—
154	C₈	#13063290	1	2	yes	—
155	C₈	#13063305	1	2	yes	—
156	C₂²	C₂×#13063245	1	1	yes	characteristic
157	C₂²	#14583245	1	1	yes	characteristic
158	C₂²	#14583302	1	1	yes	characteristic
159	C₂²	#14583329	1	4	yes	—
160	C₄	—	2	4	no	—
161	C₄	—	16	2	no	—
162	C₄	—	344	4	no	—
163	C₄	#14582804	1	1	yes	characteristic
164	C₄	C₂×#13063248	1	1	yes	characteristic
165	C₄	#14583249	1	1	yes	characteristic
166	C₄	#14583295	1	1	yes	characteristic
167	C₂	#19135555	1	1	yes	characteristic
168	C₂	#19136300	1	2	yes	—
169	C₂	#19136826	1	2	yes	—
170	C₂	#19136871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583301

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic

#19132686

C₂

yes

maximal

C₄₃×#1846

C₂

yes

maximal, characteristic

C₂×#13063317

—

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

#14583301

—

#14583301

C₄

yes

complemented

C₂×#13063259

C₂²

yes

—

C₂×#13063257

—

C₄×C₄₃⋊C₁₆

—

C₂×#107×C₄₃

C₂²

yes

—

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#382

—

#13063317

—

#13063317

—

C₂×C₄⋊Dic₄₃

—

C₂×Dic₄₃⋊C₄

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

Q₈

yes

—

#13063259

—

#13063259

C₂×C₄

yes

—

#13063257

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#106×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

D₄

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

C₂²⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

Q₁₆

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂²⋊C₄

yes

—

C₄²×C₄₃

SD₁₆

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#102

yes

—

C₂×C₄×C₄₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#103

yes

—

C₈×C₄₃

C₄×Q₈

yes

—

C₈×C₄₃

C₂²⋊Q₈

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

C₂×#499

—

C₂×C₄×C₁₆

—

#1846

D₄₃

yes

characteristic

#1380

—

C₂×C₄₃

#1990

yes

characteristic

C₂×C₄₃

#2138

yes

—

C₂×C₄₃

#2242

yes

—

C₂×C₄₃

#2253

yes

—

C₂×#125

—

C₂²×C₁₆

—

—

—

—

C₂×D₄₃

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#382

—

100

C₄×C₁₆

—

101

C₄₃

#13281

yes

characteristic, complemented, Sylow

102

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

103

C₂²×C₈

C₂²×D₄₃

yes

characteristic

104

#125

—

105

#125

—

172

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄⋊C₄

—

108

C₂×C₄⋊C₄

—

172

—

109

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

110

C₂×C₁₆

—

111

C₂×C₁₆

—

112

C₂×C₁₆

—

113

#107

—

114

#107

—

115

#107

—

116

#106

—

117

#106

—

118

C₄×C₈

C₄×D₄₃

yes

—

119

C₄×C₈

D₁₇₂

yes

—

120

#95

—

121

C₂²×C₄

—

122

C₂²×C₄

—

172

—

123

C₂²×C₄

#10621882

yes

—

124

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

125

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

—

126

C₂×C₈

#10621883

yes

—

127

C₂×C₈

#10621885

yes

—

128

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

129

C₂×C₈

C₂×D₁₇₂

yes

characteristic

130

C₂×C₈

#10621900

yes

—

131

C₂×C₈

Q₈×D₄₃

yes

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

172

—

136

C₄⋊C₄

—

172

—

137

C₄²

(C₂×D₄₃)⋊C₄

yes

—

138

C₄²

#10621884

yes

—

139

C₁₆

—

140

C₂³

C₂×#10621882

yes

characteristic

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

344

—

145

C₂×C₄

—

172

—

146

C₂×C₄

#13063245

yes

—

147

C₂×C₄

C₂×(C₂×D₄₃)⋊C₄

yes

characteristic

148

C₂×C₄

#13063312

yes

characteristic

149

C₂×C₄

#13063323

yes

characteristic

150

C₂×C₄

C₂×#10621884

yes

characteristic

151

C₂×C₄

C₂×#10621883

yes

characteristic

152

C₂×C₄

C₂×#10621885

yes

characteristic

153

C₈

#13063248

yes

—

154

C₈

#13063290

yes

—

155

C₈

#13063305

yes

—

156

C₂²

C₂×#13063245

yes

characteristic

157

C₂²

#14583245

yes

characteristic

158

C₂²

#14583302

yes

characteristic

159

C₂²

#14583329

yes

—

160

C₄

—

161

C₄

—

162

C₄

—

344

—

163

C₄

#14582804

yes

characteristic

164

C₄

C₂×#13063248

yes

characteristic

165

C₄

#14583249

yes

characteristic

166

C₄

#14583295

yes

characteristic

167

C₂

#19135555

yes

characteristic

168

C₂

#19136300

yes

—

169

C₂

#19136826

yes

—

170

C₂

#19136871

yes

—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1846	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#13281
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#10621882

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1846

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
8. #14583301	C₄	semidirect
101. C₄₃	#13281	semidirect

H ≅ G/N

type

8. #14583301

C₄

semidirect

101. C₄₃

#13281

semidirect

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

subgroup	index
C₂×#14583301	2
C₂×C₄×C₄₃⋊C₁₆	2
#19132686	2
C₄₃×#1846	2
#13281	43

subgroup

index

C₂×#14583301

C₂×C₄×C₄₃⋊C₁₆

#19132686

C₄₃×#1846

#13281

Subgroups — #25015193

Conjugacy classes of subgroups

Complements

Maximal subgroups