Group #25110174 — subgroups

Subgroups — #25110174

Order 11008.

666 conjugacy classes of subgroups; only the 259 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 259 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×#14583030	C₂	1	2	yes	maximal
2	#19135536	C₂	1	2	yes	maximal
3	#19135161	C₂	1	8	yes	maximal
4	#19134715	C₂	1	1	yes	maximal, characteristic
5	#19134347	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#2399	C₂	1	1	yes	maximal, characteristic
7	#14583030	—	2	8	no	—
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	4	yes	—
10	C₂×#13063236	C₂²	1	1	yes	characteristic
11	C₂×#13063236	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	#14582654	C₂²	1	8	yes	—
14	#14582654	C₂²	1	4	yes	—
15	#14582654	C₂²	1	4	yes	—
16	C₂×C₄×Q₈×C₄₃	C₂²	1	2	yes	—
17	C₄₃×#398	C₄	1	4	yes	—
18	C₄₃×#395	C₄	1	4	yes	—
19	C₄×C₄⋊C₄×C₄₃	C₂²	1	4	yes	—
20	C₄₃×#376	C₂²	1	1	yes	characteristic
21	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
22	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
23	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
24	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
25	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
26	#13063236	—	2	2	no	—
27	#13063236	—	2	8	no	—
28	C₄×C₄₃⋊C₈	D₄	1	4	yes	—
29	C₂²×Q₈×C₄₃	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₄₃	—	2	4	no	—
31	C₄×Q₈×C₄₃	—	2	4	no	—
32	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
35	C₂×C₄²×C₄₃	C₂³	1	4	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
37	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
38	#95×C₄₃	C₂×C₄	1	4	yes	—
39	#95×C₄₃	C₂×C₄	1	4	yes	—
40	#95×C₄₃	C₂×C₄	1	4	yes	—
41	C₂×Q₈×C₄₃	—	2	4	no	—
42	C₂×Q₈×C₄₃	—	4	4	no	—
43	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
45	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
46	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
47	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
48	C₄⋊C₄×C₄₃	—	2	4	no	—
49	C₄⋊C₄×C₄₃	—	2	8	no	—
50	C₄⋊C₄×C₄₃	—	4	8	no	—
51	C₄²×C₄₃	—	2	4	no	—
52	C₄²×C₄₃	—	2	4	no	—
53	C₄²×C₄₃	—	2	4	no	—
54	C₄²×C₄₃	C₂×D₄	1	2	yes	—
55	C₄²×C₄₃	#41	1	2	yes	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	—	2	4	no	—
58	C₂×C₄₃⋊C₈	—	2	2	no	—
59	C₂×C₄₃⋊C₈	—	2	8	no	—
60	C₂×C₄₃⋊C₈	—	2	4	no	—
61	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
62	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
63	Q₈×C₄₃	—	4	8	no	—
64	C₂×C₄×C₄₃	—	2	4	no	—
65	C₂×C₄×C₄₃	—	2	4	no	—
66	C₂×C₄×C₄₃	—	2	4	no	—
67	C₂×C₄×C₄₃	—	2	4	no	—
68	C₂×C₄×C₄₃	—	2	4	no	—
69	C₂×C₄×C₄₃	—	2	8	no	—
70	C₂×C₄×C₄₃	—	2	8	no	—
71	C₂×C₄×C₄₃	—	2	4	no	—
72	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
73	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
74	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
75	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
76	C₄₃⋊C₈	—	2	4	no	—
77	C₄₃⋊C₈	—	4	4	no	—
78	C₄₃⋊C₈	—	4	2	no	—
79	#22243	—	43	1	no	maximal, Sylow
80	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
81	C₂²×C₄₃	#519	1	2	yes	—
82	C₂²×C₄₃	#568	1	4	yes	—
83	C₄×C₄₃	—	2	4	no	—
84	C₄×C₄₃	—	4	4	no	—
85	C₄×C₄₃	—	4	4	no	—
86	C₄×C₄₃	—	4	4	no	—
87	C₄×C₄₃	#536	1	2	yes	—
88	C₄×C₄₃	#541	1	2	yes	—
89	C₂×#446	—	43	2	no	—
90	#2399	D₄₃	1	1	yes	characteristic
91	#1985	—	43	2	no	—
92	#1917	—	43	8	no	—
93	#1832	—	43	1	no	—
94	#1746	—	43	1	no	—
95	C₂×C₄₃	#2327	1	1	yes	characteristic
96	C₂×C₄₃	#2930	1	2	yes	—
97	C₂×C₄₃	#2987	1	4	yes	—
98	C₂×C₄×Q₈	C₂×D₄₃	1	2	yes	—
99	#446	—	86	8	no	—
100	C₂×C₄⋊C₈	—	43	2	no	—
101	C₂×C₄⋊C₈	—	43	4	no	—
102	C₂×#97	—	43	4	no	—
103	C₂×#97	—	43	1	no	—
104	C₂×C₄×C₈	—	43	1	no	—
105	#398	Dic₄₃	1	4	yes	—
106	#395	Dic₄₃	1	4	yes	—
107	C₄×C₄⋊C₄	C₂×D₄₃	1	4	yes	—
108	#376	C₂×D₄₃	1	1	yes	characteristic
109	#336	—	43	4	no	—
110	#336	—	43	8	no	—
111	#336	—	43	4	no	—
112	C₄₃	#22243	1	1	yes	characteristic, complemented, Sylow
113	C₂²×Q₈	C₂×Dic₄₃	1	2	yes	—
114	C₂²×C₈	—	43	2	no	—
115	C₂²×C₈	—	43	2	no	—
116	C₂²×C₈	—	43	4	no	—
117	C₄×Q₈	—	2	4	no	—
118	C₄×Q₈	—	2	4	no	—
119	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
120	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
121	C₂×C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
122	C₂×C₄²	C₂²×D₄₃	1	4	yes	—
123	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
124	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
125	C₄⋊C₈	—	86	8	no	—
126	C₄⋊C₈	—	86	4	no	—
127	#97	—	86	8	no	—
128	#97	—	86	2	no	—
129	C₄×C₈	—	43	4	no	—
130	#95	C₂×Dic₄₃	1	4	yes	—
131	#95	C₂×Dic₄₃	1	4	yes	—
132	#95	C₂×Dic₄₃	1	4	yes	—
133	C₂×Q₈	—	2	4	no	—
134	C₂×Q₈	—	4	4	no	—
135	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
136	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
137	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
138	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
139	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
140	C₂×C₈	—	43	4	no	—
141	C₂×C₈	—	86	4	no	—
142	C₂×C₈	—	86	4	no	—
143	C₂×C₈	—	86	2	no	—
144	C₂×C₈	—	86	8	no	—
145	C₂×C₈	—	86	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	8	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄²	—	2	4	no	—
150	C₄²	—	2	4	no	—
151	C₄²	—	2	4	no	—
152	C₄²	D₄×D₄₃	1	2	yes	—
153	C₄²	#10621900	1	2	yes	—
154	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
155	Q₈	—	4	8	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	D₄×Dic₄₃	1	4	yes	—
165	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
166	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
167	C₂×C₄	#13063373	1	4	yes	—
168	C₈	—	86	4	no	—
169	C₈	—	172	4	no	—
170	C₈	—	172	2	no	—
171	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
172	C₂²	#14583377	1	2	yes	—
173	C₂²	#14583666	1	4	yes	—
174	C₄	—	2	4	no	—
175	C₄	—	4	4	no	—
176	C₄	—	4	4	no	—
177	C₄	—	4	4	no	—
178	C₄	#14583474	1	2	yes	—
179	C₄	#14583510	1	2	yes	—
180	C₂	#19137315	1	1	yes	characteristic
181	C₂	#19143299	1	2	yes	—
182	C₂	#19143701	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14583030

C₂

yes

maximal

#19135536

C₂

yes

maximal

#19135161

C₂

yes

maximal

#19134715

C₂

yes

maximal, characteristic

#19134347

C₂

yes

maximal, characteristic

C₄₃×#2399

C₂

yes

maximal, characteristic

#14583030

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₂×#13063236

C₂²

yes

characteristic

C₂×#13063236

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

characteristic

#14582654

C₂²

yes

—

#14582654

C₂²

yes

—

#14582654

C₂²

yes

—

C₂×C₄×Q₈×C₄₃

C₂²

yes

—

C₄₃×#398

C₄

yes

—

C₄₃×#395

C₄

yes

—

C₄×C₄⋊C₄×C₄₃

C₂²

yes

—

C₄₃×#376

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

#13063236

—

C₄×C₄₃⋊C₈

D₄

yes

—

C₂²×Q₈×C₄₃

C₂×C₄

yes

—

C₄×Q₈×C₄₃

—

C₄×Q₈×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

#95×C₄₃

C₂×C₄

yes

—

#95×C₄₃

C₂×C₄

yes

—

#95×C₄₃

C₂×C₄

yes

—

C₂×Q₈×C₄₃

—

C₂×Q₈×C₄₃

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂³×C₄₃

C₂³×C₄

yes

characteristic

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

2₋¹⁺⁴

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#22243

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#519

yes

—

C₂²×C₄₃

#568

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#536

yes

—

C₄×C₄₃

#541

yes

—

C₂×#446

—

#2399

D₄₃

yes

—

—

—

—

C₂×C₄₃

#2327

yes

characteristic

C₂×C₄₃

#2930

yes

—

C₂×C₄₃

#2987

yes

—

C₂×C₄×Q₈

C₂×D₄₃

yes

—

#446

—

100

C₂×C₄⋊C₈

—

101

C₂×C₄⋊C₈

—

102

C₂×#97

—

103

C₂×#97

—

104

C₂×C₄×C₈

—

105

#398

Dic₄₃

yes

—

106

#395

Dic₄₃

yes

—

107

C₄×C₄⋊C₄

C₂×D₄₃

yes

—

108

#376

C₂×D₄₃

yes

characteristic

109

#336

—

110

#336

—

111

#336

—

112

C₄₃

#22243

yes

characteristic, complemented, Sylow

113

C₂²×Q₈

C₂×Dic₄₃

yes

—

114

C₂²×C₈

—

115

C₂²×C₈

—

116

C₂²×C₈

—

117

C₄×Q₈

—

118

C₄×Q₈

—

119

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

120

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

121

C₂×C₄⋊C₄

C₂×Dic₄₃

yes

—

122

C₂×C₄²

C₂²×D₄₃

yes

—

123

C₂×C₄²

C₂²×D₄₃

yes

—

124

C₂×C₄²

C₂²×D₄₃

yes

characteristic

125

C₄⋊C₈

—

126

C₄⋊C₈

—

127

#97

—

128

#97

—

129

C₄×C₈

—

130

#95

C₂×Dic₄₃

yes

—

131

#95

C₂×Dic₄₃

yes

—

132

#95

C₂×Dic₄₃

yes

—

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂²×C₄

C₂²×Dic₄₃

yes

—

136

C₂²×C₄

C₂²×Dic₄₃

yes

—

137

C₂²×C₄

C₂²×Dic₄₃

yes

—

138

C₂²×C₄

C₂²×Dic₄₃

yes

—

139

C₂²×C₄

C₂³×D₄₃

yes

characteristic

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₂×C₈

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

D₄×D₄₃

yes

—

153

C₄²

#10621900

yes

—

154

C₂³

C₂³×Dic₄₃

yes

characteristic

155

Q₈

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

—

160

C₂×C₄

—

161

C₂×C₄

—

162

C₂×C₄

—

163

C₂×C₄

—

164

C₂×C₄

D₄×Dic₄₃

yes

—

165

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

166

C₂×C₄

C₂×#10621900

yes

characteristic

167

C₂×C₄

#13063373

yes

—

168

C₈

—

169

C₈

—

172

—

170

C₈

—

172

—

171

C₂²

C₂×D₄×Dic₄₃

yes

characteristic

172

C₂²

#14583377

yes

—

173

C₂²

#14583666

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

#14583474

yes

—

179

C₄

#14583510

yes

—

180

C₂

#19137315

yes

characteristic

181

C₂

#19143299

yes

—

182

C₂

#19143701

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator, socle, prime radical	C₂³×C₄₃	C₂³×C₄
Fitting	C₄₃×#2399	C₂
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2399	D₄₃
hypocenter	C₄₃	#22243
LCS	G ▷ C₂³×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2399

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₄₃

commutator, socle, prime radical

C₂³×C₄₃

C₂³×C₄

Fitting

C₄₃×#2399

C₂

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂³×C₄₃ ▷ C₄₃

C₁ ◁ C₂²×C₄ ◁ #2399

N	H ≅ G/N	type
112. C₄₃	#22243	semidirect

H ≅ G/N

type

112. C₄₃

#22243

semidirect

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

subgroup	index
C₂×#14583030	2
#19135536	2
#19135161	2
#19134715	2
#19134347	2
C₄₃×#2399	2
#22243	43

subgroup

index

C₄₃×#2399

Subgroups — #25110174

Conjugacy classes of subgroups

Complements

Maximal subgroups