Group #24995218 — subgroups

Subgroups — #24995218

Order 11008.

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

normal subgroups only

This group has 294 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	C₄×#13063261	C₂	1	1	yes	maximal, characteristic
2	#19133348	C₂	1	2	yes	maximal
3	C₂×#14582650	C₂	1	1	yes	maximal, characteristic
4	#19132444	C₂	1	2	yes	maximal
5	C₄₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063261	C₄	1	2	yes	—
7	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
8	C₄×#10621877	C₂²	1	2	yes	—
9	#14582664	C₄	1	1	yes	characteristic
10	#14582664	C₄	1	1	yes	characteristic
11	#14582651	—	2	2	no	—
12	#14582650	—	2	2	no	—
13	#14582649	—	2	2	no	—
14	C₄₃×#431	C₂²	1	1	yes	characteristic
15	C₄₃×#431	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
17	C₄₃×#322	—	2	2	no	—
18	C₂×#10621877	C₂×C₄	1	2	yes	—
19	C₂×#10621877	C₂×C₄	1	2	yes	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	#13063261	—	2	4	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	#13063236	—	2	2	no	—
25	C₄×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₄₃	—	2	1	no	—
30	C₄⋊C₈×C₄₃	D₄	1	2	yes	—
31	C₂²⋊C₈×C₄₃	—	2	1	no	—
32	C₂²⋊C₈×C₄₃	Q₈	1	2	yes	—
33	C₄×C₈×C₄₃	—	2	1	no	—
34	C₄×C₈×C₄₃	—	2	1	no	—
35	C₄×C₈×C₄₃	D₄	1	2	yes	—
36	C₄×C₈×C₄₃	Q₈	1	2	yes	—
37	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
40	C₂×C₈×C₄₃	—	2	2	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
45	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
46	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
49	C₄²×C₄₃	#41	1	1	yes	characteristic
50	#10621877	—	2	2	no	—
51	#10621877	—	2	2	no	—
52	#10621877	—	2	2	no	—
53	#10621877	M₁₆	1	4	yes	—
54	C₂×C₄₃⋊C₈	—	2	2	no	—
55	C₂×C₄₃⋊C₈	—	2	2	no	—
56	C₂×C₄₃⋊C₈	—	4	2	no	—
57	C₂×C₄₃⋊C₈	—	8	2	no	—
58	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	2	2	no	—
60	C₂×C₄×C₄₃	—	4	1	no	—
61	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
64	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
66	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
68	C₂×C₄×C₄₃	#131	1	2	yes	—
69	C₈×C₄₃	—	2	2	no	—
70	C₈×C₄₃	—	2	2	no	—
71	C₈×C₄₃	—	4	2	no	—
72	C₈×C₄₃	—	4	2	no	—
73	C₄₃⋊C₈	—	2	4	no	—
74	C₄₃⋊C₈	—	4	2	no	—
75	C₄₃⋊C₈	—	8	2	no	—
76	#9609	—	43	1	no	maximal, Sylow
77	C₂²×C₄₃	—	2	1	no	—
78	C₂²×C₄₃	—	2	1	no	—
79	C₂²×C₄₃	#389	1	1	yes	characteristic
80	C₂²×C₄₃	#446	1	2	yes	—
81	C₄×C₄₃	—	2	1	no	—
82	C₄×C₄₃	—	2	1	no	—
83	C₄×C₄₃	—	2	1	no	—
84	C₄×C₄₃	#423	1	1	yes	characteristic
85	C₄×C₄₃	#424	1	1	yes	characteristic
86	C₄×C₄₃	#435	1	2	yes	—
87	C₄×C₄₃	#471	1	2	yes	—
88	C₄×#108	—	43	1	no	—
89	#1565	—	43	2	no	—
90	C₂×#334	—	43	1	no	—
91	#1452	D₄₃	1	1	yes	characteristic
92	#1283	—	43	2	no	—
93	C₂×C₄₃	—	2	2	no	—
94	C₂×C₄₃	#1940	1	1	yes	characteristic
95	C₂×C₄₃	#1951	1	1	yes	characteristic
96	C₂×C₄₃	#1953	1	1	yes	characteristic
97	#431	C₂×D₄₃	1	1	yes	characteristic
98	#431	C₂×D₄₃	1	1	yes	characteristic
99	C₂×#108	—	43	2	no	—
100	C₂×C₄⋊C₈	—	43	2	no	—
101	C₄×M₁₆	—	43	2	no	—
102	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
103	#341	—	43	1	no	—
104	#341	—	43	1	no	—
105	#335	—	86	2	no	—
106	#334	—	86	2	no	—
107	#334	—	86	2	no	—
108	#322	—	2	2	no	—
109	C₄₃	#9609	1	1	yes	characteristic, complemented, Sylow
110	C₂×M₁₆	—	43	2	no	—
111	C₂×M₁₆	—	43	2	no	—
112	C₂²×C₈	—	86	2	no	—
113	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
114	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
115	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
116	#108	—	86	4	no	—
117	C₄⋊C₈	—	2	1	no	—
118	C₄⋊C₈	—	86	2	no	—
119	C₄⋊C₈	—	86	2	no	—
120	C₄⋊C₈	D₁₇₂	1	2	yes	—
121	C₂²⋊C₈	—	2	1	no	—
122	C₂²⋊C₈	Dic₈₆	1	2	yes	—
123	#97	—	86	2	no	—
124	C₄×C₈	—	2	1	no	—
125	C₄×C₈	—	2	1	no	—
126	C₄×C₈	—	86	2	no	—
127	C₄×C₈	Dic₈₆	1	2	yes	—
128	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
129	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
130	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
131	C₂²×C₄	#10621898	1	1	yes	characteristic
132	M₁₆	—	43	4	no	—
133	M₁₆	—	86	2	no	—
134	M₁₆	—	86	2	no	—
135	M₁₆	—	86	2	no	—
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	2	no	—
140	C₂×C₈	—	86	2	no	—
141	C₂×C₈	—	86	2	no	—
142	C₂×C₈	—	344	2	no	—
143	C₂×C₈	—	172	2	no	—
144	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
145	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
146	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
147	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
148	C₄²	#10621898	1	1	yes	characteristic
149	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
150	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
154	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
155	C₂×C₄	#13063299	1	1	yes	characteristic
156	C₂×C₄	#13063304	1	1	yes	characteristic
157	C₂×C₄	#13063317	1	1	yes	characteristic
158	C₂×C₄	#13063325	1	1	yes	characteristic
159	C₂×C₄	M₁₆×D₄₃	1	2	yes	—
160	C₂×C₄	#13063333	1	2	yes	—
161	C₈	—	2	2	no	—
162	C₈	—	2	2	no	—
163	C₈	—	4	2	no	—
164	C₈	—	4	2	no	—
165	C₈	—	86	4	no	—
166	C₈	—	344	2	no	—
167	C₈	—	172	2	no	—
168	C₂²	—	2	1	no	—
169	C₂²	—	2	1	no	—
170	C₂²	#14582797	1	1	yes	characteristic
171	C₂²	#14583027	1	2	yes	—
172	C₄	—	2	1	no	—
173	C₄	—	2	1	no	—
174	C₄	—	2	1	no	—
175	C₄	#14582922	1	1	yes	characteristic
176	C₄	#14582927	1	1	yes	characteristic
177	C₄	#14582972	1	2	yes	—
178	C₄	#14583162	1	2	yes	—
179	C₂	—	2	2	no	—
180	C₂	#19135318	1	1	yes	characteristic
181	C₂	#19135375	1	1	yes	characteristic
182	C₂	#19135385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063261

C₂

yes

maximal, characteristic

#19133348

C₂

yes

maximal

C₂×#14582650

C₂

yes

maximal, characteristic

#19132444

C₂

yes

maximal

C₄₃×#1452

C₂

yes

maximal, characteristic

C₂×#13063261

C₄

yes

—

C₂×C₄⋊(C₄₃⋊C₈)

C₂²

yes

—

C₄×#10621877

C₂²

yes

—

#14582664

C₄

yes

characteristic

#14582664

C₄

yes

—

—

—

C₄₃×#431

C₂²

yes

characteristic

C₄₃×#431

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#322

—

C₂×#10621877

C₂×C₄

yes

—

C₂×#10621877

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

#13063236

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

D₄

yes

—

C₂²⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

Q₈

yes

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

C₄²×C₄₃

yes

—

—

—

M₁₆

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄×Q₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#125

yes

characteristic

C₂×C₄×C₄₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#9609

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#389

yes

characteristic

C₂²×C₄₃

#446

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

#471

yes

—

—

—

—

D₄₃

yes

characteristic

#1283

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×D₄₃

yes

characteristic

#431

C₂×D₄₃

yes

characteristic

C₂×#108

—

100

C₂×C₄⋊C₈

—

101

C₄×M₁₆

—

102

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

103

#341

—

104

#341

—

105

#335

—

106

#334

—

107

#334

—

108

#322

—

109

C₄₃

#9609

yes

characteristic, complemented, Sylow

110

C₂×M₁₆

—

111

C₂×M₁₆

—

112

C₂²×C₈

—

113

C₂²×C₈

C₄×D₄₃

yes

characteristic

114

C₂²×C₈

C₄×D₄₃

yes

characteristic

115

C₂×C₄²

C₂²×D₄₃

yes

characteristic

116

#108

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

C₄⋊C₈

D₁₇₂

yes

—

121

C₂²⋊C₈

—

122

C₂²⋊C₈

Dic₈₆

yes

—

123

#97

—

124

C₄×C₈

—

125

C₄×C₈

—

126

C₄×C₈

—

127

C₄×C₈

Dic₈₆

yes

—

128

C₄×C₈

C₂²⋊D₄₃

yes

—

129

C₂²×C₄

C₂×Dic₈₆

yes

characteristic

130

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

131

C₂²×C₄

#10621898

yes

characteristic

132

M₁₆

—

133

M₁₆

—

134

M₁₆

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₂×C₈

—

344

—

143

C₂×C₈

—

172

—

144

C₂×C₈

Dic₄₃⋊C₄

yes

—

145

C₂×C₈

Dic₄₃⋊C₄

yes

—

146

C₄²

C₂×Dic₈₆

yes

characteristic

147

C₄²

C₂×D₁₇₂

yes

characteristic

148

C₄²

#10621898

yes

characteristic

149

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

150

C₂³

C₄×Dic₈₆

yes

characteristic

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

154

C₂×C₄

C₄×D₁₇₂

yes

characteristic

155

C₂×C₄

#13063299

yes

characteristic

156

C₂×C₄

#13063304

yes

characteristic

157

C₂×C₄

#13063317

yes

characteristic

158

C₂×C₄

#13063325

yes

characteristic

159

C₂×C₄

M₁₆×D₄₃

yes

—

160

C₂×C₄

#13063333

yes

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

344

—

167

C₈

—

172

—

168

C₂²

—

169

C₂²

—

170

C₂²

#14582797

yes

characteristic

171

C₂²

#14583027

yes

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#14582922

yes

characteristic

176

C₄

#14582927

yes

characteristic

177

C₄

#14582972

yes

—

178

C₄

#14583162

yes

—

179

C₂

—

180

C₂

#19135318

yes

characteristic

181

C₂

#19135375

yes

characteristic

182

C₂

#19135385

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063299
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1452	C₂
socle	C₂²×C₄₃	#389
prime radical	C₂³×C₄₃	C₄×Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1452	D₄₃
hypocenter	C₄₃	#9609
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂×C₄

#13063299

commutator

C₂²×C₄×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1452

C₂

socle

C₂²×C₄₃

#389

prime radical

C₂³×C₄₃

C₄×Q₈

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
109. C₄₃	#9609	semidirect

H ≅ G/N

type

109. C₄₃

#9609

semidirect

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

subgroup	index
C₄×#13063261	2
#19133348	2
C₂×#14582650	2
#19132444	2
C₄₃×#1452	2
#9609	43

subgroup

index

C₄₃×#1452

Subgroups — #24995218

Conjugacy classes of subgroups

Complements

Maximal subgroups