Group #24999721 — subgroups

Subgroups — #24999721

Order 11008.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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	#19142982	C₂	1	1	yes	maximal, characteristic
2	#19135572	C₂	1	1	yes	maximal, characteristic
3	#19133160	C₂	1	2	yes	maximal
4	#19132454	C₂	1	2	yes	maximal
5	C₄₃×#1276	C₂	1	1	yes	maximal, characteristic
6	#14583456	C₂²	1	1	yes	characteristic
7	#14582923	C₂²	1	1	yes	characteristic
8	#14582797	—	2	2	no	—
9	#14582783	—	2	4	no	—
10	#14582765	C₂²	1	2	yes	—
11	#14582659	—	2	2	no	—
12	#14582637	—	2	4	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₄³×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#10621877	—	2	1	no	—
16	#13063325	Q₈	1	2	yes	—
17	#13063317	—	2	1	no	—
18	#13063304	—	2	2	no	—
19	C₄×Dic₈₆	—	2	2	no	—
20	#13063284	—	2	1	no	—
21	C₂×C₄⋊Dic₄₃	—	2	2	no	—
22	C₂×Dic₄₃⋊C₄	—	2	4	no	—
23	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
24	C₄₃⋊#95	—	2	4	no	—
25	C₄₃⋊#95	—	2	2	no	—
26	C₄₃⋊#95	D₄	1	4	yes	—
27	C₂×C₄²×C₄₃	—	2	2	no	—
28	C₂×C₄²×C₄₃	D₄	1	2	yes	—
29	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₃	—	2	2	no	—
31	C₂²⋊C₈×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	—	2	2	no	—
33	C₂²×C₄×C₄₃	—	2	2	no	—
34	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
36	C₂²×Dic₄₃	—	2	4	no	—
37	C₂×Dic₈₆	—	2	2	no	—
38	C₂×C₈×C₄₃	—	4	2	no	—
39	C₄²×C₄₃	—	2	2	no	—
40	C₄²×C₄₃	—	2	2	no	—
41	C₄²×C₄₃	—	4	4	no	—
42	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
43	C₄²×C₄₃	#41	1	2	yes	—
44	C₄²×C₄₃	#41	1	1	yes	characteristic
45	C₂²⋊Dic₄₃	—	4	1	no	—
46	C₄⋊Dic₄₃	—	2	2	no	—
47	C₄⋊Dic₄₃	—	2	2	no	—
48	C₄⋊Dic₄₃	—	4	2	no	—
49	C₄⋊Dic₄₃	—	4	2	no	—
50	Dic₄₃⋊C₄	—	4	8	no	—
51	Dic₄₃⋊C₄	—	4	2	no	—
52	C₄×Dic₄₃	—	4	1	no	—
53	#10621877	—	4	2	no	—
54	C₂×C₄₃⋊C₈	—	4	1	no	—
55	C₂³×C₄₃	#120	1	1	yes	characteristic
56	C₂×C₄×C₄₃	—	2	4	no	—
57	C₂×C₄×C₄₃	—	4	2	no	—
58	C₂×C₄×C₄₃	—	4	2	no	—
59	C₂×C₄×C₄₃	—	4	2	no	—
60	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
61	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#124	1	2	yes	—
63	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
64	C₂×C₄×C₄₃	#135	1	2	yes	—
65	C₂×C₄×C₄₃	#137	1	2	yes	—
66	C₂×Dic₄₃	—	4	8	no	—
67	C₂×Dic₄₃	—	4	2	no	—
68	C₂×Dic₄₃	—	4	4	no	—
69	Dic₈₆	—	8	2	no	—
70	C₈×C₄₃	—	8	2	no	—
71	C₄₃⋊C₈	—	8	1	no	—
72	#10369	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	#393	1	1	yes	characteristic
75	C₂²×C₄₃	#452	1	2	yes	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	—	2	2	no	—
78	C₄×C₄₃	—	4	4	no	—
79	C₄×C₄₃	#479	1	2	yes	—
80	C₄×C₄₃	#488	1	1	yes	characteristic
81	C₄×C₄₃	#491	1	1	yes	characteristic
82	Dic₄₃	—	8	4	no	—
83	Dic₄₃	—	8	2	no	—
84	#2864	—	43	1	no	—
85	#1994	—	43	1	no	—
86	#1518	—	43	2	no	—
87	#1286	—	43	2	no	—
88	#1276	D₄₃	1	1	yes	characteristic
89	C₂×C₄₃	—	2	2	no	—
90	C₂×C₄₃	#2026	1	1	yes	characteristic
91	C₂×C₄₃	#2031	1	2	yes	—
92	#533	—	43	1	no	—
93	#431	C₂×D₄₃	1	2	yes	—
94	#423	—	43	1	no	—
95	#389	—	86	2	no	—
96	#387	—	86	4	no	—
97	#382	—	43	2	no	—
98	C₄³	C₂×D₄₃	1	1	yes	characteristic
99	#339	—	86	2	no	—
100	#328	—	86	4	no	—
101	C₄₃	#10369	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	86	1	no	—
103	C₄⋊Q₈	—	43	2	no	—
104	#125	—	86	1	no	—
105	C₂²⋊Q₈	—	86	2	no	—
106	C₄×Q₈	—	86	2	no	—
107	#117	—	86	1	no	—
108	C₂×C₄⋊C₄	—	86	4	no	—
109	C₂×C₄⋊C₄	—	86	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	D₁₇₂	1	2	yes	—
112	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
113	C₄⋊C₈	—	2	2	no	—
114	C₄⋊C₈	—	86	2	no	—
115	C₂²⋊C₈	—	2	2	no	—
116	C₄×C₈	—	2	2	no	—
117	#95	—	43	4	no	—
118	#95	—	86	4	no	—
119	#95	—	86	2	no	—
120	C₂×Q₈	—	86	2	no	—
121	C₂²×C₄	—	2	2	no	—
122	C₂²×C₄	—	86	4	no	—
123	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
124	C₂²×C₄	D₄×D₄₃	1	2	yes	—
125	M₁₆	—	172	2	no	—
126	C₂×C₈	—	4	2	no	—
127	C₂×C₈	—	172	1	no	—
128	C₄⋊C₄	—	86	2	no	—
129	C₄⋊C₄	—	86	2	no	—
130	C₄⋊C₄	—	172	8	no	—
131	C₄⋊C₄	—	172	2	no	—
132	C₄⋊C₄	—	172	2	no	—
133	C₄⋊C₄	—	172	2	no	—
134	C₂²⋊C₄	—	172	1	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	2	2	no	—
137	C₄²	—	4	4	no	—
138	C₄²	—	172	1	no	—
139	C₄²	#10621898	1	2	yes	—
140	C₄²	#10621900	1	1	yes	characteristic
141	C₄²	Q₈×D₄₃	1	1	yes	characteristic
142	C₂³	#13063295	1	1	yes	characteristic
143	Q₈	—	344	2	no	—
144	C₂×C₄	—	2	4	no	—
145	C₂×C₄	—	4	2	no	—
146	C₂×C₄	—	4	2	no	—
147	C₂×C₄	—	4	2	no	—
148	C₂×C₄	—	172	8	no	—
149	C₂×C₄	—	172	4	no	—
150	C₂×C₄	—	172	2	no	—
151	C₂×C₄	#13063303	1	2	yes	—
152	C₂×C₄	#13063305	1	1	yes	characteristic
153	C₂×C₄	#13063311	1	2	yes	—
154	C₂×C₄	#13063313	1	1	yes	characteristic
155	C₂×C₄	#13063347	1	2	yes	—
156	C₂×C₄	#13063358	1	2	yes	—
157	C₈	—	8	2	no	—
158	C₈	—	344	1	no	—
159	C₂²	—	2	2	no	—
160	C₂²	#14582812	1	1	yes	characteristic
161	C₂²	#14583063	1	2	yes	—
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	4	4	no	—
165	C₄	—	344	4	no	—
166	C₄	—	344	2	no	—
167	C₄	#14583207	1	2	yes	—
168	C₄	#14583254	1	1	yes	characteristic
169	C₄	#14583268	1	1	yes	characteristic
170	C₂	—	2	2	no	—
171	C₂	#19135747	1	1	yes	characteristic
172	C₂	#19135768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19142982

C₂

yes

maximal, characteristic

#19135572

C₂

yes

maximal, characteristic

#19133160

C₂

yes

maximal

#19132454

C₂

yes

maximal

C₄₃×#1276

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

characteristic

#14582923

C₂²

yes

—

—

C₂²

yes

—

#14582659

—

#14582637

—

C₄₃×#431

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

#13063325

Q₈

yes

—

#13063317

—

#13063304

—

C₄×Dic₈₆

—

#13063284

—

C₂×C₄⋊Dic₄₃

—

C₂×Dic₄₃⋊C₄

—

C₄⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₄₃⋊#95

—

C₄₃⋊#95

D₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

C₂×D₄

yes

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#120

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₃

#124

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

#137

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10369

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

#393

yes

characteristic

C₂²×C₄₃

#452

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#479

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

#2026

yes

characteristic

C₂×C₄₃

#2031

yes

—

#533

—

#431

C₂×D₄₃

yes

—

—

—

—

—

C₄³

C₂×D₄₃

yes

characteristic

#339

—

100

#328

—

101

C₄₃

#10369

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₄⋊Q₈

—

104

#125

—

105

C₂²⋊Q₈

—

106

C₄×Q₈

—

107

#117

—

108

C₂×C₄⋊C₄

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄²

—

111

C₂×C₄²

D₁₇₂

yes

—

112

C₂×C₄²

C₂²×D₄₃

yes

characteristic

113

C₄⋊C₈

—

114

C₄⋊C₈

—

115

C₂²⋊C₈

—

116

C₄×C₈

—

117

#95

—

118

#95

—

119

#95

—

120

C₂×Q₈

—

121

C₂²×C₄

—

122

C₂²×C₄

—

123

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

124

C₂²×C₄

D₄×D₄₃

yes

—

125

M₁₆

—

172

—

126

C₂×C₈

—

127

C₂×C₈

—

172

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

172

—

131

C₄⋊C₄

—

172

—

132

C₄⋊C₄

—

172

—

133

C₄⋊C₄

—

172

—

134

C₂²⋊C₄

—

172

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

172

—

139

C₄²

#10621898

yes

—

140

C₄²

#10621900

yes

characteristic

141

C₄²

Q₈×D₄₃

yes

characteristic

142

C₂³

#13063295

yes

characteristic

143

Q₈

—

344

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

172

—

149

C₂×C₄

—

172

—

150

C₂×C₄

—

172

—

151

C₂×C₄

#13063303

yes

—

152

C₂×C₄

#13063305

yes

characteristic

153

C₂×C₄

#13063311

yes

—

154

C₂×C₄

#13063313

yes

characteristic

155

C₂×C₄

#13063347

yes

—

156

C₂×C₄

#13063358

yes

—

157

C₈

—

158

C₈

—

344

—

159

C₂²

—

160

C₂²

#14582812

yes

characteristic

161

C₂²

#14583063

yes

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

344

—

166

C₄

—

344

—

167

C₄

#14583207

yes

—

168

C₄

#14583254

yes

characteristic

169

C₄

#14583268

yes

characteristic

170

C₂

—

171

C₂

#19135747

yes

characteristic

172

C₂

#19135768

yes

—

	subgroup	G/H
center	C₂²	#14582812
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1276	C₂
socle	C₂²×C₄₃	#393
prime radical	C₂³×C₄₃	#120
hypercenter	#1276	D₄₃
hypocenter	C₄₃	#10369
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

subgroup

G/H

center

C₂²

#14582812

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1276

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1276

N	H ≅ G/N	type
101. C₄₃	#10369	semidirect

H ≅ G/N

type

101. C₄₃

#10369

semidirect

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

subgroup	index
#19142982	2
#19135572	2
#19133160	2
#19132454	2
C₄₃×#1276	2
#10369	43

subgroup

index

C₄₃×#1276

Subgroups — #24999721

Conjugacy classes of subgroups

Complements

Maximal subgroups