Group #24990300 — subgroups

Subgroups — #24990300

Order 11008.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#19134759	C₂	1	1	yes	maximal, characteristic
2	#19134733	C₂	1	1	yes	maximal, characteristic
3	#19132454	C₂	1	2	yes	maximal
4	#19132454	C₂	1	2	yes	maximal
5	C₄₃×#1452	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063259	—	2	1	no	—
7	C₂×#13063257	—	2	1	no	—
8	#14582923	C₂²	1	1	yes	characteristic
9	#14582923	C₂²	1	1	yes	characteristic
10	#14582765	C₂²	1	2	yes	—
11	#14582664	—	2	1	no	—
12	#14582664	—	2	1	no	—
13	C₄₃×#431	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×#322	C₄	1	4	yes	complemented
16	C₂×#10621877	—	2	1	no	—
17	C₂×#10621877	—	2	1	no	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	#13063259	—	4	2	no	—
20	#13063257	—	4	2	no	—
21	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄₃⋊#95	—	2	4	no	—
24	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
26	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₃	—	2	2	no	—
28	C₂²⋊C₈×C₄₃	—	2	2	no	—
29	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₃	#41	1	2	yes	—
34	C₂²×Dic₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	—	2	1	no	—
36	C₂×C₈×C₄₃	—	2	1	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	—	2	1	no	—
39	C₂×C₈×C₄₃	—	2	4	no	—
40	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₄₃	#41	1	2	yes	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	C₄⋊Dic₄₃	—	4	2	no	—
47	#10621877	—	4	1	no	—
48	#10621877	—	4	1	no	—
49	#10621877	—	4	1	no	—
50	#10621877	—	4	1	no	—
51	C₂×C₄₃⋊C₈	—	4	1	no	—
52	C₂×C₄₃⋊C₈	—	4	1	no	—
53	C₂³×C₄₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₃	—	4	2	no	—
55	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#117	1	2	yes	—
57	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#125	1	2	yes	—
59	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
62	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
63	C₂×Dic₄₃	—	8	4	no	—
64	C₂×Dic₄₃	—	8	2	no	—
65	C₈×C₄₃	—	2	2	no	—
66	C₈×C₄₃	—	2	2	no	—
67	C₈×C₄₃	—	4	4	no	—
68	C₄₃⋊C₈	—	8	1	no	—
69	C₄₃⋊C₈	—	8	1	no	—
70	#8795	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	—	2	2	no	—
72	C₂²×C₄₃	#382	1	1	yes	characteristic
73	C₂²×C₄₃	#490	1	1	yes	characteristic
74	C₂²×C₄₃	#491	1	1	yes	characteristic
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	#428	1	1	yes	characteristic
78	C₄×C₄₃	#428	1	1	yes	characteristic
79	C₄×C₄₃	#489	1	1	yes	characteristic
80	C₄×C₄₃	#489	1	1	yes	characteristic
81	Dic₄₃	—	16	2	no	—
82	#1846	—	43	1	no	—
83	#1837	—	43	1	no	—
84	#1452	D₄₃	1	1	yes	characteristic
85	#1286	—	43	2	no	—
86	#1286	—	43	2	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	—	2	1	no	—
89	C₂×C₄₃	#1833	1	2	yes	—
90	C₂×C₄₃	#1836	1	1	yes	characteristic
91	#431	C₂×D₄₃	1	2	yes	—
92	C₂×#107	—	86	1	no	—
93	C₂×#106	—	86	1	no	—
94	#423	—	43	1	no	—
95	#423	—	43	1	no	—
96	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
97	#382	—	43	2	no	—
98	#341	—	86	1	no	—
99	#341	—	86	1	no	—
100	#322	Dic₄₃	1	4	yes	complemented
101	C₄₃	#8795	1	1	yes	characteristic, complemented, Sylow
102	C₂×M₁₆	—	86	1	no	—
103	C₂×M₁₆	—	86	1	no	—
104	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
105	C₂²×C₈	D₁₇₂	1	1	yes	characteristic
106	C₂×C₄⋊C₄	—	86	2	no	—
107	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
108	#107	—	172	2	no	—
109	#106	—	172	2	no	—
110	C₄⋊C₈	—	2	2	no	—
111	C₄⋊C₈	—	86	2	no	—
112	C₄⋊C₈	—	86	2	no	—
113	C₂²⋊C₈	—	2	2	no	—
114	C₄×C₈	Dic₈₆	1	2	yes	—
115	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
116	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
117	#95	—	86	4	no	—
118	C₂²×C₄	—	172	2	no	—
119	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
120	C₂²×C₄	#10621898	1	2	yes	—
121	M₁₆	—	172	1	no	—
122	M₁₆	—	172	1	no	—
123	M₁₆	—	172	1	no	—
124	M₁₆	—	172	1	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	172	1	no	—
131	C₂×C₈	—	172	1	no	—
132	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
133	C₂×C₈	C₄⋊Dic₄₃	1	2	yes	—
134	C₄⋊C₄	—	172	2	no	—
135	C₄⋊C₄	—	172	2	no	—
136	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
137	C₄²	#10621898	1	2	yes	—
138	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
139	C₂³	#13063313	1	1	yes	characteristic
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	344	4	no	—
142	C₂×C₄	—	344	2	no	—
143	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
144	C₂×C₄	#13063284	1	2	yes	—
145	C₂×C₄	#13063313	1	1	yes	characteristic
146	C₂×C₄	#13063317	1	2	yes	—
147	C₂×C₄	#13063353	1	1	yes	characteristic
148	C₂×C₄	#13063353	1	1	yes	characteristic
149	C₂×C₄	#13063358	1	1	yes	characteristic
150	C₂×C₄	#13063358	1	1	yes	characteristic
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	4	4	no	—
154	C₈	—	344	1	no	—
155	C₈	—	344	1	no	—
156	C₂²	—	2	2	no	—
157	C₂²	#14582765	1	1	yes	characteristic
158	C₂²	#14583264	1	1	yes	characteristic
159	C₂²	#14583268	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	688	2	no	—
163	C₄	#14582941	1	1	yes	characteristic
164	C₄	#14582941	1	1	yes	characteristic
165	C₄	#14583259	1	1	yes	characteristic
166	C₄	#14583259	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#19134719	1	2	yes	—
170	C₂	#19134729	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134759

C₂

yes

maximal, characteristic

#19134733

C₂

yes

maximal, characteristic

#19132454

C₂

yes

maximal

#19132454

C₂

yes

maximal

C₄₃×#1452

C₂

yes

maximal, characteristic

C₂×#13063259

—

C₂×#13063257

—

#14582923

C₂²

yes

characteristic

#14582923

C₂²

yes

characteristic

#14582765

C₂²

yes

—

#14582664

—

#14582664

—

C₄₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#322

C₄

yes

complemented

C₂×#10621877

—

C₂×#10621877

—

C₂×C₄⋊Dic₄₃

—

#13063259

—

#13063257

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

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

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

—

—

—

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

#125

yes

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

#8795

—

maximal, Sylow

C₂²×C₄₃

—

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

—

#1846

—

#1837

—

#1452

D₄₃

yes

characteristic

#1286

—

#1286

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1833

yes

—

C₂×C₄₃

#1836

yes

characteristic

#431

C₂×D₄₃

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

—

—

100

#322

Dic₄₃

yes

complemented

101

C₄₃

#8795

yes

characteristic, complemented, Sylow

102

C₂×M₁₆

—

103

C₂×M₁₆

—

104

C₂²×C₈

D₁₇₂

yes

characteristic

105

C₂²×C₈

D₁₇₂

yes

characteristic

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₄₃

yes

characteristic

108

#107

—

172

—

109

#106

—

172

—

110

C₄⋊C₈

—

111

C₄⋊C₈

—

112

C₄⋊C₈

—

113

C₂²⋊C₈

—

114

C₄×C₈

Dic₈₆

yes

—

115

C₄×C₈

C₂×Dic₄₃

yes

—

116

C₄×C₈

C₂×Dic₄₃

yes

—

117

#95

—

118

C₂²×C₄

—

172

—

119

C₂²×C₄

C₂×D₁₇₂

yes

characteristic

120

C₂²×C₄

#10621898

yes

—

121

M₁₆

—

172

—

122

M₁₆

—

172

—

123

M₁₆

—

172

—

124

M₁₆

—

172

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

172

—

131

C₂×C₈

—

172

—

132

C₂×C₈

C₄⋊Dic₄₃

yes

—

133

C₂×C₈

C₄⋊Dic₄₃

yes

—

134

C₄⋊C₄

—

172

—

135

C₄⋊C₄

—

172

—

136

C₄²

C₂×Dic₈₆

yes

characteristic

137

C₄²

#10621898

yes

—

138

C₄²

C₂²×Dic₄₃

yes

characteristic

139

C₂³

#13063313

yes

characteristic

140

C₂×C₄

—

141

C₂×C₄

—

344

—

142

C₂×C₄

—

344

—

143

C₂×C₄

C₂×C₄⋊Dic₄₃

yes

characteristic

144

C₂×C₄

#13063284

yes

—

145

C₂×C₄

#13063313

yes

characteristic

146

C₂×C₄

#13063317

yes

—

147

C₂×C₄

#13063353

yes

characteristic

148

C₂×C₄

#13063353

yes

characteristic

149

C₂×C₄

#13063358

yes

characteristic

150

C₂×C₄

#13063358

yes

characteristic

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

344

—

155

C₈

—

344

—

156

C₂²

—

157

C₂²

#14582765

yes

characteristic

158

C₂²

#14583264

yes

characteristic

159

C₂²

#14583268

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

688

—

163

C₄

#14582941

yes

characteristic

164

C₄

#14582941

yes

characteristic

165

C₄

#14583259

yes

characteristic

166

C₄

#14583259

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#19134719

yes

—

170

C₂

#19134729

yes

characteristic

	subgroup	G/H
center	C₂²	#14582765
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1452	C₂
socle	C₂²×C₄₃	#382
prime radical	C₂³×C₄₃	#124
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1452	D₄₃
hypocenter	C₄₃	#8795
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#14582765

commutator

C₄²×C₄₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1452

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
15. C₄₃×#322	C₄	semidirect
100. #322	Dic₄₃	semidirect
101. C₄₃	#8795	semidirect

H ≅ G/N

type

15. C₄₃×#322

C₄

semidirect

100. #322

Dic₄₃

semidirect

101. C₄₃

#8795

semidirect

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

subgroup	index
#19134759	2
#19134733	2
#19132454	2
C₄₃×#1452	2
#8795	43

subgroup

index

C₄₃×#1452

Subgroups — #24990300

Conjugacy classes of subgroups

Complements

Maximal subgroups