Group #24983649 — subgroups

Subgroups — #24983649

Order 11008.

214 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 87 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	#19134723	C₂	1	1	yes	maximal, characteristic
2	#19133358	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#2163	C₂	1	1	yes	maximal, characteristic
4	#14582956	—	2	1	no	—
5	#14582769	—	2	1	no	—
6	#14582662	—	2	1	no	—
7	#14582662	C₄	1	2	yes	—
8	#14582652	C₄	1	2	yes	complemented
9	#14582650	—	2	1	no	—
10	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
11	C₄₃×C₄⋊C₁₆	—	2	1	no	—
12	C₄₃×C₂²⋊C₁₆	—	2	1	no	—
13	C₄₃×#346	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₄₃	—	2	2	no	—
15	C₂×C₄⋊Dic₄₃	—	4	1	no	—
16	C₂×C₄⋊Dic₄₃	—	4	1	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₃⋊C₈)	—	4	1	no	—
19	#13063236	—	4	1	no	—
20	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₄₃	—	2	2	no	—
24	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
31	C₂²×Dic₄₃	—	4	2	no	—
32	C₂×C₈×C₄₃	—	4	1	no	—
33	C₂×C₈×C₄₃	—	4	1	no	—
34	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
38	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
41	C₄²×C₄₃	Q₁₆	1	1	yes	characteristic
42	C₄⋊Dic₄₃	—	4	2	no	—
43	C₄⋊Dic₄₃	—	4	2	no	—
44	C₄⋊Dic₄₃	—	8	2	no	—
45	C₄⋊Dic₄₃	—	8	2	no	—
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₁₆×C₄₃	—	4	2	no	—
48	C₂³×C₄₃	#102	1	1	yes	characteristic
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	—	2	1	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	1	no	—
53	C₂×C₄×C₄₃	—	2	2	no	—
54	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
57	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
59	C₂×C₄×C₄₃	#107	1	1	yes	characteristic
60	C₂×Dic₄₃	—	8	2	no	—
61	C₂×Dic₄₃	—	8	2	no	—
62	C₂×Dic₄₃	—	8	2	no	—
63	C₈×C₄₃	—	2	1	no	—
64	C₈×C₄₃	—	2	1	no	—
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	—	2	1	no	—
67	C₄₃⋊C₈	—	8	2	no	—
68	#7222	—	43	1	no	maximal, Sylow
69	C₂²×C₄₃	—	2	1	no	—
70	C₂²×C₄₃	—	2	1	no	—
71	C₂²×C₄₃	#340	1	1	yes	characteristic
72	C₂²×C₄₃	#361	1	1	yes	characteristic
73	C₂²×C₄₃	#362	1	1	yes	characteristic
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	2	1	no	—
76	C₄×C₄₃	—	2	1	no	—
77	C₄×C₄₃	—	2	1	no	—
78	C₄×C₄₃	H(3,ℤ/4)	1	1	yes	characteristic
79	C₄×C₄₃	#344	1	1	yes	characteristic
80	C₄×C₄₃	#360	1	2	yes	—
81	Dic₄₃	—	16	2	no	—
82	#2163	D₄₃	1	1	yes	characteristic
83	#1834	—	43	1	no	—
84	#1568	—	43	1	no	—
85	C₂×C₄₃	—	2	1	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	#1286	1	1	yes	characteristic
88	C₂×C₄₃	#1386	1	1	yes	characteristic
89	C₂×C₄₃	#1387	1	1	yes	characteristic
90	#432	—	86	1	no	—
91	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
92	#384	—	86	1	no	—
93	C₄⋊C₁₆	—	2	1	no	—
94	C₂²⋊C₁₆	—	2	1	no	—
95	#346	Dic₄₃	1	2	yes	complemented
96	#340	—	43	2	no	—
97	#340	—	86	1	no	—
98	#335	—	43	2	no	—
99	#334	—	86	1	no	—
100	C₄₃	#7222	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	86	2	no	—
104	C₂×C₄⋊C₄	—	172	1	no	—
105	C₂×C₄⋊C₄	—	172	1	no	—
106	C₂×C₄²	D₁₇₂	1	1	yes	characteristic
107	C₂×C₁₆	—	2	2	no	—
108	C₄⋊C₈	—	86	2	no	—
109	C₂²⋊C₈	—	172	1	no	—
110	#97	—	172	1	no	—
111	C₄×C₈	Dic₈₆	1	1	yes	characteristic
112	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
113	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
114	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
115	C₂²×C₄	—	172	2	no	—
116	C₂²×C₄	#10621873	1	1	yes	characteristic
117	C₂²×C₄	D₃₄₄	1	1	yes	characteristic
118	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
119	C₂×C₈	—	4	1	no	—
120	C₂×C₈	—	4	1	no	—
121	C₂×C₈	—	172	2	no	—
122	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
123	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
124	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
125	C₂×C₈	C₂²⋊Dic₄₃	1	1	yes	characteristic
126	C₄⋊C₄	—	344	2	no	—
127	C₄⋊C₄	—	344	2	no	—
128	C₄⋊C₄	—	172	2	no	—
129	C₄⋊C₄	—	172	2	no	—
130	C₄²	#10621873	1	1	yes	characteristic
131	C₄²	Dic₁₇₂	1	1	yes	characteristic
132	C₄²	C₄⋊Dic₄₃	1	1	yes	characteristic
133	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
134	C₁₆	—	4	2	no	—
135	C₂³	#13063247	1	1	yes	characteristic
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	344	2	no	—
142	C₂×C₄	—	344	2	no	—
143	C₂×C₄	—	344	2	no	—
144	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
145	C₂×C₄	#13063247	1	1	yes	characteristic
146	C₂×C₄	#13063250	1	1	yes	characteristic
147	C₂×C₄	#13063250	1	1	yes	characteristic
148	C₂×C₄	#13063257	1	1	yes	characteristic
149	C₂×C₄	#13063259	1	1	yes	characteristic
150	C₈	—	2	1	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	344	2	no	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#14582662	1	1	yes	characteristic
158	C₂²	#14582713	1	1	yes	characteristic
159	C₂²	#14582716	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	688	2	no	—
165	C₄	#14582655	1	1	yes	characteristic
166	C₄	#14582670	1	1	yes	characteristic
167	C₄	#14582710	1	2	yes	—
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#19132454	1	1	yes	characteristic
171	C₂	#19132701	1	1	yes	characteristic
172	C₂	#19132704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134723

C₂

yes

maximal, characteristic

#19133358

C₂

yes

maximal, characteristic

C₄₃×#2163

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#14582652

C₄

yes

complemented

#14582650

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×C₄⋊C₁₆

—

C₄₃×C₂²⋊C₁₆

—

C₄₃×#346

C₄

yes

complemented

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄²

yes

characteristic

C₂×C₈×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

Q₁₆

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₁₆×C₄₃

—

C₂³×C₄₃

#102

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#7222

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

Dic₄₃

—

#2163

D₄₃

yes

characteristic

#1834

—

#1568

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

Dic₄₃

yes

—

—

—

—

100

C₄₃

#7222

yes

characteristic, complemented, Sylow

101

C₂²×C₈

C₄×D₄₃

yes

characteristic

102

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

172

—

105

C₂×C₄⋊C₄

—

172

—

106

C₂×C₄²

D₁₇₂

yes

characteristic

107

C₂×C₁₆

—

108

C₄⋊C₈

—

109

C₂²⋊C₈

—

172

—

110

#97

—

172

—

111

C₄×C₈

Dic₈₆

yes

characteristic

112

C₄×C₈

C₄×D₄₃

yes

characteristic

113

C₄×C₈

C₂×Dic₄₃

yes

characteristic

114

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

115

C₂²×C₄

—

172

—

116

C₂²×C₄

#10621873

yes

characteristic

117

C₂²×C₄

D₃₄₄

yes

characteristic

118

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

172

—

122

C₂×C₈

C₄×Dic₄₃

yes

characteristic

123

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

124

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

125

C₂×C₈

C₂²⋊Dic₄₃

yes

characteristic

126

C₄⋊C₄

—

344

—

127

C₄⋊C₄

—

344

—

128

C₄⋊C₄

—

172

—

129

C₄⋊C₄

—

172

—

130

C₄²

#10621873

yes

characteristic

131

C₄²

Dic₁₇₂

yes

characteristic

132

C₄²

C₄⋊Dic₄₃

yes

characteristic

133

C₄²

(C₂×D₄₃)⋊C₄

yes

characteristic

134

C₁₆

—

135

C₂³

#13063247

yes

characteristic

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

344

—

142

C₂×C₄

—

344

—

143

C₂×C₄

—

344

—

144

C₂×C₄

C₄₃⋊#95

yes

characteristic

145

C₂×C₄

#13063247

yes

characteristic

146

C₂×C₄

#13063250

yes

characteristic

147

C₂×C₄

#13063250

yes

characteristic

148

C₂×C₄

#13063257

yes

characteristic

149

C₂×C₄

#13063259

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

344

—

155

C₂²

—

156

C₂²

—

157

C₂²

#14582662

yes

characteristic

158

C₂²

#14582713

yes

characteristic

159

C₂²

#14582716

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

688

—

165

C₄

#14582655

yes

characteristic

166

C₄

#14582670

yes

characteristic

167

C₄

#14582710

yes

—

168

C₂

—

169

C₂

—

170

C₂

#19132454

yes

characteristic

171

C₂

#19132701

yes

characteristic

172

C₂

#19132704

yes

characteristic

	subgroup	G/H
center	C₂²	#14582662
commutator	C₂×C₈×C₄₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₄₃×#2163	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#102
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	#2163	D₄₃
hypocenter	C₄₃	#7222
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

subgroup

G/H

center

C₂²

#14582662

commutator

C₂×C₈×C₄₃

C₄²

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₄₃×#2163

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2163

N	H ≅ G/N	type
8. #14582652	C₄	semidirect
13. C₄₃×#346	C₄	semidirect
95. #346	Dic₄₃	semidirect
100. C₄₃	#7222	semidirect

H ≅ G/N

type

8. #14582652

C₄

semidirect

13. C₄₃×#346

C₄

semidirect

95. #346

Dic₄₃

semidirect

100. C₄₃

#7222

semidirect

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

subgroup	index
#19134723	2
#19133358	2
C₄₃×#2163	2
#7222	43

subgroup

index

#19134723

#19133358

C₄₃×#2163

#7222

Subgroups — #24983649

Conjugacy classes of subgroups

Complements

Maximal subgroups