Group #24983476 — subgroups

Subgroups — #24983476

Order 11008.

222 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 119 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×#13063265	C₂	1	1	yes	maximal, characteristic
2	#19133343	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄₃×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#13063265	C₄	1	2	yes	—
5	#14582952	—	2	1	no	—
6	#14582675	—	2	1	no	—
7	C₄×C₄₃⋊C₁₆	—	2	1	no	—
8	#14582652	C₄	1	2	yes	—
9	#14582650	—	2	1	no	—
10	C₂×C₄⋊C₈×C₄₃	—	2	1	no	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#334	—	2	1	no	—
13	C₄₃×#334	C₄	1	2	yes	—
14	#13063265	—	2	2	no	—
15	#13063265	C₈	1	4	yes	complemented
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₄₃⋊C₈)	—	4	1	no	—
19	C₄×C₄₃⋊C₈	—	4	1	no	—
20	C₂²×C₈×C₄₃	—	4	1	no	—
21	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
24	C₄⋊C₈×C₄₃	—	2	2	no	—
25	C₄⋊C₈×C₄₃	—	4	1	no	—
26	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
33	C₂×C₈×C₄₃	—	2	2	no	—
34	C₂×C₈×C₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	—	8	2	no	—
36	C₂×C₈×C₄₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
41	C₂×C₈×C₄₃	M₁₆	1	2	yes	—
42	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₄²×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
45	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₄₃⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₃	#107	1	1	yes	characteristic
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	—	4	1	no	—
51	C₂×C₄×C₄₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₄⋊C₈	1	2	yes	—
57	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
58	C₂×C₄×C₄₃	#108	1	2	yes	—
59	C₈×C₄₃	—	2	1	no	—
60	C₈×C₄₃	—	2	1	no	—
61	C₈×C₄₃	—	8	2	no	—
62	C₈×C₄₃	C₄×C₈	1	1	yes	characteristic
63	C₈×C₄₃	#97	1	1	yes	characteristic
64	C₈×C₄₃	C₂²⋊C₈	1	2	yes	—
65	C₄₃⋊C₈	—	8	2	no	—
66	#7158	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	—	2	1	no	—
68	C₂²×C₄₃	—	2	1	no	—
69	C₂²×C₄₃	#335	1	2	yes	—
70	C₂²×C₄₃	#340	1	1	yes	characteristic
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	—	2	1	no	—
73	C₄×C₄₃	—	2	1	no	—
74	C₄×C₄₃	#334	1	2	yes	—
75	C₄×C₄₃	#336	1	1	yes	characteristic
76	C₄×C₄₃	#341	1	1	yes	characteristic
77	C₄×C₄₃	#360	1	2	yes	—
78	C₄×M₃₂	—	43	1	no	—
79	#1564	—	43	1	no	—
80	C₂×#334	D₄₃	1	1	yes	characteristic
81	C₂×C₄₃	—	2	2	no	—
82	C₂×C₄₃	#1277	1	1	yes	characteristic
83	C₂×C₄₃	#1386	1	1	yes	characteristic
84	C₂×C₄₃	#1388	1	1	yes	characteristic
85	C₂×M₃₂	—	43	2	no	—
86	#431	—	86	1	no	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
89	#346	—	86	1	no	—
90	C₄×C₁₆	—	86	1	no	—
91	#335	—	43	2	no	—
92	#334	—	2	1	no	—
93	#334	—	86	1	no	—
94	#334	Dic₄₃	1	2	yes	—
95	C₄₃	#7158	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	4	1	no	—
97	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
98	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
99	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
100	M₃₂	—	43	4	no	—
101	M₃₂	—	86	2	no	—
102	C₂×C₁₆	—	86	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	—	4	1	no	—
105	C₄⋊C₈	—	86	2	no	—
106	C₂²⋊C₈	—	172	1	no	—
107	C₄×C₈	—	172	1	no	—
108	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
109	C₄×C₈	D₁₇₂	1	1	yes	characteristic
110	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
111	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
112	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
113	C₂²×C₄	#10621882	1	1	yes	characteristic
114	C₂²×C₄	#10621885	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	4	2	no	—
117	C₂×C₈	—	8	2	no	—
118	C₂×C₈	—	172	2	no	—
119	C₂×C₈	C₈×D₄₃	1	2	yes	—
120	C₂×C₈	#10621872	1	2	yes	—
121	C₂×C₈	C₄×Dic₄₃	1	1	yes	characteristic
122	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
123	C₂×C₈	C₄⋊Dic₄₃	1	1	yes	characteristic
124	C₂×C₈	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
125	C₄²	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
126	C₄²	#10621883	1	1	yes	characteristic
127	C₄²	#10621884	1	1	yes	characteristic
128	C₄²	C₂²⋊Dic₄₃	1	1	yes	characteristic
129	C₁₆	—	86	4	no	—
130	C₂³	#13063258	1	1	yes	characteristic
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	C₄₃⋊#95	1	1	yes	characteristic
134	C₂×C₄	#13063245	1	1	yes	characteristic
135	C₂×C₄	#13063246	1	1	yes	characteristic
136	C₂×C₄	#13063248	1	1	yes	characteristic
137	C₂×C₄	#13063249	1	1	yes	characteristic
138	C₂×C₄	Dic₄₃⋊C₈	1	2	yes	—
139	C₂×C₄	#13063256	1	1	yes	characteristic
140	C₂×C₄	#13063260	1	2	yes	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	8	2	no	—
144	C₈	—	344	2	no	—
145	C₈	C₈×Dic₄₃	1	1	yes	characteristic
146	C₈	#13063235	1	1	yes	characteristic
147	C₈	(C₂×D₄₃)⋊C₈	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#14582651	1	2	yes	—
151	C₂²	#14582661	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	#14582649	1	2	yes	—
156	C₄	#14582653	1	1	yes	characteristic
157	C₄	#14582663	1	1	yes	characteristic
158	C₄	#14582708	1	2	yes	—
159	C₂	—	2	2	no	—
160	C₂	#19132429	1	1	yes	characteristic
161	C₂	#19132702	1	1	yes	characteristic
162	C₂	#19132708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13063265

C₂

yes

maximal, characteristic

#19133343

C₂

yes

maximal, characteristic

C₂×C₄₃×#334

C₂

yes

maximal, characteristic

C₂×#13063265

C₄

yes

—

#14582952

—

#14582675

—

C₄×C₄₃⋊C₁₆

—

#14582652

C₄

yes

—

#14582650

—

C₂×C₄⋊C₈×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#334

—

C₄₃×#334

C₄

yes

—

#13063265

—

#13063265

C₈

yes

complemented

C₂×C₄₃⋊C₁₆

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

D₈

yes

characteristic

C₂²×C₄×C₄₃

Q₁₆

yes

characteristic

C₂×C₈×C₄₃

—

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

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

M₁₆

yes

—

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

C₂²⋊C₄

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#107

yes

characteristic

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

C₄⋊C₈

yes

—

C₂×C₄×C₄₃

#106

yes

characteristic

C₂×C₄×C₄₃

#108

yes

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

C₄×C₈

yes

characteristic

C₈×C₄₃

#97

yes

characteristic

C₈×C₄₃

C₂²⋊C₈

yes

—

C₄₃⋊C₈

—

#7158

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#335

yes

—

C₂²×C₄₃

#340

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#334

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₄×M₃₂

—

#1564

—

C₂×#334

D₄₃

yes

characteristic

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×M₃₂

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#346

—

C₄×C₁₆

—

—

—

—

Dic₄₃

yes

—

C₄₃

#7158

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

Dic₈₆

yes

characteristic

C₂²×C₈

C₄×D₄₃

yes

characteristic

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

100

M₃₂

—

101

M₃₂

—

102

C₂×C₁₆

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₂²⋊C₈

—

172

—

107

C₄×C₈

—

172

—

108

C₄×C₈

C₄×D₄₃

yes

characteristic

109

C₄×C₈

D₁₇₂

yes

characteristic

110

C₄×C₈

C₂×Dic₄₃

yes

characteristic

111

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

112

C₂²×C₄

Dic₄₃⋊C₄

yes

characteristic

113

C₂²×C₄

#10621882

yes

characteristic

114

C₂²×C₄

#10621885

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

172

—

119

C₂×C₈

C₈×D₄₃

yes

—

120

C₂×C₈

#10621872

yes

—

121

C₂×C₈

C₄×Dic₄₃

yes

characteristic

122

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

123

C₂×C₈

C₄⋊Dic₄₃

yes

characteristic

124

C₂×C₈

(C₂×D₄₃)⋊C₄

yes

characteristic

125

C₄²

(C₂×D₄₃)⋊C₄

yes

characteristic

126

C₄²

#10621883

yes

characteristic

127

C₄²

#10621884

yes

characteristic

128

C₄²

C₂²⋊Dic₄₃

yes

characteristic

129

C₁₆

—

130

C₂³

#13063258

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₄₃⋊#95

yes

characteristic

134

C₂×C₄

#13063245

yes

characteristic

135

C₂×C₄

#13063246

yes

characteristic

136

C₂×C₄

#13063248

yes

characteristic

137

C₂×C₄

#13063249

yes

characteristic

138

C₂×C₄

Dic₄₃⋊C₈

yes

—

139

C₂×C₄

#13063256

yes

characteristic

140

C₂×C₄

#13063260

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

344

—

145

C₈

C₈×Dic₄₃

yes

characteristic

146

C₈

#13063235

yes

characteristic

147

C₈

(C₂×D₄₃)⋊C₈

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

#14582651

yes

—

151

C₂²

#14582661

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#14582649

yes

—

156

C₄

#14582653

yes

characteristic

157

C₄

#14582663

yes

characteristic

158

C₄

#14582708

yes

—

159

C₂

—

160

C₂

#19132429

yes

characteristic

161

C₂

#19132702

yes

characteristic

162

C₂

#19132708

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13063245
commutator	C₈×C₄₃	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₃
Fitting	C₂×C₄₃×#334	C₂
socle	C₂²×C₄₃	#340
prime radical	C₂³×C₄₃	#107
cosocle	C₂×C₄×C₈×C₄₃	C₂²
hypercenter	C₂×#334	D₄₃
hypocenter	C₄₃	#7158
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

subgroup

G/H

center

C₂×C₄

#13063245

commutator

C₈×C₄₃

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₃

Fitting

C₂×C₄₃×#334

C₂

C₂²×C₄₃

C₂³×C₄₃

C₂×C₄×C₈×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#334

N	H ≅ G/N	type
15. #13063265	C₈	semidirect
95. C₄₃	#7158	semidirect

H ≅ G/N

type

15. #13063265

C₈

semidirect

95. C₄₃

#7158

semidirect

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

subgroup	index
C₄×#13063265	2
#19133343	2
C₂×C₄₃×#334	2
#7158	43

subgroup

index

C₄×#13063265

#19133343

C₂×C₄₃×#334

#7158

Subgroups — #24983476

Conjugacy classes of subgroups

Complements

Maximal subgroups