Group #24983203 — subgroups

Subgroups — #24983203

Order 11008.

364 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 91 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	#19134357	C₂	1	2	yes	maximal
2	C₄₃×#1817	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
4	C₂×#13063236	—	2	2	no	—
5	#14582654	C₄	1	4	yes	—
6	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
8	C₄₃×#336	—	2	2	no	—
9	C₂²×C₄₃⋊C₈	—	2	4	no	—
10	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
11	#13063236	—	4	4	no	—
12	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₃	Q₈	1	1	yes	characteristic
15	C₂²×C₈×C₄₃	—	2	2	no	—
16	C₂×C₄²×C₄₃	—	4	2	no	—
17	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₄₃	D₄	1	2	yes	—
19	C₂²⋊C₈×C₄₃	—	4	2	no	—
20	C₂⁴×C₄₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₃	—	2	2	no	—
22	C₂²×C₄×C₄₃	—	2	4	no	—
23	C₂²×C₄×C₄₃	—	2	2	no	—
24	C₂²×C₄×C₄₃	—	4	1	no	—
25	C₂²×C₄×C₄₃	—	4	1	no	—
26	C₂²×C₄×C₄₃	—	4	2	no	—
27	C₂²×C₄×C₄₃	—	4	1	no	—
28	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
30	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
32	C₂×C₈×C₄₃	—	4	4	no	—
33	C₂×C₈×C₄₃	—	4	2	no	—
34	C₄²×C₄₃	—	2	4	no	—
35	C₄²×C₄₃	—	4	2	no	—
36	C₂×C₄₃⋊C₈	—	2	8	no	—
37	C₂×C₄₃⋊C₈	—	4	4	no	—
38	C₂×C₄₃⋊C₈	—	4	4	no	—
39	C₂³×C₄₃	—	2	1	no	—
40	C₂³×C₄₃	—	2	2	no	—
41	C₂³×C₄₃	—	2	1	no	—
42	C₂³×C₄₃	—	2	2	no	—
43	C₂³×C₄₃	#95	1	1	yes	characteristic
44	C₂³×C₄₃	#108	1	2	yes	—
45	C₂×C₄×C₄₃	—	2	2	no	—
46	C₂×C₄×C₄₃	—	2	2	no	—
47	C₂×C₄×C₄₃	—	2	2	no	—
48	C₂×C₄×C₄₃	—	2	2	no	—
49	C₂×C₄×C₄₃	—	2	2	no	—
50	C₂×C₄×C₄₃	—	2	2	no	—
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	—	2	2	no	—
53	C₂×C₄×C₄₃	—	4	4	no	—
54	C₂×C₄×C₄₃	—	4	4	no	—
55	C₂×C₄×C₄₃	—	4	4	no	—
56	C₂×C₄×C₄₃	#99	1	2	yes	—
57	C₂×C₄×C₄₃	#104	1	4	yes	—
58	C₈×C₄₃	—	8	2	no	—
59	C₄₃⋊C₈	—	4	8	no	—
60	#7046	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	—	2	2	no	—
62	C₂²×C₄₃	—	2	4	no	—
63	C₂²×C₄₃	—	4	2	no	—
64	C₂²×C₄₃	—	4	1	no	—
65	C₂²×C₄₃	—	4	1	no	—
66	C₂²×C₄₃	#338	1	1	yes	characteristic
67	C₂²×C₄₃	#339	1	2	yes	—
68	C₂²×C₄₃	#341	1	1	yes	characteristic
69	C₂²×C₄₃	#342	1	1	yes	characteristic
70	C₂²×C₄₃	#344	1	2	yes	—
71	C₄×C₄₃	—	2	4	no	—
72	C₄×C₄₃	—	2	2	no	—
73	C₄×C₄₃	—	4	2	no	—
74	C₄×C₄₃	—	4	1	no	—
75	#1817	D₄₃	1	1	yes	characteristic
76	#1749	—	43	2	no	—
77	C₂×C₄₃	—	4	2	no	—
78	C₂×C₄₃	#1291	1	2	yes	—
79	C₂×C₄₃	#1293	1	1	yes	characteristic
80	C₂×C₄₃	#1297	1	1	yes	characteristic
81	C₂×C₄₃	#1302	1	2	yes	—
82	C₂×C₄₃	#1306	1	1	yes	characteristic
83	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
84	C₂×C₂²⋊C₈	—	86	2	no	—
85	C₂×C₂²⋊C₈	Dic₄₃	1	2	yes	—
86	C₂×#97	—	86	2	no	—
87	#336	—	2	2	no	—
88	#336	—	43	4	no	—
89	C₄₃	#7046	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	Dic₈₆	1	1	yes	characteristic
91	C₂³×C₄	D₁₇₂	1	1	yes	characteristic
92	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	86	4	no	—
95	C₂×C₄²	—	4	2	no	—
96	C₂×C₄²	C₄×D₄₃	1	2	yes	—
97	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
98	C₂²⋊C₈	—	4	2	no	—
99	C₂²⋊C₈	—	86	8	no	—
100	#97	—	172	4	no	—
101	C₂⁴	C₄⋊Dic₄₃	1	1	yes	characteristic
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	4	1	no	—
106	C₂²×C₄	—	4	1	no	—
107	C₂²×C₄	—	4	1	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
110	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
111	C₂²×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
112	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	—	4	2	no	—
115	C₂×C₈	—	86	8	no	—
116	C₂×C₈	—	172	4	no	—
117	C₂×C₈	—	172	4	no	—
118	C₄²	—	2	4	no	—
119	C₄²	—	4	2	no	—
120	C₂³	—	2	1	no	—
121	C₂³	—	2	2	no	—
122	C₂³	—	2	1	no	—
123	C₂³	—	2	2	no	—
124	C₂³	C₄₃⋊#95	1	1	yes	characteristic
125	C₂³	#13063261	1	2	yes	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	#13063240	1	2	yes	—
138	C₂×C₄	#13063251	1	4	yes	—
139	C₈	—	8	2	no	—
140	C₈	—	172	8	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	4	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	1	no	—
145	C₂²	—	4	1	no	—
146	C₂²	C₄₃⋊#338	1	1	yes	characteristic
147	C₂²	#14582660	1	2	yes	—
148	C₂²	#14582664	1	1	yes	characteristic
149	C₂²	#14582666	1	1	yes	characteristic
150	C₂²	#14582669	1	2	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	4	1	no	—
155	C₂	—	4	2	no	—
156	C₂	#19132468	1	2	yes	—
157	C₂	#19132472	1	1	yes	characteristic
158	C₂	#19132480	1	1	yes	characteristic
159	C₂	#19132495	1	2	yes	—
160	C₂	#19132503	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134357

C₂

yes

maximal

C₄₃×#1817

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₃⋊C₈)

—

C₂×#13063236

—

#14582654

C₄

yes

—

C₂²×C₄²×C₄₃

C₂²

yes

characteristic

C₂×C₂²⋊C₈×C₄₃

C₄

yes

—

C₄₃×#336

—

C₂²×C₄₃⋊C₈

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₂³×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₃

D₄

yes

characteristic

C₂³×C₄×C₄₃

Q₈

yes

characteristic

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

D₄

yes

—

C₂²⋊C₈×C₄₃

—

C₂⁴×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₄²

yes

characteristic

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

—

C₂³×C₄₃

#95

yes

characteristic

C₂³×C₄₃

#108

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#99

yes

—

C₂×C₄×C₄₃

#104

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#7046

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#338

yes

characteristic

C₂²×C₄₃

#339

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

#1817

D₄₃

yes

characteristic

#1749

—

C₂×C₄₃

—

C₂×C₄₃

#1291

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#1306

yes

characteristic

C₂²×C₄²

C₂×D₄₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

Dic₄₃

yes

—

C₂×#97

—

#336

—

#336

—

C₄₃

#7046

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₈₆

yes

characteristic

C₂³×C₄

D₁₇₂

yes

characteristic

C₂³×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₄×D₄₃

yes

—

C₂×C₄²

C₂²⋊D₄₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

172

—

101

C₂⁴

C₄⋊Dic₄₃

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×Dic₄₃

yes

characteristic

110

C₂²×C₄

Dic₄₃⋊C₄

yes

—

111

C₂²×C₄

(C₂×D₄₃)⋊C₄

yes

—

112

C₂²×C₄

C₂²⋊Dic₄₃

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

172

—

117

C₂×C₈

—

172

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

C₄₃⋊#95

yes

characteristic

125

C₂³

#13063261

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13063240

yes

—

138

C₂×C₄

#13063251

yes

—

139

C₈

—

140

C₈

—

172

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₄₃⋊#338

yes

characteristic

147

C₂²

#14582660

yes

—

148

C₂²

#14582664

yes

characteristic

149

C₂²

#14582666

yes

characteristic

150

C₂²

#14582669

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₂

—

156

C₂

#19132468

yes

—

157

C₂

#19132472

yes

characteristic

158

C₂

#19132480

yes

characteristic

159

C₂

#19132495

yes

—

160

C₂

#19132503

yes

characteristic

	subgroup	G/H
center	C₂³	C₄₃⋊#95
commutator	C₂²×C₄×C₄₃	C₄²
Frattini	C₂²×C₄²	C₂×D₄₃
Fitting	C₄₃×#1817	C₂
socle	C₂³×C₄₃	#95
prime radical	C₂⁴×C₄₃	C₄⋊C₄
cosocle	C₂²×C₄²×C₄₃	C₂²
hypercenter	#1817	D₄₃
hypocenter	C₄₃	#7046
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

subgroup

G/H

center

C₂³

C₄₃⋊#95

commutator

C₂²×C₄×C₄₃

C₄²

Frattini

C₂²×C₄²

C₂×D₄₃

Fitting

C₄₃×#1817

C₂

socle

C₂³×C₄₃

#95

prime radical

C₂⁴×C₄₃

C₄⋊C₄

cosocle

C₂²×C₄²×C₄₃

C₂²

D₄₃

C₄₃

G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1817

N	H ≅ G/N	type
89. C₄₃	#7046	semidirect

H ≅ G/N

type

89. C₄₃

#7046

semidirect

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

subgroup	index
#19134357	2
C₄₃×#1817	2
#7046	43

subgroup

index

#19134357

C₄₃×#1817

#7046

Subgroups — #24983203

Conjugacy classes of subgroups

Complements

Maximal subgroups