Group #24987027 — subgroups

Subgroups — #24987027

Order 11008.

446 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 227 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	#19134341	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄₃⋊#1273	C₂	1	4	yes	maximal
4	C₄₃×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
6	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	—
7	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#14582654	C₄	1	4	yes	—
10	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
11	C₄³×C₄₃	C₂²	1	1	yes	characteristic
12	C₄₃×#336	C₄	1	4	yes	—
13	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
15	C₄×C₄₃⋊C₈	—	2	4	no	—
16	C₄×C₄₃⋊C₈	—	2	4	no	—
17	C₄×C₄₃⋊C₈	—	2	4	no	—
18	C₄×C₄₃⋊C₈	C₈	1	8	yes	complemented
19	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
20	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
25	C₄×C₈×C₄₃	—	2	4	no	—
26	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
27	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₈×C₄₃	—	2	4	no	—
32	C₂×C₈×C₄₃	—	2	8	no	—
33	C₄²×C₄₃	—	2	4	no	—
34	C₄²×C₄₃	—	2	4	no	—
35	C₄²×C₄₃	C₂×C₈	1	4	yes	—
36	C₄²×C₄₃	M₁₆	1	4	yes	—
37	C₄²×C₄₃	#41	1	4	yes	—
38	C₂×C₄₃⋊C₈	—	2	4	no	—
39	C₂×C₄₃⋊C₈	—	2	4	no	—
40	C₂×C₄₃⋊C₈	—	2	4	no	—
41	C₂×C₄₃⋊C₈	—	2	8	no	—
42	C₂×C₄₃⋊C₈	C₂×C₈	1	8	yes	—
43	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
44	C₂×C₄×C₄₃	—	2	2	no	—
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	C₄×C₈	1	4	yes	—
49	C₂×C₄×C₄₃	#97	1	4	yes	—
50	C₂×C₄×C₄₃	#117	1	2	yes	—
51	C₂×C₄×C₄₃	#117	1	2	yes	—
52	C₂×C₄×C₄₃	#117	1	2	yes	—
53	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
54	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
55	C₂×C₄×C₄₃	#131	1	4	yes	—
56	C₂×C₄×C₄₃	#131	1	4	yes	—
57	C₈×C₄₃	—	4	4	no	—
58	C₄₃⋊C₈	—	2	8	no	—
59	C₄₃⋊C₈	—	4	4	no	—
60	#8133	—	43	1	no	maximal, Sylow
61	C₂²×C₄₃	#376	1	1	yes	characteristic
62	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
63	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
64	C₂²×C₄₃	#405	1	2	yes	—
65	C₂²×C₄₃	#405	1	2	yes	—
66	C₄×C₄₃	—	2	4	no	—
67	C₄×C₄₃	—	2	4	no	—
68	C₄×C₄₃	#431	1	4	yes	—
69	C₄×C₄₃	#433	1	4	yes	—
70	C₄×C₄₃	#433	1	4	yes	—
71	#1744	—	43	1	no	—
72	#1744	D₄₃	1	1	yes	characteristic
73	C₄²×C₈	—	43	1	no	—
74	#1273	—	43	4	no	—
75	C₂×C₄₃	#1450	1	2	yes	—
76	C₂×C₄₃	#1452	1	2	yes	—
77	C₂×C₄₃	#1744	1	1	yes	characteristic
78	C₂×C₄₃	#1746	1	1	yes	characteristic
79	C₂×C₄₃	#1746	1	1	yes	characteristic
80	C₂×C₄×C₈	—	43	2	no	—
81	C₂×C₄×C₈	—	43	2	no	—
82	C₂×C₄×C₈	—	43	2	no	—
83	C₂×C₄×C₈	—	43	2	no	—
84	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
85	C₄³	C₂×D₄₃	1	1	yes	characteristic
86	#336	—	43	4	no	—
87	#336	Dic₄₃	1	4	yes	—
88	C₄₃	#8133	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₈	—	43	4	no	—
90	C₂²×C₈	—	43	4	no	—
91	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
92	C₂×C₄²	C₄×D₄₃	1	2	yes	—
93	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
94	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
95	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
96	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	43	8	no	—
99	C₄×C₈	—	86	4	no	—
100	C₄×C₈	—	86	4	no	—
101	C₄×C₈	—	86	4	no	—
102	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
103	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
104	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
105	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
106	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	8	no	—
109	C₂×C₈	—	43	8	no	—
110	C₂×C₈	—	86	4	no	—
111	C₂×C₈	—	86	4	no	—
112	C₂×C₈	—	86	4	no	—
113	C₂×C₈	—	86	8	no	—
114	C₄²	—	2	4	no	—
115	C₄²	—	2	4	no	—
116	C₄²	C₈×D₄₃	1	4	yes	—
117	C₄²	#10621872	1	4	yes	—
118	C₄²	#10621898	1	4	yes	—
119	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	C₈×Dic₄₃	1	4	yes	—
125	C₂×C₄	#13063235	1	4	yes	—
126	C₂×C₄	#13063282	1	2	yes	—
127	C₂×C₄	#13063282	1	2	yes	—
128	C₂×C₄	#13063284	1	2	yes	—
129	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
130	C₂×C₄	C₂×#10621872	1	2	yes	—
131	C₂×C₄	#13063332	1	4	yes	—
132	C₂×C₄	#13063333	1	4	yes	—
133	C₈	—	4	4	no	—
134	C₈	—	86	8	no	—
135	C₈	—	172	4	no	—
136	C₂²	#14582745	1	1	yes	characteristic
137	C₂²	C₂×C₈×Dic₄₃	1	1	yes	characteristic
138	C₂²	C₂×#13063235	1	1	yes	characteristic
139	C₂²	#14582855	1	2	yes	—
140	C₂²	#14582857	1	2	yes	—
141	C₄	—	2	4	no	—
142	C₄	—	2	4	no	—
143	C₄	#14582950	1	4	yes	—
144	C₄	#14582960	1	4	yes	—
145	C₄	#14582961	1	4	yes	—
146	C₂	#19132879	1	2	yes	—
147	C₂	#19132888	1	2	yes	—
148	C₂	#19134340	1	1	yes	characteristic
149	C₂	#19134347	1	1	yes	characteristic
150	C₂	#19134348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134341

C₂

yes

maximal, characteristic

C₄²×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₄₃⋊#1273

C₂

yes

maximal

C₄₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582654

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₄₃×#336

C₄

yes

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₈

yes

complemented

C₂²×C₈×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₄₃

C₂×C₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₄²

yes

—

C₂²×C₄×C₄₃

C₄²

yes

—

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₈

yes

—

C₄²×C₄₃

M₁₆

yes

—

C₄²×C₄₃

#41

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₂³×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×C₈

yes

—

C₂×C₄×C₄₃

#97

yes

—

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

#117

yes

—

C₂×C₄×C₄₃

C₂²×C₈

yes

—

C₂×C₄×C₄₃

C₂×M₁₆

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₂×C₄×C₄₃

#131

yes

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₄₃

C₂×#97

yes

characteristic

C₂²×C₄₃

#405

yes

—

C₂²×C₄₃

#405

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#431

yes

—

C₄×C₄₃

#433

yes

—

C₄×C₄₃

#433

yes

—

#1744

—

#1744

D₄₃

yes

characteristic

C₄²×C₈

—

#1273

—

C₂×C₄₃

#1450

yes

—

C₂×C₄₃

#1452

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₄³

C₂×D₄₃

yes

characteristic

#336

—

#336

Dic₄₃

yes

—

C₄₃

#8133

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₄₃

yes

—

C₂×C₄²

C₄×D₄₃

yes

—

C₂×C₄²

C₄×D₄₃

yes

characteristic

C₂×C₄²

C₄×D₄₃

yes

characteristic

C₂×C₄²

C₂×Dic₄₃

yes

—

C₂×C₄²

C₂²×D₄₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₂²×C₄

C₄×Dic₄₃

yes

—

103

C₂²×C₄

C₄×Dic₄₃

yes

—

104

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

105

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

106

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₈×D₄₃

yes

—

117

C₄²

#10621872

yes

—

118

C₄²

#10621898

yes

—

119

C₂³

C₂×C₄×Dic₄₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₈×Dic₄₃

yes

—

125

C₂×C₄

#13063235

yes

—

126

C₂×C₄

#13063282

yes

—

127

C₂×C₄

#13063282

yes

—

128

C₂×C₄

#13063284

yes

—

129

C₂×C₄

C₂×C₈×D₄₃

yes

—

130

C₂×C₄

C₂×#10621872

yes

—

131

C₂×C₄

#13063332

yes

—

132

C₂×C₄

#13063333

yes

—

133

C₈

—

134

C₈

—

135

C₈

—

172

—

136

C₂²

#14582745

yes

characteristic

137

C₂²

C₂×C₈×Dic₄₃

yes

characteristic

138

C₂²

C₂×#13063235

yes

characteristic

139

C₂²

#14582855

yes

—

140

C₂²

#14582857

yes

—

141

C₄

—

142

C₄

—

143

C₄

#14582950

yes

—

144

C₄

#14582960

yes

—

145

C₄

#14582961

yes

—

146

C₂

#19132879

yes

—

147

C₂

#19132888

yes

—

148

C₂

#19134340

yes

characteristic

149

C₂

#19134347

yes

characteristic

150

C₂

#19134348

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₄₃×#1744	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1744	D₄₃
hypocenter	C₄₃	#8133
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂×C₄×C₈

Fitting

C₄₃×#1744

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₄²

cosocle

C₂×C₄²×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
18. C₄×C₄₃⋊C₈	C₈	semidirect
88. C₄₃	#8133	semidirect

H ≅ G/N

type

18. C₄×C₄₃⋊C₈

C₈

semidirect

88. C₄₃

#8133

semidirect

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

subgroup	index
#19134341	2
C₄²×C₄₃⋊C₈	2
C₄₃⋊#1273	2
C₄₃×#1744	2
#8133	43

subgroup

index

#19134341

C₄²×C₄₃⋊C₈

C₄₃⋊#1273

C₄₃×#1744

#8133

Subgroups — #24987027

Conjugacy classes of subgroups

Complements

Maximal subgroups