Group #24987023 — subgroups

Subgroups — #24987023

Order 11008.

478 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 243 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	C₄×#13063236	C₂	1	1	yes	maximal, characteristic
3	C₄₃⋊#1273	C₂	1	4	yes	maximal
4	C₄²×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063236	C₄	1	2	yes	—
6	C₂×#13063236	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	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₄₃	C₂²	1	2	yes	—
12	C₄³×C₄₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
14	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
15	#13063236	—	2	4	no	—
16	#13063236	C₈	1	8	yes	complemented
17	C₄×C₄₃⋊C₈	—	2	4	no	—
18	C₄×C₄₃⋊C₈	—	2	4	no	—
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	8	no	—
26	C₄×C₈×C₄₃	—	2	4	no	—
27	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
28	C₂²×C₄×C₄₃	C₄²	1	2	yes	—
29	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₈×C₄₃	—	2	4	no	—
33	C₂×C₈×C₄₃	—	2	4	no	—
34	C₂×C₈×C₄₃	M₁₆	1	8	yes	—
35	C₄²×C₄₃	—	2	4	no	—
36	C₄²×C₄₃	—	2	4	no	—
37	C₄²×C₄₃	C₂×C₈	1	4	yes	—
38	C₄²×C₄₃	M₁₆	1	4	yes	—
39	C₄²×C₄₃	#41	1	4	yes	—
40	C₂×C₄₃⋊C₈	—	2	4	no	—
41	C₂×C₄₃⋊C₈	—	2	4	no	—
42	C₂×C₄₃⋊C₈	—	2	4	no	—
43	C₂×C₄₃⋊C₈	—	2	8	no	—
44	C₂×C₄₃⋊C₈	C₂×C₈	1	8	yes	—
45	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
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	C₄×C₈	1	4	yes	—
51	C₂×C₄×C₄₃	#97	1	4	yes	—
52	C₂×C₄×C₄₃	#117	1	2	yes	—
53	C₂×C₄×C₄₃	#117	1	2	yes	—
54	C₂×C₄×C₄₃	#117	1	2	yes	—
55	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
56	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₄₃	C₂×M₁₆	1	4	yes	—
58	C₂×C₄×C₄₃	#131	1	4	yes	—
59	C₈×C₄₃	—	2	8	no	—
60	C₄₃⋊C₈	—	2	8	no	—
61	C₄₃⋊C₈	—	4	4	no	—
62	#8132	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	#376	1	1	yes	characteristic
64	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
65	C₂²×C₄₃	C₂×#97	1	1	yes	characteristic
66	C₂²×C₄₃	C₄×M₁₆	1	2	yes	—
67	C₂²×C₄₃	#405	1	2	yes	—
68	C₄×C₄₃	—	2	4	no	—
69	C₄×C₄₃	—	2	4	no	—
70	C₄×C₄₃	#431	1	4	yes	—
71	C₄×C₄₃	#432	1	4	yes	—
72	C₄×C₄₃	#433	1	4	yes	—
73	#1744	—	43	1	no	—
74	C₄×#97	—	43	1	no	—
75	C₄²×C₈	D₄₃	1	1	yes	characteristic
76	#1273	—	43	4	no	—
77	C₂×C₄₃	C₈×M₁₆	1	2	yes	—
78	C₂×C₄₃	#1451	1	2	yes	—
79	C₂×C₄₃	#1744	1	1	yes	characteristic
80	C₂×C₄₃	#1745	1	1	yes	characteristic
81	C₂×C₄₃	#1746	1	1	yes	characteristic
82	C₂×#97	—	43	2	no	—
83	C₂×#97	—	43	2	no	—
84	C₂×C₄×C₈	—	43	2	no	—
85	C₂×C₄×C₈	—	43	2	no	—
86	C₂×C₄×C₈	Dic₄₃	1	4	yes	—
87	C₂×C₄×C₈	C₂×D₄₃	1	2	yes	—
88	C₄³	C₂×D₄₃	1	1	yes	characteristic
89	#336	—	43	4	no	—
90	C₄₃	#8132	1	1	yes	characteristic, complemented, Sylow
91	C₂²×C₈	—	43	4	no	—
92	C₂²×C₈	—	43	4	no	—
93	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
94	C₂×C₄²	C₄×D₄₃	1	2	yes	—
95	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
96	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
97	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
98	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
99	#97	—	43	8	no	—
100	#97	—	86	4	no	—
101	C₄×C₈	—	2	8	no	—
102	C₄×C₈	—	2	4	no	—
103	C₄×C₈	—	86	4	no	—
104	C₄×C₈	—	86	4	no	—
105	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
106	C₂²×C₄	C₄×Dic₄₃	1	2	yes	—
107	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
108	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
109	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	2	4	no	—
112	C₂×C₈	—	43	8	no	—
113	C₂×C₈	—	86	4	no	—
114	C₂×C₈	—	86	4	no	—
115	C₂×C₈	—	86	4	no	—
116	C₂×C₈	—	86	8	no	—
117	C₂×C₈	#10621877	1	8	yes	—
118	C₄²	—	2	4	no	—
119	C₄²	—	2	4	no	—
120	C₄²	C₈×D₄₃	1	4	yes	—
121	C₄²	#10621872	1	4	yes	—
122	C₄²	#10621898	1	4	yes	—
123	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	C₈×Dic₄₃	1	4	yes	—
129	C₂×C₄	#13063235	1	4	yes	—
130	C₂×C₄	#13063282	1	2	yes	—
131	C₂×C₄	#13063282	1	2	yes	—
132	C₂×C₄	#13063284	1	2	yes	—
133	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
134	C₂×C₄	C₂×#10621872	1	2	yes	—
135	C₂×C₄	C₂×#10621877	1	4	yes	—
136	C₂×C₄	#13063331	1	4	yes	—
137	C₈	—	2	8	no	—
138	C₈	—	86	8	no	—
139	C₈	—	172	4	no	—
140	C₂²	#14582745	1	1	yes	characteristic
141	C₂²	C₂×C₈×Dic₄₃	1	1	yes	characteristic
142	C₂²	C₂×#13063235	1	1	yes	characteristic
143	C₂²	C₄×#10621877	1	2	yes	—
144	C₂²	#14582854	1	2	yes	—
145	C₄	—	2	4	no	—
146	C₄	—	2	4	no	—
147	C₄	#14582950	1	4	yes	—
148	C₄	#14582956	1	4	yes	—
149	C₄	#14582958	1	4	yes	—
150	C₂	C₈×#10621877	1	2	yes	—
151	C₂	#19132884	1	2	yes	—
152	C₂	#19134340	1	1	yes	characteristic
153	C₂	#19134344	1	1	yes	characteristic
154	C₂	#19134346	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19134341

C₂

yes

maximal, characteristic

C₄×#13063236

C₂

yes

maximal, characteristic

C₄₃⋊#1273

C₂

yes

maximal

C₄²×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂×#13063236

C₄

yes

—

C₂×#13063236

C₄

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

C₂×C₄×C₄₃⋊C₈

C₂²

yes

—

#14582654

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₄

yes

—

C₂×C₄×C₈×C₄₃

C₂²

yes

—

C₄³×C₄₃

C₂²

yes

characteristic

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

C₂²×C₄₃⋊C₈

C₂×C₄

yes

—

#13063236

—

#13063236

C₈

yes

complemented

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

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

—

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

—

C₂×C₈×C₄₃

M₁₆

yes

—

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

C₂×M₁₆

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

C₄×M₁₆

yes

—

C₂²×C₄₃

#405

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#431

yes

—

C₄×C₄₃

#432

yes

—

C₄×C₄₃

#433

yes

—

#1744

—

C₄×#97

—

C₄²×C₈

D₄₃

yes

characteristic

#1273

—

C₂×C₄₃

C₈×M₁₆

yes

—

C₂×C₄₃

#1451

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

Dic₄₃

yes

—

C₂×C₄×C₈

C₂×D₄₃

yes

—

C₄³

C₂×D₄₃

yes

characteristic

#336

—

C₄₃

#8132

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

#97

—

100

#97

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₂²×C₄

C₄×Dic₄₃

yes

—

106

C₂²×C₄

C₄×Dic₄₃

yes

—

107

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

108

C₂²×C₄

C₂×C₄×D₄₃

yes

characteristic

109

C₂²×C₄

C₂²×Dic₄₃

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

#10621877

yes

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₈×D₄₃

yes

—

121

C₄²

#10621872

yes

—

122

C₄²

#10621898

yes

—

123

C₂³

C₂×C₄×Dic₄₃

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₈×Dic₄₃

yes

—

129

C₂×C₄

#13063235

yes

—

130

C₂×C₄

#13063282

yes

—

131

C₂×C₄

#13063282

yes

—

132

C₂×C₄

#13063284

yes

—

133

C₂×C₄

C₂×C₈×D₄₃

yes

—

134

C₂×C₄

C₂×#10621872

yes

—

135

C₂×C₄

C₂×#10621877

yes

—

136

C₂×C₄

#13063331

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

172

—

140

C₂²

#14582745

yes

characteristic

141

C₂²

C₂×C₈×Dic₄₃

yes

characteristic

142

C₂²

C₂×#13063235

yes

characteristic

143

C₂²

C₄×#10621877

yes

—

144

C₂²

#14582854

yes

—

145

C₄

—

146

C₄

—

147

C₄

#14582950

yes

—

148

C₄

#14582956

yes

—

149

C₄

#14582958

yes

—

150

C₂

C₈×#10621877

yes

—

151

C₂

#19132884

yes

—

152

C₂

#19134340

yes

characteristic

153

C₂

#19134344

yes

characteristic

154

C₂

#19134346

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Fitting	C₄²×C₈×C₄₃	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₄²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₄²×C₈	D₄₃
hypocenter	C₄₃	#8132
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄² ◁ C₄²×C₈

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₄₃

commutator

C₂²×C₄₃

C₂×C₄×C₈

Fitting

C₄²×C₈×C₄₃

C₂

socle, prime radical

C₂³×C₄₃

C₂×C₄²

cosocle

C₂×C₄²×C₄₃

C₂³

hypercenter

C₄²×C₈

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₄₃

C₁ ◁ C₂×C₄² ◁ C₄²×C₈

N	H ≅ G/N	type
16. #13063236	C₈	semidirect
90. C₄₃	#8132	semidirect

H ≅ G/N

type

16. #13063236

C₈

semidirect

90. C₄₃

#8132

semidirect

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

subgroup	index
#19134341	2
C₄×#13063236	2
C₄₃⋊#1273	2
C₄²×C₈×C₄₃	2
#8132	43

subgroup

index

#19134341

C₄×#13063236

C₄₃⋊#1273

C₄²×C₈×C₄₃

#8132

Subgroups — #24987023

Conjugacy classes of subgroups

Complements

Maximal subgroups