Group #25012989 — subgroups

Subgroups — #25012989

Order 11008.

274 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₂×#14582675	C₂	1	1	yes	maximal, characteristic
2	#19134767	C₂	1	1	yes	maximal, characteristic
3	#19132688	C₂	1	4	yes	maximal
4	C₄₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13063261	—	2	1	no	—
7	C₂×#13063261	C₂²	1	2	yes	—
8	#14582923	—	2	1	no	—
9	#14582675	C₄	1	4	yes	—
10	C₂×#108×C₄₃	—	2	1	no	—
11	C₂×#108×C₄₃	C₂²	1	2	yes	—
12	C₄₃×#423	—	2	1	no	—
13	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
14	C₂×#10621877	—	2	2	no	—
15	C₂×C₄₃⋊C₁₆	—	2	2	no	—
16	C₂×C₄₃⋊C₁₆	—	2	2	no	—
17	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
18	#13063261	—	2	4	no	—
19	#13063261	—	4	2	no	—
20	C₄⋊(C₄₃⋊C₈)	—	4	2	no	—
21	C₂×M₁₆×C₄₃	—	2	2	no	—
22	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
25	#108×C₄₃	—	2	4	no	—
26	#108×C₄₃	—	4	2	no	—
27	C₄⋊C₈×C₄₃	—	4	2	no	—
28	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₄₃	Q₈	1	2	yes	—
30	C₂²×C₄×C₄₃	Q₁₆	1	2	yes	—
31	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
32	M₁₆×C₄₃	—	4	2	no	—
33	M₁₆×C₄₃	—	4	2	no	—
34	C₂×C₈×C₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
38	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	#41	1	2	yes	—
41	C₂×C₈×C₄₃	#41	1	2	yes	—
42	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
43	C₄²×C₄₃	D₈	1	2	yes	—
44	#10621877	—	4	2	no	—
45	#10621877	—	4	2	no	—
46	C₂×C₄₃⋊C₈	—	4	2	no	—
47	C₄₃⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₂×C₄×C₄₃	—	4	2	no	—
50	C₂×C₄×C₄₃	#107	1	4	yes	—
51	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₈×C₄₃	—	8	2	no	—
58	C₈×C₄₃	#106	1	4	yes	—
59	C₈×C₄₃	#117	1	2	yes	—
60	C₈×C₄₃	#125	1	2	yes	—
61	C₄₃⋊C₈	—	8	2	no	—
62	#12892	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	—	2	2	no	—
64	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
65	C₂²×C₄₃	#487	1	1	yes	characteristic
66	C₂²×C₄₃	#487	1	1	yes	characteristic
67	C₄×C₄₃	—	2	4	no	—
68	C₄×C₄₃	#382	1	1	yes	characteristic
69	C₄×C₄₃	C₂×#106	1	1	yes	characteristic
70	C₄×C₄₃	#486	1	1	yes	characteristic
71	C₄×C₄₃	#486	1	1	yes	characteristic
72	C₂×#346	—	43	1	no	—
73	#1848	—	43	1	no	—
74	#1848	D₄₃	1	1	yes	characteristic
75	#1381	—	43	4	no	—
76	C₂×C₄₃	—	2	1	no	—
77	C₂×C₄₃	—	2	1	no	—
78	C₂×C₄₃	#1834	1	1	yes	characteristic
79	C₂×C₄₃	#2161	1	2	yes	—
80	C₂²×C₁₆	—	43	2	no	—
81	C₂×#108	—	2	1	no	—
82	C₂×#108	—	43	2	no	—
83	C₂×#108	—	86	1	no	—
84	C₂×#108	C₂×D₄₃	1	2	yes	—
85	#423	—	2	1	no	—
86	#423	—	86	1	no	—
87	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
88	#346	—	43	4	no	—
89	C₄₃	#12892	1	1	yes	characteristic, complemented, Sylow
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	—	86	2	no	—
92	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
93	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
94	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
95	C₂×C₁₆	—	43	4	no	—
96	C₂×C₁₆	—	86	2	no	—
97	C₂×C₁₆	—	86	2	no	—
98	#108	—	2	4	no	—
99	#108	—	4	2	no	—
100	#108	—	86	4	no	—
101	#108	—	172	2	no	—
102	C₄⋊C₈	—	4	2	no	—
103	C₄⋊C₈	—	172	2	no	—
104	C₄×C₈	Dic₈₆	1	2	yes	—
105	C₄×C₈	C₄×D₄₃	1	2	yes	—
106	C₂²×C₄	#10621885	1	2	yes	—
107	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
108	M₁₆	—	4	2	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	172	2	no	—
111	M₁₆	—	172	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	172	2	no	—
114	C₂×C₈	Dic₄₃⋊C₄	1	2	yes	—
115	C₂×C₈	#10621883	1	2	yes	—
116	C₂×C₈	#10621884	1	2	yes	—
117	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
118	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
119	C₂×C₈	#10621900	1	2	yes	—
120	C₂×C₈	#10621902	1	2	yes	—
121	C₄²	Dic₄₃⋊C₄	1	2	yes	—
122	C₄²	#10621882	1	2	yes	—
123	C₁₆	—	86	4	no	—
124	C₂³	C₂×#10621885	1	1	yes	characteristic
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#13063258	1	4	yes	—
127	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
128	C₂×C₄	#13063312	1	1	yes	characteristic
129	C₂×C₄	#13063314	1	1	yes	characteristic
130	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
131	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
132	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	344	2	no	—
135	C₈	#13063256	1	4	yes	—
136	C₈	#13063283	1	2	yes	—
137	C₈	#13063316	1	2	yes	—
138	C₂²	—	2	2	no	—
139	C₂²	C₂×#13063258	1	1	yes	characteristic
140	C₂²	#14583249	1	1	yes	characteristic
141	C₂²	#14583251	1	1	yes	characteristic
142	C₄	—	2	4	no	—
143	C₄	#14582764	1	1	yes	characteristic
144	C₄	C₂×#13063256	1	1	yes	characteristic
145	C₄	#14583245	1	1	yes	characteristic
146	C₄	#14583247	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#19134722	1	1	yes	characteristic
150	C₂	#19136389	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582675

C₂

yes

maximal, characteristic

#19134767

C₂

yes

maximal, characteristic

#19132688

C₂

yes

maximal

C₄₃×#1848

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

C₂×#13063261

—

C₂×#13063261

C₂²

yes

—

#14582923

—

#14582675

C₄

yes

—

C₂×#108×C₄₃

—

C₂×#108×C₄₃

C₂²

yes

—

C₄₃×#423

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₂×#10621877

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

#13063261

—

#13063261

—

C₄⋊(C₄₃⋊C₈)

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

#108×C₄₃

—

#108×C₄₃

—

C₄⋊C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

C₂²×C₄×C₄₃

Q₁₆

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

SD₁₆

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₃

#41

yes

—

C₂×C₈×C₄₃

#41

yes

—

C₄²×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

D₈

yes

—

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#107

yes

—

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

#106

yes

—

C₈×C₄₃

#117

yes

—

C₈×C₄₃

#125

yes

—

C₄₃⋊C₈

—

#12892

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

—

D₄₃

yes

characteristic

#1381

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#1834

yes

characteristic

C₂×C₄₃

#2161

yes

—

C₂²×C₁₆

—

—

—

—

C₂×D₄₃

yes

—

#423

—

#423

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#346

—

C₄₃

#12892

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

100

#108

—

101

#108

—

172

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

172

—

104

C₄×C₈

Dic₈₆

yes

—

105

C₄×C₈

C₄×D₄₃

yes

—

106

C₂²×C₄

#10621885

yes

—

107

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

172

—

111

M₁₆

—

172

—

112

C₂×C₈

—

113

C₂×C₈

—

172

—

114

C₂×C₈

Dic₄₃⋊C₄

yes

—

115

C₂×C₈

#10621883

yes

—

116

C₂×C₈

#10621884

yes

—

117

C₂×C₈

C₂×Dic₈₆

yes

characteristic

118

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

119

C₂×C₈

#10621900

yes

—

120

C₂×C₈

#10621902

yes

—

121

C₄²

Dic₄₃⋊C₄

yes

—

122

C₄²

#10621882

yes

—

123

C₁₆

—

124

C₂³

C₂×#10621885

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

#13063258

yes

—

127

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

128

C₂×C₄

#13063312

yes

characteristic

129

C₂×C₄

#13063314

yes

characteristic

130

C₂×C₄

C₂×#10621882

yes

characteristic

131

C₂×C₄

C₂×#10621884

yes

characteristic

132

C₂×C₄

C₂×#10621883

yes

characteristic

133

C₈

—

134

C₈

—

344

—

135

C₈

#13063256

yes

—

136

C₈

#13063283

yes

—

137

C₈

#13063316

yes

—

138

C₂²

—

139

C₂²

C₂×#13063258

yes

characteristic

140

C₂²

#14583249

yes

characteristic

141

C₂²

#14583251

yes

characteristic

142

C₄

—

143

C₄

#14582764

yes

characteristic

144

C₄

C₂×#13063256

yes

characteristic

145

C₄

#14583245

yes

characteristic

146

C₄

#14583247

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#19134722

yes

characteristic

150

C₂

#19136389

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1848	C₂
socle	C₂²×C₄₃	C₂×#107
prime radical	C₂³×C₄₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1848	D₄₃
hypocenter	C₄₃	#12892
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

subgroup

G/H

center

C₂×C₄

C₂×#10621882

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1848

C₂

socle

C₂²×C₄₃

C₂×#107

prime radical

C₂³×C₄₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

N	H ≅ G/N	type
89. C₄₃	#12892	semidirect

H ≅ G/N

type

89. C₄₃

#12892

semidirect

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

subgroup	index
C₂×#14582675	2
#19134767	2
#19132688	2
C₄₃×#1848	2
#12892	43

subgroup

index

C₄₃×#1848

Subgroups — #25012989

Conjugacy classes of subgroups

Complements

Maximal subgroups