Group #25012979 — subgroups

Subgroups — #25012979

Order 11008.

310 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 155 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×C₄×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#19134759	C₂	1	1	yes	maximal, characteristic
3	#19132686	C₂	1	4	yes	maximal
4	C₄₃×#1846	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#13063259	C₂²	1	2	yes	—
7	C₂×#13063257	—	2	1	no	—
8	#14582765	—	2	1	no	—
9	C₄×C₄₃⋊C₁₆	C₄	1	4	yes	complemented
10	C₂×#107×C₄₃	C₂²	1	2	yes	—
11	C₂×#106×C₄₃	—	2	1	no	—
12	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#382	—	2	1	no	—
14	C₂×C₄⋊Dic₄₃	—	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	#13063259	—	2	4	no	—
19	#13063257	—	4	2	no	—
20	C₄₃⋊#95	—	4	2	no	—
21	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
23	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
24	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
25	#107×C₄₃	—	2	4	no	—
26	#106×C₄₃	—	4	2	no	—
27	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
28	C₄×C₈×C₄₃	Q₈	1	2	yes	—
29	#95×C₄₃	—	4	2	no	—
30	C₂²×C₄×C₄₃	—	4	2	no	—
31	C₂²×C₄×C₄₃	D₈	1	2	yes	—
32	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₄₃	—	4	2	no	—
34	C₂×C₈×C₄₃	C₄⋊C₄	1	2	yes	—
35	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
36	C₂×C₈×C₄₃	SD₁₆	1	2	yes	—
37	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	#41	1	2	yes	—
40	C₂×C₈×C₄₃	#41	1	2	yes	—
41	C₄⋊C₄×C₄₃	—	4	2	no	—
42	C₄⋊C₄×C₄₃	—	4	2	no	—
43	C₄²×C₄₃	C₄⋊C₄	1	2	yes	—
44	C₄²×C₄₃	Q₁₆	1	2	yes	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	C₄⋊Dic₄₃	—	4	2	no	—
47	C₄₃⋊C₁₆	—	2	4	no	—
48	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₃	—	2	4	no	—
50	C₂×C₄×C₄₃	—	8	2	no	—
51	C₂×C₄×C₄₃	—	8	4	no	—
52	C₂×C₄×C₄₃	#107	1	4	yes	—
53	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₄₃	—	8	2	no	—
60	C₂×Dic₄₃	—	8	4	no	—
61	C₈×C₄₃	#106	1	4	yes	—
62	C₈×C₄₃	#117	1	2	yes	—
63	C₈×C₄₃	#125	1	2	yes	—
64	#12889	—	43	1	no	maximal, Sylow
65	C₂²×C₄₃	C₂×#107	1	1	yes	characteristic
66	C₂²×C₄₃	#486	1	1	yes	characteristic
67	C₂²×C₄₃	#486	1	1	yes	characteristic
68	C₂²×C₄₃	#508	1	4	yes	—
69	C₄×C₄₃	—	2	4	no	—
70	C₄×C₄₃	—	16	2	no	—
71	C₄×C₄₃	#382	1	1	yes	characteristic
72	C₄×C₄₃	C₂×#106	1	1	yes	characteristic
73	C₄×C₄₃	#487	1	1	yes	characteristic
74	C₄×C₄₃	#487	1	1	yes	characteristic
75	Dic₄₃	—	16	2	no	—
76	C₂×C₄×C₁₆	—	43	1	no	—
77	#1846	—	43	1	no	—
78	#1846	D₄₃	1	1	yes	characteristic
79	#1380	—	43	4	no	—
80	C₂×C₄₃	#1834	1	1	yes	characteristic
81	C₂×C₄₃	#2158	1	2	yes	—
82	C₂×C₄₃	#2242	1	2	yes	—
83	C₂×C₄₃	#2242	1	2	yes	—
84	C₂²×C₁₆	—	43	2	no	—
85	C₂×#107	—	43	2	no	—
86	C₂×#107	C₂×D₄₃	1	2	yes	—
87	C₂×#106	—	2	1	no	—
88	C₂×#106	—	86	1	no	—
89	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
90	#382	—	2	1	no	—
91	#382	—	86	1	no	—
92	C₄×C₁₆	—	43	4	no	—
93	C₄₃	#12889	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	86	2	no	—
98	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
99	C₂×C₁₆	—	43	4	no	—
100	C₂×C₁₆	—	86	2	no	—
101	C₂×C₁₆	—	86	2	no	—
102	#107	—	2	4	no	—
103	#107	—	86	4	no	—
104	#106	—	4	2	no	—
105	#106	—	172	2	no	—
106	C₄×C₈	Dic₈₆	1	2	yes	—
107	C₄×C₈	C₄×D₄₃	1	2	yes	—
108	#95	—	4	2	no	—
109	#95	—	172	2	no	—
110	C₂²×C₄	—	4	2	no	—
111	C₂²×C₄	—	172	2	no	—
112	C₂²×C₄	#10621882	1	2	yes	—
113	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
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	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	2	no	—
123	C₄⋊C₄	—	172	2	no	—
124	C₄⋊C₄	—	172	2	no	—
125	C₄²	Dic₄₃⋊C₄	1	2	yes	—
126	C₄²	#10621885	1	2	yes	—
127	C₁₆	—	86	4	no	—
128	C₂³	C₂×#10621882	1	1	yes	characteristic
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	8	2	no	—
132	C₂×C₄	—	344	4	no	—
133	C₂×C₄	—	344	2	no	—
134	C₂×C₄	#13063258	1	4	yes	—
135	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#13063312	1	1	yes	characteristic
137	C₂×C₄	#13063314	1	1	yes	characteristic
138	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
139	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
140	C₂×C₄	C₂×#10621885	1	1	yes	characteristic
141	C₈	#13063256	1	4	yes	—
142	C₈	#13063283	1	2	yes	—
143	C₈	#13063316	1	2	yes	—
144	C₂²	C₂×#13063258	1	1	yes	characteristic
145	C₂²	#14583245	1	1	yes	characteristic
146	C₂²	#14583247	1	1	yes	characteristic
147	C₂²	#14583329	1	4	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	16	2	no	—
150	C₄	—	688	2	no	—
151	C₄	#14582764	1	1	yes	characteristic
152	C₄	C₂×#13063256	1	1	yes	characteristic
153	C₄	#14583249	1	1	yes	characteristic
154	C₄	#14583251	1	1	yes	characteristic
155	C₂	#19134722	1	1	yes	characteristic
156	C₂	#19136377	1	2	yes	—
157	C₂	#19136826	1	2	yes	—
158	C₂	#19136829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic

#19134759

C₂

yes

maximal, characteristic

#19132686

C₂

yes

maximal

C₄₃×#1846

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₁₆

C₂²

yes

—

C₂×#13063259

C₂²

yes

—

C₂×#13063257

—

#14582765

—

C₄×C₄₃⋊C₁₆

C₄

yes

complemented

C₂×#107×C₄₃

C₂²

yes

—

C₂×#106×C₄₃

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#382

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

#13063259

—

#13063257

—

C₄₃⋊#95

—

C₂²×C₈×C₄₃

D₄

yes

characteristic

C₂²×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄²×C₄₃

D₄

yes

characteristic

#107×C₄₃

—

#106×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

Q₈

yes

—

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

D₈

yes

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×Dic₄₃

—

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

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₄⋊C₄

yes

—

C₄²×C₄₃

Q₁₆

yes

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

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×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×Q₁₆

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

#106

yes

—

C₈×C₄₃

#117

yes

—

C₈×C₄₃

#125

yes

—

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

C₂×C₄×C₁₆

—

#1846

—

#1846

D₄₃

yes

characteristic

#1380

—

C₂×C₄₃

#1834

yes

characteristic

C₂×C₄₃

#2158

yes

—

C₂×C₄₃

#2242

yes

—

C₂×C₄₃

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#107

C₂×D₄₃

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

#382

—

#382

—

C₄×C₁₆

—

C₄₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₃

yes

characteristic

C₂²×C₈

C₂²×D₄₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

#107

—

103

#107

—

104

#106

—

105

#106

—

172

—

106

C₄×C₈

Dic₈₆

yes

—

107

C₄×C₈

C₄×D₄₃

yes

—

108

#95

—

109

#95

—

172

—

110

C₂²×C₄

—

111

C₂²×C₄

—

172

—

112

C₂²×C₄

#10621882

yes

—

113

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

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

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

172

—

124

C₄⋊C₄

—

172

—

125

C₄²

Dic₄₃⋊C₄

yes

—

126

C₄²

#10621885

yes

—

127

C₁₆

—

128

C₂³

C₂×#10621882

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

344

—

133

C₂×C₄

—

344

—

134

C₂×C₄

#13063258

yes

—

135

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

136

C₂×C₄

#13063312

yes

characteristic

137

C₂×C₄

#13063314

yes

characteristic

138

C₂×C₄

C₂×#10621884

yes

characteristic

139

C₂×C₄

C₂×#10621883

yes

characteristic

140

C₂×C₄

C₂×#10621885

yes

characteristic

141

C₈

#13063256

yes

—

142

C₈

#13063283

yes

—

143

C₈

#13063316

yes

—

144

C₂²

C₂×#13063258

yes

characteristic

145

C₂²

#14583245

yes

characteristic

146

C₂²

#14583247

yes

characteristic

147

C₂²

#14583329

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

688

—

151

C₄

#14582764

yes

characteristic

152

C₄

C₂×#13063256

yes

characteristic

153

C₄

#14583249

yes

characteristic

154

C₄

#14583251

yes

characteristic

155

C₂

#19134722

yes

characteristic

156

C₂

#19136377

yes

—

157

C₂

#19136826

yes

—

158

C₂

#19136829

yes

—

	subgroup	G/H
center	C₂³	C₂×#10621882
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#1846	C₂
socle, prime radical	C₂³×C₄₃	C₂×D₈
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#1846	D₄₃
hypocenter	C₄₃	#12889
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

subgroup

G/H

center

C₂³

C₂×#10621882

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₃

Fitting

C₄₃×#1846

C₂

socle, prime radical

C₂³×C₄₃

C₂×D₈

cosocle

C₂²×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1846

N	H ≅ G/N	type
9. C₄×C₄₃⋊C₁₆	C₄	semidirect
93. C₄₃	#12889	semidirect

H ≅ G/N

type

9. C₄×C₄₃⋊C₁₆

C₄

semidirect

93. C₄₃

#12889

semidirect

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

subgroup	index
C₂×C₄×C₄₃⋊C₁₆	2
#19134759	2
#19132686	2
C₄₃×#1846	2
#12889	43

subgroup

index

C₂×C₄×C₄₃⋊C₁₆

#19134759

#19132686

C₄₃×#1846

#12889

Subgroups — #25012979

Conjugacy classes of subgroups

Complements

Maximal subgroups