Group #25110303 — subgroups

Subgroups — #25110303

Order 11008.

594 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 203 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	#19145119	C₂	1	1	yes	maximal, characteristic
2	#19143532	C₂	1	2	yes	maximal
3	#19135571	C₂	1	2	yes	maximal
4	#19135193	C₂	1	8	yes	maximal
5	#19134343	C₂	1	1	yes	maximal, characteristic
6	C₄₃×#1838	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂²	1	2	yes	—
8	#14583308	C₄	1	8	yes	complemented
9	#14583028	—	2	8	no	—
10	#14582937	C₂²	1	4	yes	—
11	#14582927	C₂²	1	4	yes	—
12	#14582857	C₂²	1	4	yes	—
13	C₂×#13063236	C₂²	1	1	yes	characteristic
14	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
15	#14582660	C₂²	1	8	yes	—
16	#14582653	C₂²	1	4	yes	—
17	C₄₃×#423	C₂²	1	2	yes	—
18	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#341	C₂²	1	4	yes	—
20	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
21	#13063325	C₂×C₄	1	4	yes	—
22	#13063317	C₂×C₄	1	4	yes	—
23	#13063304	—	2	4	no	—
24	C₄×Dic₈₆	—	2	4	no	—
25	#13063284	C₂³	1	4	yes	—
26	C₂×C₄×Dic₄₃	—	2	2	no	—
27	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
28	#13063259	C₂×C₄	1	8	yes	—
29	#13063257	C₂×C₄	1	8	yes	—
30	Dic₄₃⋊C₈	—	2	8	no	—
31	#13063236	D₄	1	4	yes	—
32	#13063235	—	2	4	no	—
33	C₈×Dic₄₃	—	2	4	no	—
34	C₂×M₁₆×C₄₃	C₂³	1	4	yes	—
35	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
36	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₄₃	—	2	4	no	—
38	#97×C₄₃	C₂×C₄	1	4	yes	—
39	C₂²×C₄×C₄₃	#41	1	2	yes	—
40	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
41	C₂²×Dic₄₃	—	2	2	no	—
42	C₂×Dic₈₆	—	2	4	no	—
43	M₁₆×C₄₃	—	2	8	no	—
44	C₂×C₈×C₄₃	—	2	4	no	—
45	C₂×C₈×C₄₃	—	2	4	no	—
46	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
47	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
48	C₄²×C₄₃	C₂×D₄	1	2	yes	—
49	C₂²⋊Dic₄₃	—	2	4	no	—
50	C₄⋊Dic₄₃	—	2	4	no	—
51	C₄⋊Dic₄₃	C₂²×C₄	1	8	yes	—
52	Dic₄₃⋊C₄	—	2	8	no	—
53	C₄×Dic₄₃	—	2	4	no	—
54	C₄×Dic₄₃	—	2	4	no	—
55	C₄×Dic₄₃	—	4	4	no	—
56	C₂×C₄₃⋊C₈	—	2	4	no	—
57	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
58	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
59	C₂×C₄×C₄₃	—	4	2	no	—
60	C₂×C₄×C₄₃	C₄×D₄	1	4	yes	—
61	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₄×C₄₃	2₋¹⁺⁴	1	4	yes	—
64	C₂×Dic₄₃	—	2	4	no	—
65	C₂×Dic₄₃	—	2	8	no	—
66	C₂×Dic₄₃	—	4	2	no	—
67	C₂×Dic₄₃	—	4	2	no	—
68	Dic₈₆	—	8	4	no	—
69	C₈×C₄₃	—	2	4	no	—
70	C₈×C₄₃	—	4	4	no	—
71	C₄₃⋊C₈	—	2	4	no	—
72	#22259	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	—	2	2	no	—
74	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
75	C₂²×C₄₃	#541	1	2	yes	—
76	C₄×C₄₃	—	2	4	no	—
77	C₄×C₄₃	#519	1	2	yes	—
78	C₄×C₄₃	#536	1	2	yes	—
79	Dic₄₃	—	4	4	no	—
80	Dic₄₃	—	4	8	no	—
81	#3162	—	43	1	no	—
82	#2961	—	43	2	no	—
83	#1994	—	43	2	no	—
84	#1922	—	43	8	no	—
85	#1838	D₄₃	1	1	yes	characteristic
86	#1745	—	43	1	no	—
87	C₂×C₄₃	—	2	2	no	—
88	C₂×C₄₃	#2327	1	1	yes	characteristic
89	C₂×C₄₃	#2956	1	2	yes	—
90	#533	—	43	2	no	—
91	#501	—	43	8	no	—
92	#446	—	86	8	no	—
93	#427	—	43	4	no	—
94	#424	—	43	4	no	—
95	#423	C₂×D₄₃	1	2	yes	—
96	#405	—	43	4	no	—
97	C₂×#97	—	43	1	no	—
98	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
99	C₄³	—	43	1	no	—
100	#341	C₂×D₄₃	1	4	yes	—
101	#339	—	43	8	no	—
102	#336	—	43	4	no	—
103	C₄₃	#22259	1	1	yes	characteristic, complemented, Sylow
104	C₂×M₁₆	C₂²×D₄₃	1	4	yes	—
105	C₂²×C₈	—	43	2	no	—
106	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
107	C₄⋊Q₈	—	43	4	no	—
108	#125	—	43	4	no	—
109	C₂²⋊Q₈	—	86	4	no	—
110	C₄×Q₈	—	86	4	no	—
111	#117	—	43	4	no	—
112	C₂×C₄²	—	43	2	no	—
113	C₂×C₄²	—	86	2	no	—
114	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
115	#107	—	43	8	no	—
116	#106	—	43	8	no	—
117	C₄⋊C₈	—	2	4	no	—
118	C₄⋊C₈	—	86	8	no	—
119	#97	—	43	4	no	—
120	#97	—	86	4	no	—
121	#97	C₄×D₄₃	1	4	yes	—
122	C₄×C₈	—	86	4	no	—
123	C₂×Q₈	—	86	4	no	—
124	C₂²×C₄	—	86	2	no	—
125	C₂²×C₄	#10621902	1	2	yes	—
126	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
127	M₁₆	—	2	8	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	43	4	no	—
131	C₂×C₈	—	86	4	no	—
132	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
133	C₄⋊C₄	—	43	8	no	—
134	C₄⋊C₄	—	86	8	no	—
135	C₄⋊C₄	—	86	4	no	—
136	C₂²⋊C₄	—	86	4	no	—
137	C₄²	—	86	4	no	—
138	C₄²	—	86	4	no	—
139	C₄²	—	172	4	no	—
140	C₄²	C₂×C₄×D₄₃	1	2	yes	—
141	C₄²	D₄×D₄₃	1	2	yes	—
142	C₂³	C₂×#10621902	1	1	yes	characteristic
143	Q₈	—	344	4	no	—
144	C₂×C₄	—	4	2	no	—
145	C₂×C₄	—	86	4	no	—
146	C₂×C₄	—	86	8	no	—
147	C₂×C₄	—	172	2	no	—
148	C₂×C₄	—	172	2	no	—
149	C₂×C₄	#13063287	1	4	yes	—
150	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
151	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
152	C₂×C₄	#13063372	1	4	yes	—
153	C₈	—	2	4	no	—
154	C₈	—	4	4	no	—
155	C₈	—	86	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂×#13063287	1	1	yes	characteristic
158	C₂²	#14583512	1	2	yes	—
159	C₄	—	2	4	no	—
160	C₄	—	172	4	no	—
161	C₄	—	172	8	no	—
162	C₄	#14583376	1	2	yes	—
163	C₄	#14583469	1	2	yes	—
164	C₂	—	2	2	no	—
165	C₂	#19137312	1	1	yes	characteristic
166	C₂	#19143504	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19145119

C₂

yes

maximal, characteristic

#19143532

C₂

yes

maximal

#19135571

C₂

yes

maximal

#19135193

C₂

yes

maximal

#19134343

C₂

yes

maximal, characteristic

C₄₃×#1838

C₂

yes

maximal, characteristic

#14583456

C₂²

yes

—

#14583308

C₄

yes

complemented

#14583028

—

#14582937

C₂²

yes

—

#14582927

C₂²

yes

—

#14582857

C₂²

yes

—

C₂×#13063236

C₂²

yes

characteristic

C₄²×Dic₄₃

C₂²

yes

characteristic

#14582660

C₂²

yes

—

#14582653

C₂²

yes

—

C₄₃×#423

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄₃×#341

C₂²

yes

—

C₂²×C₄₃⋊C₈

C₂³

yes

—

#13063325

C₂×C₄

yes

—

#13063317

C₂×C₄

yes

—

#13063304

—

C₄×Dic₈₆

—

#13063284

C₂³

yes

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₂³

yes

—

#13063259

C₂×C₄

yes

—

#13063257

C₂×C₄

yes

—

Dic₄₃⋊C₈

—

#13063236

D₄

yes

—

#13063235

—

C₈×Dic₄₃

—

C₂×M₁₆×C₄₃

C₂³

yes

—

C₂²×C₈×C₄₃

C₂³

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

#97×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

#41

yes

—

C₂²×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂×D₄

yes

—

C₂²⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

C₂²×C₄

yes

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₂×D₄

yes

—

C₂³×C₄₃

C₂×#41

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

—

C₂×C₄×C₄₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₃

2₋¹⁺⁴

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#22259

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₄×D₄

yes

characteristic

C₂²×C₄₃

#541

yes

—

C₄×C₄₃

—

C₄×C₄₃

#519

yes

—

C₄×C₄₃

#536

yes

—

Dic₄₃

—

Dic₄₃

—

—

—

—

—

D₄₃

yes

characteristic

#1745

—

C₂×C₄₃

—

C₂×C₄₃

#2327

yes

characteristic

C₂×C₄₃

#2956

yes

—

—

—

—

—

—

C₂×D₄₃

yes

—

#405

—

C₂×#97

—

C₂×#97

C₂×D₄₃

yes

characteristic

C₄³

—

100

#341

C₂×D₄₃

yes

—

101

#339

—

102

#336

—

103

C₄₃

#22259

yes

characteristic, complemented, Sylow

104

C₂×M₁₆

C₂²×D₄₃

yes

—

105

C₂²×C₈

—

106

C₂²×C₈

C₂²×D₄₃

yes

—

107

C₄⋊Q₈

—

108

#125

—

109

C₂²⋊Q₈

—

110

C₄×Q₈

—

111

#117

—

112

C₂×C₄²

—

113

C₂×C₄²

—

114

C₂×C₄²

C₂²×D₄₃

yes

characteristic

115

#107

—

116

#106

—

117

C₄⋊C₈

—

118

C₄⋊C₈

—

119

#97

—

120

#97

—

121

#97

C₄×D₄₃

yes

—

122

C₄×C₈

—

123

C₂×Q₈

—

124

C₂²×C₄

—

125

C₂²×C₄

#10621902

yes

—

126

C₂²×C₄

C₂³×D₄₃

yes

characteristic

127

M₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

131

C₂×C₈

—

132

C₂×C₈

C₂×C₄×D₄₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₂²⋊C₄

—

137

C₄²

—

138

C₄²

—

139

C₄²

—

172

—

140

C₄²

C₂×C₄×D₄₃

yes

—

141

C₄²

D₄×D₄₃

yes

—

142

C₂³

C₂×#10621902

yes

characteristic

143

Q₈

—

344

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

172

—

148

C₂×C₄

—

172

—

149

C₂×C₄

#13063287

yes

—

150

C₂×C₄

C₂²×C₄×D₄₃

yes

characteristic

151

C₂×C₄

C₂×D₄×D₄₃

yes

characteristic

152

C₂×C₄

#13063372

yes

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₂²

—

157

C₂²

C₂×#13063287

yes

characteristic

158

C₂²

#14583512

yes

—

159

C₄

—

160

C₄

—

172

—

161

C₄

—

172

—

162

C₄

#14583376

yes

—

163

C₄

#14583469

yes

—

164

C₂

—

165

C₂

#19137312

yes

characteristic

166

C₂

#19143504

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1838	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂³×C₄₃	C₂×#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1838	D₄₃
hypocenter	C₄₃	#22259
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂×C₄

C₂×D₄×D₄₃

commutator

C₂×C₄×C₄₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₄₃

Fitting

C₄₃×#1838

C₂

socle

C₂²×C₄₃

C₂×C₄×D₄

prime radical

C₂³×C₄₃

C₂×#41

cosocle

C₂²×C₄×C₄₃

C₂⁴

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
8. #14583308	C₄	semidirect
103. C₄₃	#22259	semidirect

H ≅ G/N

type

8. #14583308

C₄

semidirect

103. C₄₃

#22259

semidirect

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

subgroup	index
#19145119	2
#19143532	2
#19135571	2
#19135193	2
#19134343	2
C₄₃×#1838	2
#22259	43

subgroup

index

C₄₃×#1838

Subgroups — #25110303

Conjugacy classes of subgroups

Complements

Maximal subgroups