Group #25001408 — subgroups

Subgroups — #25001408

Order 11008.

294 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 85 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	#19135415	C₂	1	1	yes	maximal, characteristic
2	#19133268	C₂	1	2	yes	maximal
3	#19132455	C₂	1	2	yes	maximal
4	#19132440	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1838	C₂	1	1	yes	maximal, characteristic
6	C₂×#13063325	C₂²	1	1	yes	characteristic
7	#14582952	C₂²	1	2	yes	—
8	C₂×#13063250	—	2	2	no	—
9	#14582765	C₂²	1	1	yes	characteristic
10	#14582646	—	2	4	no	—
11	C₄₃×#423	C₂²	1	2	yes	—
12	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#341	—	2	2	no	—
14	C₂²×Dic₈₆	—	2	1	no	—
15	#13063325	—	2	2	no	—
16	#13063325	—	2	1	no	—
17	#13063325	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₄₃	—	2	2	no	—
19	C₂×C₄⋊Dic₄₃	—	2	1	no	—
20	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
21	#13063250	—	4	4	no	—
22	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
23	C₄×C₄₃⋊C₈	—	2	2	no	—
24	C₄₃⋊#95	—	2	2	no	—
25	C₂×M₁₆×C₄₃	—	2	2	no	—
26	C₂²×C₈×C₄₃	D₄	1	2	yes	—
27	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₃	—	2	2	no	—
29	C₄⋊C₈×C₄₃	—	2	2	no	—
30	#97×C₄₃	—	2	2	no	—
31	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₃	#41	1	2	yes	—
33	C₂²×Dic₄₃	—	4	1	no	—
34	C₂²×Dic₄₃	—	4	1	no	—
35	C₂×Dic₈₆	—	2	2	no	—
36	C₂×Dic₈₆	—	4	2	no	—
37	C₂×Dic₈₆	—	4	2	no	—
38	C₂×Dic₈₆	—	4	1	no	—
39	M₁₆×C₄₃	—	4	2	no	—
40	M₁₆×C₄₃	—	4	2	no	—
41	C₂×C₈×C₄₃	—	2	2	no	—
42	C₂×C₈×C₄₃	—	2	2	no	—
43	C₂×C₈×C₄₃	—	2	2	no	—
44	C₂×C₈×C₄₃	—	4	2	no	—
45	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₄₃	#41	1	2	yes	—
47	C₄²×C₄₃	#41	1	1	yes	characteristic
48	C₄⋊Dic₄₃	—	2	4	no	—
49	C₄⋊Dic₄₃	—	4	2	no	—
50	C₄⋊Dic₄₃	—	4	2	no	—
51	C₂×C₄₃⋊C₈	—	4	2	no	—
52	C₂³×C₄₃	#124	1	1	yes	characteristic
53	C₂×C₄×C₄₃	—	4	2	no	—
54	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
55	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
56	C₂×C₄×C₄₃	#123	1	2	yes	—
57	C₂×C₄×C₄₃	#126	1	2	yes	—
58	C₂×C₄×C₄₃	#135	1	2	yes	—
59	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×Dic₄₃	—	4	2	no	—
61	C₂×Dic₄₃	—	8	1	no	—
62	C₂×Dic₄₃	—	8	1	no	—
63	C₂×Dic₄₃	—	8	2	no	—
64	C₂×Dic₄₃	—	8	1	no	—
65	Dic₈₆	—	4	4	no	—
66	Dic₈₆	—	8	2	no	—
67	C₈×C₄₃	—	4	2	no	—
68	C₈×C₄₃	—	8	2	no	—
69	C₄₃⋊C₈	—	8	2	no	—
70	#10635	—	43	1	no	maximal, Sylow
71	C₂²×C₄₃	—	2	2	no	—
72	C₂²×C₄₃	#397	1	1	yes	characteristic
73	C₂²×C₄₃	#488	1	1	yes	characteristic
74	C₂²×C₄₃	#490	1	1	yes	characteristic
75	C₄×C₄₃	—	2	2	no	—
76	C₄×C₄₃	—	2	2	no	—
77	C₄×C₄₃	#463	1	2	yes	—
78	C₄×C₄₃	#482	1	2	yes	—
79	Dic₄₃	—	8	2	no	—
80	Dic₄₃	—	16	1	no	—
81	#1960	—	43	1	no	—
82	#1838	D₄₃	1	1	yes	characteristic
83	#1546	—	43	2	no	—
84	#1286	—	43	2	no	—
85	#1281	—	43	1	no	—
86	C₂×C₄₃	—	2	1	no	—
87	C₂×C₄₃	—	2	1	no	—
88	C₂×C₄₃	#2073	1	1	yes	characteristic
89	C₂×C₄₃	#2082	1	2	yes	—
90	C₂×C₄⋊Q₈	—	43	1	no	—
91	#431	—	43	2	no	—
92	#423	C₂×D₄₃	1	2	yes	—
93	C₂×#103	—	86	2	no	—
94	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
95	#382	—	43	1	no	—
96	#341	—	2	2	no	—
97	#332	—	86	4	no	—
98	C₄₃	#10635	1	1	yes	characteristic, complemented, Sylow
99	C₂²×Q₈	—	86	1	no	—
100	C₂×M₁₆	—	2	2	no	—
101	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
102	C₄⋊Q₈	—	43	2	no	—
103	C₄⋊Q₈	—	86	2	no	—
104	C₄⋊Q₈	—	86	1	no	—
105	C₂×C₄⋊C₄	—	86	2	no	—
106	C₂×C₄⋊C₄	—	86	1	no	—
107	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	2	no	—
110	C₄⋊C₈	—	86	2	no	—
111	#103	—	172	4	no	—
112	C₂²⋊C₈	—	86	2	no	—
113	#97	—	2	2	no	—
114	C₄×C₈	—	86	2	no	—
115	#95	—	86	2	no	—
116	C₂×Q₈	—	86	2	no	—
117	C₂×Q₈	—	172	2	no	—
118	C₂×Q₈	—	172	2	no	—
119	C₂×Q₈	—	172	1	no	—
120	C₂²×C₄	—	172	1	no	—
121	C₂²×C₄	—	172	1	no	—
122	C₂²×C₄	#10621902	1	2	yes	—
123	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	4	2	no	—
130	C₂×C₈	—	172	2	no	—
131	C₄⋊C₄	—	86	4	no	—
132	C₄⋊C₄	—	172	2	no	—
133	C₄⋊C₄	—	172	2	no	—
134	C₄²	#10621898	1	2	yes	—
135	C₄²	D₄×D₄₃	1	1	yes	characteristic
136	C₄²	#10621900	1	1	yes	characteristic
137	C₂³	#13063314	1	1	yes	characteristic
138	Q₈	—	344	2	no	—
139	Q₈	—	172	4	no	—
140	C₂×C₄	—	4	2	no	—
141	C₂×C₄	—	344	1	no	—
142	C₂×C₄	—	344	1	no	—
143	C₂×C₄	—	344	2	no	—
144	C₂×C₄	—	344	1	no	—
145	C₂×C₄	—	172	2	no	—
146	C₂×C₄	#13063296	1	1	yes	characteristic
147	C₂×C₄	#13063308	1	1	yes	characteristic
148	C₂×C₄	#13063310	1	2	yes	—
149	C₂×C₄	#13063320	1	2	yes	—
150	C₂×C₄	#13063345	1	2	yes	—
151	C₂×C₄	#13063350	1	2	yes	—
152	C₈	—	4	2	no	—
153	C₈	—	8	2	no	—
154	C₈	—	344	2	no	—
155	C₂²	—	2	2	no	—
156	C₂²	#14582825	1	1	yes	characteristic
157	C₂²	#14583256	1	1	yes	characteristic
158	C₂²	#14583265	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	344	2	no	—
162	C₄	—	688	1	no	—
163	C₄	#14583111	1	2	yes	—
164	C₄	#14583223	1	2	yes	—
165	C₂	—	2	1	no	—
166	C₂	—	2	1	no	—
167	C₂	#19136009	1	1	yes	characteristic
168	C₂	#19136060	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19135415

C₂

yes

maximal, characteristic

#19133268

C₂

yes

maximal

#19132455

C₂

yes

maximal

#19132440

C₂

yes

maximal, characteristic

C₄₃×#1838

C₂

yes

maximal, characteristic

C₂×#13063325

C₂²

yes

characteristic

#14582952

C₂²

yes

—

C₂×#13063250

—

#14582765

C₂²

yes

characteristic

#14582646

—

C₄₃×#423

C₂²

yes

—

C₂×#97×C₄₃

C₂²

yes

characteristic

C₄₃×#341

—

C₂²×Dic₈₆

—

#13063325

—

#13063325

—

#13063325

D₄

yes

—

C₂×C₄⋊Dic₄₃

—

C₂×C₄⋊Dic₄₃

—

C₄⋊(C₄₃⋊C₈)

—

#13063250

—

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₄₃⋊#95

—

C₂×M₁₆×C₄₃

—

C₂²×C₈×C₄₃

D₄

yes

—

C₂×C₄²×C₄₃

C₂³

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

—

#97×C₄₃

—

C₂²×C₄×C₄₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₃

#41

yes

—

C₂²×Dic₄₃

—

C₂²×Dic₄₃

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

C₂×Dic₈₆

—

M₁₆×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄²×C₄₃

#41

yes

—

C₄²×C₄₃

#41

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

#124

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

—

C₂×C₄×C₄₃

#126

yes

—

C₂×C₄×C₄₃

#135

yes

—

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

#10635

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#463

yes

—

C₄×C₄₃

#482

yes

—

Dic₄₃

—

Dic₄₃

—

#1960

—

#1838

D₄₃

yes

—

—

—

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

#2073

yes

characteristic

C₂×C₄₃

#2082

yes

—

C₂×C₄⋊Q₈

—

#431

—

#423

C₂×D₄₃

yes

—

C₂×#103

—

C₂×#97

C₂×D₄₃

yes

—

—

—

C₄₃

#10635

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

100

C₂×M₁₆

—

101

C₂²×C₈

C₂²⋊D₄₃

yes

—

102

C₄⋊Q₈

—

103

C₄⋊Q₈

—

104

C₄⋊Q₈

—

105

C₂×C₄⋊C₄

—

106

C₂×C₄⋊C₄

—

107

C₂×C₄²

C₂²×D₄₃

yes

characteristic

108

C₄⋊C₈

—

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

—

172

—

112

C₂²⋊C₈

—

113

#97

—

114

C₄×C₈

—

115

#95

—

116

C₂×Q₈

—

117

C₂×Q₈

—

172

—

118

C₂×Q₈

—

172

—

119

C₂×Q₈

—

172

—

120

C₂²×C₄

—

172

—

121

C₂²×C₄

—

172

—

122

C₂²×C₄

#10621902

yes

—

123

C₂²×C₄

C₂×C₂²⋊D₄₃

yes

characteristic

124

M₁₆

—

125

M₁₆

—

126

C₂×C₈

—

127

C₂×C₈

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

—

172

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

172

—

133

C₄⋊C₄

—

172

—

134

C₄²

#10621898

yes

—

135

C₄²

D₄×D₄₃

yes

characteristic

136

C₄²

#10621900

yes

characteristic

137

C₂³

#13063314

yes

characteristic

138

Q₈

—

344

—

139

Q₈

—

172

—

140

C₂×C₄

—

141

C₂×C₄

—

344

—

142

C₂×C₄

—

344

—

143

C₂×C₄

—

344

—

144

C₂×C₄

—

344

—

145

C₂×C₄

—

172

—

146

C₂×C₄

#13063296

yes

characteristic

147

C₂×C₄

#13063308

yes

characteristic

148

C₂×C₄

#13063310

yes

—

149

C₂×C₄

#13063320

yes

—

150

C₂×C₄

#13063345

yes

—

151

C₂×C₄

#13063350

yes

—

152

C₈

—

153

C₈

—

154

C₈

—

344

—

155

C₂²

—

156

C₂²

#14582825

yes

characteristic

157

C₂²

#14583256

yes

characteristic

158

C₂²

#14583265

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

344

—

162

C₄

—

688

—

163

C₄

#14583111

yes

—

164

C₄

#14583223

yes

—

165

C₂

—

166

C₂

—

167

C₂

#19136009

yes

characteristic

168

C₂

#19136060

yes

—

	subgroup	G/H
center	C₂²	#14582825
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1838	C₂
socle	C₂²×C₄₃	#397
prime radical	C₂³×C₄₃	#124
hypercenter	#1838	D₄₃
hypocenter	C₄₃	#10635
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

subgroup

G/H

center

C₂²

#14582825

commutator, cosocle

C₂×C₄²×C₄₃

C₂³

Frattini

C₂×C₄²

C₂²×D₄₃

Fitting

C₄₃×#1838

C₂

C₂²×C₄₃

C₂³×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1838

N	H ≅ G/N	type
98. C₄₃	#10635	semidirect

H ≅ G/N

type

98. C₄₃

#10635

semidirect

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

subgroup	index
#19135415	2
#19133268	2
#19132455	2
#19132440	2
C₄₃×#1838	2
#10635	43

subgroup

index

C₄₃×#1838

Subgroups — #25001408

Conjugacy classes of subgroups

Complements

Maximal subgroups