Group #25011226 — subgroups

Subgroups — #25011226

Order 11008.

226 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 71 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	#19136742	C₂	1	2	yes	maximal
2	#19134213	C₂	1	1	yes	maximal, characteristic, complemented
3	#19132659	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132654	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1589	C₂	1	1	yes	maximal, characteristic
6	#14583304	—	2	1	no	—
7	#14583304	C₂²	1	2	yes	—
8	#14583279	—	2	1	no	—
9	#14582726	—	2	2	no	—
10	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	2	yes	—
11	#14582700	—	2	2	no	—
12	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#436	—	2	1	no	—
14	C₄₃×#335	C₂²	1	1	yes	characteristic
15	C₄₃×D₄⋊C₈	—	2	1	no	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	4	2	no	—
18	#13063325	—	2	2	no	—
19	C₂×C₄₃⋊C₁₆	—	2	2	no	—
20	#13063259	—	2	2	no	—
21	#13063259	—	2	2	no	—
22	#13063259	—	4	1	no	—
23	#13063259	—	4	1	no	—
24	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₄₃	—	4	1	no	—
26	C₄×D₄×C₄₃	—	2	1	no	—
27	#107×C₄₃	D₄	1	1	yes	characteristic
28	C₄⋊C₈×C₄₃	—	2	1	no	—
29	#102×C₄₃	—	2	1	no	—
30	C₂²⋊C₈×C₄₃	—	4	1	no	—
31	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
34	C₂×D₄×C₄₃	—	2	1	no	—
35	C₂²×C₄×C₄₃	—	4	1	no	—
36	C₂×Dic₈₆	—	4	2	no	—
37	D₈×C₄₃	—	2	2	no	—
38	M₁₆×C₄₃	—	8	2	no	—
39	C₂×C₈×C₄₃	—	2	1	no	—
40	C₂×C₈×C₄₃	—	4	1	no	—
41	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
42	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
43	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
45	C₄⋊C₄×C₄₃	—	2	1	no	—
46	C₂²⋊C₄×C₄₃	—	4	1	no	—
47	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
48	C₄⋊Dic₄₃	—	4	2	no	—
49	C₄⋊Dic₄₃	—	4	2	no	—
50	Dic₁₇₂	—	8	4	no	—
51	C₄₃⋊C₁₆	—	4	2	no	—
52	C₂³×C₄₃	—	4	1	no	—
53	D₄×C₄₃	—	4	1	no	—
54	D₄×C₄₃	—	4	1	no	—
55	C₂×C₄×C₄₃	—	4	1	no	—
56	C₂×C₄×C₄₃	—	8	1	no	—
57	C₂×C₄×C₄₃	—	8	1	no	—
58	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
59	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×Dic₄₃	—	8	2	no	—
62	Dic₈₆	—	8	4	no	—
63	C₈×C₄₃	—	2	2	no	—
64	C₈×C₄₃	—	2	1	no	—
65	C₈×C₄₃	—	2	1	no	—
66	C₈×C₄₃	—	8	1	no	—
67	C₈×C₄₃	SD₃₂	1	2	yes	—
68	C₈×C₄₃	#137	1	2	yes	—
69	#12499	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	—	8	1	no	—
71	C₂²×C₄₃	—	8	1	no	—
72	C₂²×C₄₃	—	8	1	no	—
73	C₂²×C₄₃	#459	1	1	yes	characteristic
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	—	8	1	no	—
76	C₄×C₄₃	#470	1	1	yes	characteristic
77	C₄×C₄₃	#473	1	1	yes	characteristic
78	C₄×C₄₃	C₂×SD₃₂	1	1	yes	characteristic
79	C₄×C₄₃	#510	1	1	yes	characteristic
80	Dic₄₃	—	16	2	no	—
81	#2226	—	43	2	no	—
82	#1713	—	43	1	no	—
83	#1589	D₄₃	1	1	yes	characteristic
84	#1367	—	43	1	no	—
85	#1365	—	43	2	no	—
86	C₂×C₄₃	—	16	1	no	—
87	C₂×C₄₃	#1689	1	1	yes	characteristic
88	C₂×C₄₃	#2208	1	1	yes	characteristic
89	C₂×C₄₃	#2213	1	1	yes	characteristic
90	#500	—	43	2	no	—
91	#500	—	86	1	no	—
92	#494	—	86	1	no	—
93	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
94	#436	—	2	1	no	—
95	#367	—	86	2	no	—
96	C₄⋊C₁₆	—	43	2	no	—
97	#357	—	86	2	no	—
98	#335	C₂×D₄₃	1	1	yes	characteristic
99	D₄⋊C₈	—	2	1	no	—
100	C₈²	C₂×D₄₃	1	1	yes	characteristic
101	C₄₃	#12499	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	172	2	no	—
103	C₂×D₈	D₁₇₂	1	1	yes	characteristic
104	C₂×M₁₆	—	4	1	no	—
105	C₄⋊Q₈	—	86	2	no	—
106	C₄×D₄	—	2	1	no	—
107	C₂×C₁₆	—	86	2	no	—
108	#107	—	86	2	no	—
109	#107	—	86	2	no	—
110	#107	—	172	1	no	—
111	#107	—	172	1	no	—
112	#107	D₁₇₂	1	1	yes	characteristic
113	C₄⋊C₈	—	2	1	no	—
114	#102	—	2	1	no	—
115	C₂²⋊C₈	—	4	1	no	—
116	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
117	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
118	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
119	C₂×Q₈	—	172	2	no	—
120	C₂×D₄	—	2	1	no	—
121	C₂²×C₄	—	4	1	no	—
122	Q₁₆	—	344	4	no	—
123	D₈	—	2	2	no	—
124	M₁₆	—	8	2	no	—
125	C₂×C₈	—	2	1	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	#10621882	1	1	yes	characteristic
128	C₂×C₈	#10621882	1	1	yes	characteristic
129	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
130	C₂×C₈	#10621898	1	1	yes	characteristic
131	C₄⋊C₄	—	2	1	no	—
132	C₄⋊C₄	—	172	2	no	—
133	C₄⋊C₄	—	172	2	no	—
134	C₂²⋊C₄	—	4	1	no	—
135	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
136	C₁₆	—	172	2	no	—
137	C₂³	—	4	1	no	—
138	Q₈	—	344	4	no	—
139	D₄	—	4	1	no	—
140	D₄	—	4	1	no	—
141	C₂×C₄	—	4	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	8	1	no	—
144	C₂×C₄	—	344	2	no	—
145	C₂×C₄	#13063299	1	1	yes	characteristic
146	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
147	C₂×C₄	#13063350	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	8	1	no	—
152	C₈	#13063269	1	2	yes	—
153	C₈	#13063358	1	2	yes	—
154	C₂²	—	8	1	no	—
155	C₂²	—	8	1	no	—
156	C₂²	—	8	1	no	—
157	C₂²	#14583094	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	8	1	no	—
160	C₄	—	688	2	no	—
161	C₄	#14583157	1	1	yes	characteristic
162	C₄	#14583174	1	1	yes	characteristic
163	C₄	C₂×#13063269	1	1	yes	characteristic
164	C₄	#14583339	1	1	yes	characteristic
165	C₂	—	16	1	no	—
166	C₂	#19134084	1	1	yes	characteristic
167	C₂	#19136640	1	1	yes	characteristic
168	C₂	#19136671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136742

C₂

yes

maximal

#19134213

C₂

yes

maximal, characteristic, complemented

#19132659

C₂

yes

maximal, characteristic, complemented

#19132654

C₂

yes

maximal, complemented

C₄₃×#1589

C₂

yes

maximal, characteristic

#14583304

—

#14583304

C₂²

yes

—

#14583279

—

#14582726

—

C₄⋊(C₄₃⋊C₁₆)

C₂²

yes

—

#14582700

—

C₄×D₈×C₄₃

C₂²

yes

characteristic

C₄₃×#436

—

C₄₃×#335

C₂²

yes

characteristic

C₄₃×D₄⋊C₈

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

#13063325

—

C₂×C₄₃⋊C₁₆

—

—

—

—

—

C₂×D₈×C₄₃

D₄

yes

characteristic

C₂×M₁₆×C₄₃

—

C₄×D₄×C₄₃

—

#107×C₄₃

D₄

yes

characteristic

C₄⋊C₈×C₄₃

—

#102×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×D₄×C₄₃

—

C₂²×C₄×C₄₃

—

C₂×Dic₈₆

—

D₈×C₄₃

—

M₁₆×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

D₈

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₂×C₈×C₄₃

#41

yes

characteristic

C₄⋊C₄×C₄₃

—

C₂²⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₁₇₂

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

#121

yes

characteristic

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×C₄×C₄₃

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₄₃

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

SD₃₂

yes

—

C₈×C₄₃

#137

yes

—

#12499

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#459

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×SD₃₂

yes

characteristic

C₄×C₄₃

#510

yes

characteristic

Dic₄₃

—

#2226

—

#1713

—

#1589

D₄₃

yes

characteristic

#1367

—

#1365

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

—

C₄×D₈

C₂×D₄₃

yes

characteristic

#436

—

#367

—

C₄⋊C₁₆

—

#357

—

#335

C₂×D₄₃

yes

characteristic

D₄⋊C₈

—

100

C₈²

C₂×D₄₃

yes

characteristic

101

C₄₃

#12499

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

172

—

103

C₂×D₈

D₁₇₂

yes

characteristic

104

C₂×M₁₆

—

105

C₄⋊Q₈

—

106

C₄×D₄

—

107

C₂×C₁₆

—

108

#107

—

109

#107

—

110

#107

—

172

—

111

#107

—

172

—

112

#107

D₁₇₂

yes

characteristic

113

C₄⋊C₈

—

114

#102

—

115

C₂²⋊C₈

—

116

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

117

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

118

C₄×C₈

C₂²×D₄₃

yes

characteristic

119

C₂×Q₈

—

172

—

120

C₂×D₄

—

121

C₂²×C₄

—

122

Q₁₆

—

344

—

123

D₈

—

124

M₁₆

—

125

C₂×C₈

—

126

C₂×C₈

—

127

C₂×C₈

#10621882

yes

characteristic

128

C₂×C₈

#10621882

yes

characteristic

129

C₂×C₈

C₂×D₁₇₂

yes

characteristic

130

C₂×C₈

#10621898

yes

characteristic

131

C₄⋊C₄

—

132

C₄⋊C₄

—

172

—

133

C₄⋊C₄

—

172

—

134

C₂²⋊C₄

—

135

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

136

C₁₆

—

172

—

137

C₂³

—

138

Q₈

—

344

—

139

D₄

—

140

D₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

344

—

145

C₂×C₄

#13063299

yes

characteristic

146

C₂×C₄

C₂×#10621882

yes

characteristic

147

C₂×C₄

#13063350

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#13063269

yes

—

153

C₈

#13063358

yes

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#14583094

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

688

—

161

C₄

#14583157

yes

characteristic

162

C₄

#14583174

yes

characteristic

163

C₄

C₂×#13063269

yes

characteristic

164

C₄

#14583339

yes

characteristic

165

C₂

—

166

C₂

#19134084

yes

characteristic

167

C₂

#19136640

yes

characteristic

168

C₂

#19136671

yes

characteristic

	subgroup	G/H
center	C₂²	#14583094
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1589	C₂
socle	C₂²×C₄₃	#459
prime radical	C₂×D₈×C₄₃	D₄
hypercenter	#1589	D₄₃
hypocenter	C₄₃	#12499
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1589

subgroup

G/H

center

C₂²

#14583094

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1589

C₂

socle

C₂²×C₄₃

#459

prime radical

C₂×D₈×C₄₃

D₄

D₄₃

C₄₃

G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1589

N	H ≅ G/N	type
2. #19134213	C₂	semidirect
3. #19132659	C₂	semidirect
4. #19132654	C₂	semidirect
101. C₄₃	#12499	semidirect

H ≅ G/N

type

2. #19134213

C₂

semidirect

3. #19132659

C₂

semidirect

4. #19132654

C₂

semidirect

101. C₄₃

#12499

semidirect

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

subgroup	index
#19136742	2
#19134213	2
#19132659	2
#19132654	2
C₄₃×#1589	2
#12499	43

subgroup

index

C₄₃×#1589

Subgroups — #25011226

Conjugacy classes of subgroups

Complements

Maximal subgroups