Group #25010785 — subgroups

Subgroups — #25010785

Order 11008.

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

normal subgroups only

This group has 230 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	#19136646	C₂	1	2	yes	maximal
2	#19134213	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic
4	#19132645	C₂	1	2	yes	maximal
5	C₄₃×#1370	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₄	—	2	2	no	—
7	#14583304	—	2	1	no	—
8	#14583304	—	2	1	no	—
9	#14583279	C₂²	1	2	yes	—
10	C₈×Dic₈₆	—	2	1	no	—
11	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
12	#14582704	—	2	2	no	—
13	#14582652	C₂²	1	1	yes	characteristic
14	#14582631	—	2	1	no	—
15	C₄₃×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	—	2	2	no	—
18	C₂×Dic₁₇₂	—	2	2	no	—
19	C₂×Dic₁₇₂	D₄	1	1	yes	characteristic
20	#13063325	—	2	2	no	—
21	C₄×Dic₈₆	—	2	1	no	—
22	Dic₃₄₄	—	4	4	no	—
23	#13063259	—	4	2	no	—
24	#13063259	—	4	1	no	—
25	#13063259	—	4	1	no	—
26	#13063259	D₄	1	1	yes	characteristic
27	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
28	Dic₄₃⋊C₈	—	4	1	no	—
29	#13063250	—	2	1	no	—
30	C₈×Dic₄₃	—	4	1	no	—
31	C₂×C₁₆×C₄₃	—	2	2	no	—
32	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
33	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
34	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₈₆	—	2	1	no	—
36	C₂×Dic₈₆	—	4	2	no	—
37	C₂×C₈×C₄₃	—	2	1	no	—
38	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
39	C₂×C₈×C₄₃	D₈	1	1	yes	characteristic
40	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
42	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₄⋊Dic₄₃	—	2	1	no	—
44	C₄⋊Dic₄₃	—	4	2	no	—
45	C₄⋊Dic₄₃	—	4	2	no	—
46	Dic₄₃⋊C₄	—	4	1	no	—
47	C₄×Dic₄₃	—	4	1	no	—
48	C₂×C₄₃⋊C₈	—	4	1	no	—
49	Dic₁₇₂	—	2	1	no	—
50	Dic₁₇₂	—	4	2	no	—
51	Dic₁₇₂	—	4	2	no	—
52	Dic₁₇₂	—	8	2	no	—
53	Dic₁₇₂	D₈	1	2	yes	—
54	C₁₆×C₄₃	—	4	2	no	—
55	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
56	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
57	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	C₂×Dic₄₃	—	4	1	no	—
59	C₂×Dic₄₃	—	4	1	no	—
60	C₂×Dic₄₃	—	8	2	no	—
61	Dic₈₆	—	2	2	no	—
62	Dic₈₆	—	4	1	no	—
63	Dic₈₆	—	8	2	no	—
64	Dic₈₆	—	8	2	no	—
65	C₈×C₄₃	—	2	2	no	—
66	C₈×C₄₃	—	2	1	no	—
67	C₈×C₄₃	—	2	1	no	—
68	C₈×C₄₃	Q₃₂	1	2	yes	complemented
69	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
70	C₈×C₄₃	#135	1	1	yes	characteristic
71	C₄₃⋊C₈	—	8	1	no	—
72	#12427	—	43	1	no	maximal, Sylow
73	C₂²×C₄₃	#459	1	1	yes	characteristic
74	C₄×C₄₃	—	2	1	no	—
75	C₄×C₄₃	#466	1	1	yes	characteristic
76	C₄×C₄₃	#472	1	1	yes	characteristic
77	C₄×C₄₃	C₂×Q₃₂	1	1	yes	characteristic
78	C₄×C₄₃	#510	1	1	yes	characteristic
79	Dic₄₃	—	4	2	no	—
80	Dic₄₃	—	8	1	no	—
81	Dic₄₃	—	16	2	no	—
82	#2209	—	43	2	no	—
83	#1713	—	43	1	no	—
84	C₈×Q₁₆	—	43	1	no	—
85	#1370	D₄₃	1	1	yes	characteristic
86	#1362	—	43	2	no	—
87	C₂×C₄₃	#1667	1	1	yes	characteristic
88	C₂×C₄₃	#2209	1	1	yes	characteristic
89	C₂×C₄₃	#2214	1	1	yes	characteristic
90	C₂×Q₃₂	—	86	2	no	—
91	#500	—	86	1	no	—
92	#500	—	86	1	no	—
93	#494	—	43	2	no	—
94	C₈×Q₈	—	86	1	no	—
95	C₄×Q₁₆	—	43	1	no	—
96	C₄⋊C₁₆	C₂×D₄₃	1	2	yes	—
97	#358	—	86	2	no	—
98	#335	—	43	1	no	—
99	Q₈⋊C₈	—	86	1	no	—
100	C₈²	C₂×D₄₃	1	1	yes	characteristic
101	C₄₃	#12427	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	43	1	no	—
103	C₂×Q₁₆	—	86	2	no	—
104	C₂×Q₁₆	—	86	2	no	—
105	C₄⋊Q₈	—	86	2	no	—
106	C₄×Q₈	—	86	1	no	—
107	Q₃₂	—	172	4	no	—
108	C₂×C₁₆	—	2	2	no	—
109	#107	—	43	1	no	—
110	#107	—	172	2	no	—
111	#107	—	172	1	no	—
112	#107	—	172	1	no	—
113	C₄⋊C₈	—	86	1	no	—
114	C₄⋊C₈	—	172	1	no	—
115	#103	—	86	1	no	—
116	C₄×C₈	—	172	1	no	—
117	C₄×C₈	D₁₇₂	1	1	yes	characteristic
118	C₄×C₈	D₁₇₂	1	1	yes	characteristic
119	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
120	C₂×Q₈	—	86	1	no	—
121	C₂×Q₈	—	172	2	no	—
122	Q₁₆	—	43	2	no	—
123	Q₁₆	—	86	1	no	—
124	Q₁₆	—	344	2	no	—
125	Q₁₆	—	172	2	no	—
126	Q₁₆	—	172	2	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	172	1	no	—
129	C₂×C₈	D₃₄₄	1	1	yes	characteristic
130	C₂×C₈	D₃₄₄	1	1	yes	characteristic
131	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
132	C₂×C₈	#10621902	1	1	yes	characteristic
133	C₄⋊C₄	—	86	1	no	—
134	C₄⋊C₄	—	172	2	no	—
135	C₄⋊C₄	—	172	2	no	—
136	C₄⋊C₄	—	172	1	no	—
137	C₄²	—	172	1	no	—
138	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
139	C₁₆	—	4	2	no	—
140	Q₈	—	86	2	no	—
141	Q₈	—	344	2	no	—
142	Q₈	—	344	2	no	—
143	Q₈	—	172	1	no	—
144	C₂×C₄	—	344	2	no	—
145	C₂×C₄	—	172	1	no	—
146	C₂×C₄	—	172	1	no	—
147	C₂×C₄	#13063300	1	1	yes	characteristic
148	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
149	C₂×C₄	#13063353	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	—	2	1	no	—
152	C₈	—	2	1	no	—
153	C₈	—	344	1	no	—
154	C₈	Dic₃₄₄	1	2	yes	complemented
155	C₈	D₈×D₄₃	1	1	yes	characteristic
156	C₈	#13063348	1	1	yes	characteristic
157	C₂²	#14583095	1	1	yes	characteristic
158	C₄	—	2	1	no	—
159	C₄	—	344	1	no	—
160	C₄	—	172	2	no	—
161	C₄	—	688	2	no	—
162	C₄	#14583136	1	1	yes	characteristic
163	C₄	#14583170	1	1	yes	characteristic
164	C₄	C₂×Dic₃₄₄	1	1	yes	characteristic
165	C₄	#14583342	1	1	yes	characteristic
166	C₂	#19133951	1	1	yes	characteristic
167	C₂	#19136646	1	1	yes	characteristic
168	C₂	#19136677	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136646

C₂

yes

maximal

#19134213

C₂

yes

maximal, characteristic

C₈×Dic₁₇₂

C₂

yes

maximal, characteristic

#19132645

C₂

yes

maximal

C₄₃×#1370

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

—

#14583304

—

#14583304

—

#14583279

C₂²

yes

—

C₈×Dic₈₆

—

C₄×Dic₁₇₂

C₂²

yes

characteristic

#14582704

—

#14582652

C₂²

yes

characteristic

#14582631

—

C₄₃×C₄⋊C₁₆

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

—

C₂×Dic₁₇₂

D₄

yes

characteristic

#13063325

—

C₄×Dic₈₆

—

Dic₃₄₄

—

—

—

—

D₄

yes

characteristic

C₄⋊(C₄₃⋊C₈)

—

Dic₄₃⋊C₈

—

#13063250

—

C₈×Dic₄₃

—

C₂×C₁₆×C₄₃

—

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×Dic₈₆

—

C₂×Dic₈₆

—

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

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

—

Dic₁₇₂

D₈

yes

—

C₁₆×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

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

Q₃₂

yes

complemented

C₈×C₄₃

C₂×D₈

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₂×Q₃₂

yes

characteristic

C₄×C₄₃

#510

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

#2209

—

#1713

—

C₈×Q₁₆

—

#1370

D₄₃

yes

characteristic

#1362

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×Q₃₂

—

#500

—

#500

—

#494

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄⋊C₁₆

C₂×D₄₃

yes

—

#358

—

#335

—

Q₈⋊C₈

—

100

C₈²

C₂×D₄₃

yes

characteristic

101

C₄₃

#12427

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂×Q₁₆

—

105

C₄⋊Q₈

—

106

C₄×Q₈

—

107

Q₃₂

—

172

—

108

C₂×C₁₆

—

109

#107

—

110

#107

—

172

—

111

#107

—

172

—

112

#107

—

172

—

113

C₄⋊C₈

—

114

C₄⋊C₈

—

172

—

115

#103

—

116

C₄×C₈

—

172

—

117

C₄×C₈

D₁₇₂

yes

characteristic

118

C₄×C₈

D₁₇₂

yes

characteristic

119

C₄×C₈

C₂²×D₄₃

yes

characteristic

120

C₂×Q₈

—

121

C₂×Q₈

—

172

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

344

—

125

Q₁₆

—

172

—

126

Q₁₆

—

172

—

127

C₂×C₈

—

128

C₂×C₈

—

172

—

129

C₂×C₈

D₃₄₄

yes

characteristic

130

C₂×C₈

D₃₄₄

yes

characteristic

131

C₂×C₈

D₄×D₄₃

yes

characteristic

132

C₂×C₈

#10621902

yes

characteristic

133

C₄⋊C₄

—

134

C₄⋊C₄

—

172

—

135

C₄⋊C₄

—

172

—

136

C₄⋊C₄

—

172

—

137

C₄²

—

172

—

138

C₄²

C₂×D₁₇₂

yes

characteristic

139

C₁₆

—

140

Q₈

—

141

Q₈

—

344

—

142

Q₈

—

344

—

143

Q₈

—

172

—

144

C₂×C₄

—

344

—

145

C₂×C₄

—

172

—

146

C₂×C₄

—

172

—

147

C₂×C₄

#13063300

yes

characteristic

148

C₂×C₄

C₂×D₃₄₄

yes

characteristic

149

C₂×C₄

#13063353

yes

characteristic

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

344

—

154

C₈

Dic₃₄₄

yes

complemented

155

C₈

D₈×D₄₃

yes

characteristic

156

C₈

#13063348

yes

characteristic

157

C₂²

#14583095

yes

characteristic

158

C₄

—

159

C₄

—

344

—

160

C₄

—

172

—

161

C₄

—

688

—

162

C₄

#14583136

yes

characteristic

163

C₄

#14583170

yes

characteristic

164

C₄

C₂×Dic₃₄₄

yes

characteristic

165

C₄

#14583342

yes

characteristic

166

C₂

#19133951

yes

characteristic

167

C₂

#19136646

yes

characteristic

168

C₂

#19136677

yes

characteristic

	subgroup	G/H
center	C₂²	#14583095
commutator, cosocle	C₄×C₈×C₄₃	C₂³
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1370	C₂
socle, prime radical	C₂²×C₄₃	#459
hypercenter	#1370	D₄₃
hypocenter	C₄₃	#12427
LCS	G ▷ C₄×C₈×C₄₃ ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

subgroup

G/H

center

C₂²

#14583095

commutator, cosocle

C₄×C₈×C₄₃

C₂³

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1370

C₂

C₂²×C₄₃

D₄₃

C₄₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1370

N	H ≅ G/N	type
68. C₈×C₄₃	Q₃₂	semidirect
101. C₄₃	#12427	semidirect
154. C₈	Dic₃₄₄	semidirect

H ≅ G/N

type

68. C₈×C₄₃

Q₃₂

semidirect

101. C₄₃

#12427

semidirect

154. C₈

Dic₃₄₄

semidirect

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

subgroup	index
#19136646	2
#19134213	2
C₈×Dic₁₇₂	2
#19132645	2
C₄₃×#1370	2
#12427	43

subgroup

index

C₈×Dic₁₇₂

C₄₃×#1370

Subgroups — #25010785

Conjugacy classes of subgroups

Complements

Maximal subgroups