Group #25017492 — subgroups

Subgroups — #25017492

Order 11008.

164 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 59 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	#19136869	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19132795	C₂	1	1	yes	maximal, characteristic
4	#19132794	C₂	1	1	yes	maximal, characteristic
5	Dic₄₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	#19132776	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1416	C₂	1	1	yes	maximal, characteristic
8	#14583303	—	2	1	no	—
9	Q₁₆×Dic₄₃	—	2	1	no	—
10	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
11	#14582725	C₂²	1	1	yes	characteristic
12	#14582725	C₂²	1	1	yes	characteristic
13	#14582724	—	2	1	no	—
14	#14582703	—	2	1	no	—
15	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
16	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#366	C₂²	1	1	yes	characteristic
19	#13063324	—	4	1	no	—
20	Q₈×Dic₄₃	—	4	1	no	—
21	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
22	#13063259	—	2	1	no	—
23	#13063259	—	2	1	no	—
24	#13063258	—	4	1	no	—
25	#13063249	—	4	1	no	—
26	C₈×Dic₄₃	D₄	1	1	yes	characteristic
27	C₄₃⋊C₃₂	—	2	1	no	—
28	C₂×Q₁₆×C₄₃	—	2	1	no	—
29	Q₃₂×C₄₃	—	2	1	no	—
30	Q₃₂×C₄₃	Q₈	1	2	yes	complemented
31	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
32	#107×C₄₃	—	2	1	no	—
33	C₃₂×C₄₃	—	2	1	no	—
34	C₂×Q₈×C₄₃	—	4	1	no	—
35	C₂×Dic₈₆	—	8	1	no	—
36	Q₁₆×C₄₃	—	2	2	no	—
37	Q₁₆×C₄₃	—	4	1	no	—
38	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄×C₄₃	—	4	1	no	—
40	C₄⋊Dic₄₃	—	4	1	no	—
41	C₄⋊Dic₄₃	—	4	1	no	—
42	C₄⋊Dic₄₃	—	8	1	no	—
43	Dic₄₃⋊C₄	—	8	1	no	—
44	C₄×Dic₄₃	—	8	1	no	—
45	C₄×Dic₄₃	D₈	1	1	yes	characteristic
46	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
47	C₁₆×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₁₆×C₄₃	#41	1	1	yes	characteristic
49	C₄₃⋊C₁₆	—	2	1	no	—
50	Q₈×C₄₃	—	4	2	no	—
51	Q₈×C₄₃	—	8	1	no	—
52	C₂×C₄×C₄₃	—	8	1	no	—
53	C₂×C₄×C₄₃	—	8	1	no	—
54	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₄₃	—	8	1	no	—
56	C₂×Dic₄₃	—	8	1	no	—
57	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
58	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
59	Dic₈₆	—	16	2	no	—
60	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
61	C₈×C₄₃	#137	1	1	yes	characteristic
62	C₄₃⋊C₈	—	2	1	no	—
63	#13706	—	43	1	no	maximal, Sylow
64	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
65	C₄×C₄₃	—	8	2	no	—
66	C₄×C₄₃	—	16	1	no	—
67	C₄×C₄₃	#474	1	1	yes	characteristic
68	C₄×C₄₃	#510	1	1	yes	characteristic
69	Dic₄₃	—	2	1	no	—
70	Dic₄₃	—	16	1	no	—
71	Dic₄₃	—	16	1	no	—
72	Dic₄₃	Q₆₄	1	2	yes	—
73	#2252	—	43	1	no	—
74	C₄×Q₃₂	—	43	1	no	—
75	#1423	—	43	1	no	—
76	#1423	—	43	1	no	—
77	C₄⋊C₃₂	—	43	1	no	—
78	#1416	—	43	1	no	—
79	#1416	D₄₃	1	1	yes	characteristic
80	C₂×C₄₃	#2224	1	1	yes	characteristic
81	C₂×C₄₃	C₂×Q₆₄	1	1	yes	characteristic
82	C₂×C₄₃	#2263	1	1	yes	characteristic
83	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
84	#500	—	86	1	no	—
85	C₄×Q₁₆	—	86	1	no	—
86	C₂×C₃₂	—	43	1	no	—
87	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
88	#366	—	43	1	no	—
89	#366	—	43	1	no	—
90	#366	—	86	1	no	—
91	#366	C₂×D₄₃	1	1	yes	characteristic
92	#358	—	86	1	no	—
93	C₄×C₁₆	—	43	1	no	—
94	C₄₃	#13706	1	1	yes	characteristic, complemented, Sylow
95	C₂×Q₁₆	—	2	1	no	—
96	C₄⋊Q₈	—	172	1	no	—
97	C₄×Q₈	—	172	1	no	—
98	Q₃₂	—	2	1	no	—
99	Q₃₂	Dic₈₆	1	2	yes	complemented
100	C₂×C₁₆	—	43	1	no	—
101	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
102	#107	—	2	1	no	—
103	#107	—	86	1	no	—
104	#107	—	86	1	no	—
105	#107	—	172	1	no	—
106	#103	—	172	1	no	—
107	C₄×C₈	—	43	1	no	—
108	C₃₂	—	2	1	no	—
109	C₃₂	—	86	1	no	—
110	C₂×Q₈	—	4	1	no	—
111	C₂×Q₈	—	344	1	no	—
112	Q₁₆	—	2	2	no	—
113	Q₁₆	—	4	1	no	—
114	C₂×C₈	—	43	1	no	—
115	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	344	1	no	—
118	C₄⋊C₄	—	344	1	no	—
119	C₄⋊C₄	—	172	1	no	—
120	C₄⋊C₄	—	172	1	no	—
121	C₄²	—	43	1	no	—
122	C₄²	—	344	1	no	—
123	C₁₆	—	86	1	no	—
124	C₁₆	C₂×Dic₈₆	1	1	yes	characteristic
125	C₁₆	#10621900	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	8	1	no	—
128	Q₈	—	688	2	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	43	1	no	—
132	C₂×C₄	—	43	1	no	—
133	C₂×C₄	—	344	1	no	—
134	C₂×C₄	—	344	1	no	—
135	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
136	C₈	—	86	1	no	—
137	C₈	#13063302	1	1	yes	characteristic
138	C₈	#13063354	1	1	yes	characteristic
139	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
140	C₄	—	8	2	no	—
141	C₄	—	16	1	no	—
142	C₄	—	43	2	no	—
143	C₄	—	86	1	no	—
144	C₄	—	688	1	no	—
145	C₄	—	688	1	no	—
146	C₄	#14583177	1	1	yes	characteristic
147	C₄	#14583338	1	1	yes	characteristic
148	C₂	#19136730	1	1	yes	characteristic
149	C₂	Q₆₄×D₄₃	1	1	yes	characteristic
150	C₂	#19136909	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136869

C₂

yes

maximal, characteristic

Q₃₂×Dic₄₃

C₂

yes

maximal, characteristic

#19132795

C₂

yes

maximal, characteristic

#19132794

C₂

yes

maximal, characteristic

Dic₄₃⋊C₃₂

C₂

yes

maximal, characteristic

#19132776

C₂

yes

maximal, characteristic

C₄₃×#1416

C₂

yes

maximal, characteristic

#14583303

—

Q₁₆×Dic₄₃

—

C₂×C₄₃⋊C₃₂

C₂²

yes

characteristic

#14582725

C₂²

yes

characteristic

#14582725

C₂²

yes

characteristic

#14582724

—

#14582703

—

C₁₆×Dic₄₃

C₂²

yes

characteristic

C₂×Q₃₂×C₄₃

C₂²

yes

characteristic

C₂×C₃₂×C₄₃

C₂²

yes

characteristic

C₄₃×#366

C₂²

yes

characteristic

#13063324

—

Q₈×Dic₄₃

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

—

—

—

C₈×Dic₄₃

D₄

yes

characteristic

C₄₃⋊C₃₂

—

C₂×Q₁₆×C₄₃

—

Q₃₂×C₄₃

—

Q₃₂×C₄₃

Q₈

yes

complemented

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

#107×C₄₃

—

C₃₂×C₄₃

—

C₂×Q₈×C₄₃

—

C₂×Dic₈₆

—

Q₁₆×C₄₃

—

Q₁₆×C₄₃

—

C₂×C₈×C₄₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

C₄⋊Dic₄₃

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₈

yes

characteristic

C₂×C₄₃⋊C₈

D₈

yes

characteristic

C₁₆×C₄₃

C₂×Q₈

yes

characteristic

C₁₆×C₄₃

#41

yes

characteristic

C₄₃⋊C₁₆

—

Q₈×C₄₃

—

Q₈×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

D₁₆

yes

characteristic

C₂×Dic₄₃

D₁₆

yes

characteristic

Dic₈₆

—

C₈×C₄₃

C₂²⋊Q₈

yes

characteristic

C₈×C₄₃

#137

yes

characteristic

C₄₃⋊C₈

—

#13706

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

Q₆₄

yes

—

#2252

—

C₄×Q₃₂

—

#1423

—

#1423

—

C₄⋊C₃₂

—

#1416

—

#1416

D₄₃

yes

characteristic

C₂×C₄₃

#2224

yes

characteristic

C₂×C₄₃

C₂×Q₆₄

yes

characteristic

C₂×C₄₃

#2263

yes

characteristic

C₂×Q₃₂

C₂×D₄₃

yes

characteristic

#500

—

C₄×Q₁₆

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₃

yes

—

—

—

C₂×D₄₃

yes

characteristic

#358

—

C₄×C₁₆

—

C₄₃

#13706

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

172

—

C₄×Q₈

—

172

—

Q₃₂

—

Q₃₂

Dic₈₆

yes

complemented

100

C₂×C₁₆

—

101

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

102

#107

—

103

#107

—

104

#107

—

105

#107

—

172

—

106

#103

—

172

—

107

C₄×C₈

—

108

C₃₂

—

109

C₃₂

—

110

C₂×Q₈

—

111

C₂×Q₈

—

344

—

112

Q₁₆

—

113

Q₁₆

—

114

C₂×C₈

—

115

C₂×C₈

D₄×D₄₃

yes

characteristic

116

C₄⋊C₄

—

117

C₄⋊C₄

—

344

—

118

C₄⋊C₄

—

344

—

119

C₄⋊C₄

—

172

—

120

C₄⋊C₄

—

172

—

121

C₄²

—

122

C₄²

—

344

—

123

C₁₆

—

124

C₁₆

C₂×Dic₈₆

yes

characteristic

125

C₁₆

#10621900

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

688

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

344

—

134

C₂×C₄

—

344

—

135

C₂×C₄

D₈×D₄₃

yes

characteristic

136

C₈

—

137

C₈

#13063302

yes

characteristic

138

C₈

#13063354

yes

characteristic

139

C₂²

D₁₆×D₄₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

688

—

145

C₄

—

688

—

146

C₄

#14583177

yes

characteristic

147

C₄

#14583338

yes

characteristic

148

C₂

#19136730

yes

characteristic

149

C₂

Q₆₄×D₄₃

yes

characteristic

150

C₂

#19136909

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₃
commutator, cosocle	C₂×C₁₆×C₄₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1416	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
hypercenter	#1416	D₄₃
hypocenter	C₄₃	#13706
LCS	G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

subgroup

G/H

center

C₂²

D₁₆×D₄₃

commutator, cosocle

C₂×C₁₆×C₄₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#1416

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

D₄₃

C₄₃

G ▷ C₂×C₁₆×C₄₃ ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1416

N	H ≅ G/N	type
30. Q₃₂×C₄₃	Q₈	semidirect
94. C₄₃	#13706	semidirect
99. Q₃₂	Dic₈₆	semidirect

H ≅ G/N

type

30. Q₃₂×C₄₃

Q₈

semidirect

94. C₄₃

#13706

semidirect

99. Q₃₂

Dic₈₆

semidirect

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

subgroup	index
#19136869	2
Q₃₂×Dic₄₃	2
#19132795	2
#19132794	2
Dic₄₃⋊C₃₂	2
#19132776	2
C₄₃×#1416	2
#13706	43

subgroup

index

Q₃₂×Dic₄₃

Dic₄₃⋊C₃₂

C₄₃×#1416

Subgroups — #25017492

Conjugacy classes of subgroups

Complements

Maximal subgroups