Group #25017180 — subgroups

Subgroups — #25017180

Order 11008.

190 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂×#14582739	C₂	1	1	yes	maximal, characteristic
2	Q₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	#19136453	C₂	1	1	yes	maximal, characteristic
4	#19132794	C₂	1	1	yes	maximal, characteristic
5	#19132777	C₂	1	1	yes	maximal, characteristic
6	C₃₂×Dic₄₃	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1416	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₄₄	C₂²	1	1	yes	characteristic
9	Q₁₆×Dic₄₃	—	2	1	no	—
10	#14582993	—	2	1	no	—
11	#14582739	C₄	1	4	yes	complemented
12	C₂×C₄₃⋊C₃₂	C₂²	1	1	yes	characteristic
13	#14582725	C₂²	1	1	yes	characteristic
14	#14582703	—	2	1	no	—
15	#14582702	—	2	1	no	—
16	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
17	C₂×Q₃₂×C₄₃	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#366	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₇₂	—	2	1	no	—
21	Q₈×Dic₄₃	—	4	1	no	—
22	#13063290	—	4	1	no	—
23	Dic₃₄₄	—	2	1	no	—
24	Dic₃₄₄	C₂×C₄	1	2	yes	—
25	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
26	#13063259	—	2	1	no	—
27	#13063249	—	4	1	no	—
28	#13063248	—	4	1	no	—
29	C₈×Dic₄₃	D₄	1	1	yes	characteristic
30	C₄₃⋊C₃₂	C₂×C₄	1	2	yes	—
31	C₂×Q₁₆×C₄₃	—	2	1	no	—
32	Q₃₂×C₄₃	—	2	1	no	—
33	Q₃₂×C₄₃	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
35	#107×C₄₃	—	2	1	no	—
36	C₃₂×C₄₃	—	2	1	no	—
37	C₂×Q₈×C₄₃	—	4	1	no	—
38	C₂×Dic₈₆	—	4	1	no	—
39	Q₁₆×C₄₃	—	2	2	no	—
40	Q₁₆×C₄₃	—	4	1	no	—
41	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
42	C₄⋊C₄×C₄₃	—	4	1	no	—
43	C₄⋊Dic₄₃	—	4	1	no	—
44	C₄⋊Dic₄₃	—	8	1	no	—
45	Dic₄₃⋊C₄	—	8	1	no	—
46	C₄×Dic₄₃	—	8	1	no	—
47	C₄×Dic₄₃	—	8	1	no	—
48	C₄×Dic₄₃	D₈	1	1	yes	characteristic
49	C₂×C₄₃⋊C₈	D₈	1	1	yes	characteristic
50	Dic₁₇₂	—	2	2	no	—
51	Dic₁₇₂	—	4	1	no	—
52	C₁₆×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₄₃	#41	1	1	yes	characteristic
54	C₄₃⋊C₁₆	—	2	1	no	—
55	Q₈×C₄₃	—	4	2	no	—
56	Q₈×C₄₃	—	8	1	no	—
57	C₂×C₄×C₄₃	—	8	1	no	—
58	C₂×C₄×C₄₃	—	8	1	no	—
59	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
60	C₂×Dic₄₃	—	8	1	no	—
61	C₂×Dic₄₃	—	8	1	no	—
62	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
63	C₂×Dic₄₃	D₁₆	1	1	yes	characteristic
64	Dic₈₆	—	4	2	no	—
65	Dic₈₆	—	8	1	no	—
66	C₈×C₄₃	C₄×D₄	1	1	yes	characteristic
67	C₈×C₄₃	#135	1	1	yes	characteristic
68	C₄₃⋊C₈	—	2	1	no	—
69	C₄×Q₆₄	—	43	1	no	maximal, Sylow
70	C₂²×C₄₃	C₂×D₁₆	1	1	yes	characteristic
71	C₄×C₄₃	—	8	2	no	—
72	C₄×C₄₃	—	16	1	no	—
73	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
74	C₄×C₄₃	#508	1	1	yes	characteristic
75	Dic₄₃	—	2	1	no	—
76	Dic₄₃	—	8	2	no	—
77	Dic₄₃	—	16	1	no	—
78	Dic₄₃	Q₆₄	1	2	yes	complemented
79	C₂×Q₆₄	—	43	1	no	—
80	C₄×Q₃₂	—	43	1	no	—
81	C₄×Q₃₂	—	43	1	no	—
82	#1423	—	43	1	no	—
83	#1416	—	43	1	no	—
84	#1416	D₄₃	1	1	yes	characteristic
85	C₄×C₃₂	—	43	1	no	—
86	C₂×C₄₃	C₄×D₁₆	1	1	yes	characteristic
87	C₂×C₄₃	C₂×Q₆₄	1	1	yes	characteristic
88	C₂×C₄₃	#2262	1	1	yes	characteristic
89	C₂×Q₃₂	—	43	1	no	—
90	C₂×Q₃₂	C₂×D₄₃	1	1	yes	characteristic
91	C₄×Q₁₆	—	86	1	no	—
92	C₄×Q₁₆	—	86	1	no	—
93	Q₆₄	—	43	4	no	—
94	C₂×C₃₂	—	43	1	no	—
95	C₂×C₃₂	C₂×D₄₃	1	1	yes	characteristic
96	#366	—	43	1	no	—
97	#366	C₂×D₄₃	1	1	yes	characteristic
98	#358	—	86	1	no	—
99	#358	—	86	1	no	—
100	C₄×C₁₆	—	43	1	no	—
101	C₄₃	C₄×Q₆₄	1	1	yes	characteristic, complemented, Sylow
102	C₂×Q₁₆	—	2	1	no	—
103	C₂×Q₁₆	—	86	1	no	—
104	C₄×Q₈	—	172	1	no	—
105	C₄×Q₈	—	172	1	no	—
106	Q₃₂	—	2	1	no	—
107	Q₃₂	—	43	2	no	—
108	Q₃₂	—	86	1	no	—
109	Q₃₂	C₄×D₄₃	1	2	yes	—
110	C₂×C₁₆	—	43	1	no	—
111	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
112	#107	—	2	1	no	—
113	#107	—	86	1	no	—
114	#103	—	172	1	no	—
115	#103	—	172	1	no	—
116	C₄×C₈	—	43	1	no	—
117	C₃₂	—	2	1	no	—
118	C₃₂	—	43	2	no	—
119	C₂×Q₈	—	4	1	no	—
120	C₂×Q₈	—	172	1	no	—
121	Q₁₆	—	2	2	no	—
122	Q₁₆	—	4	1	no	—
123	Q₁₆	—	86	2	no	—
124	Q₁₆	—	172	1	no	—
125	C₂×C₈	—	43	1	no	—
126	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
127	C₄⋊C₄	—	4	1	no	—
128	C₄⋊C₄	—	344	1	no	—
129	C₄⋊C₄	—	344	1	no	—
130	C₄⋊C₄	—	172	1	no	—
131	C₄²	—	43	1	no	—
132	C₄²	—	344	1	no	—
133	C₄²	—	344	1	no	—
134	C₁₆	—	86	1	no	—
135	C₁₆	C₂×C₄×D₄₃	1	1	yes	characteristic
136	C₁₆	#10621900	1	1	yes	characteristic
137	Q₈	—	4	2	no	—
138	Q₈	—	8	1	no	—
139	Q₈	—	344	1	no	—
140	Q₈	—	172	2	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	—	8	1	no	—
143	C₂×C₄	—	43	1	no	—
144	C₂×C₄	—	43	1	no	—
145	C₂×C₄	—	344	1	no	—
146	C₂×C₄	—	344	1	no	—
147	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
148	C₈	—	86	1	no	—
149	C₈	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
150	C₈	#13063346	1	1	yes	characteristic
151	C₂²	D₁₆×D₄₃	1	1	yes	characteristic
152	C₄	—	8	2	no	—
153	C₄	—	16	1	no	—
154	C₄	—	43	2	no	—
155	C₄	—	86	1	no	—
156	C₄	—	344	2	no	—
157	C₄	—	688	1	no	—
158	C₄	#14582979	1	1	yes	characteristic
159	C₄	#14583330	1	1	yes	characteristic
160	C₂	#19136439	1	1	yes	characteristic
161	C₂	Q₆₄×D₄₃	1	1	yes	characteristic
162	C₂	#19136903	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14582739

C₂

yes

maximal, characteristic

Q₃₂×Dic₄₃

C₂

yes

maximal, characteristic

#19136453

C₂

yes

maximal, characteristic

#19132794

C₂

yes

maximal, characteristic

#19132777

C₂

yes

maximal, characteristic

C₃₂×Dic₄₃

C₂

yes

maximal, characteristic

C₄₃×#1416

C₂

yes

maximal, characteristic

C₂×Dic₃₄₄

C₂²

yes

characteristic

Q₁₆×Dic₄₃

—

#14582993

—

#14582739

C₄

yes

complemented

C₂×C₄₃⋊C₃₂

C₂²

yes

characteristic

#14582725

C₂²

yes

characteristic

#14582703

—

#14582702

—

C₁₆×Dic₄₃

C₂²

yes

characteristic

C₂×Q₃₂×C₄₃

C₂²

yes

characteristic

C₂×C₃₂×C₄₃

C₂²

yes

characteristic

C₄₃×#366

C₂²

yes

characteristic

C₂×Dic₁₇₂

—

Q₈×Dic₄₃

—

#13063290

—

Dic₃₄₄

—

Dic₃₄₄

C₂×C₄

yes

—

C₂×C₄₃⋊C₁₆

D₄

yes

—

—

—

C₈×Dic₄₃

D₄

yes

characteristic

C₄₃⋊C₃₂

C₂×C₄

yes

—

C₂×Q₁₆×C₄₃

—

Q₃₂×C₄₃

—

Q₃₂×C₄₃

C₂×C₄

yes

—

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

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

D₈

yes

characteristic

C₂×C₄₃⋊C₈

D₈

yes

characteristic

Dic₁₇₂

—

Dic₁₇₂

—

C₁₆×C₄₃

C₂²×C₄

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

—

Dic₈₆

—

C₈×C₄₃

C₄×D₄

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

C₄×Q₆₄

—

maximal, Sylow

C₂²×C₄₃

C₂×D₁₆

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×D₈

yes

characteristic

C₄×C₄₃

#508

yes

characteristic

Dic₄₃

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

Q₆₄

yes

complemented

C₂×Q₆₄

—

C₄×Q₃₂

—

C₄×Q₃₂

—

#1423

—

#1416

—

#1416

D₄₃

yes

characteristic

C₄×C₃₂

—

C₂×C₄₃

C₄×D₁₆

yes

characteristic

C₂×C₄₃

C₂×Q₆₄

yes

characteristic

C₂×C₄₃

#2262

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₄₃

yes

characteristic

C₄×Q₁₆

—

C₄×Q₁₆

—

Q₆₄

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₃

yes

characteristic

#366

—

#366

C₂×D₄₃

yes

characteristic

#358

—

#358

—

100

C₄×C₁₆

—

101

C₄₃

C₄×Q₆₄

yes

characteristic, complemented, Sylow

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₄×Q₈

—

172

—

105

C₄×Q₈

—

172

—

106

Q₃₂

—

107

Q₃₂

—

108

Q₃₂

—

109

Q₃₂

C₄×D₄₃

yes

—

110

C₂×C₁₆

—

111

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

112

#107

—

113

#107

—

114

#103

—

172

—

115

#103

—

172

—

116

C₄×C₈

—

117

C₃₂

—

118

C₃₂

—

119

C₂×Q₈

—

120

C₂×Q₈

—

172

—

121

Q₁₆

—

122

Q₁₆

—

123

Q₁₆

—

124

Q₁₆

—

172

—

125

C₂×C₈

—

126

C₂×C₈

D₄×D₄₃

yes

characteristic

127

C₄⋊C₄

—

128

C₄⋊C₄

—

344

—

129

C₄⋊C₄

—

344

—

130

C₄⋊C₄

—

172

—

131

C₄²

—

132

C₄²

—

344

—

133

C₄²

—

344

—

134

C₁₆

—

135

C₁₆

C₂×C₄×D₄₃

yes

characteristic

136

C₁₆

#10621900

yes

characteristic

137

Q₈

—

138

Q₈

—

139

Q₈

—

344

—

140

Q₈

—

172

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

344

—

146

C₂×C₄

—

344

—

147

C₂×C₄

D₈×D₄₃

yes

characteristic

148

C₈

—

149

C₈

(C₂²⋊D₄₃)⋊C₄

yes

characteristic

150

C₈

#13063346

yes

characteristic

151

C₂²

D₁₆×D₄₃

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

344

—

157

C₄

—

688

—

158

C₄

#14582979

yes

characteristic

159

C₄

#14583330

yes

characteristic

160

C₂

#19136439

yes

characteristic

161

C₂

Q₆₄×D₄₃

yes

characteristic

162

C₂

#19136903

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₃
commutator	C₁₆×C₄₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1416	C₂
socle, prime radical	C₂²×C₄₃	C₂×D₁₆
cosocle	C₂×C₁₆×C₄₃	C₂³
hypercenter	#1416	D₄₃
hypocenter	C₄₃	C₄×Q₆₄
LCS	G ▷ 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

C₁₆×C₄₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#1416

C₂

socle, prime radical

C₂²×C₄₃

C₂×D₁₆

cosocle

C₂×C₁₆×C₄₃

C₂³

hypercenter

#1416

D₄₃

hypocenter

C₄₃

C₄×Q₆₄

LCS

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

UCS

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

N	H ≅ G/N	type
11. #14582739	C₄	semidirect
78. Dic₄₃	Q₆₄	semidirect
101. C₄₃	C₄×Q₆₄	semidirect

H ≅ G/N

type

11. #14582739

C₄

semidirect

78. Dic₄₃

Q₆₄

semidirect

101. C₄₃

C₄×Q₆₄

semidirect

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

subgroup	index
C₂×#14582739	2
Q₃₂×Dic₄₃	2
#19136453	2
#19132794	2
#19132777	2
C₃₂×Dic₄₃	2
C₄₃×#1416	2
C₄×Q₆₄	43

subgroup

index

Q₃₂×Dic₄₃

C₃₂×Dic₄₃

C₄₃×#1416

C₄×Q₆₄

Subgroups — #25017180

Conjugacy classes of subgroups

Complements

Maximal subgroups