Group #25017088 — subgroups

Subgroups — #25017088

Order 11008.

198 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 99 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	M₆₄×D₄₃	C₂	1	2	yes	maximal
2	#19136190	C₂	1	1	yes	maximal, characteristic
3	#19132793	C₂	1	1	yes	maximal, characteristic, complemented
4	#19132792	C₂	1	2	yes	maximal, complemented
5	C₄₃×#1422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13063264	C₄	1	1	yes	characteristic
7	C₂×C₁₆×D₄₃	C₂²	1	1	yes	characteristic
8	#14582853	C₄	1	1	yes	characteristic
9	#14582733	C₂²	1	2	yes	—
10	#14582732	—	2	2	no	—
11	C₃₂×D₄₃	—	2	2	no	—
12	#14582675	C₄	1	1	yes	characteristic
13	#14582674	C₄	1	1	yes	characteristic
14	C₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
15	C₄₃×M₆₄	C₂²	1	2	yes	—
16	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621872	C₈	1	1	yes	characteristic
18	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
19	#13063282	C₈	1	1	yes	characteristic
20	#13063264	—	2	2	no	—
21	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄₃⋊C₁₆	C₂×C₄	1	1	yes	characteristic
23	C₁₆×D₄₃	D₄	1	2	yes	—
24	C₁₆×D₄₃	Q₈	1	2	yes	—
25	#13063236	C₈	1	1	yes	characteristic
26	#13063235	C₈	1	1	yes	characteristic
27	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
28	C₄₃⋊C₃₂	—	2	2	no	—
29	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
32	C₃₂×C₄₃	—	2	2	no	—
33	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₄₃	C₂×C₈	1	1	yes	characteristic
37	(C₂×D₄₃)⋊C₄	—	2	1	no	—
38	Dic₄₃⋊C₄	—	2	1	no	—
39	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
40	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄₃⋊C₈	C₂×C₈	1	1	yes	characteristic
42	#10621872	C₁₆	1	4	yes	—
43	C₈×D₄₃	C₄⋊C₄	1	2	yes	—
44	C₈×D₄₃	M₁₆	1	2	yes	—
45	C₁₆×C₄₃	—	2	1	no	—
46	C₁₆×C₄₃	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₄₃	C₂×Q₈	1	1	yes	characteristic
48	C₄₃⋊C₁₆	—	2	1	no	—
49	C₄₃⋊C₁₆	C₄⋊C₄	1	2	yes	—
50	C₂²×D₄₃	M₃₂	1	1	yes	characteristic
51	C₂×C₄×C₄₃	—	2	1	no	—
52	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×Dic₄₃	—	2	1	no	—
54	C₂×Dic₄₃	M₃₂	1	1	yes	characteristic
55	C₄×D₄₃	C₄⋊C₈	1	2	yes	—
56	C₄×D₄₃	C₂×C₁₆	1	2	yes	—
57	C₈×C₄₃	C₂×C₁₆	1	2	yes	—
58	C₈×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₈×C₄₃	C₂×M₁₆	1	1	yes	characteristic
60	C₄₃⋊C₈	C₄⋊C₈	1	2	yes	—
61	C₄₃⋊C₈	C₂×C₁₆	1	2	yes	—
62	#13630	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	C₂×M₃₂	1	1	yes	characteristic
64	C₂×D₄₃	—	4	1	no	—
65	C₂×D₄₃	C₄⋊C₁₆	1	2	yes	—
66	C₄×C₄₃	—	4	1	no	—
67	C₄×C₄₃	C₂×C₄⋊C₈	1	1	yes	characteristic
68	C₄×C₄₃	C₂²×C₁₆	1	1	yes	characteristic
69	Dic₄₃	—	4	1	no	—
70	Dic₄₃	C₄⋊C₁₆	1	2	yes	—
71	C₂×M₆₄	—	43	2	no	—
72	#2108	—	43	1	no	—
73	#1422	—	43	2	no	—
74	#1422	—	43	1	no	—
75	#1422	D₄₃	1	1	yes	characteristic, complemented
76	C₂×C₄₃	—	2	1	no	—
77	C₂×C₄₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
78	D₄₃	—	2	2	no	—
79	C₂×M₃₂	—	43	1	no	—
80	C₂²×C₁₆	—	43	1	no	—
81	#405	—	43	1	no	—
82	M₆₄	—	43	2	no	—
83	M₆₄	—	86	2	no	—
84	M₆₄	C₂×D₄₃	1	2	yes	—
85	C₂×C₃₂	—	86	2	no	—
86	#346	—	43	1	no	—
87	#346	—	43	1	no	—
88	C₄×C₁₆	—	43	1	no	—
89	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
90	C₄₃	#13630	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	43	1	no	—
92	C₂²×C₈	—	43	1	no	—
93	#117	—	43	1	no	—
94	M₃₂	—	86	2	no	—
95	C₂×C₁₆	—	43	1	no	—
96	C₂×C₁₆	—	43	2	no	—
97	C₂×C₁₆	—	43	2	no	—
98	C₂×C₁₆	—	43	1	no	—
99	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
100	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
101	#97	—	43	1	no	—
102	#97	—	43	1	no	—
103	C₄×C₈	—	43	1	no	—
104	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
105	C₃₂	—	2	2	no	—
106	C₃₂	—	86	2	no	—
107	C₂²×C₄	—	43	1	no	—
108	M₁₆	—	43	4	no	—
109	C₂×C₈	—	43	1	no	—
110	C₂×C₈	—	43	1	no	—
111	C₂×C₈	—	43	2	no	—
112	C₂×C₈	—	43	2	no	—
113	C₂×C₈	C₈×D₄₃	1	1	yes	characteristic
114	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
115	C₄⋊C₄	—	86	1	no	—
116	C₂²⋊C₄	—	86	1	no	—
117	C₄²	—	43	1	no	—
118	C₄²	C₈×D₄₃	1	1	yes	characteristic
119	C₁₆	—	2	1	no	—
120	C₁₆	—	43	2	no	—
121	C₁₆	—	86	1	no	—
122	C₁₆	D₄×D₄₃	1	1	yes	characteristic
123	C₁₆	Q₈×D₄₃	1	1	yes	characteristic
124	C₂³	—	43	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	43	1	no	—
127	C₂×C₄	—	43	2	no	—
128	C₂×C₄	—	43	2	no	—
129	C₂×C₄	—	86	1	no	—
130	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
131	C₈	—	43	2	no	—
132	C₈	—	43	2	no	—
133	C₈	C₁₆×D₄₃	1	2	yes	—
134	C₈	C₄⋊C₄×D₄₃	1	1	yes	characteristic
135	C₈	M₁₆×D₄₃	1	1	yes	characteristic
136	C₂²	—	43	2	no	—
137	C₂²	—	172	1	no	—
138	C₂²	M₃₂×D₄₃	1	1	yes	characteristic
139	C₄	—	4	1	no	—
140	C₄	—	43	2	no	—
141	C₄	—	172	1	no	—
142	C₄	C₄⋊C₈×D₄₃	1	1	yes	characteristic
143	C₄	C₂×C₁₆×D₄₃	1	1	yes	characteristic
144	C₂	—	2	1	no	—
145	C₂	—	86	2	no	—
146	C₂	C₄⋊C₁₆×D₄₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₆₄×D₄₃

C₂

yes

maximal

#19136190

C₂

yes

maximal, characteristic

#19132793

C₂

yes

maximal, characteristic, complemented

#19132792

C₂

yes

maximal, complemented

C₄₃×#1422

C₂

yes

maximal, characteristic, complemented

C₂×#13063264

C₄

yes

characteristic

C₂×C₁₆×D₄₃

C₂²

yes

characteristic

#14582853

C₄

yes

characteristic

#14582733

C₂²

yes

—

#14582732

—

C₃₂×D₄₃

—

#14582675

C₄

yes

characteristic

#14582674

C₄

yes

characteristic

C₁₆×Dic₄₃

C₂²

yes

characteristic

C₄₃×M₆₄

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

C₂×#10621872

C₈

yes

characteristic

C₂×C₈×D₄₃

C₂×C₄

yes

characteristic

#13063282

C₈

yes

characteristic

#13063264

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

characteristic

C₁₆×D₄₃

D₄

yes

—

C₁₆×D₄₃

Q₈

yes

—

#13063236

C₈

yes

characteristic

#13063235

C₈

yes

characteristic

C₈×Dic₄₃

C₂×C₄

yes

characteristic

C₄₃⋊C₃₂

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂³

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₃₂×C₄₃

—

C₂×C₄×D₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₈

yes

characteristic

(C₂×D₄₃)⋊C₄

—

Dic₄₃⋊C₄

—

C₄×Dic₄₃

C₂×C₈

yes

characteristic

C₂×C₄₃⋊C₈

C₂×C₈

yes

characteristic

C₂×C₄₃⋊C₈

C₂×C₈

yes

characteristic

#10621872

C₁₆

yes

—

C₈×D₄₃

C₄⋊C₄

yes

—

C₈×D₄₃

M₁₆

yes

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×D₄

yes

characteristic

C₁₆×C₄₃

C₂×Q₈

yes

characteristic

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

C₄⋊C₄

yes

—

C₂²×D₄₃

M₃₂

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

M₃₂

yes

characteristic

C₄×D₄₃

C₄⋊C₈

yes

—

C₄×D₄₃

C₂×C₁₆

yes

—

C₈×C₄₃

C₂×C₁₆

yes

—

C₈×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₄₃

C₂×M₁₆

yes

characteristic

C₄₃⋊C₈

C₄⋊C₈

yes

—

C₄₃⋊C₈

C₂×C₁₆

yes

—

#13630

—

maximal, Sylow

C₂²×C₄₃

C₂×M₃₂

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

C₄⋊C₁₆

yes

—

C₄×C₄₃

—

C₄×C₄₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₄₃

C₂²×C₁₆

yes

characteristic

Dic₄₃

—

Dic₄₃

C₄⋊C₁₆

yes

—

C₂×M₆₄

—

—

—

—

D₄₃

yes

characteristic, complemented

C₂×C₄₃

—

C₂×C₄₃

C₂×C₄⋊C₁₆

yes

characteristic

D₄₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

M₆₄

—

M₆₄

C₂×D₄₃

yes

—

C₂×C₃₂

—

#346

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₄₃

#13630

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

100

C₂×C₁₆

C₂²×D₄₃

yes

characteristic

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

C₄×D₄₃

yes

characteristic

105

C₃₂

—

106

C₃₂

—

107

C₂²×C₄

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₈×D₄₃

yes

characteristic

114

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₈×D₄₃

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

D₄×D₄₃

yes

characteristic

123

C₁₆

Q₈×D₄₃

yes

characteristic

124

C₂³

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

C₁₆×D₄₃

yes

—

134

C₈

C₄⋊C₄×D₄₃

yes

characteristic

135

C₈

M₁₆×D₄₃

yes

characteristic

136

C₂²

—

137

C₂²

—

172

—

138

C₂²

M₃₂×D₄₃

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

172

—

142

C₄

C₄⋊C₈×D₄₃

yes

characteristic

143

C₄

C₂×C₁₆×D₄₃

yes

characteristic

144

C₂

—

145

C₂

—

146

C₂

C₄⋊C₁₆×D₄₃

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₄₃
commutator	C₄×C₄₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₄₃
Fitting	C₄₃×#1422	C₂
socle	C₂×C₄₃	C₂×C₄⋊C₁₆
prime radical	C₂²×D₄₃	M₃₂
cosocle	C₂×C₁₆×C₄₃	C₂³
hypercenter	#1422	D₄₃
hypocenter	C₄₃	#13630
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

subgroup

G/H

center

C₁₆

D₄×D₄₃

commutator

C₄×C₄₃

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₄₃

Fitting

C₄₃×#1422

C₂

socle

C₂×C₄₃

C₂×C₄⋊C₁₆

prime radical

C₂²×D₄₃

M₃₂

cosocle

C₂×C₁₆×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #1422

N	H ≅ G/N	type
3. #19132793	C₂	semidirect
4. #19132792	C₂	semidirect
5. C₄₃×#1422	C₂	semidirect
75. #1422	D₄₃	semidirect
90. C₄₃	#13630	semidirect

H ≅ G/N

type

3. #19132793

C₂

semidirect

4. #19132792

C₂

semidirect

5. C₄₃×#1422

C₂

semidirect

75. #1422

D₄₃

semidirect

90. C₄₃

#13630

semidirect

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

subgroup	index
M₆₄×D₄₃	2
#19136190	2
#19132793	2
#19132792	2
C₄₃×#1422	2
#13630	43

subgroup

index

M₆₄×D₄₃

C₄₃×#1422

Subgroups — #25017088

Conjugacy classes of subgroups

Complements

Maximal subgroups