Group #25005625 — subgroups

Subgroups — #25005625

Order 11008.

306 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 167 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	#19136392	C₂	1	2	yes	maximal
2	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#19132514	C₂	1	2	yes	maximal, complemented
4	C₈×C₄₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×D₄₃	C₄	1	1	yes	characteristic
7	C₄×C₈×D₄₃	C₄	1	1	yes	characteristic
8	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
9	Dic₄₃⋊C₁₆	C₄	1	4	yes	—
10	(C₂×D₄₃)⋊C₁₆	C₄	1	4	yes	—
11	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	—
12	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
13	C₈×C₄₃⋊C₈	C₄	1	1	yes	characteristic
14	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₁₆×C₄₃	C₂²	1	2	yes	—
16	C₈²×C₄₃	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₄₃	C₈	1	2	yes	—
18	C₂×C₈×D₄₃	C₈	1	2	yes	—
19	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
20	C₄²×D₄₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
22	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
24	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
25	C₈×Dic₄₃	C₈	1	2	yes	—
26	C₈×Dic₄₃	C₈	1	2	yes	—
27	C₈×Dic₄₃	C₂×C₄	1	2	yes	—
28	C₂×C₁₆×C₄₃	C₂×C₄	1	4	yes	—
29	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×D₄₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×D₄₃	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₄₃	C₂×C₈	1	2	yes	—
37	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
38	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
39	C₄×Dic₄₃	C₄²	1	1	yes	characteristic
40	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
41	C₄×Dic₄₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
43	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
44	C₂×C₄₃⋊C₈	C₂×C₈	1	2	yes	—
45	C₈×D₄₃	—	2	4	no	—
46	C₈×D₄₃	—	2	4	no	—
47	C₈×D₄₃	C₁₆	1	8	yes	complemented
48	C₁₆×C₄₃	—	2	4	no	—
49	C₄₃⋊C₁₆	—	2	4	no	—
50	C₂²×D₄₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
54	C₂×Dic₄₃	C₄×C₈	1	1	yes	characteristic
55	C₂×Dic₄₃	C₄×C₈	1	1	yes	characteristic
56	C₂×Dic₄₃	C₄×C₈	1	1	yes	characteristic
57	C₄×D₄₃	—	2	2	no	—
58	C₄×D₄₃	—	2	2	no	—
59	C₄×D₄₃	C₂×C₁₆	1	4	yes	—
60	C₈×C₄₃	C₂×C₁₆	1	4	yes	—
61	C₈×C₄₃	M₃₂	1	4	yes	—
62	C₈×C₄₃	#131	1	4	yes	—
63	C₄₃⋊C₈	—	2	2	no	—
64	C₄₃⋊C₈	—	2	2	no	—
65	C₄₃⋊C₈	C₂×C₁₆	1	4	yes	—
66	#11410	—	43	1	no	maximal, Sylow
67	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
68	C₂×D₄₃	—	2	1	no	—
69	C₂×D₄₃	—	2	1	no	—
70	C₂×D₄₃	C₄×C₁₆	1	2	yes	—
71	C₄×C₄₃	#405	1	2	yes	—
72	C₄×C₄₃	C₂²×C₁₆	1	2	yes	—
73	C₄×C₄₃	C₂×M₃₂	1	2	yes	—
74	Dic₄₃	—	2	1	no	—
75	Dic₄₃	—	2	1	no	—
76	Dic₄₃	C₄×C₁₆	1	2	yes	—
77	#2162	—	43	2	no	—
78	C₂×C₈²	—	43	1	no	—
79	#1312	—	43	2	no	—
80	C₈×C₁₆	—	43	1	no	—
81	C₈×C₁₆	D₄₃	1	1	yes	characteristic, complemented
82	C₂×C₄₃	#1450	1	1	yes	characteristic
83	C₂×C₄₃	C₂×C₄×C₁₆	1	1	yes	characteristic
84	C₂×C₄₃	C₄×M₃₂	1	1	yes	characteristic
85	D₄₃	—	2	2	no	—
86	C₂×C₄×C₈	—	43	1	no	—
87	C₂×C₄×C₈	—	43	1	no	—
88	C₂×C₄×C₈	—	43	1	no	—
89	C₄⋊C₁₆	—	43	4	no	—
90	C₂²⋊C₁₆	—	43	4	no	—
91	C₄×C₁₆	—	43	2	no	—
92	C₄×C₁₆	C₂×D₄₃	1	2	yes	—
93	C₈²	—	43	1	no	—
94	C₈²	—	43	1	no	—
95	C₈²	—	43	1	no	—
96	C₈²	C₂×D₄₃	1	1	yes	characteristic
97	C₄₃	#11410	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₈	—	43	2	no	—
99	C₂²×C₈	—	43	2	no	—
100	C₂²×C₈	—	43	2	no	—
101	C₂×C₄²	—	43	1	no	—
102	C₂×C₁₆	—	43	4	no	—
103	C₂×C₁₆	C₄×D₄₃	1	4	yes	—
104	C₄×C₈	—	43	2	no	—
105	C₄×C₈	—	43	2	no	—
106	C₄×C₈	—	43	2	no	—
107	C₄×C₈	—	43	1	no	—
108	C₄×C₈	—	43	1	no	—
109	C₄×C₈	—	43	1	no	—
110	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
111	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
112	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
113	C₂²×C₄	—	43	1	no	—
114	C₂²×C₄	—	43	1	no	—
115	C₂²×C₄	—	43	1	no	—
116	C₂×C₈	—	43	2	no	—
117	C₂×C₈	—	43	8	no	—
118	C₂×C₈	—	43	2	no	—
119	C₂×C₈	—	43	2	no	—
120	C₂×C₈	—	86	4	no	—
121	C₂×C₈	—	86	4	no	—
122	C₂×C₈	C₈×D₄₃	1	2	yes	—
123	C₂×C₈	C₈×D₄₃	1	2	yes	—
124	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
125	C₄²	—	43	1	no	—
126	C₄²	—	43	1	no	—
127	C₄²	—	43	1	no	—
128	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
129	C₁₆	—	2	4	no	—
130	C₁₆	—	86	4	no	—
131	C₂³	—	43	1	no	—
132	C₂×C₄	—	43	1	no	—
133	C₂×C₄	—	43	4	no	—
134	C₂×C₄	—	43	1	no	—
135	C₂×C₄	—	43	1	no	—
136	C₂×C₄	—	86	2	no	—
137	C₂×C₄	—	86	2	no	—
138	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
141	C₈	—	43	4	no	—
142	C₈	—	86	2	no	—
143	C₈	—	86	2	no	—
144	C₈	C₁₆×D₄₃	1	4	yes	—
145	C₈	#13063264	1	4	yes	—
146	C₈	#13063331	1	4	yes	—
147	C₂²	—	43	2	no	—
148	C₂²	—	86	1	no	—
149	C₂²	—	86	1	no	—
150	C₂²	C₄×C₈×D₄₃	1	1	yes	characteristic
151	C₄	—	43	2	no	—
152	C₄	—	86	1	no	—
153	C₄	—	86	1	no	—
154	C₄	#14582853	1	2	yes	—
155	C₄	C₂×C₁₆×D₄₃	1	2	yes	—
156	C₄	C₂×#13063264	1	2	yes	—
157	C₂	—	86	2	no	—
158	C₂	#19132877	1	1	yes	characteristic
159	C₂	C₄×C₁₆×D₄₃	1	1	yes	characteristic
160	C₂	C₄×#13063264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136392

C₂

yes

maximal

C₈²×D₄₃

C₂

yes

maximal, characteristic

#19132514

C₂

yes

maximal, complemented

C₈×C₄₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×C₁₆×C₄₃

C₂

yes

maximal, characteristic, complemented

C₄×C₈×D₄₃

C₄

yes

characteristic

C₄×C₈×D₄₃

C₄

yes

characteristic

C₄×C₈×D₄₃

C₂²

yes

characteristic

Dic₄₃⋊C₁₆

C₄

yes

—

(C₂×D₄₃)⋊C₁₆

C₄

yes

—

C₄×C₄₃⋊C₁₆

C₂²

yes

—

C₈×C₄₃⋊C₈

C₄

yes

characteristic

C₈×C₄₃⋊C₈

C₄

yes

characteristic

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₄×C₁₆×C₄₃

C₂²

yes

—

C₈²×C₄₃

C₂²

yes

characteristic

C₂×C₈×D₄₃

C₈

yes

—

C₂×C₈×D₄₃

C₈

yes

—

C₂×C₈×D₄₃

C₂×C₄

yes

—

C₄²×D₄₃

C₂×C₄

yes

characteristic

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₄×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₄×C₄₃⋊C₈

C₂×C₄

yes

characteristic

C₈×Dic₄₃

C₈

yes

—

C₈×Dic₄₃

C₈

yes

—

C₈×Dic₄₃

C₂×C₄

yes

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂×C₄×D₄₃

C₄²

yes

characteristic

C₂×C₄×D₄₃

C₂×C₈

yes

characteristic

C₂×C₄×D₄₃

C₂×C₈

yes

characteristic

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂×C₈

yes

—

C₂×C₈×C₄₃

C₂²×C₄

yes

—

C₄²×C₄₃

C₂²×C₄

yes

characteristic

C₄×Dic₄₃

C₄²

yes

characteristic

C₄×Dic₄₃

C₂×C₈

yes

characteristic

C₄×Dic₄₃

C₂×C₈

yes

characteristic

C₂×C₄₃⋊C₈

C₄²

yes

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₂×C₄₃⋊C₈

C₂×C₈

yes

—

C₈×D₄₃

—

C₈×D₄₃

—

C₈×D₄₃

C₁₆

yes

complemented

C₁₆×C₄₃

—

C₄₃⋊C₁₆

—

C₂²×D₄₃

C₄×C₈

yes

characteristic

C₂×C₄×C₄₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₄₃

C₂²×C₈

yes

characteristic

C₂×Dic₄₃

C₄×C₈

yes

characteristic

C₂×Dic₄₃

C₄×C₈

yes

characteristic

C₂×Dic₄₃

C₄×C₈

yes

characteristic

C₄×D₄₃

—

C₄×D₄₃

—

C₄×D₄₃

C₂×C₁₆

yes

—

C₈×C₄₃

C₂×C₁₆

yes

—

C₈×C₄₃

M₃₂

yes

—

C₈×C₄₃

#131

yes

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₄₃⋊C₈

C₂×C₁₆

yes

—

#11410

—

maximal, Sylow

C₂²×C₄₃

C₂×C₄×C₈

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

C₄×C₁₆

yes

—

C₄×C₄₃

#405

yes

—

C₄×C₄₃

C₂²×C₁₆

yes

—

C₄×C₄₃

C₂×M₃₂

yes

—

Dic₄₃

—

Dic₄₃

—

Dic₄₃

C₄×C₁₆

yes

—

#2162

—

C₂×C₈²

—

#1312

—

C₈×C₁₆

—

C₈×C₁₆

D₄₃

yes

characteristic, complemented

C₂×C₄₃

#1450

yes

characteristic

C₂×C₄₃

C₂×C₄×C₁₆

yes

characteristic

C₂×C₄₃

C₄×M₃₂

yes

characteristic

D₄₃

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

—

C₈²

—

C₈²

—

C₈²

—

C₈²

C₂×D₄₃

yes

characteristic

C₄₃

#11410

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

100

C₂²×C₈

—

101

C₂×C₄²

—

102

C₂×C₁₆

—

103

C₂×C₁₆

C₄×D₄₃

yes

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

C₄×D₄₃

yes

characteristic

111

C₄×C₈

C₄×D₄₃

yes

characteristic

112

C₄×C₈

C₂²×D₄₃

yes

characteristic

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₈×D₄₃

yes

—

123

C₂×C₈

C₈×D₄₃

yes

—

124

C₂×C₈

C₂×C₄×D₄₃

yes

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×C₄×D₄₃

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂³

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

C₄²×D₄₃

yes

characteristic

139

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

140

C₂×C₄

C₂×C₈×D₄₃

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

C₁₆×D₄₃

yes

—

145

C₈

#13063264

yes

—

146

C₈

#13063331

yes

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₄×C₈×D₄₃

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#14582853

yes

—

155

C₄

C₂×C₁₆×D₄₃

yes

—

156

C₄

C₂×#13063264

yes

—

157

C₂

—

158

C₂

#19132877

yes

characteristic

159

C₂

C₄×C₁₆×D₄₃

yes

characteristic

160

C₂

C₄×#13063264

yes

characteristic

	subgroup	G/H
center	C₈²	C₂×D₄₃
commutator	C₂×C₄₃	C₂×C₄×C₁₆
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₈×C₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×C₈
prime radical	C₂²×D₄₃	C₄×C₈
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	C₈×C₁₆	D₄₃
hypocenter	C₄₃	#11410
LCS	G ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₈²

C₂×D₄₃

commutator

C₂×C₄₃

C₂×C₄×C₁₆

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₈×C₁₆×C₄₃

C₂

socle

C₂²×C₄₃

C₂×C₄×C₈

prime radical

C₂²×D₄₃

C₄×C₈

cosocle

C₄×C₈×C₄₃

C₂³

hypercenter

C₈×C₁₆

D₄₃

C₄₃

G ▷ C₂×C₄₃ ▷ C₄₃

C₁ ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
3. #19132514	C₂	semidirect
4. C₈×C₄₃⋊C₁₆	C₂	semidirect
5. C₈×C₁₆×C₄₃	C₂	semidirect
47. C₈×D₄₃	C₁₆	semidirect
81. C₈×C₁₆	D₄₃	semidirect
97. C₄₃	#11410	semidirect

H ≅ G/N

type

3. #19132514

C₂

semidirect

4. C₈×C₄₃⋊C₁₆

C₂

semidirect

5. C₈×C₁₆×C₄₃

C₂

semidirect

47. C₈×D₄₃

C₁₆

semidirect

81. C₈×C₁₆

D₄₃

semidirect

97. C₄₃

#11410

semidirect

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

subgroup	index
#19136392	2
C₈²×D₄₃	2
#19132514	2
C₈×C₄₃⋊C₁₆	2
C₈×C₁₆×C₄₃	2
#11410	43

subgroup

index

#19136392

C₈²×D₄₃

#19132514

C₈×C₄₃⋊C₁₆

C₈×C₁₆×C₄₃

#11410

Subgroups — #25005625

Conjugacy classes of subgroups

Complements

Maximal subgroups