Group #25008273 — subgroups

Subgroups — #25008273

Order 11008.

200 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 95 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	#19136398	C₂	1	1	yes	maximal, characteristic
2	#19133358	C₂	1	1	yes	maximal, characteristic
3	#19132660	C₂	1	4	yes	maximal, complemented
4	C₄₃×#1564	C₂	1	1	yes	maximal, characteristic
5	#14582956	—	2	1	no	—
6	C₄⋊(C₄₃⋊C₁₆)	C₄	1	2	yes	—
7	C₂²⋊(C₄₃⋊C₁₆)	C₄	1	2	yes	—
8	#14582675	C₂²	1	2	yes	—
9	#14582652	—	2	1	no	—
10	#14582650	C₂²	1	2	yes	—
11	C₄₃×#431	—	2	1	no	—
12	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
13	C₄₃×#335	—	2	1	no	—
14	C₄₃×#334	C₂²	1	2	yes	—
15	C₂×C₄₃⋊C₁₆	C₂×C₄	1	4	yes	—
16	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
17	C₂²⋊(C₄₃⋊C₈)	—	4	1	no	—
18	#13063236	—	4	1	no	—
19	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₄₃	Q₈	1	1	yes	characteristic
21	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
22	C₄⋊C₈×C₄₃	—	2	2	no	—
23	C₂²⋊C₈×C₄₃	—	4	1	no	—
24	C₄×C₈×C₄₃	—	4	1	no	—
25	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
27	C₄×C₈×C₄₃	Q₈	1	1	yes	characteristic
28	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
31	C₂²×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₈×C₄₃	—	2	2	no	—
33	C₂×C₈×C₄₃	—	4	1	no	—
34	C₂×C₈×C₄₃	—	4	2	no	—
35	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
41	C₄²×C₄₃	SD₁₆	1	1	yes	characteristic
42	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄₃⋊C₈	—	4	2	no	—
44	C₄₃⋊C₁₆	—	2	4	no	—
45	C₂³×C₄₃	#106	1	1	yes	characteristic
46	C₂×C₄×C₄₃	—	2	1	no	—
47	C₂×C₄×C₄₃	—	2	1	no	—
48	C₂×C₄×C₄₃	—	4	1	no	—
49	C₂×C₄×C₄₃	—	4	1	no	—
50	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
51	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
52	C₂×C₄×C₄₃	#106	1	1	yes	characteristic
53	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
56	C₈×C₄₃	—	2	1	no	—
57	C₈×C₄₃	—	2	1	no	—
58	C₈×C₄₃	—	2	1	no	—
59	C₈×C₄₃	—	8	2	no	—
60	C₈×C₄₃	#131	1	2	yes	—
61	C₄₃⋊C₈	—	8	2	no	—
62	#12031	—	43	1	no	maximal, Sylow
63	C₂²×C₄₃	—	2	1	no	—
64	C₂²×C₄₃	—	2	1	no	—
65	C₂²×C₄₃	—	2	1	no	—
66	C₂²×C₄₃	C₂×#106	1	1	yes	characteristic
67	C₄×C₄₃	—	2	1	no	—
68	C₄×C₄₃	—	2	1	no	—
69	C₄×C₄₃	—	2	1	no	—
70	C₄×C₄₃	—	2	1	no	—
71	C₄×C₄₃	—	2	1	no	—
72	C₄×C₄₃	#424	1	1	yes	characteristic
73	C₄×C₄₃	#430	1	1	yes	characteristic
74	#2163	—	43	1	no	—
75	#1568	—	43	1	no	—
76	#1564	D₄₃	1	1	yes	characteristic
77	#1368	—	43	4	no	—
78	C₂×C₄₃	—	4	1	no	—
79	C₂×C₄₃	#1570	1	1	yes	characteristic
80	C₂×C₄₃	#2155	1	1	yes	characteristic
81	C₂×C₄₃	#2155	1	1	yes	characteristic
82	#432	—	86	1	no	—
83	#431	—	2	1	no	—
84	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
85	C₄⋊C₁₆	—	43	2	no	—
86	C₂²⋊C₁₆	—	43	2	no	—
87	#346	—	43	2	no	—
88	#335	—	2	1	no	—
89	#335	—	86	1	no	—
90	#334	—	43	2	no	—
91	#334	C₂×D₄₃	1	2	yes	—
92	C₄₃	#12031	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	Dic₈₆	1	1	yes	characteristic
94	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
95	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
96	C₂×C₁₆	—	43	4	no	—
97	C₄⋊C₈	—	2	2	no	—
98	C₄⋊C₈	—	86	2	no	—
99	C₂²⋊C₈	—	4	1	no	—
100	C₂²⋊C₈	—	172	1	no	—
101	#97	—	172	1	no	—
102	C₄×C₈	—	4	1	no	—
103	C₄×C₈	Dic₈₆	1	1	yes	characteristic
104	C₄×C₈	C₄×D₄₃	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
107	C₂²×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
108	C₂²×C₄	#10621883	1	1	yes	characteristic
109	C₂²×C₄	#10621884	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	172	2	no	—
114	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
115	C₂×C₈	Dic₄₃⋊C₄	1	1	yes	characteristic
116	C₂×C₈	C₂×Dic₈₆	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
118	C₄²	Dic₄₃⋊C₄	1	1	yes	characteristic
119	C₄²	#10621883	1	1	yes	characteristic
120	C₄²	#10621884	1	1	yes	characteristic
121	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
122	C₁₆	—	86	4	no	—
123	C₂³	#13063256	1	1	yes	characteristic
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#13063256	1	1	yes	characteristic
129	C₂×C₄	#13063256	1	1	yes	characteristic
130	C₂×C₄	#13063256	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
132	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
133	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	8	2	no	—
138	C₈	—	344	2	no	—
139	C₈	#13063331	1	2	yes	—
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#13063256	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#14582925	1	1	yes	characteristic
150	C₄	#14582947	1	1	yes	characteristic
151	C₂	—	4	1	no	—
152	C₂	#19133365	1	1	yes	characteristic
153	C₂	#19136367	1	1	yes	characteristic
154	C₂	#19136369	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136398

C₂

yes

maximal, characteristic

#19133358

C₂

yes

maximal, characteristic

#19132660

C₂

yes

maximal, complemented

C₄₃×#1564

C₂

yes

maximal, characteristic

#14582956

—

C₄⋊(C₄₃⋊C₁₆)

C₄

yes

—

C₂²⋊(C₄₃⋊C₁₆)

C₄

yes

—

#14582675

C₂²

yes

—

#14582652

—

#14582650

C₂²

yes

—

C₄₃×#431

—

C₂×C₄×C₈×C₄₃

C₂²

yes

characteristic

C₄₃×#335

—

C₄₃×#334

C₂²

yes

—

C₂×C₄₃⋊C₁₆

C₂×C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₂²⋊(C₄₃⋊C₈)

—

#13063236

—

C₂²×C₈×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₃

Q₈

yes

characteristic

C₂×C₄²×C₄₃

D₄

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₄×C₈×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

D₄

yes

characteristic

C₄×C₈×C₄₃

Q₈

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

C₂²×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

characteristic

C₂²×C₄×C₄₃

SD₁₆

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×Q₈

yes

characteristic

C₄²×C₄₃

C₄⋊C₄

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

SD₁₆

yes

characteristic

C₄²×C₄₃

C₂×D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₄₃⋊C₁₆

—

C₂³×C₄₃

#106

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

yes

C₂×C₄×C₄₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₃

C₂×SD₁₆

yes

characteristic

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

—

C₈×C₄₃

#131

yes

—

C₄₃⋊C₈

—

#12031

—

maximal, Sylow

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×#106

yes

characteristic

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

—

D₄₃

yes

characteristic

#1368

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

—

C₂×C₄×C₈

C₂×D₄₃

yes

characteristic

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

—

—

—

—

C₂×D₄₃

yes

—

C₄₃

#12031

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₈₆

yes

characteristic

C₂²×C₈

C₄×D₄₃

yes

characteristic

C₂×C₄²

C₂²⋊D₄₃

yes

characteristic

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

100

C₂²⋊C₈

—

172

—

101

#97

—

172

—

102

C₄×C₈

—

103

C₄×C₈

Dic₈₆

yes

characteristic

104

C₄×C₈

C₄×D₄₃

yes

characteristic

105

C₄×C₈

C₂²⋊D₄₃

yes

characteristic

106

C₄×C₈

C₂²×D₄₃

yes

characteristic

107

C₂²×C₄

Dic₄₃⋊C₄

yes

characteristic

108

C₂²×C₄

#10621883

yes

characteristic

109

C₂²×C₄

#10621884

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

172

—

114

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

115

C₂×C₈

Dic₄₃⋊C₄

yes

characteristic

116

C₂×C₈

C₂×Dic₈₆

yes

characteristic

117

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

118

C₄²

Dic₄₃⋊C₄

yes

characteristic

119

C₄²

#10621883

yes

characteristic

120

C₄²

#10621884

yes

characteristic

121

C₄²

C₂×C₂²⋊D₄₃

yes

characteristic

122

C₁₆

—

123

C₂³

#13063256

yes

characteristic

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#13063256

yes

characteristic

129

C₂×C₄

#13063256

yes

characteristic

130

C₂×C₄

#13063256

yes

characteristic

131

C₂×C₄

C₂×Dic₄₃⋊C₄

yes

characteristic

132

C₂×C₄

C₂×#10621884

yes

characteristic

133

C₂×C₄

C₂×#10621883

yes

characteristic

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

344

—

139

C₈

#13063331

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#13063256

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#14582925

yes

characteristic

150

C₄

#14582947

yes

characteristic

151

C₂

—

152

C₂

#19133365

yes

characteristic

153

C₂

#19136367

yes

characteristic

154

C₂

#19136369

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#13063256
commutator	C₂×C₈×C₄₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1564	C₂
socle	C₂²×C₄₃	C₂×#106
prime radical	C₂³×C₄₃	#106
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1564	D₄₃
hypocenter	C₄₃	#12031
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

subgroup

G/H

center

C₂²

C₂×#13063256

commutator

C₂×C₈×C₄₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1564

C₂

C₂²×C₄₃

C₂³×C₄₃

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

N	H ≅ G/N	type
3. #19132660	C₂	semidirect
92. C₄₃	#12031	semidirect

H ≅ G/N

type

3. #19132660

C₂

semidirect

92. C₄₃

#12031

semidirect

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

subgroup	index
#19136398	2
#19133358	2
#19132660	2
C₄₃×#1564	2
#12031	43

subgroup

index

C₄₃×#1564

Subgroups — #25008273

Conjugacy classes of subgroups

Complements

Maximal subgroups