Group #25008357 — subgroups

Subgroups — #25008357

Order 11008.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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	#19136477	C₂	1	1	yes	maximal, characteristic
2	#19136431	C₂	1	2	yes	maximal, complemented
3	#19132567	C₂	1	2	yes	maximal, complemented
4	#19132511	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583331	C₄	1	1	yes	characteristic
7	#14583317	—	2	2	no	—
8	#14583017	C₂²	1	2	yes	—
9	#14582729	C₄	1	1	yes	characteristic
10	#14582707	C₄	1	2	yes	—
11	#14582684	—	2	2	no	—
12	C₄×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
14	C₄₃×#364	C₂²	1	2	yes	—
15	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	#13063347	C₂×C₄	1	2	yes	—
17	#13063331	—	2	2	no	—
18	Dic₃₄₄	C₈	1	1	yes	characteristic, complemented
19	#13063270	C₈	1	2	yes	complemented
20	D₆₈₈	C₈	1	1	yes	characteristic, complemented
21	#13063264	—	4	2	no	—
22	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
23	C₂×C₄₃⋊C₁₆	D₄	1	1	yes	characteristic
24	C₁₆×D₄₃	—	4	2	no	—
25	#13063261	C₂×C₄	1	2	yes	—
26	#13063253	—	2	2	no	—
27	C₄×C₄₃⋊C₈	—	2	1	no	—
28	M₃₂×C₄₃	—	2	2	no	—
29	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
32	#10621898	—	2	2	no	—
33	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₄₃	#41	1	1	yes	characteristic
36	#10621877	—	2	2	no	—
37	C₂×C₄₃⋊C₈	—	2	2	no	—
38	Dic₁₇₂	C₂×C₈	1	2	yes	—
39	D₃₄₄	C₂×C₈	1	2	yes	—
40	#10621873	—	2	2	no	—
41	#10621872	—	4	2	no	—
42	C₈×D₄₃	—	4	2	no	—
43	C₁₆×C₄₃	—	2	1	no	—
44	C₁₆×C₄₃	—	4	2	no	—
45	C₁₆×C₄₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₄₃	C₂×C₈	1	1	yes	characteristic
47	C₄₃⋊C₁₆	—	2	1	no	—
48	C₄₃⋊C₁₆	D₈	1	2	yes	—
49	C₂×C₄×C₄₃	—	2	1	no	—
50	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₄₃	—	4	2	no	—
52	D₁₇₂	—	2	2	no	—
53	C₄×D₄₃	—	4	2	no	—
54	Dic₈₆	—	2	2	no	—
55	C₈×C₄₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₄₃	#131	1	1	yes	characteristic
57	C₈×C₄₃	C₂×D₈	1	1	yes	characteristic
58	C₈×C₄₃	#135	1	1	yes	characteristic
59	C₄₃⋊C₈	—	4	2	no	—
60	C₄₃⋊C₈	—	4	2	no	—
61	#12047	—	43	1	no	maximal, Sylow
62	C₂²×C₄₃	#443	1	1	yes	characteristic
63	C₂×D₄₃	—	4	2	no	—
64	C₄×C₄₃	—	2	2	no	—
65	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
66	C₄×C₄₃	C₄×D₈	1	1	yes	characteristic
67	Dic₄₃	—	4	2	no	—
68	#2178	—	43	1	no	—
69	#2170	—	43	2	no	—
70	#1373	D₄₃	1	1	yes	characteristic, complemented
71	#1335	—	43	2	no	—
72	#1311	—	43	1	no	—
73	C₂×C₄₃	—	2	1	no	—
74	C₂×C₄₃	C₈×D₈	1	1	yes	characteristic
75	D₄₃	—	8	2	no	—
76	#508	—	43	1	no	—
77	#504	—	86	2	no	—
78	#443	—	43	2	no	—
79	#368	—	43	1	no	—
80	#364	C₂×D₄₃	1	2	yes	—
81	#359	—	43	2	no	—
82	#350	—	86	2	no	—
83	C₄×C₁₆	—	43	1	no	—
84	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
85	C₈²	—	43	1	no	—
86	C₄₃	#12047	1	1	yes	characteristic, complemented, Sylow
87	#135	—	43	2	no	—
88	#131	—	86	2	no	—
89	Q₃₂	—	43	1	no	—
90	SD₃₂	—	43	2	no	—
91	D₁₆	—	43	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	172	2	no	—
94	C₂×C₁₆	—	43	1	no	—
95	C₂×C₁₆	—	43	1	no	—
96	C₂×C₁₆	—	172	2	no	—
97	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₄₃	1	1	yes	characteristic
99	#108	—	43	2	no	—
100	#104	—	86	2	no	—
101	C₄×C₈	—	86	1	no	—
102	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
103	#41	—	86	2	no	—
104	Q₁₆	—	43	2	no	—
105	SD₁₆	—	86	2	no	—
106	D₈	—	43	2	no	—
107	M₁₆	—	86	2	no	—
108	M₁₆	—	172	2	no	—
109	C₂×C₈	—	86	2	no	—
110	C₂×C₈	—	172	2	no	—
111	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
112	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
113	C₄²	#10621902	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	43	2	no	—
117	C₁₆	—	86	1	no	—
118	C₁₆	C₈×D₄₃	1	1	yes	characteristic
119	C₁₆	C₈×D₄₃	1	1	yes	characteristic
120	Q₈	—	86	2	no	—
121	D₄	—	86	2	no	—
122	D₄	—	172	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	172	2	no	—
125	C₂×C₄	#13063287	1	1	yes	characteristic
126	C₈	—	172	2	no	—
127	C₈	—	172	2	no	—
128	C₈	C₂×C₈×D₄₃	1	1	yes	characteristic
129	C₈	#13063333	1	1	yes	characteristic
130	C₈	D₈×D₄₃	1	1	yes	characteristic
131	C₈	#13063348	1	1	yes	characteristic
132	C₂²	—	172	2	no	—
133	C₂²	#14583015	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	172	2	no	—
136	C₄	#14582964	1	1	yes	characteristic
137	C₄	#14582981	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	344	2	no	—
140	C₂	#19133389	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19136477

C₂

yes

maximal, characteristic

#19136431

C₂

yes

maximal, complemented

#19132567

C₂

yes

maximal, complemented

#19132511

C₂

yes

maximal, characteristic, complemented

C₄₃×#1373

C₂

yes

maximal, characteristic, complemented

#14583331

C₄

yes

characteristic

#14583317

—

#14583017

C₂²

yes

—

#14582729

C₄

yes

characteristic

#14582707

C₄

yes

—

#14582684

—

C₄×C₄₃⋊C₁₆

C₂²

yes

characteristic

C₈×C₄₃⋊C₈

C₂²

yes

characteristic

C₄₃×#364

C₂²

yes

—

C₄×C₁₆×C₄₃

C₂²

yes

characteristic

#13063347

C₂×C₄

yes

—

#13063331

—

Dic₃₄₄

C₈

yes

characteristic, complemented

#13063270

C₈

yes

complemented

D₆₈₈

C₈

yes

characteristic, complemented

#13063264

—

C₂×C₄₃⋊C₁₆

D₄

yes

characteristic

C₂×C₄₃⋊C₁₆

D₄

yes

characteristic

C₁₆×D₄₃

—

#13063261

C₂×C₄

yes

—

#13063253

—

C₄×C₄₃⋊C₈

—

M₃₂×C₄₃

—

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₃

C₂×C₄

yes

characteristic

C₄×C₈×C₄₃

C₂³

yes

characteristic

#10621898

—

C₂×C₈×C₄₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₃

C₂×D₄

yes

C₄²×C₄₃

yes

—

C₂×C₄₃⋊C₈

—

Dic₁₇₂

C₂×C₈

yes

—

D₃₄₄

C₂×C₈

yes

—

#10621873

—

#10621872

—

C₈×D₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

—

C₁₆×C₄₃

C₂×C₈

yes

characteristic

C₁₆×C₄₃

C₂×C₈

yes

characteristic

C₄₃⋊C₁₆

—

C₄₃⋊C₁₆

D₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄×D₄

yes

characteristic

C₂²⋊D₄₃

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂²×C₈

yes

characteristic

C₈×C₄₃

#131

yes

characteristic

C₈×C₄₃

C₂×D₈

yes

characteristic

C₈×C₄₃

#135

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

—

C₂²×C₄₃

yes

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₈×D₄

yes

characteristic

C₄×C₄₃

C₄×D₈

yes

characteristic

Dic₄₃

—

#2178

—

#2170

—

#1373

D₄₃

yes

characteristic, complemented

#1335

—

#1311

—

C₂×C₄₃

—

C₂×C₄₃

C₈×D₈

yes

characteristic

D₄₃

—

—

—

—

—

C₂×D₄₃

yes

—

#359

—

#350

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₄₃

yes

characteristic

C₈²

—

C₄₃

#12047

yes

characteristic, complemented, Sylow

#135

—

#131

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

172

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

172

—

C₂×C₁₆

C₄×D₄₃

yes

characteristic

C₂×C₁₆

C₄×D₄₃

yes

characteristic

#108

—

100

#104

—

101

C₄×C₈

—

102

C₄×C₈

C₂²×D₄₃

yes

characteristic

103

#41

—

104

Q₁₆

—

105

SD₁₆

—

106

D₈

—

107

M₁₆

—

108

M₁₆

—

172

—

109

C₂×C₈

—

110

C₂×C₈

—

172

—

111

C₂×C₈

C₂×C₄×D₄₃

yes

characteristic

112

C₂×C₈

D₄×D₄₃

yes

characteristic

113

C₄²

#10621902

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

C₈×D₄₃

yes

characteristic

119

C₁₆

C₈×D₄₃

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

172

—

123

C₂×C₄

—

124

C₂×C₄

—

172

—

125

C₂×C₄

#13063287

yes

characteristic

126

C₈

—

172

—

127

C₈

—

172

—

128

C₈

C₂×C₈×D₄₃

yes

characteristic

129

C₈

#13063333

yes

characteristic

130

C₈

D₈×D₄₃

yes

characteristic

131

C₈

#13063348

yes

characteristic

132

C₂²

—

172

—

133

C₂²

#14583015

yes

characteristic

134

C₄

—

135

C₄

—

172

—

136

C₄

#14582964

yes

characteristic

137

C₄

#14582981

yes

characteristic

138

C₂

—

139

C₂

—

344

—

140

C₂

#19133389

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₄₃
commutator	C₈×C₄₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₄₃
Fitting	C₄₃×#1373	C₂
socle	C₂×C₄₃	C₈×D₈
prime radical	#14583331	C₄
cosocle	C₄×C₈×C₄₃	C₂³
hypercenter	#1373	D₄₃
hypocenter	C₄₃	#12047
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₄₃

commutator

C₈×C₄₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₄₃

Fitting

C₄₃×#1373

C₂

socle

C₂×C₄₃

C₈×D₈

prime radical

#14583331

C₄

cosocle

C₄×C₈×C₄₃

C₂³

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #19136431	C₂	semidirect
3. #19132567	C₂	semidirect
4. #19132511	C₂	semidirect
5. C₄₃×#1373	C₂	semidirect
18. Dic₃₄₄	C₈	semidirect
19. #13063270	C₈	semidirect
20. D₆₈₈	C₈	semidirect
70. #1373	D₄₃	semidirect
86. C₄₃	#12047	semidirect

H ≅ G/N

type

2. #19136431

C₂

semidirect

3. #19132567

C₂

semidirect

4. #19132511

C₂

semidirect

5. C₄₃×#1373

C₂

semidirect

18. Dic₃₄₄

C₈

semidirect

19. #13063270

C₈

semidirect

20. D₆₈₈

C₈

semidirect

70. #1373

D₄₃

semidirect

86. C₄₃

#12047

semidirect

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

subgroup	index
#19136477	2
#19136431	2
#19132567	2
#19132511	2
C₄₃×#1373	2
#12047	43

subgroup

index

C₄₃×#1373

Subgroups — #25008357

Conjugacy classes of subgroups

Complements

Maximal subgroups