Group #19147734 — subgroups

Subgroups — #19147734

Order 5504.

370 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 129 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	#14583704	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583335	C₂	1	3	yes	maximal, complemented
3	#14583334	C₂	1	3	yes	maximal, complemented
4	#14583329	C₂	1	3	yes	maximal, complemented
5	C₂×#13063266	C₂	1	3	yes	maximal, complemented
6	#14583316	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#13063371	—	2	1	no	—
9	#13063353	—	2	3	no	—
10	#13063347	C₂²	1	3	yes	complemented
11	C₂×D₃₄₄	C₂²	1	3	yes	complemented
12	#13063271	C₂²	1	1	yes	characteristic, complemented
13	#13063269	C₂²	1	3	yes	complemented
14	#13063268	C₂²	1	3	yes	complemented
15	#13063266	C₂²	1	3	yes	complemented
16	#13063266	C₂²	1	6	yes	complemented
17	#13063265	C₂²	1	3	yes	complemented
18	C₂×C₄₃⋊C₁₆	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₄₃	—	2	1	no	—
20	AGL(1,ℤ/8)×C₄₃	—	2	3	no	—
21	#135×C₄₃	C₂²	1	3	yes	complemented
22	C₂×D₈×C₄₃	C₂²	1	3	yes	complemented
23	#131×C₄₃	C₂²	1	1	yes	characteristic
24	#41×C₄₃	—	2	3	no	—
25	#41×C₄₃	—	4	1	no	—
26	#41×C₄₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₄₃	—	2	3	no	—
28	C₂×D₄×C₄₃	—	4	3	no	—
29	#10621902	—	4	1	no	—
30	D₄×D₄₃	—	4	3	no	—
31	#10621898	—	2	3	no	—
32	C₂×D₁₇₂	—	2	3	no	—
33	Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
34	SD₁₆×C₄₃	—	2	3	no	—
35	D₈×C₄₃	—	2	3	no	—
36	D₈×C₄₃	C₂³	1	3	yes	—
37	M₁₆×C₄₃	D₄	1	3	yes	—
38	C₂×C₈×C₄₃	C₂³	1	3	yes	—
39	Dic₁₇₂	C₂³	1	1	yes	characteristic
40	D₃₄₄	—	2	3	no	—
41	D₃₄₄	C₂³	1	3	yes	—
42	#10621873	—	2	3	no	—
43	C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₄₃⋊C₁₆	C₂³	1	3	yes	—
45	C₂³×C₄₃	—	4	3	no	—
46	C₂²×D₄₃	—	4	3	no	—
47	Q₈×C₄₃	—	2	1	no	—
48	Q₈×C₄₃	D₈	1	1	yes	characteristic
49	D₄×C₄₃	—	2	3	no	—
50	D₄×C₄₃	—	4	3	no	—
51	D₄×C₄₃	—	4	3	no	—
52	D₄×C₄₃	D₈	1	3	yes	—
53	C₂×C₄×C₄₃	—	4	3	no	—
54	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
55	C₂²⋊D₄₃	—	4	3	no	—
56	D₁₇₂	—	2	3	no	—
57	D₁₇₂	—	4	3	no	—
58	C₄×D₄₃	—	4	3	no	—
59	Dic₈₆	—	2	1	no	—
60	C₈×C₄₃	C₂×D₄	1	3	yes	—
61	C₈×C₄₃	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄₃	—	4	3	no	—
63	C₂²×C₄₃	—	16	3	no	—
64	C₂²×C₄₃	C₂×D₈	1	3	yes	—
65	C₂×D₄₃	—	4	3	no	—
66	C₂×D₄₃	—	16	3	no	—
67	C₄×C₄₃	—	4	1	no	—
68	C₄×C₄₃	C₂×D₈	1	3	yes	—
69	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
70	Dic₄₃	—	4	1	no	—
71	#3415	—	43	1	no	maximal, Sylow
72	C₂×C₄₃	—	2	3	no	—
73	C₂×C₄₃	—	8	3	no	—
74	C₂×C₄₃	C₂²×D₈	1	1	yes	characteristic
75	D₄₃	—	8	3	no	—
76	#576	—	43	1	no	—
77	#576	D₄₃	1	1	yes	characteristic, complemented
78	#509	—	43	3	no	—
79	#509	—	43	3	no	—
80	#508	—	43	3	no	—
81	C₂×D₁₆	—	43	3	no	—
82	#504	—	43	1	no	—
83	C₄₃	#3415	1	1	yes	characteristic, complemented, Sylow
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	86	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	86	3	no	—
88	#135	—	43	3	no	—
89	#135	C₂×D₄₃	1	3	yes	complemented
90	C₂×D₈	—	43	3	no	—
91	C₂×D₈	C₂×D₄₃	1	3	yes	complemented
92	#131	C₂×D₄₃	1	1	yes	characteristic
93	Q₃₂	—	43	1	no	—
94	SD₃₂	—	43	3	no	—
95	SD₃₂	—	43	3	no	—
96	D₁₆	—	43	3	no	—
97	D₁₆	—	43	6	no	—
98	M₃₂	—	43	3	no	—
99	C₂×C₁₆	—	43	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	86	3	no	—
103	#41	—	172	1	no	—
104	#41	C₂²⋊D₄₃	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	86	3	no	—
108	C₂×D₄	—	172	3	no	—
109	Q₁₆	—	43	1	no	—
110	Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	86	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	43	3	no	—
115	D₈	—	86	3	no	—
116	D₈	C₂²×D₄₃	1	3	yes	—
117	M₁₆	C₂²⋊D₄₃	1	3	yes	—
118	C₂×C₈	C₂²×D₄₃	1	3	yes	—
119	C₁₆	—	43	1	no	—
120	C₁₆	—	43	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	172	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	86	1	no	—
125	Q₈	#10621882	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	86	3	no	—
130	D₄	—	172	3	no	—
131	D₄	—	172	3	no	—
132	D₄	#10621882	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	172	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
136	C₈	C₂×C₂²⋊D₄₃	1	3	yes	—
137	C₈	C₂³×D₄₃	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	172	3	no	—
141	C₂²	—	688	3	no	—
142	C₂²	C₂×#10621882	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	172	1	no	—
145	C₄	C₂×#10621882	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	344	3	no	—
150	C₂	C₂²×#10621882	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14583704

C₂

yes

maximal, characteristic, complemented

#14583335

C₂

yes

maximal, complemented

#14583334

C₂

yes

maximal, complemented

#14583329

C₂

yes

maximal, complemented

C₂×#13063266

C₂

yes

maximal, complemented

#14583316

C₂

yes

maximal, characteristic, complemented

C₄₃×#576

C₂

yes

maximal, characteristic, complemented

#13063371

—

#13063353

—

#13063347

C₂²

yes

complemented

C₂×D₃₄₄

C₂²

yes

complemented

#13063271

C₂²

yes

characteristic, complemented

#13063269

C₂²

yes

complemented

#13063268

C₂²

yes

complemented

#13063266

C₂²

yes

complemented

#13063266

C₂²

yes

complemented

#13063265

C₂²

yes

complemented

C₂×C₄₃⋊C₁₆

C₂²

yes

complemented

H(5,𝔽₂)×C₄₃

—

AGL(1,ℤ/8)×C₄₃

—

#135×C₄₃

C₂²

yes

complemented

C₂×D₈×C₄₃

C₂²

yes

complemented

#131×C₄₃

C₂²

yes

characteristic

#41×C₄₃

—

#41×C₄₃

—

#41×C₄₃

D₄

yes

characteristic

C₂×D₄×C₄₃

—

C₂×D₄×C₄₃

—

#10621902

—

D₄×D₄₃

—

#10621898

—

C₂×D₁₇₂

—

Q₁₆×C₄₃

C₂³

yes

characteristic

SD₁₆×C₄₃

—

D₈×C₄₃

—

D₈×C₄₃

C₂³

yes

—

M₁₆×C₄₃

D₄

yes

—

C₂×C₈×C₄₃

C₂³

yes

—

Dic₁₇₂

C₂³

yes

characteristic

D₃₄₄

—

D₃₄₄

C₂³

yes

—

#10621873

—

C₄₃⋊C₁₆

C₂³

yes

characteristic

C₄₃⋊C₁₆

C₂³

yes

—

C₂³×C₄₃

—

C₂²×D₄₃

—

Q₈×C₄₃

—

Q₈×C₄₃

D₈

yes

characteristic

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

—

D₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂²⋊D₄₃

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

C₈×C₄₃

C₂×D₄

yes

—

C₈×C₄₃

C₂⁴

yes

characteristic

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×D₈

yes

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₂×D₈

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

Dic₄₃

—

#3415

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×D₈

yes

characteristic

D₄₃

—

#576

—

#576

D₄₃

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₄₃

#3415

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₄₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₄₃

yes

complemented

#131

C₂×D₄₃

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

100

#41

—

101

#41

—

102

#41

—

103

#41

—

172

—

104

#41

C₂²⋊D₄₃

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

172

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₄₃

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

113

D₈

—

114

D₈

—

115

D₈

—

116

D₈

C₂²×D₄₃

yes

—

117

M₁₆

C₂²⋊D₄₃

yes

—

118

C₂×C₈

C₂²×D₄₃

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

172

—

123

Q₈

—

124

Q₈

—

125

Q₈

#10621882

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

172

—

131

D₄

—

172

—

132

D₄

#10621882

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

172

—

135

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

136

C₈

C₂×C₂²⋊D₄₃

yes

—

137

C₈

C₂³×D₄₃

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

172

—

141

C₂²

—

688

—

142

C₂²

C₂×#10621882

yes

—

143

C₄

—

144

C₄

—

172

—

145

C₄

C₂×#10621882

yes

—

146

C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

344

—

150

C₂

C₂²×#10621882

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#10621882
commutator	C₈×C₄₃	C₂⁴
Frattini	C₈	C₂³×D₄₃
Fitting	C₄₃×#576	C₂
socle	C₂×C₄₃	C₂²×D₈
hypercenter	#576	D₄₃
hypocenter	C₄₃	#3415
LCS	G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#10621882

commutator

C₈×C₄₃

C₂⁴

Frattini

C₈

C₂³×D₄₃

Fitting

C₄₃×#576

C₂

socle

C₂×C₄₃

C₂²×D₈

D₄₃

C₄₃

G ▷ C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #14583704	C₂	semidirect
2. #14583335	C₂	semidirect
3. #14583334	C₂	semidirect
4. #14583329	C₂	semidirect
5. C₂×#13063266	C₂	semidirect
6. #14583316	C₂	semidirect
7. C₄₃×#576	C₂	semidirect
10. #13063347	C₂²	semidirect
11. C₂×D₃₄₄	C₂²	semidirect
12. #13063271	C₂²	semidirect
13. #13063269	C₂²	semidirect
14. #13063268	C₂²	semidirect
15. #13063266	C₂²	semidirect
16. #13063266	C₂²	semidirect
17. #13063265	C₂²	semidirect
18. C₂×C₄₃⋊C₁₆	C₂²	semidirect
21. #135×C₄₃	C₂²	semidirect
22. C₂×D₈×C₄₃	C₂²	semidirect
77. #576	D₄₃	semidirect
83. C₄₃	#3415	semidirect
89. #135	C₂×D₄₃	semidirect
91. C₂×D₈	C₂×D₄₃	semidirect

H ≅ G/N

type

1. #14583704

C₂

semidirect

2. #14583335

C₂

semidirect

3. #14583334

C₂

semidirect

4. #14583329

C₂

semidirect

5. C₂×#13063266

C₂

semidirect

6. #14583316

C₂

semidirect

7. C₄₃×#576

C₂

semidirect

10. #13063347

C₂²

semidirect

11. C₂×D₃₄₄

C₂²

semidirect

12. #13063271

C₂²

semidirect

13. #13063269

C₂²

semidirect

14. #13063268

C₂²

semidirect

15. #13063266

C₂²

semidirect

16. #13063266

C₂²

semidirect

17. #13063265

C₂²

semidirect

18. C₂×C₄₃⋊C₁₆

C₂²

semidirect

21. #135×C₄₃

C₂²

semidirect

22. C₂×D₈×C₄₃

C₂²

semidirect

77. #576

D₄₃

semidirect

83. C₄₃

#3415

semidirect

89. #135

C₂×D₄₃

semidirect

91. C₂×D₈

C₂×D₄₃

semidirect

Subgroups — #19147734

Conjugacy classes of subgroups

Complements