Group #19143073 — subgroups

Subgroups — #19143073

Order 5504.

756 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 249 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	C₂×#13063371	C₂	1	1	yes	maximal, characteristic
2	#14583666	C₂	1	1	yes	maximal, characteristic, complemented
3	#14582881	C₂	1	6	yes	maximal, complemented
4	C₂×#13063241	C₂	1	6	yes	maximal, complemented
5	C₄₃×#568	C₂	1	1	yes	maximal, characteristic, complemented
6	#13063371	—	2	4	no	—
7	#13063371	—	2	4	no	—
8	C₂×#10621902	C₄	1	2	yes	—
9	C₂×#10621898	C₂²	1	3	yes	—
10	C₂²×D₁₇₂	C₂²	1	3	yes	—
11	C₂×D₄×D₄₃	C₄	1	6	yes	—
12	#13063332	—	2	4	no	—
13	C₂×#10621877	C₂²	1	6	yes	complemented
14	#13063243	C₂²	1	4	yes	complemented
15	#13063241	C₂²	1	12	yes	complemented
16	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
17	#131×C₄₃	—	2	4	no	—
18	C₂×M₁₆×C₄₃	C₂²	1	6	yes	complemented
19	C₂³×D₄₃	C₂×C₄	1	6	yes	—
20	#41×C₄₃	D₄	1	4	yes	—
21	#41×C₄₃	D₄	1	4	yes	—
22	C₂×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₄₃	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
25	C₂×C₂²⋊D₄₃	C₂×C₄	1	6	yes	—
26	#10621902	—	2	8	no	—
27	D₄×D₄₃	—	2	24	no	—
28	#10621898	—	2	6	no	—
29	#10621898	—	2	6	no	—
30	C₂×D₁₇₂	—	2	6	no	—
31	C₂×D₁₇₂	—	2	6	no	—
32	C₂×D₁₇₂	C₂×C₄	1	6	yes	—
33	C₂×D₁₇₂	C₂³	1	3	yes	—
34	C₂×C₄×D₄₃	C₂×C₄	1	6	yes	—
35	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
36	M₁₆×C₄₃	—	2	6	no	—
37	M₁₆×C₄₃	C₂³	1	4	yes	—
38	C₂×C₈×C₄₃	—	2	6	no	—
39	#10621877	—	2	6	no	—
40	#10621877	C₂³	1	4	yes	—
41	C₂×C₄₃⋊C₈	—	2	6	no	—
42	C₂³×C₄₃	C₂²×C₄	1	3	yes	—
43	C₂²×D₄₃	—	2	12	no	—
44	C₂²×D₄₃	—	8	6	no	—
45	C₂²×D₄₃	C₂²×C₄	1	6	yes	—
46	Q₈×C₄₃	C₂²⋊C₄	1	4	yes	—
47	D₄×C₄₃	C₂²⋊C₄	1	12	yes	—
48	C₂×C₄×C₄₃	C₂²×C₄	1	3	yes	—
49	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
50	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
51	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
52	C₂²⋊D₄₃	—	2	12	no	—
53	C₂×Dic₄₃	C₂²×C₄	1	2	yes	—
54	D₁₇₂	—	2	12	no	—
55	D₁₇₂	—	2	3	no	—
56	D₁₇₂	—	2	3	no	—
57	C₄×D₄₃	—	2	12	no	—
58	Dic₈₆	—	2	1	no	—
59	Dic₈₆	—	2	1	no	—
60	C₈×C₄₃	—	2	4	no	—
61	C₄₃⋊C₈	—	2	4	no	—
62	C₂²×C₄₃	—	4	3	no	—
63	C₂²×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
64	C₂²×C₄₃	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₄₃	—	2	6	no	—
66	C₂×D₄₃	—	4	6	no	—
67	C₂×D₄₃	—	8	12	no	—
68	C₄×C₄₃	C₂×C₂²⋊C₄	1	6	yes	—
69	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
70	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
71	Dic₄₃	—	2	2	no	—
72	#2888	—	43	1	no	maximal, Sylow
73	C₂×C₄₃	—	2	1	no	—
74	C₂×C₄₃	—	2	6	no	—
75	C₂×C₄₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
76	D₄₃	—	4	6	no	—
77	C₂×H(5,𝔽₂)	—	43	1	no	—
78	#568	—	43	1	no	—
79	#568	D₄₃	1	1	yes	characteristic, complemented
80	#413	—	43	6	no	—
81	C₂×#100	—	43	6	no	—
82	C₄₃	#2888	1	1	yes	characteristic, complemented, Sylow
83	H(5,𝔽₂)	—	86	4	no	—
84	H(5,𝔽₂)	—	86	4	no	—
85	C₂×#41	—	43	2	no	—
86	C₂×#41	—	43	3	no	—
87	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
88	C₂²×D₄	—	43	6	no	—
89	C₂²×D₄	—	43	3	no	—
90	#131	—	2	4	no	—
91	#131	—	86	4	no	—
92	C₂×M₁₆	—	43	6	no	—
93	C₂×M₁₆	C₂×D₄₃	1	6	yes	complemented
94	#101	—	43	4	no	—
95	#100	—	43	12	no	—
96	C₂⁴	—	43	6	no	—
97	#41	—	86	8	no	—
98	#41	—	86	6	no	—
99	#41	—	86	6	no	—
100	#41	D₁₇₂	1	4	yes	—
101	#41	C₂²⋊D₄₃	1	4	yes	—
102	C₂×Q₈	—	43	1	no	—
103	C₂×Q₈	C₄×D₄₃	1	1	yes	characteristic
104	C₂×D₄	—	43	6	no	—
105	C₂×D₄	—	43	6	no	—
106	C₂×D₄	—	43	3	no	—
107	C₂×D₄	—	86	24	no	—
108	C₂×D₄	—	86	6	no	—
109	C₂×D₄	—	86	6	no	—
110	C₂×D₄	C₄×D₄₃	1	3	yes	—
111	C₂²×C₄	—	43	6	no	—
112	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
113	M₁₆	—	2	6	no	—
114	M₁₆	—	43	4	no	—
115	M₁₆	—	86	6	no	—
116	M₁₆	C₂²×D₄₃	1	4	yes	—
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	86	6	no	—
119	C₂³	—	43	6	no	—
120	C₂³	—	86	12	no	—
121	C₂³	—	344	6	no	—
122	C₂³	C₂×C₄×D₄₃	1	3	yes	—
123	Q₈	—	86	1	no	—
124	Q₈	—	86	1	no	—
125	Q₈	(C₂×D₄₃)⋊C₄	1	4	yes	—
126	D₄	—	86	12	no	—
127	D₄	—	86	12	no	—
128	D₄	—	86	3	no	—
129	D₄	—	86	3	no	—
130	D₄	(C₂×D₄₃)⋊C₄	1	12	yes	—
131	C₂×C₄	—	43	2	no	—
132	C₂×C₄	—	86	12	no	—
133	C₂×C₄	C₂×C₄×D₄₃	1	3	yes	—
134	C₂×C₄	C₂×D₁₇₂	1	6	yes	—
135	C₂×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
136	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
137	C₈	—	2	4	no	—
138	C₈	—	86	4	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	86	6	no	—
141	C₂²	—	344	12	no	—
142	C₂²	—	172	6	no	—
143	C₂²	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
144	C₂²	C₂²×C₄×D₄₃	1	1	yes	characteristic
145	C₄	—	86	2	no	—
146	C₄	C₂×(C₂×D₄₃)⋊C₄	1	6	yes	—
147	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
148	C₄	C₂²×D₁₇₂	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	6	no	—
151	C₂	—	172	6	no	—
152	C₂	C₂²×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13063371

C₂

yes

maximal, characteristic

#14583666

C₂

yes

maximal, characteristic, complemented

#14582881

C₂

yes

maximal, complemented

C₂×#13063241

C₂

yes

maximal, complemented

C₄₃×#568

C₂

yes

maximal, characteristic, complemented

#13063371

—

#13063371

—

C₂×#10621902

C₄

yes

—

C₂×#10621898

C₂²

yes

—

C₂²×D₁₇₂

C₂²

yes

—

C₂×D₄×D₄₃

C₄

yes

—

#13063332

—

C₂×#10621877

C₂²

yes

complemented

#13063243

C₂²

yes

complemented

#13063241

C₂²

yes

complemented

C₂×#41×C₄₃

C₂²

yes

characteristic

#131×C₄₃

—

C₂×M₁₆×C₄₃

C₂²

yes

complemented

C₂³×D₄₃

C₂×C₄

yes

—

#41×C₄₃

D₄

yes

—

#41×C₄₃

D₄

yes

—

C₂×Q₈×C₄₃

C₂×C₄

yes

characteristic

C₂×D₄×C₄₃

C₂×C₄

yes

—

C₂²×C₄×C₄₃

C₂³

yes

—

C₂×C₂²⋊D₄₃

C₂×C₄

yes

—

#10621902

—

D₄×D₄₃

—

#10621898

—

#10621898

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₂×C₄

yes

—

C₂×D₁₇₂

C₂³

yes

—

C₂×C₄×D₄₃

C₂×C₄

yes

—

C₂×Dic₈₆

C₂³

yes

characteristic

M₁₆×C₄₃

—

M₁₆×C₄₃

C₂³

yes

—

C₂×C₈×C₄₃

—

#10621877

—

#10621877

C₂³

yes

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²×C₄

yes

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

C₂²×C₄

yes

—

Q₈×C₄₃

C₂²⋊C₄

yes

—

D₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²×C₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂×D₄

yes

—

C₂×C₄×C₄₃

C₂⁴

yes

characteristic

C₂²⋊D₄₃

—

C₂×Dic₄₃

C₂²×C₄

yes

—

D₁₇₂

—

D₁₇₂

—

D₁₇₂

—

C₄×D₄₃

—

Dic₈₆

—

Dic₈₆

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₄₃

C₂³×C₄

yes

characteristic

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

C₂×C₂²⋊C₄

yes

—

C₄×C₄₃

C₂²×D₄

yes

characteristic

C₄×C₄₃

C₂²×D₄

yes

characteristic

Dic₄₃

—

#2888

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

C₂²×C₂²⋊C₄

yes

characteristic

D₄₃

—

C₂×H(5,𝔽₂)

—

#568

—

#568

D₄₃

yes

characteristic, complemented

#413

—

C₂×#100

—

C₄₃

#2888

yes

characteristic, complemented, Sylow

H(5,𝔽₂)

—

H(5,𝔽₂)

—

C₂×#41

—

C₂×#41

—

C₂×#41

C₂×D₄₃

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₃

yes

complemented

#101

—

#100

—

C₂⁴

—

#41

—

#41

—

#41

—

100

#41

D₁₇₂

yes

—

101

#41

C₂²⋊D₄₃

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

C₄×D₄₃

yes

characteristic

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

C₄×D₄₃

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₄₃

yes

—

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

C₂²×D₄₃

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

344

—

122

C₂³

C₂×C₄×D₄₃

yes

—

123

Q₈

—

124

Q₈

—

125

Q₈

(C₂×D₄₃)⋊C₄

yes

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

(C₂×D₄₃)⋊C₄

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄×D₄₃

yes

—

134

C₂×C₄

C₂×D₁₇₂

yes

—

135

C₂×C₄

C₂×C₂²⋊D₄₃

yes

—

136

C₂×C₄

C₂³×D₄₃

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

344

—

142

C₂²

—

172

—

143

C₂²

C₂×(C₂×D₄₃)⋊C₄

yes

—

144

C₂²

C₂²×C₄×D₄₃

yes

characteristic

145

C₄

—

146

C₄

C₂×(C₂×D₄₃)⋊C₄

yes

—

147

C₄

C₂²×C₂²⋊D₄₃

yes

characteristic

148

C₄

C₂²×D₁₇₂

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

—

172

—

152

C₂

C₂²×(C₂×D₄₃)⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×(C₂×D₄₃)⋊C₄
commutator	C₂²×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#568	C₂
socle	C₂×C₄₃	C₂²×C₂²⋊C₄
hypercenter	#568	D₄₃
hypocenter	C₄₃	#2888
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₂

C₂²×(C₂×D₄₃)⋊C₄

commutator

C₂²×C₄₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₄₃

Fitting

C₄₃×#568

C₂

socle

C₂×C₄₃

C₂²×C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
2. #14583666	C₂	semidirect
3. #14582881	C₂	semidirect
4. C₂×#13063241	C₂	semidirect
5. C₄₃×#568	C₂	semidirect
13. C₂×#10621877	C₂²	semidirect
14. #13063243	C₂²	semidirect
15. #13063241	C₂²	semidirect
18. C₂×M₁₆×C₄₃	C₂²	semidirect
79. #568	D₄₃	semidirect
82. C₄₃	#2888	semidirect
93. C₂×M₁₆	C₂×D₄₃	semidirect

H ≅ G/N

type

2. #14583666

C₂

semidirect

3. #14582881

C₂

semidirect

4. C₂×#13063241

C₂

semidirect

5. C₄₃×#568

C₂

semidirect

13. C₂×#10621877

C₂²

semidirect

14. #13063243

C₂²

semidirect

15. #13063241

C₂²

semidirect

18. C₂×M₁₆×C₄₃

C₂²

semidirect

79. #568

D₄₃

semidirect

82. C₄₃

#2888

semidirect

93. C₂×M₁₆

C₂×D₄₃

semidirect

Subgroups — #19143073

Conjugacy classes of subgroups

Complements