Group #19132526 — subgroups

Subgroups — #19132526

Order 5504.

170 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 37 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	#14582800	C₂	1	1	yes	maximal, characteristic
2	#14582625	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄₃×#324	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₁₇₂	C₄	1	1	yes	characteristic
5	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
6	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
7	C₂×C₄×Dic₄₃	C₄	1	1	yes	characteristic
8	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
10	C₂²⋊C₈×C₄₃	—	2	1	no	—
11	C₂³×D₄₃	—	2	1	no	—
12	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
15	C₂²×Dic₄₃	—	2	1	no	—
16	C₂×D₁₇₂	—	2	1	no	—
17	C₂×D₁₇₂	—	4	1	no	—
18	C₂×D₁₇₂	—	4	1	no	—
19	C₂×D₁₇₂	C₈	1	2	yes	complemented
20	C₂×C₈×C₄₃	—	4	1	no	—
21	C₄⋊C₄×C₄₃	—	4	1	no	—
22	(C₂×D₄₃)⋊C₄	—	4	2	no	—
23	(C₂×D₄₃)⋊C₄	—	4	2	no	—
24	C₄×Dic₄₃	—	4	1	no	—
25	C₂×C₄₃⋊C₈	—	4	1	no	—
26	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×D₄₃	—	2	2	no	—
28	C₂²×D₄₃	—	4	1	no	—
29	C₂²×D₄₃	—	4	1	no	—
30	C₂²×D₄₃	—	4	1	no	—
31	C₂²×D₄₃	—	4	1	no	—
32	C₂²×D₄₃	—	4	1	no	—
33	C₂²×D₄₃	—	4	1	no	—
34	C₂×C₄×C₄₃	—	2	1	no	—
35	C₂×C₄×C₄₃	—	2	1	no	—
36	C₂×C₄×C₄₃	—	2	1	no	—
37	C₂×C₄×C₄₃	—	2	1	no	—
38	C₂×C₄×C₄₃	—	4	1	no	—
39	C₂×C₄×C₄₃	—	4	1	no	—
40	C₂×C₄×C₄₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₄₃	M₁₆	1	1	yes	characteristic
42	C₂×Dic₄₃	—	2	2	no	—
43	C₂×Dic₄₃	—	4	1	no	—
44	C₂×Dic₄₃	—	4	1	no	—
45	D₁₇₂	—	4	2	no	—
46	D₁₇₂	—	4	2	no	—
47	C₈×C₄₃	—	8	1	no	—
48	C₄₃⋊C₈	—	8	1	no	—
49	C₂²×C₄₃	—	2	1	no	—
50	C₂²×C₄₃	—	2	1	no	—
51	C₂²×C₄₃	C₂²⋊C₈	1	1	yes	characteristic
52	C₂²×C₄₃	#99	1	1	yes	characteristic
53	C₂²×C₄₃	#100	1	1	yes	characteristic
54	C₂×D₄₃	—	4	2	no	—
55	C₂×D₄₃	—	4	2	no	—
56	C₂×D₄₃	—	4	2	no	—
57	C₂×D₄₃	—	8	1	no	—
58	C₂×D₄₃	—	8	1	no	—
59	C₂×D₄₃	—	8	1	no	—
60	C₂×D₄₃	—	8	1	no	—
61	C₄×C₄₃	—	2	1	no	—
62	C₄×C₄₃	—	2	1	no	—
63	C₄×C₄₃	—	4	1	no	—
64	C₄×C₄₃	—	4	1	no	—
65	Dic₄₃	—	4	2	no	—
66	#1318	—	43	1	no	maximal, Sylow
67	C₂×C₄₃	—	2	1	no	—
68	C₂×C₄₃	—	2	1	no	—
69	C₂×C₄₃	#323	1	1	yes	characteristic
70	C₂×C₄₃	#353	1	1	yes	characteristic
71	C₂×C₄₃	#355	1	1	yes	characteristic
72	D₄₃	—	8	2	no	—
73	#390	—	43	1	no	—
74	#324	—	43	1	no	—
75	#324	D₄₃	1	1	yes	characteristic, complemented
76	C₄₃	#1318	1	1	yes	characteristic, complemented, Sylow
77	C₂²×D₄	—	43	1	no	—
78	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
79	C₂×C₂²⋊C₄	—	86	1	no	—
80	C₂×C₂²⋊C₄	—	86	1	no	—
81	C₂×C₄²	—	43	1	no	—
82	C₂²⋊C₈	—	2	1	no	—
83	C₂²⋊C₈	—	86	1	no	—
84	C₂⁴	—	86	1	no	—
85	C₂×D₄	—	43	2	no	—
86	C₂×D₄	—	86	1	no	—
87	C₂×D₄	—	172	1	no	—
88	C₂×D₄	—	172	1	no	—
89	C₂²×C₄	—	86	1	no	—
90	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
91	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
92	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	172	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₂²⋊C₄	—	172	2	no	—
97	C₂²⋊C₄	—	172	2	no	—
98	C₄²	—	172	1	no	—
99	C₂³	—	86	2	no	—
100	C₂³	—	172	1	no	—
101	C₂³	—	172	1	no	—
102	C₂³	—	172	1	no	—
103	C₂³	—	172	1	no	—
104	C₂³	—	172	1	no	—
105	C₂³	—	172	1	no	—
106	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
107	D₄	—	172	2	no	—
108	D₄	—	172	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	86	2	no	—
116	C₂×C₄	—	172	1	no	—
117	C₂×C₄	—	172	1	no	—
118	C₂×C₄	C₈×D₄₃	1	1	yes	characteristic
119	C₂×C₄	#10621872	1	1	yes	characteristic
120	C₈	—	8	1	no	—
121	C₈	—	344	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	344	1	no	—
125	C₂²	—	344	1	no	—
126	C₂²	—	344	1	no	—
127	C₂²	—	344	1	no	—
128	C₂²	—	172	2	no	—
129	C₂²	—	172	2	no	—
130	C₂²	—	172	2	no	—
131	C₂²	(C₂×D₄₃)⋊C₈	1	1	yes	characteristic
132	C₂²	#13063239	1	1	yes	characteristic
133	C₂²	#13063241	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	1	no	—
138	C₄	—	172	2	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	—	344	2	no	—
142	C₂	#14582622	1	1	yes	characteristic
143	C₂	#14582691	1	1	yes	characteristic
144	C₂	#14582695	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582800

C₂

yes

maximal, characteristic

#14582625

C₂

yes

maximal, characteristic, complemented

C₄₃×#324

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₇₂

C₄

yes

characteristic

C₂×(C₂×D₄₃)⋊C₄

—

C₂×(C₂×D₄₃)⋊C₄

—

C₂×C₄×Dic₄₃

C₄

yes

characteristic

C₂²⋊(C₄₃⋊C₈)

—

C₂×C₄⋊C₄×C₄₃

C₂²

yes

characteristic

C₂²⋊C₈×C₄₃

—

C₂³×D₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

—

C₂×D₁₇₂

C₈

yes

complemented

C₂×C₈×C₄₃

—

C₄⋊C₄×C₄₃

—

(C₂×D₄₃)⋊C₄

—

(C₂×D₄₃)⋊C₄

—

C₄×Dic₄₃

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂²×D₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₂×C₈

yes

characteristic

C₂×C₄×C₄₃

M₁₆

yes

characteristic

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

D₁₇₂

—

D₁₇₂

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

C₂²⋊C₈

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₂×D₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

Dic₄₃

—

#1318

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

—

#390

—

#324

—

#324

D₄₃

yes

characteristic, complemented

C₄₃

#1318

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×C₄⋊C₄

C₂×D₄₃

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

172

—

C₂×D₄

—

172

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₃

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

172

—

C₄⋊C₄

—

C₂²⋊C₄

—

172

—

C₂²⋊C₄

—

172

—

C₄²

—

172

—

C₂³

—

100

C₂³

—

172

—

101

C₂³

—

172

—

102

C₂³

—

172

—

103

C₂³

—

172

—

104

C₂³

—

172

—

105

C₂³

—

172

—

106

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

107

D₄

—

172

—

108

D₄

—

172

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

172

—

117

C₂×C₄

—

172

—

118

C₂×C₄

C₈×D₄₃

yes

characteristic

119

C₂×C₄

#10621872

yes

characteristic

120

C₈

—

121

C₈

—

344

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

344

—

125

C₂²

—

344

—

126

C₂²

—

344

—

127

C₂²

—

344

—

128

C₂²

—

172

—

129

C₂²

—

172

—

130

C₂²

—

172

—

131

C₂²

(C₂×D₄₃)⋊C₈

yes

characteristic

132

C₂²

#13063239

yes

characteristic

133

C₂²

#13063241

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

172

—

139

C₂

—

140

C₂

—

141

C₂

—

344

—

142

C₂

#14582622

yes

characteristic

143

C₂

#14582691

yes

characteristic

144

C₂

#14582695

yes

characteristic

	subgroup	G/H
center	C₂²	#13063239
commutator	C₂×C₄×C₄₃	C₂×C₈
Frattini	C₂×C₄⋊C₄	C₂×D₄₃
Fitting	C₄₃×#324	C₂
socle	C₂²×C₄₃	#99
hypercenter	#324	D₄₃
hypocenter	C₄₃	#1318
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

subgroup

G/H

center

C₂²

#13063239

commutator

C₂×C₄×C₄₃

C₂×C₈

Frattini

C₂×C₄⋊C₄

C₂×D₄₃

Fitting

C₄₃×#324

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄⋊C₄ ◁ #324

N	H ≅ G/N	type
2. #14582625	C₂	semidirect
3. C₄₃×#324	C₂	semidirect
19. C₂×D₁₇₂	C₈	semidirect
75. #324	D₄₃	semidirect
76. C₄₃	#1318	semidirect

H ≅ G/N

type

2. #14582625

C₂

semidirect

3. C₄₃×#324

C₂

semidirect

19. C₂×D₁₇₂

C₈

semidirect

75. #324

D₄₃

semidirect

76. C₄₃

#1318

semidirect

Subgroups — #19132526

Conjugacy classes of subgroups

Complements