Group #19132436 — subgroups

Subgroups — #19132436

Order 5504.

208 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₈	—	2	1	no	—
5	C₂²×C₄₃⋊C₈	—	2	1	no	—
6	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	complemented
8	C₄×C₄₃⋊C₈	—	2	1	no	—
9	C₄×C₄₃⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
11	C₂×C₄²×C₄₃	—	2	1	no	—
12	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
13	#95×C₄₃	—	2	1	no	—
14	C₂²×C₄×C₄₃	—	2	2	no	—
15	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₄₃	—	2	2	no	—
19	C₄⋊C₄×C₄₃	C₈	1	4	yes	complemented
20	C₄²×C₄₃	—	4	1	no	—
21	C₄²×C₄₃	—	4	1	no	—
22	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₄₃	D₄	1	1	yes	characteristic
25	C₄²×C₄₃	Q₈	1	1	yes	characteristic
26	C₂×C₄₃⋊C₈	—	2	4	no	—
27	C₂×C₄₃⋊C₈	—	2	2	no	—
28	C₂×C₄₃⋊C₈	—	2	2	no	—
29	C₂×C₄₃⋊C₈	—	4	1	no	—
30	C₂×C₄₃⋊C₈	—	4	1	no	—
31	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₄₃	—	2	2	no	—
33	C₂×C₄×C₄₃	—	2	4	no	—
34	C₂×C₄×C₄₃	—	4	2	no	—
35	C₂×C₄×C₄₃	—	4	2	no	—
36	C₂×C₄×C₄₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
45	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
47	C₂×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
48	C₄₃⋊C₈	—	2	4	no	—
49	C₄₃⋊C₈	—	4	2	no	—
50	C₂²×C₄₃	#95	1	1	yes	characteristic
51	C₂²×C₄₃	C₂²⋊C₈	1	2	yes	—
52	C₂²×C₄₃	#102	1	1	yes	characteristic
53	C₂²×C₄₃	#103	1	1	yes	characteristic
54	C₂²×C₄₃	#104	1	2	yes	—
55	C₄×C₄₃	—	2	1	no	—
56	C₄×C₄₃	—	2	1	no	—
57	C₄×C₄₃	—	4	4	no	—
58	C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
59	C₄×C₄₃	#97	1	1	yes	characteristic
60	C₄×C₄₃	#102	1	1	yes	characteristic
61	C₄×C₄₃	#103	1	1	yes	characteristic
62	C₄×C₄₃	C₄⋊C₈	1	2	yes	—
63	C₄×C₄₃	#106	1	1	yes	characteristic
64	C₄×C₄₃	#107	1	1	yes	characteristic
65	#1279	—	43	1	no	maximal, Sylow
66	C₂×C₄₃	D₄⋊C₈	1	2	yes	—
67	C₂×C₄₃	Q₈⋊C₈	1	2	yes	—
68	C₂×C₄₃	#336	1	1	yes	characteristic
69	C₂×C₄₃	#339	1	1	yes	characteristic
70	C₂×C₄₃	#340	1	1	yes	characteristic
71	C₂×C₄⋊C₈	—	43	1	no	—
72	C₂×C₄×C₈	—	43	1	no	—
73	C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
74	C₄₃	#1279	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	86	1	no	—
76	C₂²×C₈	—	86	1	no	—
77	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
80	C₄⋊C₈	—	43	2	no	—
81	C₄⋊C₈	—	86	1	no	—
82	C₄×C₈	—	43	2	no	—
83	C₄×C₈	—	86	1	no	—
84	#95	—	2	1	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
89	C₂×C₈	—	86	4	no	—
90	C₂×C₈	—	86	2	no	—
91	C₂×C₈	—	86	2	no	—
92	C₂×C₈	—	172	1	no	—
93	C₂×C₈	—	172	1	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	C₄₃⋊C₈	1	4	yes	complemented
96	C₄²	—	4	1	no	—
97	C₄²	—	4	1	no	—
98	C₄²	Dic₈₆	1	1	yes	characteristic
99	C₄²	C₄×D₄₃	1	1	yes	characteristic
100	C₄²	C₄×D₄₃	1	1	yes	characteristic
101	C₄²	D₁₇₂	1	1	yes	characteristic
102	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
108	C₂×C₄	#10621877	1	2	yes	—
109	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
110	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
111	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
112	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
113	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
114	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
115	C₂×C₄	#10621882	1	1	yes	characteristic
116	C₂×C₄	#10621883	1	1	yes	characteristic
117	C₂×C₄	#10621884	1	1	yes	characteristic
118	C₂×C₄	#10621885	1	1	yes	characteristic
119	C₈	—	86	4	no	—
120	C₈	—	172	2	no	—
121	C₂²	C₄₃⋊#95	1	1	yes	characteristic
122	C₂²	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
123	C₂²	#13063246	1	1	yes	characteristic
124	C₂²	#13063249	1	1	yes	characteristic
125	C₂²	#13063252	1	2	yes	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	4	4	no	—
129	C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
130	C₄	#13063236	1	1	yes	characteristic
131	C₄	#13063245	1	1	yes	characteristic
132	C₄	#13063248	1	1	yes	characteristic
133	C₄	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
134	C₄	#13063256	1	1	yes	characteristic
135	C₄	#13063258	1	1	yes	characteristic
136	C₂	D₄⋊(C₄₃⋊C₈)	1	2	yes	—
137	C₂	Q₈⋊(C₄₃⋊C₈)	1	2	yes	—
138	C₂	#14582654	1	1	yes	characteristic
139	C₂	#14582658	1	1	yes	characteristic
140	C₂	#14582661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×C₄×C₄₃⋊C₈

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

complemented

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

#95×C₄₃

—

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

C₈

yes

complemented

C₄²×C₄₃

—

C₄²×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

Q₈

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

SD₁₆

yes

characteristic

C₂×C₄×C₄₃

SD₁₆

yes

characteristic

C₂×C₄×C₄₃

Q₁₆

yes

characteristic

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#95

yes

characteristic

C₂²×C₄₃

C₂²⋊C₈

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×C₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

C₄⋊C₈

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

D₄⋊C₈

yes

—

C₂×C₄₃

Q₈⋊C₈

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₄×C₄⋊C₄

D₄₃

yes

characteristic

C₄₃

#1279

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄⋊C₄

—

C₄⋊C₄

C₄₃⋊C₈

yes

complemented

C₄²

—

C₄²

—

C₄²

Dic₈₆

yes

characteristic

C₄²

C₄×D₄₃

yes

characteristic

100

C₄²

C₄×D₄₃

yes

characteristic

101

C₄²

D₁₇₂

yes

characteristic

102

C₂³

C₂²⋊Dic₄₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

108

C₂×C₄

#10621877

yes

—

109

C₂×C₄

C₄×Dic₄₃

yes

characteristic

110

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

111

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

112

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

113

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

114

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

115

C₂×C₄

#10621882

yes

characteristic

116

C₂×C₄

#10621883

yes

characteristic

117

C₂×C₄

#10621884

yes

characteristic

118

C₂×C₄

#10621885

yes

characteristic

119

C₈

—

120

C₈

—

172

—

121

C₂²

C₄₃⋊#95

yes

characteristic

122

C₂²

C₂²⋊(C₄₃⋊C₈)

yes

—

123

C₂²

#13063246

yes

characteristic

124

C₂²

#13063249

yes

characteristic

125

C₂²

#13063252

yes

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₄×C₄₃⋊C₈

yes

characteristic

130

C₄

#13063236

yes

characteristic

131

C₄

#13063245

yes

characteristic

132

C₄

#13063248

yes

characteristic

133

C₄

C₄⋊(C₄₃⋊C₈)

yes

—

134

C₄

#13063256

yes

characteristic

135

C₄

#13063258

yes

characteristic

136

C₂

D₄⋊(C₄₃⋊C₈)

yes

—

137

C₂

Q₈⋊(C₄₃⋊C₈)

yes

—

138

C₂

#14582654

yes

characteristic

139

C₂

#14582658

yes

characteristic

140

C₂

#14582661

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₃
Fitting	C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂²⋊C₄
hypercenter	C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#1279
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₃

commutator

C₄×C₄₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₃

Fitting

C₄×C₄⋊C₄×C₄₃

C₂

socle

C₂³×C₄₃

C₂²⋊C₄

hypercenter

C₄×C₄⋊C₄

D₄₃

hypocenter

C₄₃

#1279

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
7. C₄⋊(C₄₃⋊C₈)	C₄	semidirect
9. C₄×C₄₃⋊C₈	C₄	semidirect
19. C₄⋊C₄×C₄₃	C₈	semidirect
74. C₄₃	#1279	semidirect
95. C₄⋊C₄	C₄₃⋊C₈	semidirect

H ≅ G/N

type

7. C₄⋊(C₄₃⋊C₈)

C₄

semidirect

9. C₄×C₄₃⋊C₈

C₄

semidirect

19. C₄⋊C₄×C₄₃

C₈

semidirect

74. C₄₃

#1279

semidirect

95. C₄⋊C₄

C₄₃⋊C₈

semidirect

Subgroups — #19132436

Conjugacy classes of subgroups

Complements