Group #19132429 — subgroups

Subgroups — #19132429

Order 5504.

180 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂×C₄⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₃⋊C₈	—	2	1	no	—
5	C₂²×C₄₃⋊C₈	C₄	1	2	yes	—
6	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	—
8	C₄×C₄₃⋊C₈	—	2	1	no	—
9	C₄×C₄₃⋊C₈	—	2	1	no	—
10	C₂²×C₈×C₄₃	—	2	1	no	—
11	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₄₃	—	2	1	no	—
13	C₄⋊C₈×C₄₃	C₄	1	2	yes	—
14	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₃	Q₈	1	1	yes	characteristic
17	C₂×C₈×C₄₃	—	2	2	no	—
18	C₂×C₈×C₄₃	—	4	1	no	—
19	C₂×C₈×C₄₃	—	4	1	no	—
20	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₄₃	D₄	1	1	yes	characteristic
23	C₄²×C₄₃	D₄	1	1	yes	characteristic
24	C₂×C₄₃⋊C₈	—	2	2	no	—
25	C₂×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄₃⋊C₈	—	2	2	no	—
27	C₂×C₄₃⋊C₈	—	4	1	no	—
28	C₂×C₄₃⋊C₈	—	4	1	no	—
29	C₂×C₄₃⋊C₈	C₈	1	4	yes	complemented
30	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₄₃	—	2	2	no	—
32	C₂×C₄×C₄₃	C₄²	1	1	yes	characteristic
33	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
39	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
40	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
41	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
42	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
44	C₈×C₄₃	—	4	2	no	—
45	C₄₃⋊C₈	—	2	4	no	—
46	C₄₃⋊C₈	—	4	2	no	—
47	C₂²×C₄₃	#95	1	1	yes	characteristic
48	C₂²×C₄₃	C₄⋊C₈	1	2	yes	—
49	C₂²×C₄₃	#106	1	1	yes	characteristic
50	C₂²×C₄₃	#107	1	1	yes	characteristic
51	C₂²×C₄₃	#108	1	2	yes	—
52	C₄×C₄₃	—	2	1	no	—
53	C₄×C₄₃	—	2	1	no	—
54	C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
55	C₄×C₄₃	#97	1	1	yes	characteristic
56	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
57	C₄×C₄₃	#102	1	1	yes	characteristic
58	C₄×C₄₃	#102	1	1	yes	characteristic
59	C₄×C₄₃	#103	1	1	yes	characteristic
60	C₄×C₄₃	#103	1	1	yes	characteristic
61	#1277	—	43	1	no	maximal, Sylow
62	C₂×C₄₃	#334	1	2	yes	—
63	C₂×C₄₃	#335	1	2	yes	—
64	C₂×C₄₃	#336	1	1	yes	characteristic
65	C₂×C₄₃	#340	1	1	yes	characteristic
66	C₂×C₄₃	#341	1	1	yes	characteristic
67	C₂×C₄⋊C₈	—	43	1	no	—
68	C₂×C₄⋊C₈	D₄₃	1	1	yes	characteristic
69	C₂×C₄×C₈	—	43	1	no	—
70	C₄₃	#1277	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₈	—	2	1	no	—
72	C₂²×C₈	—	43	2	no	—
73	C₂²×C₈	—	86	1	no	—
74	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	43	2	no	—
77	C₄⋊C₈	—	86	1	no	—
78	C₄⋊C₈	Dic₄₃	1	2	yes	—
79	C₄×C₈	—	86	1	no	—
80	C₄×C₈	—	86	1	no	—
81	C₂²×C₄	Dic₈₆	1	1	yes	characteristic
82	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	43	4	no	—
88	C₂×C₈	—	86	2	no	—
89	C₂×C₈	—	86	2	no	—
90	C₂×C₈	—	86	2	no	—
91	C₂×C₈	—	172	1	no	—
92	C₂×C₈	—	172	1	no	—
93	C₄²	C₄×D₄₃	1	1	yes	characteristic
94	C₄²	D₁₇₂	1	1	yes	characteristic
95	C₄²	C₂×Dic₄₃	1	1	yes	characteristic
96	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
97	C₂³	Dic₄₃⋊C₄	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₈×D₄₃	1	2	yes	—
100	C₂×C₄	#10621872	1	2	yes	—
101	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
102	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
104	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
105	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
106	C₂×C₄	#10621882	1	1	yes	characteristic
107	C₂×C₄	#10621883	1	1	yes	characteristic
108	C₂×C₄	#10621884	1	1	yes	characteristic
109	C₂×C₄	#10621885	1	1	yes	characteristic
110	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
111	C₈	—	4	2	no	—
112	C₈	—	86	4	no	—
113	C₈	—	172	2	no	—
114	C₂²	C₄₃⋊#95	1	1	yes	characteristic
115	C₂²	Dic₄₃⋊C₈	1	2	yes	—
116	C₂²	#13063256	1	1	yes	characteristic
117	C₂²	#13063258	1	1	yes	characteristic
118	C₂²	#13063260	1	2	yes	—
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	C₈×Dic₄₃	1	1	yes	characteristic
122	C₄	#13063235	1	1	yes	characteristic
123	C₄	(C₂×D₄₃)⋊C₈	1	2	yes	—
124	C₄	#13063245	1	1	yes	characteristic
125	C₄	#13063246	1	1	yes	characteristic
126	C₄	#13063248	1	1	yes	characteristic
127	C₄	#13063249	1	1	yes	characteristic
128	C₂	#14582649	1	2	yes	—
129	C₂	#14582651	1	2	yes	—
130	C₂	#14582653	1	1	yes	characteristic
131	C₂	#14582661	1	1	yes	characteristic
132	C₂	#14582663	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

C₂×C₄⋊C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₂²×C₄₃⋊C₈

C₄

yes

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

—

C₄×C₄₃⋊C₈

—

C₄×C₄₃⋊C₈

—

C₂²×C₈×C₄₃

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₄⋊C₈×C₄₃

C₄

yes

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

Q₈

yes

characteristic

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

C₈

yes

complemented

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

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

C₂²⋊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

C₈×C₄₃

—

C₄₃⋊C₈

—

C₄₃⋊C₈

—

C₂²×C₄₃

#95

yes

characteristic

C₂²×C₄₃

C₄⋊C₈

yes

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×C₈

yes

characteristic

C₄×C₄₃

#97

yes

characteristic

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₂×C₄₃

yes

—

C₂×C₄₃

#335

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄₃

yes

characteristic

C₂×C₄×C₈

—

C₄₃

#1277

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₄₃

yes

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

Dic₈₆

yes

characteristic

C₂²×C₄

C₄×D₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄²

C₄×D₄₃

yes

characteristic

C₄²

D₁₇₂

yes

characteristic

C₄²

C₂×Dic₄₃

yes

characteristic

C₄²

C₂²⋊D₄₃

yes

characteristic

C₂³

Dic₄₃⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

C₈×D₄₃

yes

—

100

C₂×C₄

#10621872

yes

—

101

C₂×C₄

C₄×Dic₄₃

yes

characteristic

102

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

103

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

104

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

105

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

106

C₂×C₄

#10621882

yes

characteristic

107

C₂×C₄

#10621883

yes

characteristic

108

C₂×C₄

#10621884

yes

characteristic

109

C₂×C₄

#10621885

yes

characteristic

110

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

—

172

—

114

C₂²

C₄₃⋊#95

yes

characteristic

115

C₂²

Dic₄₃⋊C₈

yes

—

116

C₂²

#13063256

yes

characteristic

117

C₂²

#13063258

yes

characteristic

118

C₂²

#13063260

yes

—

119

C₄

—

120

C₄

—

121

C₄

C₈×Dic₄₃

yes

characteristic

122

C₄

#13063235

yes

characteristic

123

C₄

(C₂×D₄₃)⋊C₈

yes

—

124

C₄

#13063245

yes

characteristic

125

C₄

#13063246

yes

characteristic

126

C₄

#13063248

yes

characteristic

127

C₄

#13063249

yes

characteristic

128

C₂

#14582649

yes

—

129

C₂

#14582651

yes

—

130

C₂

#14582653

yes

characteristic

131

C₂

#14582661

yes

characteristic

132

C₂

#14582663

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₃
Fitting	C₂×C₄⋊C₈×C₄₃	C₂
socle	C₂³×C₄₃	C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₄₃
hypocenter	C₄₃	#1277
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²×C₄

C₂²⋊D₄₃

commutator

C₄×C₄₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₃

Fitting

C₂×C₄⋊C₈×C₄₃

C₂

socle

C₂³×C₄₃

C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₄₃

hypocenter

C₄₃

#1277

LCS

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
29. C₂×C₄₃⋊C₈	C₈	semidirect
70. C₄₃	#1277	semidirect

H ≅ G/N

type

29. C₂×C₄₃⋊C₈

C₈

semidirect

70. C₄₃

#1277

semidirect

Subgroups — #19132429

Conjugacy classes of subgroups

Complements