Group #19132491 — subgroups

Subgroups — #19132491

Order 5504.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#14582923	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×#10621877	—	2	1	no	—
5	C₂²×C₄₃⋊C₈	—	2	1	no	—
6	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	—
9	C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	—
10	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₃	—	2	1	no	—
12	C₂²⋊C₈×C₄₃	C₄	1	2	yes	—
13	C₄×C₈×C₄₃	—	2	1	no	—
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	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₄₃	D₄	1	1	yes	characteristic
21	C₄²×C₄₃	D₄	1	1	yes	characteristic
22	#10621877	—	4	1	no	—
23	#10621877	—	4	1	no	—
24	C₂×C₄₃⋊C₈	—	2	2	no	—
25	C₂×C₄₃⋊C₈	—	2	2	no	—
26	C₂×C₄₃⋊C₈	—	4	1	no	—
27	C₂×C₄₃⋊C₈	—	4	1	no	—
28	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₃	—	2	2	no	—
30	C₂×C₄×C₄₃	—	4	1	no	—
31	C₂×C₄×C₄₃	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₄₃	C₂²⋊C₄	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	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₄₃	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
41	C₈×C₄₃	—	4	2	no	—
42	C₄₃⋊C₈	—	4	2	no	—
43	C₄₃⋊C₈	—	4	2	no	—
44	C₂²×C₄₃	—	2	1	no	—
45	C₂²×C₄₃	—	2	1	no	—
46	C₂²×C₄₃	#95	1	1	yes	characteristic
47	C₂²×C₄₃	#106	1	1	yes	characteristic
48	C₂²×C₄₃	#107	1	1	yes	characteristic
49	C₄×C₄₃	—	2	1	no	—
50	C₄×C₄₃	—	2	1	no	—
51	C₄×C₄₃	—	2	1	no	—
52	C₄×C₄₃	#102	1	1	yes	characteristic
53	C₄×C₄₃	#102	1	1	yes	characteristic
54	C₄×C₄₃	#103	1	1	yes	characteristic
55	C₄×C₄₃	#103	1	1	yes	characteristic
56	C₄×C₄₃	#108	1	2	yes	—
57	#1301	—	43	1	no	maximal, Sylow
58	C₂×C₄₃	—	2	1	no	—
59	C₂×C₄₃	—	2	1	no	—
60	C₂×C₄₃	#340	1	1	yes	characteristic
61	C₂×C₄₃	#341	1	1	yes	characteristic
62	C₂×C₄₃	#344	1	1	yes	characteristic
63	#431	D₄₃	1	1	yes	characteristic
64	#423	—	43	1	no	—
65	C₂×C₄⋊C₈	—	43	1	no	—
66	C₄₃	#1301	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₁₆	—	86	1	no	—
68	C₂²×C₈	—	86	1	no	—
69	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	1	no	—
71	C₄⋊C₈	—	43	2	no	—
72	C₄⋊C₈	—	43	2	no	—
73	C₄⋊C₈	—	86	1	no	—
74	C₄⋊C₈	—	86	1	no	—
75	C₂²⋊C₈	Dic₄₃	1	2	yes	—
76	C₄×C₈	—	2	1	no	—
77	C₂²×C₄	Dic₈₆	1	1	yes	characteristic
78	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
79	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
80	M₁₆	—	172	1	no	—
81	M₁₆	—	172	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	86	2	no	—
84	C₂×C₈	—	86	2	no	—
85	C₂×C₈	—	172	1	no	—
86	C₂×C₈	—	172	1	no	—
87	C₄²	C₄×D₄₃	1	1	yes	characteristic
88	C₄²	C₄×D₄₃	1	1	yes	characteristic
89	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
90	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
91	C₂³	C₄⋊Dic₄₃	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	#10621873	1	1	yes	characteristic
95	C₂×C₄	#10621873	1	1	yes	characteristic
96	C₂×C₄	D₃₄₄	1	1	yes	characteristic
97	C₂×C₄	Dic₁₇₂	1	1	yes	characteristic
98	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
99	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
100	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
101	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
102	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	172	2	no	—
106	C₈	—	172	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	C₄₃⋊#95	1	1	yes	characteristic
110	C₂²	#13063257	1	1	yes	characteristic
111	C₂²	#13063259	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#13063247	1	1	yes	characteristic
116	C₄	#13063247	1	1	yes	characteristic
117	C₄	#13063250	1	1	yes	characteristic
118	C₄	#13063250	1	1	yes	characteristic
119	C₄	#13063260	1	2	yes	—
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#14582662	1	1	yes	characteristic
123	C₂	#14582663	1	1	yes	characteristic
124	C₂	#14582669	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582923

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal, characteristic

C₄₃×#431

C₂

yes

maximal, characteristic

C₂×#10621877

—

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

—

C₂²⋊C₈×C₄₃

C₄

yes

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

Q₈

yes

characteristic

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

#10621877

—

#10621877

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

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

—

C₂²×C₄₃

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

#1301

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

D₄₃

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₄₃

#1301

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

Dic₄₃

yes

—

C₄×C₈

—

C₂²×C₄

Dic₈₆

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

M₁₆

—

172

—

M₁₆

—

172

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄²

C₄×D₄₃

yes

characteristic

C₄²

C₄×D₄₃

yes

characteristic

C₄²

C₂²⋊D₄₃

yes

characteristic

C₄²

C₂²⋊D₄₃

yes

characteristic

C₂³

C₄⋊Dic₄₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

D₃₄₄

yes

characteristic

C₂×C₄

Dic₁₇₂

yes

characteristic

C₂×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

100

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

101

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

102

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

104

C₈

—

105

C₈

—

172

—

106

C₈

—

172

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₄₃⋊#95

yes

characteristic

110

C₂²

#13063257

yes

characteristic

111

C₂²

#13063259

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#13063247

yes

characteristic

116

C₄

#13063247

yes

characteristic

117

C₄

#13063250

yes

characteristic

118

C₄

#13063250

yes

characteristic

119

C₄

#13063260

yes

—

120

C₂

—

121

C₂

—

122

C₂

#14582662

yes

characteristic

123

C₂

#14582663

yes

characteristic

124

C₂

#14582669

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₃)⋊C₄
commutator	C₂×C₄×C₄₃	C₄²
Frattini	C₂×C₄²	C₂×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	#95
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1301
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

(C₂×D₄₃)⋊C₄

commutator

C₂×C₄×C₄₃

C₄²

Frattini

C₂×C₄²

C₂×D₄₃

Fitting

C₄₃×#431

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
66. C₄₃	#1301	semidirect

H ≅ G/N

type

66. C₄₃

#1301

semidirect

Subgroups — #19132491

Conjugacy classes of subgroups

Complements