Group #19132427 — subgroups

Subgroups — #19132427

Order 5504.

208 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 85 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	#14582952	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₄₃	C₂	1	1	yes	maximal, characteristic
3	C₄₃×#431	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×Dic₄₃	—	2	1	no	—
5	C₂×C₄×Dic₄₃	—	2	1	no	—
6	C₂×C₄×Dic₄₃	C₄	1	2	yes	—
7	C₄⋊(C₄₃⋊C₈)	C₄	1	2	yes	complemented
8	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
9	C₄×C₄₃⋊C₈	—	2	1	no	—
10	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₄₃	C₄	1	2	yes	complemented
12	C₂²⋊C₈×C₄₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
17	C₂²×Dic₄₃	—	2	2	no	—
18	C₂×C₈×C₄₃	—	2	2	no	—
19	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₄₃	D₄	1	1	yes	characteristic
22	C₄²×C₄₃	Q₈	1	1	yes	characteristic
23	C₄×Dic₄₃	—	2	4	no	—
24	C₄×Dic₄₃	—	2	4	no	—
25	C₄×Dic₄₃	—	2	2	no	—
26	C₄×Dic₄₃	C₈	1	4	yes	complemented
27	C₂×C₄₃⋊C₈	—	2	2	no	—
28	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₃	—	2	2	no	—
30	C₂×C₄×C₄₃	—	2	2	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	C₄⋊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	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
39	C₂×Dic₄₃	—	2	4	no	—
40	C₂×Dic₄₃	—	2	4	no	—
41	C₂×Dic₄₃	—	2	4	no	—
42	C₈×C₄₃	—	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	C₂²⋊C₈	1	2	yes	—
48	C₄×C₄₃	—	2	1	no	—
49	C₄×C₄₃	—	2	1	no	—
50	C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
51	C₄×C₄₃	#97	1	1	yes	characteristic
52	C₄×C₄₃	#104	1	2	yes	—
53	C₄×C₄₃	#104	1	2	yes	—
54	C₄×C₄₃	C₄⋊C₈	1	2	yes	—
55	Dic₄₃	—	2	8	no	—
56	#1276	—	43	1	no	maximal, Sylow
57	C₂×C₄₃	—	2	2	no	—
58	C₂×C₄₃	#336	1	1	yes	characteristic
59	C₂×C₄₃	#339	1	1	yes	characteristic
60	C₂×C₄₃	#339	1	1	yes	characteristic
61	#431	—	43	1	no	—
62	#431	D₄₃	1	1	yes	characteristic
63	C₄³	—	43	1	no	—
64	C₄₃	#1276	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄²	—	43	2	no	—
66	C₂×C₄²	—	86	1	no	—
67	C₂×C₄²	—	86	1	no	—
68	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
69	C₄⋊C₈	—	43	2	no	—
70	C₄⋊C₈	Dic₄₃	1	2	yes	complemented
71	C₂²⋊C₈	—	2	1	no	—
72	C₂²⋊C₈	—	86	1	no	—
73	C₄×C₈	—	2	1	no	—
74	C₄×C₈	—	86	1	no	—
75	C₂²×C₄	—	86	2	no	—
76	C₂²×C₄	C₄×D₄₃	1	1	yes	characteristic
77	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
78	C₂²×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	86	2	no	—
81	C₄²	—	43	4	no	—
82	C₄²	—	86	4	no	—
83	C₄²	—	86	4	no	—
84	C₄²	—	86	2	no	—
85	C₄²	Dic₈₆	1	1	yes	characteristic
86	C₄²	C₄×D₄₃	1	1	yes	characteristic
87	C₄²	C₂×Dic₄₃	1	1	yes	characteristic
88	C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
89	C₂³	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	86	4	no	—
93	C₂×C₄	—	86	4	no	—
94	C₂×C₄	—	86	4	no	—
95	C₂×C₄	C₈×D₄₃	1	2	yes	—
96	C₂×C₄	#10621872	1	2	yes	—
97	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
98	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
99	C₂×C₄	Dic₄₃⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
101	C₂×C₄	(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
102	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	172	2	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₄₃⋊#95	1	1	yes	characteristic
108	C₂²	(C₂×D₄₃)⋊C₈	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	86	8	no	—
112	C₄	C₈×Dic₄₃	1	1	yes	characteristic
113	C₄	#13063235	1	1	yes	characteristic
114	C₄	#13063251	1	2	yes	—
115	C₄	#13063252	1	2	yes	—
116	C₄	Dic₄₃⋊C₈	1	2	yes	—
117	C₂	—	2	2	no	—
118	C₂	#14582653	1	1	yes	characteristic
119	C₂	#14582658	1	1	yes	characteristic
120	C₂	#14582660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14582952

C₂

yes

maximal, characteristic

C₄²×Dic₄₃

C₂

yes

maximal, characteristic

C₄₃×#431

C₂

yes

maximal, characteristic

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

—

C₂×C₄×Dic₄₃

C₄

yes

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

complemented

C₂²⋊(C₄₃⋊C₈)

—

C₄×C₄₃⋊C₈

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₄⋊C₈×C₄₃

C₄

yes

complemented

C₂²⋊C₈×C₄₃

—

C₄×C₈×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

characteristic

C₂²×Dic₄₃

—

C₂×C₈×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

C₂×C₄

yes

characteristic

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

Q₈

yes

characteristic

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

—

C₄×Dic₄₃

C₈

yes

complemented

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₂²⋊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

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×Dic₄₃

—

C₂×Dic₄₃

—

C₂×Dic₄₃

—

C₈×C₄₃

—

C₄₃⋊C₈

—

C₂²×C₄₃

—

C₂²×C₄₃

—

C₂²×C₄₃

#95

yes

characteristic

C₂²×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

C₄×C₈

yes

C₄×C₄₃

yes

C₄×C₄₃

yes

—

C₄×C₄₃

#104

yes

—

C₄×C₄₃

C₄⋊C₈

yes

—

Dic₄₃

—

#1276

—

maximal, Sylow

C₂×C₄₃

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

—

D₄₃

yes

characteristic

C₄³

—

C₄₃

#1276

yes

characteristic, complemented, Sylow

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

Dic₄₃

yes

complemented

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₃

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₈₆

yes

characteristic

C₄²

C₄×D₄₃

yes

characteristic

C₄²

C₂×Dic₄₃

yes

characteristic

C₄²

C₂²⋊D₄₃

yes

characteristic

C₂³

(C₂×D₄₃)⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₄₃

yes

—

C₂×C₄

#10621872

yes

—

C₂×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

C₂×C₄

Dic₄₃⋊C₄

yes

characteristic

100

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

101

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

characteristic

102

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

103

C₈

—

104

C₈

—

172

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₄₃⋊#95

yes

characteristic

108

C₂²

(C₂×D₄₃)⋊C₈

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

C₈×Dic₄₃

yes

characteristic

113

C₄

#13063235

yes

characteristic

114

C₄

#13063251

yes

—

115

C₄

#13063252

yes

—

116

C₄

Dic₄₃⋊C₈

yes

—

117

C₂

—

118

C₂

#14582653

yes

characteristic

119

C₂

#14582658

yes

characteristic

120

C₂

#14582660

yes

characteristic

	subgroup	G/H
center	C₄²	C₂²⋊D₄₃
commutator	C₄×C₄₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₄₃
Fitting	C₄₃×#431	C₂
socle	C₂²×C₄₃	#95
hypercenter	#431	D₄₃
hypocenter	C₄₃	#1276
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₄²

C₂²⋊D₄₃

commutator

C₄×C₄₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₄₃

Fitting

C₄₃×#431

C₂

C₂²×C₄₃

D₄₃

C₄₃

G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
7. C₄⋊(C₄₃⋊C₈)	C₄	semidirect
11. C₄⋊C₈×C₄₃	C₄	semidirect
26. C₄×Dic₄₃	C₈	semidirect
64. C₄₃	#1276	semidirect
70. C₄⋊C₈	Dic₄₃	semidirect

H ≅ G/N

type

7. C₄⋊(C₄₃⋊C₈)

C₄

semidirect

11. C₄⋊C₈×C₄₃

C₄

semidirect

26. C₄×Dic₄₃

C₈

semidirect

64. C₄₃

#1276

semidirect

70. C₄⋊C₈

Dic₄₃

semidirect

Subgroups — #19132427

Conjugacy classes of subgroups

Complements