Group #19132430 — subgroups

Subgroups — #19132430

Order 5504.

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

normal subgroups only

This group has 180 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	2	yes	maximal
2	C₂×C₄×C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₃⋊C₈	—	2	2	no	—
4	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
5	C₄⋊(C₄₃⋊C₈)	C₄	1	4	yes	—
6	C₂²×C₈×C₄₃	C₄	1	2	yes	—
7	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₄₃	—	2	2	no	—
9	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₄₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₄₃	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₄₃	—	2	2	no	—
13	C₂×C₈×C₄₃	—	2	2	no	—
14	C₂×C₈×C₄₃	C₈	1	4	yes	complemented
15	C₄²×C₄₃	C₂×C₄	1	2	yes	—
16	C₄²×C₄₃	D₄	1	2	yes	—
17	C₂×C₄₃⋊C₈	—	2	4	no	—
18	C₂×C₄₃⋊C₈	—	4	2	no	—
19	C₂×C₄₃⋊C₈	—	4	2	no	—
20	C₂³×C₄₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₄₃	—	2	2	no	—
22	C₂×C₄×C₄₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₄₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₄₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₄₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
30	C₂×C₄×C₄₃	Q₁₆	1	1	yes	characteristic
31	C₈×C₄₃	—	2	4	no	—
32	C₄₃⋊C₈	—	4	4	no	—
33	C₂²×C₄₃	#95	1	1	yes	characteristic
34	C₂²×C₄₃	C₄⋊C₈	1	2	yes	—
35	C₂²×C₄₃	#106	1	1	yes	characteristic
36	C₂²×C₄₃	#107	1	1	yes	characteristic
37	C₂²×C₄₃	#108	1	2	yes	—
38	C₄×C₄₃	—	2	2	no	—
39	C₄×C₄₃	C₄×C₈	1	1	yes	characteristic
40	C₄×C₄₃	#97	1	1	yes	characteristic
41	C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
42	C₄×C₄₃	#102	1	2	yes	—
43	C₄×C₄₃	#103	1	2	yes	—
44	#1277	—	43	1	no	maximal, Sylow
45	C₂×C₄₃	#334	1	2	yes	—
46	C₂×C₄₃	#335	1	2	yes	—
47	C₂×C₄₃	#336	1	1	yes	characteristic
48	C₂×C₄₃	#340	1	1	yes	characteristic
49	C₂×C₄₃	#341	1	1	yes	characteristic
50	C₂×C₄⋊C₈	—	43	2	no	—
51	C₂×C₄×C₈	D₄₃	1	1	yes	characteristic
52	C₄₃	#1277	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₈	—	86	2	no	—
54	C₂²×C₈	Dic₄₃	1	2	yes	—
55	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
56	C₄⋊C₈	—	43	4	no	—
57	C₄⋊C₈	—	86	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	Dic₈₆	1	1	yes	characteristic
60	C₂²×C₄	D₁₇₂	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	86	4	no	—
65	C₂×C₈	—	172	2	no	—
66	C₂×C₈	—	172	2	no	—
67	C₂×C₈	C₄₃⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₄₃	1	2	yes	—
69	C₄²	C₂²⋊D₄₃	1	2	yes	—
70	C₂³	C₄⋊Dic₄₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	#10621873	1	2	yes	—
73	C₂×C₄	D₃₄₄	1	1	yes	characteristic
74	C₂×C₄	Dic₁₇₂	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
76	C₂×C₄	#10621877	1	2	yes	—
77	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
78	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
79	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	172	4	no	—
83	C₂²	C₄₃⋊#95	1	1	yes	characteristic
84	C₂²	C₄⋊(C₄₃⋊C₈)	1	2	yes	—
85	C₂²	#13063257	1	1	yes	characteristic
86	C₂²	#13063259	1	1	yes	characteristic
87	C₂²	#13063261	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×C₄₃⋊C₈	1	1	yes	characteristic
90	C₄	#13063236	1	1	yes	characteristic
91	C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
92	C₄	#13063247	1	2	yes	—
93	C₄	#13063250	1	2	yes	—
94	C₂	#14582650	1	2	yes	—
95	C₂	#14582652	1	2	yes	—
96	C₂	#14582654	1	1	yes	characteristic
97	C₂	#14582662	1	1	yes	characteristic
98	C₂	#14582664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₄₃

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

—

C₂²×C₈×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

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

—

C₂×C₈×C₄₃

—

C₂×C₈×C₄₃

C₈

yes

complemented

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂×C₄₃⋊C₈

—

C₂³×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

—

C₂×C₄×C₄₃

C₄²

yes

characteristic

C₂×C₄×C₄₃

C₂²⋊C₄

yes

—

C₂×C₄×C₄₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₂×C₈

yes

—

C₂×C₄×C₄₃

M₁₆

yes

—

C₂×C₄×C₄₃

D₈

yes

characteristic

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

characteristic

C₈×C₄₃

—

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×C₈

yes

characteristic

C₄×C₄₃

#97

yes

characteristic

C₄×C₄₃

C₂²⋊C₈

yes

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#103

yes

—

#1277

—

maximal, Sylow

C₂×C₄₃

#334

yes

—

C₂×C₄₃

#335

yes

—

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄₃

yes

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄₃

yes

characteristic

C₄₃

#1277

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₈₆

yes

characteristic

C₂²×C₄

D₁₇₂

yes

characteristic

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₂×C₈

C₄₃⋊C₈

yes

complemented

C₄²

C₄×D₄₃

yes

—

C₄²

C₂²⋊D₄₃

yes

—

C₂³

C₄⋊Dic₄₃

yes

characteristic

C₂×C₄

—

C₂×C₄

#10621873

yes

—

C₂×C₄

D₃₄₄

yes

characteristic

C₂×C₄

Dic₁₇₂

yes

characteristic

C₂×C₄

C₂×C₄₃⋊C₈

yes

—

C₂×C₄

#10621877

yes

—

C₂×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₄

Dic₄₃⋊C₄

yes

—

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₄₃

yes

characteristic

C₈

—

C₈

—

172

—

C₂²

C₄₃⋊#95

yes

characteristic

C₂²

C₄⋊(C₄₃⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

C₄×C₄₃⋊C₈

yes

characteristic

C₄

#13063236

yes

characteristic

C₄

C₂²⋊(C₄₃⋊C₈)

yes

—

C₄

#13063247

yes

—

C₄

#13063250

yes

—

C₂

#14582650

yes

—

C₂

#14582652

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₁₇₂
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

D₁₇₂

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
14. C₂×C₈×C₄₃	C₈	semidirect
52. C₄₃	#1277	semidirect
67. C₂×C₈	C₄₃⋊C₈	semidirect

H ≅ G/N

type

14. C₂×C₈×C₄₃

C₈

semidirect

52. C₄₃

#1277

semidirect

67. C₂×C₈

C₄₃⋊C₈

semidirect

Subgroups — #19132430

Conjugacy classes of subgroups

Complements