Group #19132478 — subgroups

Subgroups — #19132478

Order 5504.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₄₃×#384	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	complemented
6	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
7	C₂×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
8	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₄₃	—	2	2	no	—
10	C₂²×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
11	C₂²×C₄×C₄₃	D₄	1	2	yes	—
12	C₄⋊C₄×C₄₃	—	4	2	no	—
13	C₄⋊C₄×C₄₃	—	4	2	no	—
14	C₄⋊C₄×C₄₃	—	4	2	no	—
15	C₄²×C₄₃	C₂×C₄	1	2	yes	—
16	C₄²×C₄₃	D₄	1	1	yes	characteristic
17	C₄²×C₄₃	Q₈	1	1	yes	characteristic
18	C₂×C₄₃⋊C₈	—	2	4	no	—
19	C₂×C₄₃⋊C₈	—	4	2	no	—
20	C₂×C₄₃⋊C₈	—	4	2	no	—
21	C₂³×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₄×C₄₃	—	2	1	no	—
23	C₂×C₄×C₄₃	—	2	1	no	—
24	C₂×C₄×C₄₃	—	4	2	no	—
25	C₂×C₄×C₄₃	—	4	4	no	—
26	C₂×C₄×C₄₃	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₄₃	C₂²⋊C₄	1	2	yes	—
28	C₂×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₄₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₄₃	D₈	1	2	yes	—
31	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
32	C₂×C₄×C₄₃	SD₁₆	1	2	yes	—
33	C₂×C₄×C₄₃	Q₁₆	1	2	yes	—
34	C₄₃⋊C₈	—	4	4	no	—
35	C₂²×C₄₃	#95	1	1	yes	characteristic
36	C₂²×C₄₃	#100	1	1	yes	characteristic
37	C₂²×C₄₃	#101	1	1	yes	characteristic
38	C₂²×C₄₃	#102	1	2	yes	—
39	C₂²×C₄₃	#103	1	2	yes	—
40	C₄×C₄₃	—	2	1	no	—
41	C₄×C₄₃	—	2	1	no	—
42	C₄×C₄₃	—	8	2	no	—
43	C₄×C₄₃	#102	1	2	yes	—
44	C₄×C₄₃	#103	1	2	yes	—
45	C₄×C₄₃	#106	1	2	yes	—
46	C₄×C₄₃	#107	1	2	yes	—
47	#1296	—	43	1	no	maximal, Sylow
48	C₂×C₄₃	#331	1	1	yes	characteristic
49	C₂×C₄₃	#332	1	2	yes	—
50	C₂×C₄₃	#333	1	1	yes	characteristic
51	C₂×C₄₃	#340	1	2	yes	—
52	C₂×C₄₃	#343	1	1	yes	characteristic
53	C₂×C₄⋊C₈	—	43	2	no	—
54	#384	D₄₃	1	1	yes	characteristic
55	C₄₃	#1296	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₈	—	86	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₄₃	1	2	yes	—
59	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
60	C₄⋊C₈	—	43	4	no	—
61	C₄⋊C₈	—	86	2	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₄₃	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₄₃	1	2	yes	—
65	C₂×C₈	—	86	4	no	—
66	C₂×C₈	—	172	2	no	—
67	C₂×C₈	—	172	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	Dic₈₆	1	1	yes	characteristic
72	C₄²	C₄×D₄₃	1	2	yes	—
73	C₄²	D₁₇₂	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₄₃	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₄₃	1	1	yes	characteristic
80	C₂×C₄	Dic₄₃⋊C₄	1	2	yes	—
81	C₂×C₄	C₄⋊Dic₄₃	1	1	yes	characteristic
82	C₂×C₄	(C₂×D₄₃)⋊C₄	1	2	yes	—
83	C₂×C₄	#10621882	1	2	yes	—
84	C₂×C₄	#10621883	1	2	yes	—
85	C₂×C₄	#10621884	1	2	yes	—
86	C₂×C₄	#10621885	1	2	yes	—
87	C₈	—	172	4	no	—
88	C₂²	C₄₃⋊#95	1	1	yes	characteristic
89	C₂²	#13063242	1	1	yes	characteristic
90	C₂²	#13063244	1	1	yes	characteristic
91	C₂²	#13063246	1	2	yes	—
92	C₂²	#13063249	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#13063245	1	2	yes	—
97	C₄	#13063248	1	2	yes	—
98	C₄	#13063256	1	2	yes	—
99	C₄	#13063258	1	2	yes	—
100	C₂	#14582643	1	1	yes	characteristic
101	C₂	#14582645	1	2	yes	—
102	C₂	#14582648	1	1	yes	characteristic
103	C₂	#14582661	1	2	yes	—
104	C₂	#14582668	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₃⋊C₈)

C₂

yes

maximal

C₄₃×#384

C₂

yes

maximal, characteristic

C₂²×C₄₃⋊C₈

—

C₄⋊(C₄₃⋊C₈)

—

C₄⋊(C₄₃⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₃

—

C₂×C₄⋊C₄×C₄₃

C₄

yes

—

C₂×C₄²×C₄₃

C₂²

yes

characteristic

C₂²×C₄×C₄₃

—

C₂²×C₄×C₄₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₃

D₄

yes

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄⋊C₄×C₄₃

—

C₄²×C₄₃

C₂×C₄

yes

—

C₄²×C₄₃

D₄

yes

characteristic

C₄²×C₄₃

Q₈

yes

characteristic

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

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

—

C₂×C₄×C₄₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₃

D₈

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

SD₁₆

yes

—

C₂×C₄×C₄₃

Q₁₆

yes

—

C₄₃⋊C₈

—

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

C₂²×C₄₃

yes

—

C₂²×C₄₃

#103

yes

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

—

C₄×C₄₃

#102

yes

—

C₄×C₄₃

#103

yes

—

C₄×C₄₃

#106

yes

—

C₄×C₄₃

#107

yes

—

—

C₂×C₄₃

yes

C₂×C₄₃

yes

—

C₂×C₄₃

#333

yes

characteristic

C₂×C₄₃

#340

yes

—

C₂×C₄₃

#343

yes

characteristic

C₂×C₄⋊C₈

—

#384

D₄₃

yes

characteristic

C₄₃

#1296

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₃

yes

—

C₂×C₄²

C₂×D₄₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₃

yes

characteristic

C₂²×C₄

C₂²⋊D₄₃

yes

—

C₂×C₈

—

C₂×C₈

—

172

—

C₂×C₈

—

172

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₈₆

yes

characteristic

C₄²

C₄×D₄₃

yes

—

C₄²

D₁₇₂

yes

characteristic

C₂³

C₂²⋊Dic₄₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₄₃

yes

characteristic

C₂×C₄

Dic₄₃⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₄₃

yes

characteristic

C₂×C₄

(C₂×D₄₃)⋊C₄

yes

—

C₂×C₄

#10621882

yes

—

C₂×C₄

#10621883

yes

—

C₂×C₄

#10621884

yes

—

C₂×C₄

#10621885

yes

—

C₈

—

172

—

C₂²

C₄₃⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#13063249

yes

—

C₄

—

C₄

—

C₄

—

C₄

#13063245

yes

—

C₄

#13063248

yes

—

C₄

#13063256

yes

—

C₄

#13063258

yes

—

100

C₂

#14582643

yes

characteristic

101

C₂

#14582645

yes

—

102

C₂

#14582648

yes

characteristic

103

C₂

#14582661

yes

—

104

C₂

#14582668

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₃
commutator	C₂×C₄×C₄₃	C₄²
Frattini	C₂×C₄²	C₂×D₄₃
Fitting	C₄₃×#384	C₂
socle	C₂³×C₄₃	C₂²⋊C₄
hypercenter	#384	D₄₃
hypocenter	C₄₃	#1296
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₃

commutator

C₂×C₄×C₄₃

C₄²

Frattini

C₂×C₄²

C₂×D₄₃

Fitting

C₄₃×#384

C₂

socle

C₂³×C₄₃

C₂²⋊C₄

D₄₃

C₄₃

G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₄₃⋊C₈)	C₄	semidirect
55. C₄₃	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₄₃⋊C₈)

C₄

semidirect

55. C₄₃

#1296

semidirect

Subgroups — #19132478

Conjugacy classes of subgroups

Complements