Group #26020402 — subgroups

Subgroups — #26020402

Order 11392.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#19151370	C₂	1	2	yes	maximal
3	#19151359	C₂	1	2	yes	maximal
4	#19151355	C₂	1	1	yes	maximal, characteristic
5	#338×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	1	yes	characteristic
7	#15667971	C₂²	1	2	yes	—
8	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
9	#15667922	C₄	1	2	yes	complemented
10	C₈×Dic₈₉	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₈₉	C₂²	1	2	yes	—
12	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₉	—	2	1	no	—
14	#13083558	—	2	2	no	—
15	C₂×#10623494	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₈₉	—	2	2	no	—
17	C₂²⋊Dic₈₉	—	2	2	no	—
18	C₄×Dic₈₉	—	2	1	no	—
19	C₄×Dic₈₉	—	2	2	no	—
20	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
23	M₁₆×C₈₉	—	2	2	no	—
24	M₁₆×C₈₉	—	2	2	no	—
25	M₁₆×C₈₉	—	2	2	no	—
26	C₂×C₈×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₂³×C₈₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	#41	1	2	yes	—
31	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
32	C₂×C₄×C₈₉	#41	1	2	yes	—
33	C₂×Dic₈₉	—	2	1	no	—
34	C₂×Dic₈₉	—	2	1	no	—
35	C₂×Dic₈₉	—	4	2	no	—
36	C₂×Dic₈₉	—	4	2	no	—
37	C₈×C₈₉	—	2	2	no	—
38	C₈×C₈₉	—	4	2	no	—
39	#10623494	—	2	2	no	—
40	C₂²×C₈₉	—	4	1	no	—
41	C₂²×C₈₉	#117	1	1	yes	characteristic
42	C₂²×C₈₉	#126	1	2	yes	—
43	C₄×C₈₉	#117	1	1	yes	characteristic
44	C₄×C₈₉	#117	1	1	yes	characteristic
45	C₄×C₈₉	#126	1	2	yes	—
46	Dic₈₉	—	2	2	no	—
47	Dic₈₉	—	4	1	no	—
48	Dic₈₉	—	8	2	no	—
49	C₂×C₈₉	—	2	1	no	—
50	C₂×C₈₉	—	2	2	no	—
51	C₂×C₈₉	#383	1	1	yes	characteristic
52	#1841	—	89	1	no	maximal, Sylow
53	C₈₉	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	89	1	no	—
55	#344	—	89	2	no	—
56	#339	—	89	2	no	—
57	#338	D₈₉	1	1	yes	characteristic
58	H(3,ℤ/4)	—	89	1	no	—
59	C₂×M₁₆	—	89	1	no	—
60	C₂×M₁₆	C₂×D₈₉	1	2	yes	—
61	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
62	#117	—	89	2	no	—
63	C₂×C₄²	—	89	1	no	—
64	#97	—	89	2	no	—
65	C₄×C₈	—	89	2	no	—
66	C₂²×C₄	—	178	1	no	—
67	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	178	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	89	2	no	—
74	C₂×C₈	C₄×D₈₉	1	2	yes	—
75	C₄⋊C₄	—	178	2	no	—
76	C₂²⋊C₄	—	178	2	no	—
77	C₄²	—	89	1	no	—
78	C₄²	—	89	1	no	—
79	C₄²	—	178	2	no	—
80	C₄²	—	178	1	no	—
81	C₂³	#13083575	1	1	yes	characteristic
82	C₂×C₄	—	356	2	no	—
83	C₂×C₄	—	356	2	no	—
84	C₂×C₄	—	178	1	no	—
85	C₂×C₄	—	178	1	no	—
86	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
87	C₂×C₄	#13083573	1	2	yes	—
88	C₂×C₄	#13083573	1	1	yes	characteristic
89	C₂×C₄	#13083574	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	178	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#15667970	1	1	yes	characteristic
95	C₂²	#15668007	1	2	yes	—
96	C₄	—	356	1	no	—
97	C₄	—	178	2	no	—
98	C₄	—	712	2	no	—
99	C₄	#15667968	1	1	yes	characteristic
100	C₄	#15667969	1	1	yes	characteristic
101	C₄	#15668005	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#19151465	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₈₉

C₂

yes

maximal, characteristic

#19151370

C₂

yes

maximal

#19151359

C₂

yes

maximal

#19151355

C₂

yes

maximal, characteristic

#338×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

characteristic

#15667971

C₂²

yes

—

C₂×C₄×Dic₈₉

C₂²

yes

characteristic

#15667922

C₄

yes

complemented

C₈×Dic₈₉

C₄

yes

complemented

C₂×M₁₆×C₈₉

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×Dic₈₉

—

#13083558

—

C₂×#10623494

C₂×C₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₄×Dic₈₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#117

yes

characteristic

C₂²×C₈₉

#126

yes

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

Dic₈₉

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

#344

—

#339

—

#338

D₈₉

yes

characteristic

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

178

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₉

yes

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

C₄²

—

C₄²

—

178

—

C₄²

—

178

—

C₂³

#13083575

yes

characteristic

C₂×C₄

—

356

—

C₂×C₄

—

356

—

C₂×C₄

—

178

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

#13083573

yes

—

C₂×C₄

#13083573

yes

characteristic

C₂×C₄

#13083574

yes

—

C₈

—

C₈

—

C₈

—

178

—

C₂²

—

C₂²

#15667970

yes

characteristic

C₂²

#15668007

yes

—

C₄

—

356

—

C₄

—

178

—

C₄

—

712

—

C₄

#15667968

yes

characteristic

100

C₄

#15667969

yes

characteristic

101

C₄

#15668005

yes

—

102

C₂

—

103

C₂

—

104

C₂

#19151465

yes

characteristic

	subgroup	G/H
center	C₄	#15667969
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#338×C₈₉	C₂
socle	C₂×C₈₉	#383
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#338	D₈₉
hypocenter	C₈₉	#1841
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#15667969

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#338×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #15667922	C₄	semidirect
10. C₈×Dic₈₉	C₄	semidirect
53. C₈₉	#1841	semidirect

H ≅ G/N

type

9. #15667922

C₄

semidirect

10. C₈×Dic₈₉

C₄

semidirect

53. C₈₉

#1841

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
M₁₆×Dic₈₉	2
#19151370	2
#19151359	2
#19151355	2
#338×C₈₉	2
#1841	89

subgroup

index

M₁₆×Dic₈₉

#338×C₈₉

Subgroups — #26020402

Conjugacy classes of subgroups

Complements

Maximal subgroups