Group #26020401 — subgroups

Subgroups — #26020401

Order 11392.

162 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 73 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	#19151369	C₂	1	2	yes	maximal
2	#19151358	C₂	1	2	yes	maximal
3	#19151356	C₂	1	1	yes	maximal, characteristic
4	#19151355	C₂	1	1	yes	maximal, characteristic
5	C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	#15667971	C₂²	1	2	yes	—
8	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	#97×C₈₉	C₄	1	2	yes	complemented
12	C₄×C₈×C₈₉	C₄	1	2	yes	complemented
13	#13083558	—	2	2	no	—
14	#13083558	—	2	2	no	—
15	C₂×#10623494	—	2	2	no	—
16	C₄⋊Dic₈₉	—	2	2	no	—
17	C₂²⋊Dic₈₉	—	2	2	no	—
18	C₄×Dic₈₉	—	2	2	no	—
19	C₂²×C₄×C₈₉	—	2	1	no	—
20	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	M₁₆×C₈₉	—	2	2	no	—
22	M₁₆×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
25	C₄²×C₈₉	—	2	1	no	—
26	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₈₉	#41	1	1	yes	characteristic
29	C₂×C₄×C₈₉	—	2	1	no	—
30	C₂×C₄×C₈₉	—	2	1	no	—
31	C₂×C₄×C₈₉	—	4	2	no	—
32	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈₉	#41	1	2	yes	—
34	C₂×C₄×C₈₉	#41	1	2	yes	—
35	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
36	C₂×Dic₈₉	—	4	2	no	—
37	C₈×C₈₉	—	2	2	no	—
38	C₈×C₈₉	—	2	2	no	—
39	#10623494	—	4	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	—	2	2	no	—
44	C₄×C₈₉	—	4	1	no	—
45	C₄×C₈₉	#117	1	1	yes	characteristic
46	C₄×C₈₉	#117	1	1	yes	characteristic
47	C₄×C₈₉	#126	1	2	yes	—
48	Dic₈₉	—	8	2	no	—
49	C₂×C₈₉	—	2	2	no	—
50	C₂×C₈₉	—	2	1	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	D₈₉	1	1	yes	characteristic
55	#344	—	89	2	no	—
56	#339	—	89	2	no	—
57	#338	—	89	1	no	—
58	H(3,ℤ/4)	—	89	1	no	—
59	C₂×M₁₆	—	89	2	no	—
60	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
61	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
62	#117	—	89	2	no	—
63	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
64	#97	Dic₈₉	1	2	yes	complemented
65	C₄×C₈	Dic₈₉	1	2	yes	complemented
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	178	2	no	—
71	M₁₆	—	178	2	no	—
72	C₂×C₈	—	178	2	no	—
73	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
74	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
75	C₄⋊C₄	—	178	2	no	—
76	C₂²⋊C₄	—	178	2	no	—
77	C₄²	—	2	1	no	—
78	C₄²	—	178	2	no	—
79	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
80	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
81	C₂³	#13083574	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	356	2	no	—
86	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
87	C₂×C₄	#13083574	1	2	yes	—
88	C₂×C₄	#13083574	1	2	yes	—
89	C₂×C₄	#13083574	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	2	no	—
92	C₈	—	356	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#15667971	1	1	yes	characteristic
95	C₂²	#15668006	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	4	1	no	—
98	C₄	—	712	2	no	—
99	C₄	#15667971	1	1	yes	characteristic
100	C₄	#15667971	1	1	yes	characteristic
101	C₄	#15668006	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#19151466	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151369

C₂

yes

maximal

#19151358

C₂

yes

maximal

#19151356

C₂

yes

maximal, characteristic

#19151355

C₂

yes

maximal, characteristic

C₄×M₁₆×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15667971

C₂²

yes

—

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₄²×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₄

yes

complemented

C₄×C₈×C₈₉

C₄

yes

—

—

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#117

yes

characteristic

C₂²×C₈₉

#126

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

C₄×M₁₆

D₈₉

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂×M₁₆

C₂×D₈₉

yes

characteristic

#117

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

#97

Dic₈₉

yes

complemented

C₄×C₈

Dic₈₉

yes

complemented

C₂²×C₄

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

178

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

178

—

C₄²

—

C₄²

—

178

—

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₂³

#13083574

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

356

—

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₄

#13083574

yes

—

C₂×C₄

#13083574

yes

—

C₂×C₄

#13083574

yes

characteristic

C₈

—

C₈

—

C₈

—

356

—

C₂²

—

C₂²

#15667971

yes

characteristic

C₂²

#15668006

yes

—

C₄

—

C₄

—

C₄

—

712

—

C₄

#15667971

yes

characteristic

100

C₄

#15667971

yes

characteristic

101

C₄

#15668006

yes

—

102

C₂

—

103

C₂

—

104

C₂

#19151466

yes

characteristic

	subgroup	G/H
center	C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	C₄×M₁₆×C₈₉	C₂
socle	C₂×C₈₉	#383
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	C₄×M₁₆	D₈₉
hypocenter	C₈₉	#1841
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

subgroup

G/H

center

C₄

#15667971

commutator

C₂×C₄×C₈₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

C₄×M₁₆×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₄×C₈₉

C₂³

hypercenter

C₄×M₁₆

D₈₉

hypocenter

C₈₉

#1841

LCS

G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₄×M₁₆

N	H ≅ G/N	type
11. #97×C₈₉	C₄	semidirect
12. C₄×C₈×C₈₉	C₄	semidirect
53. C₈₉	#1841	semidirect
64. #97	Dic₈₉	semidirect
65. C₄×C₈	Dic₈₉	semidirect

H ≅ G/N

type

11. #97×C₈₉

C₄

semidirect

12. C₄×C₈×C₈₉

C₄

semidirect

53. C₈₉

#1841

semidirect

64. #97

Dic₈₉

semidirect

65. C₄×C₈

Dic₈₉

semidirect

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

subgroup	index
#19151369	2
#19151358	2
#19151356	2
#19151355	2
C₄×M₁₆×C₈₉	2
#1841	89

subgroup

index

C₄×M₁₆×C₈₉

Subgroups — #26020401

Conjugacy classes of subgroups

Complements

Maximal subgroups