Group #26020006 — subgroups

Subgroups — #26020006

Order 11392.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#19151556	C₂	1	1	yes	maximal, characteristic
2	#19151554	C₂	1	2	yes	maximal
3	#19151356	C₂	1	1	yes	maximal, characteristic
4	#19151354	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	C₂²×#10623494	C₂²	1	1	yes	characteristic
8	#15667971	C₂²	1	1	yes	characteristic
9	#15667923	C₄	1	4	yes	complemented
10	#15667922	C₄	1	2	yes	complemented
11	C₄×#10623494	C₄	1	4	yes	complemented
12	C₈×Dic₈₉	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	#117×C₈₉	C₂²	1	2	yes	—
15	#13083558	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	C₂×#10623494	C₂×C₄	1	4	yes	—
18	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
19	C₂×#10623494	C₂×C₄	1	2	yes	—
20	C₂×#10623494	C₂×C₄	1	1	yes	characteristic
21	C₂×#10623494	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₈₉	—	2	1	no	—
23	C₂²⋊Dic₈₉	—	2	1	no	—
24	C₄×Dic₈₉	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₈₉	—	2	2	no	—
27	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₈₉	—	2	2	no	—
29	C₂²⋊C₄×C₈₉	—	2	2	no	—
30	C₄²×C₈₉	C₂×C₄	1	4	yes	—
31	C₂³×C₈₉	#41	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	4	no	—
33	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
34	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₈₉	#41	1	2	yes	—
36	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
37	C₂×Dic₈₉	—	2	2	no	—
38	C₈×C₈₉	—	2	2	no	—
39	#10623494	—	2	4	no	—
40	#10623494	C₄²	1	1	yes	characteristic
41	#10623494	C₄²	1	1	yes	characteristic
42	#10623494	C₄²	1	2	yes	—
43	C₂²×C₈₉	—	4	1	no	—
44	C₂²×C₈₉	#117	1	2	yes	—
45	C₂²×C₈₉	#117	1	1	yes	characteristic
46	C₄×C₈₉	—	4	4	no	—
47	C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₉	#117	1	1	yes	characteristic
49	C₄×C₈₉	#117	1	2	yes	—
50	Dic₈₉	—	4	2	no	—
51	C₂×C₈₉	—	2	1	no	—
52	C₂×C₈₉	—	2	2	no	—
53	C₂×C₈₉	#376	1	1	yes	characteristic
54	#1747	—	89	1	no	maximal, Sylow
55	C₈₉	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	89	1	no	—
57	#405	—	89	2	no	—
58	#338	—	89	1	no	—
59	H(3,ℤ/4)	—	89	2	no	—
60	H(3,ℤ/4)	D₈₉	1	1	yes	characteristic
61	C₂×M₁₆	—	89	2	no	—
62	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
63	C₂²×C₈	—	89	1	no	—
64	#117	—	89	1	no	—
65	#117	C₂×D₈₉	1	2	yes	—
66	#97	—	89	2	no	—
67	#97	—	89	4	no	—
68	C₄×C₈	—	89	2	no	—
69	C₄×C₈	—	89	4	no	—
70	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	178	2	no	—
73	M₁₆	—	178	2	no	—
74	C₂×C₈	—	89	1	no	—
75	C₂×C₈	—	89	2	no	—
76	C₂×C₈	—	89	1	no	—
77	C₂×C₈	—	89	2	no	—
78	C₂×C₈	—	89	4	no	—
79	C₂×C₈	C₄×D₈₉	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	178	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	178	1	no	—
84	C₄²	—	89	2	no	—
85	C₄²	C₄×D₈₉	1	4	yes	—
86	C₂³	#13083573	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	178	2	no	—
89	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
91	C₂×C₄	#13083573	1	2	yes	—
92	C₂×C₄	#13083575	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	89	1	no	—
95	C₈	—	89	1	no	—
96	C₈	—	89	2	no	—
97	C₈	—	178	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#15667968	1	2	yes	—
100	C₂²	#15667970	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	356	2	no	—
103	C₄	C₄²×D₈₉	1	1	yes	characteristic
104	C₄	#15667968	1	1	yes	characteristic
105	C₄	#15667970	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#19151443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151556

C₂

yes

maximal, characteristic

#19151554

C₂

yes

maximal

#19151356

C₂

yes

maximal, characteristic

#19151354

C₂

yes

maximal

H(3,ℤ/4)×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

C₂²×#10623494

C₂²

yes

characteristic

#15667971

C₂²

yes

characteristic

#15667923

C₄

yes

complemented

#15667922

C₄

yes

complemented

C₄×#10623494

C₄

yes

complemented

C₈×Dic₈₉

C₄

yes

complemented

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

—

#13083558

—

#13083558

—

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

—

C₂×#10623494

C₂×C₄

yes

characteristic

C₂×#10623494

C₂×C₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

M₁₆×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

C₂×C₄

yes

—

C₂³×C₈₉

#41

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

—

C₂×C₄×C₈₉

#41

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

#10623494

—

#10623494

C₄²

yes

characteristic

#10623494

C₄²

yes

characteristic

#10623494

C₄²

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

#117

yes

—

C₂²×C₈₉

#117

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄²

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

—

—

—

—

D₈₉

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₈₉

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

C₄²

C₄×D₈₉

yes

—

C₂³

#13083573

yes

characteristic

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

characteristic

C₂×C₄

C₂×C₄×D₈₉

yes

—

C₂×C₄

#13083573

yes

—

C₂×C₄

#13083575

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

178

—

C₂²

—

C₂²

#15667968

yes

—

100

C₂²

#15667970

yes

characteristic

101

C₄

—

102

C₄

—

356

—

103

C₄

C₄²×D₈₉

yes

characteristic

104

C₄

#15667968

yes

characteristic

105

C₄

#15667970

yes

—

106

C₂

—

107

C₂

—

108

C₂

#19151443

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₈₉
commutator	C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	H(3,ℤ/4)×C₈₉	C₂
socle	C₂×C₈₉	#376
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	H(3,ℤ/4)	D₈₉
hypocenter	C₈₉	#1747
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₈₉

commutator

C₄×C₈₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

H(3,ℤ/4)×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₈₉

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #15667923	C₄	semidirect
10. #15667922	C₄	semidirect
11. C₄×#10623494	C₄	semidirect
12. C₈×Dic₈₉	C₄	semidirect
55. C₈₉	#1747	semidirect

H ≅ G/N

type

9. #15667923

C₄

semidirect

10. #15667922

C₄

semidirect

11. C₄×#10623494

C₄

semidirect

12. C₈×Dic₈₉

C₄

semidirect

55. C₈₉

#1747

semidirect

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

subgroup	index
#19151556	2
#19151554	2
#19151356	2
#19151354	2
H(3,ℤ/4)×C₈₉	2
#1747	89

subgroup

index

H(3,ℤ/4)×C₈₉

Subgroups — #26020006

Conjugacy classes of subgroups

Complements

Maximal subgroups