Group #26020007 — subgroups

Subgroups — #26020007

Order 11392.

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

normal subgroups only

This group has 200 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	#19151553	C₂	1	2	yes	maximal
2	#19151356	C₂	1	1	yes	maximal, characteristic
3	#19151355	C₂	1	1	yes	maximal, characteristic
4	#19151354	C₂	1	2	yes	maximal
5	#405×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13083558	C₂²	1	2	yes	—
7	#15667971	C₂²	1	2	yes	—
8	#15667922	C₄	1	4	yes	complemented
9	C₈×Dic₈₉	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	#117×C₈₉	C₂²	1	1	yes	characteristic
13	#97×C₈₉	C₄	1	2	yes	complemented
14	C₄×C₈×C₈₉	C₄	1	2	yes	complemented
15	#13083558	—	2	2	no	—
16	#13083558	—	2	2	no	—
17	C₂×#10623494	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₈₉	—	2	2	no	—
19	C₂²⋊Dic₈₉	—	2	2	no	—
20	C₄×Dic₈₉	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
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	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂²⋊C₄×C₈₉	—	2	1	no	—
30	C₄²×C₈₉	C₂×C₄	1	2	yes	—
31	C₂³×C₈₉	#41	1	1	yes	characteristic
32	C₂×C₄×C₈₉	—	2	2	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	1	yes	characteristic
36	C₂×C₄×C₈₉	#41	1	2	yes	—
37	C₂×Dic₈₉	—	2	4	no	—
38	C₈×C₈₉	—	2	2	no	—
39	C₈×C₈₉	C₄²	1	2	yes	—
40	C₈×C₈₉	C₄²	1	1	yes	characteristic
41	C₈×C₈₉	C₄²	1	1	yes	characteristic
42	#10623494	—	2	4	no	—
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	2	no	—
47	C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₈₉	#117	1	2	yes	—
49	C₄×C₈₉	#117	1	1	yes	characteristic
50	Dic₈₉	—	4	4	no	—
51	C₂×C₈₉	—	2	2	no	—
52	C₂×C₈₉	—	2	1	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	2	no	—
57	#405	D₈₉	1	1	yes	characteristic
58	#338	—	89	1	no	—
59	H(3,ℤ/4)	—	89	1	no	—
60	H(3,ℤ/4)	—	89	2	no	—
61	C₂×M₁₆	—	89	2	no	—
62	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
64	#117	—	89	2	no	—
65	#117	C₂×D₈₉	1	1	yes	characteristic
66	#97	—	89	4	no	—
67	#97	Dic₈₉	1	2	yes	complemented
68	C₄×C₈	—	89	4	no	—
69	C₄×C₈	Dic₈₉	1	2	yes	complemented
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	4	no	—
75	C₂×C₈	C₄×D₈₉	1	2	yes	—
76	C₂×C₈	C₄×D₈₉	1	2	yes	—
77	C₂×C₈	C₂×Dic₈₉	1	2	yes	—
78	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₈₉	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	178	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	178	2	no	—
84	C₄²	—	89	4	no	—
85	C₄²	C₂×Dic₈₉	1	2	yes	—
86	C₂³	#13083574	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	178	4	no	—
89	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
90	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
91	C₂×C₄	#13083574	1	1	yes	characteristic
92	C₂×C₄	#13083574	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	178	4	no	—
95	C₈	C₄×Dic₈₉	1	1	yes	characteristic
96	C₈	C₄×Dic₈₉	1	2	yes	—
97	C₈	C₄×Dic₈₉	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#15667969	1	2	yes	—
100	C₂²	#15667971	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	356	4	no	—
103	C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
104	C₄	#15667969	1	2	yes	—
105	C₄	#15667971	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#19151444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19151553

C₂

yes

maximal

#19151356

C₂

yes

maximal, characteristic

#19151355

C₂

yes

maximal, characteristic

#19151354

C₂

yes

maximal

#405×C₈₉

C₂

yes

maximal, characteristic

C₂×#13083558

C₂²

yes

—

#15667971

C₂²

yes

—

#15667922

C₄

yes

complemented

C₈×Dic₈₉

C₄

yes

complemented

C₂×M₁₆×C₈₉

C₂²

yes

characteristic

C₂²×C₈×C₈₉

C₂²

yes

characteristic

#117×C₈₉

C₂²

yes

characteristic

#97×C₈₉

C₄

yes

complemented

C₄×C₈×C₈₉

C₄

yes

—

—

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

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

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

C₂×C₄×C₈₉

yes

C₂×C₄×C₈₉

yes

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₄²

yes

—

C₈×C₈₉

C₄²

yes

characteristic

C₈×C₈₉

C₄²

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

#117

yes

—

C₂²×C₈₉

#117

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×C₄²

yes

characteristic

C₄×C₈₉

#117

yes

—

C₄×C₈₉

#117

yes

characteristic

Dic₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

yes

—

C₈₉

yes

characteristic, complemented, Sylow

#405

—

#405

D₈₉

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₉

yes

characteristic

C₂²×C₈

C₂×D₈₉

yes

characteristic

#117

—

#117

C₂×D₈₉

yes

characteristic

#97

—

#97

Dic₈₉

yes

complemented

C₄×C₈

—

C₄×C₈

Dic₈₉

yes

complemented

C₂²×C₄

C₂²×D₈₉

yes

characteristic

M₁₆

—

M₁₆

—

178

—

M₁₆

—

178

—

C₂×C₈

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₄×D₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

—

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₂×C₈

C₂×Dic₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

178

—

C₂²⋊C₄

—

C₂²⋊C₄

—

178

—

C₄²

—

C₄²

C₂×Dic₈₉

yes

—

C₂³

#13083574

yes

characteristic

C₂×C₄

—

C₂×C₄

—

178

—

C₂×C₄

C₂×C₄×D₈₉

yes

—

C₂×C₄

C₂²×Dic₈₉

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

178

—

C₈

C₄×Dic₈₉

yes

characteristic

C₈

C₄×Dic₈₉

yes

—

C₈

C₄×Dic₈₉

yes

characteristic

C₂²

—

C₂²

#15667969

yes

—

100

C₂²

#15667971

yes

characteristic

101

C₄

—

102

C₄

—

356

—

103

C₄

C₂×C₄×Dic₈₉

yes

characteristic

104

C₄

#15667969

yes

—

105

C₄

#15667971

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#19151444

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₈₉
commutator	C₄×C₈₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₉
Fitting	#405×C₈₉	C₂
socle	C₂×C₈₉	#376
prime radical	C₂³×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂³
hypercenter	#405	D₈₉
hypocenter	C₈₉	#1747
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₈₉

commutator

C₄×C₈₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₉

Fitting

#405×C₈₉

C₂

C₂×C₈₉

C₂³×C₈₉

C₂²×C₄×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₄×C₈₉ ▷ C₈₉

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #15667922	C₄	semidirect
9. C₈×Dic₈₉	C₄	semidirect
13. #97×C₈₉	C₄	semidirect
14. C₄×C₈×C₈₉	C₄	semidirect
55. C₈₉	#1747	semidirect
67. #97	Dic₈₉	semidirect
69. C₄×C₈	Dic₈₉	semidirect

H ≅ G/N

type

8. #15667922

C₄

semidirect

9. C₈×Dic₈₉

C₄

semidirect

13. #97×C₈₉

C₄

semidirect

14. C₄×C₈×C₈₉

C₄

semidirect

55. C₈₉

#1747

semidirect

67. #97

Dic₈₉

semidirect

69. C₄×C₈

Dic₈₉

semidirect

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

subgroup	index
#19151553	2
#19151356	2
#19151355	2
#19151354	2
#405×C₈₉	2
#1747	89

subgroup

index

#405×C₈₉

Subgroups — #26020007

Conjugacy classes of subgroups

Complements

Maximal subgroups