Group #26029358 — subgroups

Subgroups — #26029358

Order 11392.

314 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 197 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	#19151729	C₂	1	8	yes	maximal, complemented
2	#19151659	C₂	1	4	yes	maximal
3	#19151655	C₂	1	2	yes	maximal
4	#533×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#10623494	C₂²	1	4	yes	—
6	C₄⋊#10623494	C₂²	1	8	yes	—
7	C₂²⋊#10623494	C₂²	1	4	yes	—
8	#15667923	C₂²	1	4	yes	—
9	#15667923	C₂²	1	4	yes	—
10	C₄×#10623494	C₂²	1	4	yes	—
11	C₄⋊Q₈×C₈₉	C₄	1	2	yes	—
12	#125×C₈₉	C₄	1	2	yes	—
13	C₂²⋊Q₈×C₈₉	C₄	1	4	yes	—
14	C₄×Q₈×C₈₉	C₂²	1	4	yes	—
15	#117×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#10623494	C₂³	1	8	yes	—
18	C₂×Q₈×C₈₉	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
22	C₄⋊C₄×C₈₉	C₂×C₄	1	8	yes	—
23	C₄⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
24	C₂²⋊C₄×C₈₉	C₂×C₄	1	4	yes	—
25	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₄²×C₈₉	C₂³	1	2	yes	—
28	C₄²×C₈₉	C₂³	1	4	yes	—
29	C₂³×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	Q₈×C₈₉	—	4	4	no	—
31	C₂×C₄×C₈₉	—	2	2	no	—
32	C₂×C₄×C₈₉	—	4	2	no	—
33	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₈₉	C₂²×C₄	1	8	yes	—
37	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
38	#10623494	—	2	8	no	—
39	C₂²×C₈₉	—	2	1	no	—
40	C₂²×C₈₉	—	2	2	no	—
41	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₈₉	—	2	1	no	—
43	C₄×C₈₉	—	2	2	no	—
44	C₄×C₈₉	—	2	2	no	—
45	C₄×C₈₉	—	2	8	no	—
46	C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
47	C₂×C₈₉	—	4	1	no	—
48	C₂×C₈₉	#519	1	1	yes	characteristic
49	C₂×C₈₉	#568	1	2	yes	—
50	#2987	—	89	1	no	maximal, Sylow
51	C₈₉	#2987	1	1	yes	characteristic, complemented, Sylow
52	#533	D₈₉	1	1	yes	characteristic
53	#446	—	89	8	no	—
54	#433	—	89	4	no	—
55	#432	—	89	2	no	—
56	C₄⋊Q₈	Dic₈₉	1	2	yes	—
57	#125	Dic₈₉	1	2	yes	—
58	C₂²⋊Q₈	Dic₈₉	1	4	yes	—
59	C₄×Q₈	C₂×D₈₉	1	4	yes	—
60	#117	C₂×D₈₉	1	2	yes	—
61	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
62	C₄⋊C₈	—	89	4	no	—
63	C₄⋊C₈	—	89	8	no	—
64	C₂²⋊C₈	—	89	4	no	—
65	#97	—	89	4	no	—
66	#97	—	89	4	no	—
67	C₄×C₈	—	89	4	no	—
68	C₂×Q₈	C₂×Dic₈₉	1	4	yes	—
69	C₂²×C₄	C₂×Dic₈₉	1	2	yes	—
70	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
71	C₂×C₈	—	89	8	no	—
72	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
73	C₄⋊C₄	C₂×Dic₈₉	1	8	yes	—
74	C₄⋊C₄	C₂×Dic₈₉	1	4	yes	—
75	C₂²⋊C₄	C₂×Dic₈₉	1	4	yes	—
76	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
77	C₄²	C₂×Dic₈₉	1	1	yes	characteristic
78	C₄²	C₂²×D₈₉	1	4	yes	—
79	C₄²	C₂²×D₈₉	1	2	yes	—
80	C₂³	C₂²×Dic₈₉	1	1	yes	characteristic
81	Q₈	—	4	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
86	C₂×C₄	C₂²×Dic₈₉	1	2	yes	—
87	C₂×C₄	C₂²×Dic₈₉	1	8	yes	—
88	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
89	C₈	—	178	8	no	—
90	C₂²	—	2	1	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₂³×Dic₈₉	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	8	no	—
97	C₄	#15668060	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	#19152076	1	1	yes	characteristic
100	C₂	#19152365	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19151729

C₂

yes

maximal, complemented

#19151659

C₂

yes

maximal

#19151655

C₂

yes

maximal

#533×C₈₉

C₂

yes

maximal, characteristic

C₄⋊#10623494

C₂²

yes

—

C₄⋊#10623494

C₂²

yes

—

C₂²⋊#10623494

C₂²

yes

—

#15667923

C₂²

yes

—

#15667923

C₂²

yes

—

C₄×#10623494

C₂²

yes

—

C₄⋊Q₈×C₈₉

C₄

yes

—

#125×C₈₉

C₄

yes

—

C₂²⋊Q₈×C₈₉

C₄

yes

—

C₄×Q₈×C₈₉

C₂²

yes

—

#117×C₈₉

C₂²

yes

—

C₂×C₄²×C₈₉

C₂²

yes

characteristic

C₂×#10623494

C₂³

yes

—

C₂×Q₈×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂²×C₄×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₄⋊C₄×C₈₉

C₂×C₄

yes

—

C₂²⋊C₄×C₈₉

C₂×C₄

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₄²×C₈₉

C₂³

yes

—

C₄²×C₈₉

C₂³

yes

—

C₂³×C₈₉

C₂²×C₄

yes

characteristic

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂²×C₄

yes

—

C₂×C₄×C₈₉

C₂⁴

yes

characteristic

#10623494

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂³×C₄

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

2₋¹⁺⁴

yes

—

C₂×C₈₉

—

C₂×C₈₉

#519

yes

characteristic

C₂×C₈₉

#568

yes

—

#2987

—

maximal, Sylow

C₈₉

#2987

yes

characteristic, complemented, Sylow

#533

D₈₉

yes

—

—

—

C₄⋊Q₈

Dic₈₉

yes

—

#125

Dic₈₉

yes

—

C₂²⋊Q₈

Dic₈₉

yes

—

C₄×Q₈

C₂×D₈₉

yes

—

#117

C₂×D₈₉

yes

—

C₂×C₄²

C₂×D₈₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂×Dic₈₉

yes

—

C₂²×C₄

C₂²×D₈₉

yes

characteristic

C₂×C₈

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₄⋊C₄

C₂×Dic₈₉

yes

—

C₂²⋊C₄

C₂×Dic₈₉

yes

—

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂×Dic₈₉

yes

characteristic

C₄²

C₂²×D₈₉

yes

—

C₄²

C₂²×D₈₉

yes

—

C₂³

C₂²×Dic₈₉

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×Dic₈₉

yes

characteristic

C₂×C₄

C₂²×Dic₈₉

yes

—

C₂×C₄

C₂²×Dic₈₉

yes

—

C₂×C₄

C₂²×Dic₈₉

yes

—

C₂×C₄

C₂³×D₈₉

yes

characteristic

C₈

—

178

—

C₂²

—

C₂²

—

C₂²

C₂³×Dic₈₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#15668060

yes

—

C₂

—

C₂

#19152076

yes

characteristic

100

C₂

#19152365

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#533×C₈₉	C₂
prime radical	C₂³×C₈₉	C₂²×C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#533	D₈₉
hypocenter	C₈₉	#2987
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #533

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₈₉

commutator, socle

C₂²×C₈₉

C₂³×C₄

Fitting

#533×C₈₉

C₂

prime radical

C₂³×C₈₉

C₂²×C₄

cosocle

C₂×C₄×C₈₉

C₂⁴

D₈₉

C₈₉

G ▷ C₂²×C₈₉ ▷ C₈₉

C₁ ◁ C₂×C₄ ◁ #533

N	H ≅ G/N	type
1. #19151729	C₂	semidirect
51. C₈₉	#2987	semidirect

H ≅ G/N

type

1. #19151729

C₂

semidirect

51. C₈₉

#2987

semidirect

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

subgroup	index
#19151729	2
#19151659	2
#19151655	2
#533×C₈₉	2
#2987	89

subgroup

index

#533×C₈₉

Subgroups — #26029358

Conjugacy classes of subgroups

Complements

Maximal subgroups