Group #36442286 — subgroups

Subgroups — #36442286

Order 22784.

186 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 27 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	#26018415	C₂	1	2	yes	maximal, complemented
2	C₈₉×#1712	C₂	1	1	yes	maximal, characteristic
3	#19151391	—	2	2	no	—
4	#493×C₈₉	—	2	2	no	—
5	#493×C₈₉	C₄	1	1	yes	characteristic, complemented
6	#493×C₈₉	C₂²	1	1	yes	characteristic
7	C₈²×C₈₉	C₄	1	1	yes	characteristic, complemented
8	#15667927	—	4	2	no	—
9	C₂×D₈×C₈₉	—	4	1	no	—
10	C₂×D₈×C₈₉	—	4	1	no	—
11	C₂×D₈×C₈₉	—	4	4	no	—
12	#127×C₈₉	—	2	1	no	—
13	#127×C₈₉	D₄	1	2	yes	—
14	C₄×C₈×C₈₉	—	2	1	no	—
15	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
16	C₂²⋊Dic₈₉	—	8	2	no	—
17	C₂×D₄×C₈₉	—	2	2	no	—
18	C₂×D₄×C₈₉	—	4	2	no	—
19	C₂×D₄×C₈₉	—	4	2	no	—
20	C₂×D₄×C₈₉	—	4	1	no	—
21	D₈×C₈₉	—	8	2	no	—
22	D₈×C₈₉	—	8	2	no	—
23	D₈×C₈₉	—	8	8	no	—
24	C₂×C₈×C₈₉	—	2	2	no	—
25	C₂×C₈×C₈₉	—	2	1	no	—
26	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂³×C₈₉	—	4	2	no	—
28	C₂³×C₈₉	—	8	1	no	—
29	D₄×C₈₉	—	8	2	no	—
30	D₄×C₈₉	—	8	4	no	—
31	D₄×C₈₉	—	8	4	no	—
32	D₄×C₈₉	—	8	2	no	—
33	C₂×C₄×C₈₉	—	2	1	no	—
34	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
35	C₂×Dic₈₉	—	16	2	no	—
36	C₈×C₈₉	—	2	4	no	—
37	C₈×C₈₉	—	2	2	no	—
38	C₂²×C₈₉	—	8	2	no	—
39	C₂²×C₈₉	—	16	2	no	—
40	C₂²×C₈₉	—	16	2	no	—
41	C₂²×C₈₉	—	16	1	no	—
42	C₂²×C₈₉	#353	1	1	yes	characteristic
43	C₄×C₈₉	—	2	2	no	—
44	C₄×C₈₉	—	2	1	no	—
45	Dic₈₉	—	32	2	no	—
46	#7529	—	89	1	no	maximal, Sylow
47	C₂×C₈₉	—	2	1	no	—
48	C₂×C₈₉	—	16	2	no	—
49	C₂×C₈₉	—	32	1	no	—
50	C₂×C₈₉	#1408	1	1	yes	characteristic
51	#1712	D₈₉	1	1	yes	characteristic
52	#1408	—	89	2	no	—
53	C₈₉	#7529	1	1	yes	characteristic, complemented, Sylow
54	#493	—	2	2	no	—
55	#493	Dic₈₉	1	1	yes	characteristic, complemented
56	#493	C₂×D₈₉	1	1	yes	characteristic
57	#353	—	178	2	no	—
58	C₈²	Dic₈₉	1	1	yes	characteristic, complemented
59	C₂×D₈	—	4	1	no	—
60	C₂×D₈	—	4	1	no	—
61	C₂×D₈	—	4	4	no	—
62	#127	—	2	1	no	—
63	#127	C₂²⋊D₈₉	1	2	yes	—
64	#99	—	356	2	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
67	C₂×D₄	—	2	2	no	—
68	C₂×D₄	—	4	2	no	—
69	C₂×D₄	—	4	2	no	—
70	C₂×D₄	—	4	1	no	—
71	D₈	—	8	2	no	—
72	D₈	—	8	2	no	—
73	D₈	—	8	8	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂²⋊C₄	—	712	2	no	—
77	C₄²	C₂²⋊Dic₈₉	1	1	yes	characteristic
78	C₂³	—	4	2	no	—
79	C₂³	—	8	1	no	—
80	D₄	—	8	2	no	—
81	D₄	—	8	4	no	—
82	D₄	—	8	4	no	—
83	D₄	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	1424	2	no	—
86	C₂×C₄	#15667927	1	1	yes	characteristic
87	C₈	—	2	4	no	—
88	C₈	—	2	2	no	—
89	C₂²	—	8	2	no	—
90	C₂²	—	16	2	no	—
91	C₂²	—	16	2	no	—
92	C₂²	—	16	1	no	—
93	C₂²	#19151391	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	1	no	—
96	C₄	—	2848	2	no	—
97	C₂	—	2	1	no	—
98	C₂	—	16	2	no	—
99	C₂	—	32	1	no	—
100	C₂	#26018415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26018415

C₂

yes

maximal, complemented

C₈₉×#1712

C₂

yes

maximal, characteristic

#19151391

—

#493×C₈₉

—

#493×C₈₉

C₄

yes

characteristic, complemented

#493×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₄

yes

characteristic, complemented

#15667927

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

C₂×D₈×C₈₉

—

#127×C₈₉

—

#127×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂²⋊Dic₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

C₂×D₄×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

C₂²⋊C₄

yes

characteristic

C₂³×C₈₉

—

C₂³×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#99

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

#353

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

Dic₈₉

—

#7529

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

#1408

yes

characteristic

#1712

D₈₉

yes

characteristic

#1408

—

C₈₉

#7529

yes

characteristic, complemented, Sylow

#493

—

#493

Dic₈₉

yes

characteristic, complemented

#493

C₂×D₈₉

yes

characteristic

#353

—

178

—

C₈²

Dic₈₉

yes

characteristic, complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

C₂²⋊D₈₉

yes

—

#99

—

356

—

C₄×C₈

—

C₄×C₈

C₂×Dic₈₉

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂²⋊C₄

—

712

—

C₄²

C₂²⋊Dic₈₉

yes

characteristic

C₂³

—

C₂³

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

1424

—

C₂×C₄

#15667927

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#19151391

yes

characteristic

C₄

—

C₄

—

C₄

—

2848

—

C₂

—

C₂

—

C₂

—

100

C₂

#26018415

yes

characteristic

	subgroup	G/H
center	C₂	#26018415
commutator	C₄×C₈×C₈₉	C₂×C₄
Frattini	#493	C₂×D₈₉
Fitting, prime radical	C₈₉×#1712	C₂
socle	C₂×C₈₉	#1408
cosocle	#493×C₈₉	C₂²
hypercenter	#1712	D₈₉
hypocenter	C₈₉	#7529
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

subgroup

G/H

center

C₂

#26018415

commutator

C₄×C₈×C₈₉

C₂×C₄

Frattini

#493

C₂×D₈₉

Fitting, prime radical

C₈₉×#1712

C₂

socle

C₂×C₈₉

#1408

cosocle

#493×C₈₉

C₂²

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₄²×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ #1712

N	H ≅ G/N	type
1. #26018415	C₂	semidirect
5. #493×C₈₉	C₄	semidirect
7. C₈²×C₈₉	C₄	semidirect
53. C₈₉	#7529	semidirect
55. #493	Dic₈₉	semidirect
58. C₈²	Dic₈₉	semidirect

H ≅ G/N

type

1. #26018415

C₂

semidirect

5. #493×C₈₉

C₄

semidirect

7. C₈²×C₈₉

C₄

semidirect

53. C₈₉

#7529

semidirect

55. #493

Dic₈₉

semidirect

58. C₈²

Dic₈₉

semidirect

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

subgroup	index
#26018415	2
C₈₉×#1712	2
#7529	89

subgroup

index

#26018415

C₈₉×#1712

#7529

Subgroups — #36442286

Conjugacy classes of subgroups

Complements

Maximal subgroups