Group #36469132 — subgroups

Subgroups — #36469132

Order 22784.

290 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 83 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	#26022476	C₂	1	4	yes	maximal, complemented
2	#26019867	C₂	1	1	yes	maximal, characteristic
3	#26019862	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#19152003	C₂²	1	2	yes	—
6	#19151975	—	2	2	no	—
7	#19151975	C₂²	1	2	yes	—
8	#19151949	—	2	2	no	—
9	#19151400	—	2	8	no	—
10	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
11	C₈²×C₈₉	C₂²	1	1	yes	characteristic
12	C₂×D₇₁₂	—	2	4	no	—
13	C₂×D₇₁₂	—	4	4	no	—
14	#15668012	—	2	2	no	—
15	#15668009	—	2	2	no	—
16	#15667946	—	2	4	no	—
17	#15667937	—	4	4	no	—
18	C₂×C₁₆×C₈₉	D₄	1	4	yes	—
19	C₄×C₈×C₈₉	D₄	1	2	yes	—
20	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₇₈	—	4	2	no	—
22	C₂×D₃₅₆	—	4	2	no	—
23	C₂×D₃₅₆	—	4	4	no	—
24	D₇₁₂	—	4	8	no	—
25	D₇₁₂	—	8	8	no	—
26	C₄⋊Dic₈₉	—	4	2	no	—
27	C₄⋊Dic₈₉	—	4	2	no	—
28	C₂×C₈×C₈₉	D₈	1	4	yes	—
29	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
30	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
31	C₁₆×C₈₉	—	2	4	no	—
32	C₂²×D₈₉	—	8	2	no	—
33	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
35	C₂×Dic₈₉	—	8	2	no	—
36	D₃₅₆	—	8	4	no	—
37	D₃₅₆	—	8	8	no	—
38	Dic₁₇₈	—	8	4	no	—
39	C₈×C₈₉	—	2	4	no	—
40	C₈×C₈₉	#135	1	4	yes	—
41	C₂²×C₈₉	#493	1	1	yes	characteristic
42	C₂×D₈₉	—	16	2	no	—
43	C₂×D₈₉	—	16	4	no	—
44	C₄×C₈₉	#495	1	2	yes	—
45	C₄×C₈₉	#509	1	4	yes	—
46	Dic₈₉	—	16	2	no	—
47	#12513	—	89	1	no	maximal, Sylow
48	C₂×C₈₉	#1709	1	1	yes	characteristic
49	C₂×C₈₉	#2250	1	2	yes	—
50	D₈₉	—	32	2	no	—
51	#2240	—	89	4	no	—
52	#1712	—	89	1	no	—
53	#1710	—	89	1	no	—
54	#1311	D₈₉	1	1	yes	characteristic, complemented
55	C₈₉	#12513	1	1	yes	characteristic, complemented, Sylow
56	#500	—	89	2	no	—
57	#493	—	89	2	no	—
58	#493	—	178	2	no	—
59	#487	—	178	2	no	—
60	#357	—	178	8	no	—
61	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
62	C₈²	C₂×D₈₉	1	1	yes	characteristic
63	C₂×D₈	—	356	4	no	—
64	C₂×D₈	—	178	4	no	—
65	C₄⋊Q₈	—	178	2	no	—
66	#127	—	178	2	no	—
67	C₂×C₁₆	D₃₅₆	1	4	yes	—
68	#107	—	178	4	no	—
69	#103	—	356	4	no	—
70	C₄×C₈	D₃₅₆	1	2	yes	—
71	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
72	C₂×Q₈	—	356	2	no	—
73	C₂×D₄	—	356	2	no	—
74	C₂×D₄	—	356	4	no	—
75	D₈	—	356	8	no	—
76	D₈	—	712	8	no	—
77	C₂×C₈	D₇₁₂	1	4	yes	—
78	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
79	C₄⋊C₄	—	356	2	no	—
80	C₄⋊C₄	—	356	2	no	—
81	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
82	C₁₆	—	2	4	no	—
83	C₂³	—	712	2	no	—
84	Q₈	—	712	4	no	—
85	D₄	—	712	4	no	—
86	D₄	—	712	8	no	—
87	C₂×C₄	—	712	2	no	—
88	C₂×C₄	#15668009	1	1	yes	characteristic
89	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#15668034	1	4	yes	—
92	C₂²	—	1424	2	no	—
93	C₂²	—	1424	4	no	—
94	C₂²	#19151975	1	1	yes	characteristic
95	C₄	—	1424	2	no	—
96	C₄	#19151982	1	2	yes	—
97	C₄	#19152036	1	4	yes	—
98	C₂	—	2848	2	no	—
99	C₂	#26019858	1	1	yes	characteristic
100	C₂	#26022518	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022476

C₂

yes

maximal, complemented

#26019867

C₂

yes

maximal, characteristic

#26019862

C₂

yes

maximal, characteristic, complemented

C₈₉×#1311

C₂

yes

maximal, characteristic, complemented

#19152003

C₂²

yes

—

#19151975

—

#19151975

C₂²

yes

—

#19151949

—

#19151400

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₈²×C₈₉

C₂²

yes

characteristic

C₂×D₇₁₂

—

C₂×D₇₁₂

—

—

—

—

—

C₂×C₁₆×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

characteristic

C₂×Dic₁₇₈

—

C₂×D₃₅₆

—

C₂×D₃₅₆

—

D₇₁₂

—

D₇₁₂

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂×C₈×C₈₉

D₈

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₄²×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

—

C₂²×D₈₉

—

C₂×C₄×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×Dic₈₉

—

D₃₅₆

—

D₃₅₆

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

#135

yes

—

C₂²×C₈₉

#493

yes

characteristic

C₂×D₈₉

—

C₂×D₈₉

—

C₄×C₈₉

#495

yes

—

C₄×C₈₉

#509

yes

—

Dic₈₉

—

—

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

—

—

—

—

D₈₉

yes

characteristic, complemented

C₈₉

#12513

yes

characteristic, complemented, Sylow

#500

—

#493

—

#493

—

178

—

#487

—

178

—

#357

—

178

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₈²

C₂×D₈₉

yes

characteristic

C₂×D₈

—

356

—

C₂×D₈

—

178

—

C₄⋊Q₈

—

178

—

#127

—

178

—

C₂×C₁₆

D₃₅₆

yes

—

#107

—

178

—

#103

—

356

—

C₄×C₈

D₃₅₆

yes

—

C₄×C₈

C₂²×D₈₉

yes

characteristic

C₂×Q₈

—

356

—

C₂×D₄

—

356

—

C₂×D₄

—

356

—

D₈

—

356

—

D₈

—

712

—

C₂×C₈

D₇₁₂

yes

—

C₂×C₈

C₂×D₃₅₆

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

C₂×D₃₅₆

yes

characteristic

C₁₆

—

C₂³

—

712

—

Q₈

—

712

—

D₄

—

712

—

D₄

—

712

—

C₂×C₄

—

712

—

C₂×C₄

#15668009

yes

characteristic

C₂×C₄

C₂×D₇₁₂

yes

—

C₈

—

C₈

#15668034

yes

—

C₂²

—

1424

—

C₂²

—

1424

—

C₂²

#19151975

yes

characteristic

C₄

—

1424

—

C₄

#19151982

yes

—

C₄

#19152036

yes

—

C₂

—

2848

—

C₂

#26019858

yes

characteristic

100

C₂

#26022518

yes

—

	subgroup	G/H
center	C₂²	#19151975
commutator, cosocle	C₄×C₈×C₈₉	C₂³
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×#1311	C₂
socle	C₂²×C₈₉	#493
prime radical	#26019867	C₂
hypercenter	#1311	D₈₉
hypocenter	C₈₉	#12513
LCS	G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#19151975

commutator, cosocle

C₄×C₈×C₈₉

C₂³

Frattini

C₄×C₈

C₂²×D₈₉

Fitting

C₈₉×#1311

C₂

C₂²×C₈₉

C₂

D₈₉

C₈₉

G ▷ C₄×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #26022476	C₂	semidirect
3. #26019862	C₂	semidirect
4. C₈₉×#1311	C₂	semidirect
54. #1311	D₈₉	semidirect
55. C₈₉	#12513	semidirect

H ≅ G/N

type

1. #26022476

C₂

semidirect

3. #26019862

C₂

semidirect

4. C₈₉×#1311

C₂

semidirect

54. #1311

D₈₉

semidirect

55. C₈₉

#12513

semidirect

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

subgroup	index
#26022476	2
#26019867	2
#26019862	2
C₈₉×#1311	2
#12513	89

subgroup

index

C₈₉×#1311

Subgroups — #36469132

Conjugacy classes of subgroups

Complements

Maximal subgroups