Group #36448202 — subgroups

Subgroups — #36448202

Order 22784.

422 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 227 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	#26020409	C₂	1	2	yes	maximal
2	#26020380	C₂	1	4	yes	maximal
3	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667944	—	2	4	no	—
5	#19151468	C₂²	1	4	yes	—
6	#19151361	—	2	8	no	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₈²×C₈₉	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₈₉	—	2	8	no	—
11	C₂×C₄⋊Dic₈₉	—	2	4	no	—
12	#15667944	—	4	8	no	—
13	C₂²×C₈×C₈₉	D₄	1	4	yes	—
14	C₂²×C₈×C₈₉	D₄	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₈₉	Q₈	1	4	yes	—
19	C₄×C₈×C₈₉	Q₈	1	2	yes	—
20	C₂²×Dic₈₉	—	4	4	no	—
21	C₄⋊Dic₈₉	—	4	8	no	—
22	C₄⋊Dic₈₉	—	4	16	no	—
23	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
24	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
25	C₂×C₈×C₈₉	—	2	8	no	—
26	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
27	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₈₉	SD₁₆	1	8	yes	—
29	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
30	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
31	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
32	C₂³×C₈₉	#127	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
36	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
38	C₂×C₄×C₈₉	#135	1	8	yes	—
39	C₂×Dic₈₉	—	8	4	no	—
40	C₂×Dic₈₉	—	8	8	no	—
41	C₈×C₈₉	—	2	8	no	—
42	C₈×C₈₉	#106	1	8	yes	—
43	C₂²×C₈₉	#384	1	1	yes	characteristic
44	C₂²×C₈₉	#492	1	2	yes	—
45	C₂²×C₈₉	#495	1	4	yes	—
46	C₄×C₈₉	C₂×#106	1	2	yes	—
47	C₄×C₈₉	#427	1	4	yes	—
48	C₄×C₈₉	#498	1	2	yes	—
49	C₄×C₈₉	#499	1	4	yes	—
50	Dic₈₉	—	16	4	no	—
51	#8815	—	89	1	no	maximal, Sylow
52	C₂×C₈₉	#1708	1	4	yes	—
53	C₂×C₈₉	#1844	1	1	yes	characteristic
54	C₂×C₈₉	#1846	1	2	yes	—
55	#1844	—	89	2	no	—
56	#1834	—	89	4	no	—
57	C₂×C₈²	D₈₉	1	1	yes	characteristic
58	C₈₉	#8815	1	1	yes	characteristic, complemented, Sylow
59	C₂×#106	—	178	4	no	—
60	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
61	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
62	#384	—	89	4	no	—
63	#340	—	178	8	no	—
64	C₈²	Dic₈₉	1	4	yes	complemented
65	C₂²×C₈	D₃₅₆	1	4	yes	—
66	C₂²×C₈	D₃₅₆	1	2	yes	—
67	C₂×C₄⋊C₄	—	178	8	no	—
68	C₂×C₄⋊C₄	—	178	4	no	—
69	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
70	#106	—	356	8	no	—
71	C₄×C₈	Dic₁₇₈	1	4	yes	—
72	C₄×C₈	Dic₁₇₈	1	2	yes	—
73	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
74	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
75	C₂²×C₄	—	356	4	no	—
76	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
78	C₂×C₈	—	2	8	no	—
79	C₂×C₈	C₄⋊Dic₈₉	1	8	yes	—
80	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
81	C₂×C₈	#13083563	1	8	yes	—
82	C₄⋊C₄	—	356	8	no	—
83	C₄⋊C₄	—	356	16	no	—
84	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
85	C₄²	C₂×Dic₁₇₈	1	2	yes	—
86	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
87	C₂³	#15668009	1	1	yes	characteristic
88	C₂×C₄	—	712	4	no	—
89	C₂×C₄	—	712	8	no	—
90	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
91	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
92	C₂×C₄	#15668012	1	1	yes	characteristic
93	C₂×C₄	#15668012	1	2	yes	—
94	C₂×C₄	C₂×#13083563	1	4	yes	—
95	C₂×C₄	#15668034	1	8	yes	—
96	C₈	—	2	8	no	—
97	C₈	#15667944	1	8	yes	—
98	C₂²	#19151468	1	1	yes	characteristic
99	C₂²	#19151971	1	2	yes	—
100	C₂²	#19151982	1	4	yes	—
101	C₄	—	1424	4	no	—
102	C₄	C₂×#15667944	1	2	yes	—
103	C₄	#19151636	1	4	yes	—
104	C₄	#19151996	1	2	yes	—
105	C₄	#19152000	1	4	yes	—
106	C₂	#26019853	1	4	yes	—
107	C₂	#26020409	1	1	yes	characteristic
108	C₂	#26020416	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26020409

C₂

yes

maximal

#26020380

C₂

yes

maximal

C₂×C₈²×C₈₉

C₂

yes

maximal, characteristic

C₂×#15667944

—

#19151468

C₂²

yes

—

#19151361

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

C₈²×C₈₉

C₄

yes

complemented

C₂×C₄⋊Dic₈₉

—

C₂×C₄⋊Dic₈₉

—

#15667944

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂²×C₈×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

C₂²×Dic₈₉

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₄²×C₈₉

C₂²×C₄

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

characteristic

C₄²×C₈₉

C₂×Q₈

yes

—

C₂³×C₈₉

#127

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₈₉

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₈₉

C₄⋊Q₈

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₂×Dic₈₉

—

C₂×Dic₈₉

—

C₈×C₈₉

—

C₈×C₈₉

#106

yes

—

C₂²×C₈₉

#384

yes

characteristic

C₂²×C₈₉

#492

yes

—

C₂²×C₈₉

#495

yes

—

C₄×C₈₉

C₂×#106

yes

—

C₄×C₈₉

#427

yes

—

C₄×C₈₉

#498

yes

—

C₄×C₈₉

#499

yes

—

Dic₈₉

—

#8815

—

maximal, Sylow

C₂×C₈₉

#1708

yes

—

C₂×C₈₉

#1844

yes

characteristic

C₂×C₈₉

#1846

yes

—

#1844

—

#1834

—

C₂×C₈²

D₈₉

yes

characteristic

C₈₉

#8815

yes

characteristic, complemented, Sylow

C₂×#106

—

178

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₂×C₄×C₈

C₂×D₈₉

yes

characteristic

#384

—

#340

—

178

—

C₈²

Dic₈₉

yes

complemented

C₂²×C₈

D₃₅₆

yes

—

C₂²×C₈

D₃₅₆

yes

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄⋊C₄

—

178

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

#106

—

356

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

Dic₁₇₈

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₄×C₈

C₂×Dic₈₉

yes

—

C₂²×C₄

—

356

—

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×D₃₅₆

yes

—

C₂×C₈

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

#13083563

yes

—

C₄⋊C₄

—

356

—

C₄⋊C₄

—

356

—

C₄²

C₂²×Dic₈₉

yes

characteristic

C₄²

C₂×Dic₁₇₈

yes

—

C₄²

C₂×Dic₁₇₈

yes

characteristic

C₂³

#15668009

yes

characteristic

C₂×C₄

—

712

—

C₂×C₄

—

712

—

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₈₉

yes

—

C₂×C₄

#15668012

yes

characteristic

C₂×C₄

#15668012

yes

—

C₂×C₄

C₂×#13083563

yes

—

C₂×C₄

#15668034

yes

—

C₈

—

C₈

#15667944

yes

—

C₂²

#19151468

yes

characteristic

C₂²

#19151971

yes

—

100

C₂²

#19151982

yes

—

101

C₄

—

1424

—

102

C₄

C₂×#15667944

yes

—

103

C₄

#19151636

yes

—

104

C₄

#19151996

yes

—

105

C₄

#19152000

yes

—

106

C₂

#26019853

yes

—

107

C₂

#26020409

yes

characteristic

108

C₂

#26020416

yes

—

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#8815
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂³

#15668009

commutator

C₄²×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₂×C₈²×C₈₉

C₂

socle, prime radical

C₂³×C₈₉

#127

cosocle

C₂×C₄²×C₈₉

C₂³

hypercenter

C₂×C₈²

D₈₉

hypocenter

C₈₉

#8815

LCS

G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
9. C₈²×C₈₉	C₄	semidirect
58. C₈₉	#8815	semidirect
64. C₈²	Dic₈₉	semidirect

H ≅ G/N

type

9. C₈²×C₈₉

C₄

semidirect

58. C₈₉

#8815

semidirect

64. C₈²

Dic₈₉

semidirect

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

subgroup	index
#26020409	2
#26020380	2
C₂×C₈²×C₈₉	2
#8815	89

subgroup

index

#26020409

#26020380

C₂×C₈²×C₈₉

#8815

Subgroups — #36448202

Conjugacy classes of subgroups

Complements

Maximal subgroups