Group #36474931 — subgroups

Subgroups — #36474931

Order 22784.

190 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 115 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	#26022036	C₂	1	2	yes	maximal
2	#26018454	C₂	1	2	yes	maximal, complemented
3	#26018452	C₂	1	2	yes	maximal, complemented
4	C₂²×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151636	—	2	2	no	—
6	#19151426	—	2	2	no	—
7	#19151424	C₂²	1	4	yes	—
8	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₃₂×C₈₉	C₄	1	4	yes	complemented
10	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
12	#15667971	—	4	2	no	—
13	#15667946	—	2	4	no	—
14	#15667944	—	4	2	no	—
15	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
16	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₈₉	Q₈	1	2	yes	—
19	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
20	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
21	C₃₂×C₈₉	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₈₉	—	4	4	no	—
23	C₂²⋊Dic₈₉	—	8	2	no	—
24	C₄×Dic₈₉	—	8	2	no	—
25	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
26	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
28	C₂×C₈×C₈₉	Q₁₆	1	2	yes	—
29	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
30	C₁₆×C₈₉	C₄⋊C₄	1	2	yes	—
31	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
32	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
33	C₂³×C₈₉	D₁₆	1	1	yes	characteristic
34	C₂×C₄×C₈₉	#107	1	2	yes	—
35	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
36	C₂×C₄×C₈₉	Q₃₂	1	2	yes	—
37	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
38	C₂×Dic₈₉	—	8	4	no	—
39	C₈×C₈₉	#107	1	2	yes	—
40	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂²×C₈₉	—	2	2	no	—
43	C₂²×C₈₉	#366	1	2	yes	—
44	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
45	C₄×C₈₉	#366	1	2	yes	—
46	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
47	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
48	Dic₈₉	—	16	4	no	—
49	#13633	—	89	1	no	maximal, Sylow
50	C₂×C₈₉	—	2	2	no	—
51	C₂×C₈₉	C₂×#366	1	1	yes	characteristic
52	C₂×C₈₉	#2262	1	2	yes	—
53	C₂²×C₃₂	D₈₉	1	1	yes	characteristic
54	#2158	—	89	2	no	—
55	#1424	—	89	2	no	—
56	#1423	—	89	2	no	—
57	C₈₉	#13633	1	1	yes	characteristic, complemented, Sylow
58	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
59	#427	—	178	2	no	—
60	C₂×C₃₂	Dic₈₉	1	4	yes	complemented
61	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
62	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
63	#367	—	178	2	no	—
64	#366	—	89	4	no	—
65	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
66	#117	—	356	2	no	—
67	C₂×C₁₆	Dic₁₇₈	1	2	yes	—
68	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
69	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
70	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
71	#107	—	178	4	no	—
72	#106	—	356	2	no	—
73	C₃₂	C₂×Dic₈₉	1	2	yes	—
74	C₃₂	C₂×Dic₈₉	1	2	yes	—
75	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
76	C₂×C₈	C₄⋊Dic₈₉	1	2	yes	—
77	C₂×C₈	D₇₁₂	1	1	yes	characteristic
78	C₂×C₈	Dic₃₅₆	1	2	yes	—
79	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
80	C₄⋊C₄	—	356	4	no	—
81	C₂²⋊C₄	—	712	2	no	—
82	C₄²	—	712	2	no	—
83	C₁₆	C₄⋊Dic₈₉	1	2	yes	—
84	C₁₆	C₂²×Dic₈₉	1	1	yes	characteristic
85	C₁₆	C₂×Dic₁₇₈	1	1	yes	characteristic
86	C₂³	D₁₄₂₄	1	1	yes	characteristic
87	C₂×C₄	—	712	4	no	—
88	C₂×C₄	#15667946	1	2	yes	—
89	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
90	C₂×C₄	Dic₇₁₂	1	2	yes	—
91	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
92	C₈	#15667946	1	2	yes	—
93	C₈	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
94	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#19151424	1	2	yes	—
97	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
98	C₄	—	1424	4	no	—
99	C₄	#19151424	1	2	yes	—
100	C₄	C₂×#15667946	1	1	yes	characteristic
101	C₄	C₂×Dic₇₁₂	1	1	yes	characteristic
102	C₂	—	2	2	no	—
103	C₂	C₂×#19151424	1	1	yes	characteristic
104	C₂	#26022561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26022036

C₂

yes

maximal

#26018454

C₂

yes

maximal, complemented

#26018452

C₂

yes

maximal, complemented

C₂²×C₃₂×C₈₉

C₂

yes

maximal, characteristic

#19151636

—

#19151426

—

#19151424

C₂²

yes

—

C₂²×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₄

yes

complemented

C₂×C₃₂×C₈₉

C₂²

yes

characteristic

C₂×C₃₂×C₈₉

C₂²

yes

—

—

—

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

Q₈

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₃₂×C₈₉

C₂×C₄

yes

—

C₃₂×C₈₉

C₂×C₄

yes

—

C₄⋊Dic₈₉

—

C₂²⋊Dic₈₉

—

C₄×Dic₈₉

—

C₂²×C₄×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

Q₁₆

yes

—

C₂×C₈×C₈₉

C₂×D₄

yes

characteristic

C₁₆×C₈₉

C₄⋊C₄

yes

—

C₁₆×C₈₉

C₂²×C₄

yes

characteristic

C₁₆×C₈₉

C₂×Q₈

yes

characteristic

C₂³×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

D₁₆

yes

characteristic

C₂×C₄×C₈₉

Q₃₂

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

characteristic

C₂×Dic₈₉

—

C₈×C₈₉

#107

yes

—

C₈×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₈×C₈₉

C₂×Q₁₆

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

#366

yes

—

C₂²×C₈₉

C₂×D₁₆

yes

characteristic

C₄×C₈₉

#366

yes

—

C₄×C₈₉

C₂×#107

yes

characteristic

C₄×C₈₉

C₂×Q₃₂

yes

characteristic

Dic₈₉

—

#13633

—

maximal, Sylow

C₂×C₈₉

—

C₂×C₈₉

C₂×#366

yes

characteristic

C₂×C₈₉

#2262

yes

—

C₂²×C₃₂

D₈₉

yes

—

—

—

C₈₉

#13633

yes

characteristic, complemented, Sylow

C₂²×C₁₆

C₂×D₈₉

yes

characteristic

#427

—

178

—

C₂×C₃₂

Dic₈₉

yes

complemented

C₂×C₃₂

C₂×D₈₉

yes

characteristic

C₂×C₃₂

C₂×D₈₉

yes

characteristic

#367

—

178

—

#366

—

C₂²×C₈

D₃₅₆

yes

characteristic

#117

—

356

—

C₂×C₁₆

Dic₁₇₈

yes

—

C₂×C₁₆

D₃₅₆

yes

characteristic

C₂×C₁₆

C₂×Dic₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

178

—

#106

—

356

—

C₃₂

C₂×Dic₈₉

yes

—

C₃₂

C₂×Dic₈₉

yes

—

C₂²×C₄

D₇₁₂

yes

characteristic

C₂×C₈

C₄⋊Dic₈₉

yes

—

C₂×C₈

D₇₁₂

yes

characteristic

C₂×C₈

Dic₃₅₆

yes

—

C₂×C₈

C₂×D₃₅₆

yes

characteristic

C₄⋊C₄

—

356

—

C₂²⋊C₄

—

712

—

C₄²

—

712

—

C₁₆

C₄⋊Dic₈₉

yes

—

C₁₆

C₂²×Dic₈₉

yes

characteristic

C₁₆

C₂×Dic₁₇₈

yes

characteristic

C₂³

D₁₄₂₄

yes

characteristic

C₂×C₄

—

712

—

C₂×C₄

#15667946

yes

—

C₂×C₄

D₁₄₂₄

yes

characteristic

C₂×C₄

Dic₇₁₂

yes

—

C₂×C₄

C₂×D₇₁₂

yes

characteristic

C₈

#15667946

yes

—

C₈

C₂×C₄⋊Dic₈₉

yes

characteristic

C₈

C₂×Dic₃₅₆

yes

characteristic

C₂²

—

C₂²

#19151424

yes

—

C₂²

C₂×D₁₄₂₄

yes

characteristic

C₄

—

1424

—

C₄

#19151424

yes

—

100

C₄

C₂×#15667946

yes

characteristic

101

C₄

C₂×Dic₇₁₂

yes

characteristic

102

C₂

—

103

C₂

C₂×#19151424

yes

characteristic

104

C₂

#26022561

yes

—

	subgroup	G/H
center	C₂×C₄	D₁₄₂₄
commutator	C₁₆×C₈₉	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₂²×C₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₁₆
prime radical	C₂³×C₈₉	D₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	C₂²×C₃₂	D₈₉
hypocenter	C₈₉	#13633
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

subgroup

G/H

center

C₂×C₄

D₁₄₂₄

commutator

C₁₆×C₈₉

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₈₉

Fitting

C₂²×C₃₂×C₈₉

C₂

socle

C₂²×C₈₉

C₂×D₁₆

prime radical

C₂³×C₈₉

D₁₆

cosocle

C₂×C₁₆×C₈₉

C₂³

hypercenter

C₂²×C₃₂

D₈₉

hypocenter

C₈₉

#13633

LCS

G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×C₃₂

N	H ≅ G/N	type
2. #26018454	C₂	semidirect
3. #26018452	C₂	semidirect
9. C₂×C₃₂×C₈₉	C₄	semidirect
57. C₈₉	#13633	semidirect
60. C₂×C₃₂	Dic₈₉	semidirect

H ≅ G/N

type

2. #26018454

C₂

semidirect

3. #26018452

C₂

semidirect

9. C₂×C₃₂×C₈₉

C₄

semidirect

57. C₈₉

#13633

semidirect

60. C₂×C₃₂

Dic₈₉

semidirect

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

subgroup	index
#26022036	2
#26018454	2
#26018452	2
C₂²×C₃₂×C₈₉	2
#13633	89

subgroup

index

C₂²×C₃₂×C₈₉

Subgroups — #36474931

Conjugacy classes of subgroups

Complements

Maximal subgroups