Group #36448102 — subgroups

Subgroups — #36448102

Order 22784.

322 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 147 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	#26020369	C₂	1	2	yes	maximal
2	#26018087	C₂	1	4	yes	maximal
3	C₈₉×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#10623494	C₂²	1	4	yes	—
5	#19151353	—	2	4	no	—
6	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
7	C₄³×C₈₉	C₂²	1	1	yes	characteristic
8	#336×C₈₉	C₄	1	4	yes	—
9	C₂²×#10623494	—	2	4	no	—
10	C₄⋊#10623494	—	2	8	no	—
11	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₈₉	D₄	1	2	yes	—
14	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
15	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₈₉	—	2	4	no	—
17	C₂×#10623494	—	4	4	no	—
18	C₂×#10623494	—	4	8	no	—
19	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
20	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
22	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₈×C₈₉	—	2	4	no	—
24	C₂×C₈×C₈₉	—	2	8	no	—
25	C₄²×C₈₉	—	2	4	no	—
26	C₄²×C₈₉	—	2	4	no	—
27	C₄²×C₈₉	—	4	2	no	—
28	C₄²×C₈₉	#41	1	4	yes	—
29	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₈₉	—	2	2	no	—
31	C₂×C₄×C₈₉	—	2	4	no	—
32	C₂×C₄×C₈₉	—	2	4	no	—
33	C₂×C₄×C₈₉	—	2	4	no	—
34	C₂×C₄×C₈₉	#117	1	2	yes	—
35	C₂×C₄×C₈₉	#124	1	2	yes	—
36	C₂×C₄×C₈₉	#125	1	2	yes	—
37	C₂×C₄×C₈₉	#131	1	4	yes	—
38	C₂×C₄×C₈₉	#135	1	4	yes	—
39	C₈×C₈₉	—	4	4	no	—
40	#10623494	—	8	4	no	—
41	C₂²×C₈₉	#382	1	1	yes	characteristic
42	C₂²×C₈₉	#424	1	2	yes	—
43	C₂²×C₈₉	#427	1	2	yes	—
44	C₂²×C₈₉	#430	1	2	yes	—
45	C₄×C₈₉	—	2	2	no	—
46	C₄×C₈₉	—	4	4	no	—
47	C₄×C₈₉	#433	1	4	yes	—
48	C₄×C₈₉	#488	1	4	yes	—
49	#8788	—	89	1	no	maximal, Sylow
50	C₂×C₈₉	#1572	1	4	yes	—
51	C₂×C₈₉	#1832	1	1	yes	characteristic
52	C₂×C₈₉	#1835	1	1	yes	characteristic
53	C₂×C₈₉	#1838	1	1	yes	characteristic
54	#1831	—	89	2	no	—
55	#1744	D₈₉	1	1	yes	characteristic
56	#1277	—	89	4	no	—
57	C₈₉	#8788	1	1	yes	characteristic, complemented, Sylow
58	C₂×C₄⋊C₈	—	89	4	no	—
59	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
60	C₄³	C₂×D₈₉	1	1	yes	characteristic
61	#336	—	178	4	no	—
62	#336	Dic₈₉	1	4	yes	—
63	C₂²×C₈	—	178	4	no	—
64	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
65	C₂×C₄²	Dic₁₇₈	1	2	yes	—
66	C₂×C₄²	D₃₅₆	1	2	yes	—
67	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
68	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
69	C₄⋊C₈	—	178	8	no	—
70	C₄×C₈	—	2	4	no	—
71	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
72	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
73	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	8	no	—
77	C₂×C₈	—	356	4	no	—
78	C₂×C₈	—	356	8	no	—
79	C₄²	—	2	4	no	—
80	C₄²	—	2	4	no	—
81	C₄²	—	4	2	no	—
82	C₄²	#13083574	1	4	yes	—
83	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	4	no	—
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	#15667971	1	2	yes	—
89	C₂×C₄	#15668000	1	2	yes	—
90	C₂×C₄	#15668004	1	2	yes	—
91	C₂×C₄	#15668019	1	4	yes	—
92	C₂×C₄	#15668034	1	4	yes	—
93	C₈	—	4	4	no	—
94	C₈	—	712	4	no	—
95	C₂²	#19151464	1	1	yes	characteristic
96	C₂²	#19151625	1	2	yes	—
97	C₂²	#19151636	1	2	yes	—
98	C₂²	#19151647	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	4	4	no	—
101	C₄	#19151659	1	4	yes	—
102	C₄	#19151953	1	4	yes	—
103	C₂	#26019032	1	4	yes	—
104	C₂	#26020372	1	1	yes	characteristic
105	C₂	#26020383	1	1	yes	characteristic
106	C₂	#26020393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26020369

C₂

yes

maximal

#26018087

C₂

yes

maximal

C₈₉×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#10623494

C₂²

yes

—

#19151353

—

C₂×C₄×C₈×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

characteristic

#336×C₈₉

C₄

yes

—

C₂²×#10623494

—

C₄⋊#10623494

—

C₂²×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

—

C₂×C₄²×C₈₉

D₄

yes

—

C₂×C₄²×C₈₉

Q₈

yes

—

C₂×C₄²×C₈₉

C₂³

yes

characteristic

C₄×C₈×C₈₉

—

C₂×#10623494

—

C₂×#10623494

—

C₂²×C₄×C₈₉

C₄⋊C₄

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

—

C₄²×C₈₉

#41

yes

—

C₂³×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#117

yes

—

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

#125

yes

—

C₂×C₄×C₈₉

#131

yes

—

C₂×C₄×C₈₉

#135

yes

—

C₈×C₈₉

—

#10623494

—

C₂²×C₈₉

#382

yes

characteristic

C₂²×C₈₉

#424

yes

—

C₂²×C₈₉

#427

yes

—

C₂²×C₈₉

#430

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#433

yes

—

C₄×C₈₉

#488

yes

—

#8788

—

maximal, Sylow

C₂×C₈₉

#1572

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

—

D₈₉

yes

characteristic

#1277

—

C₈₉

#8788

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₈₉

yes

—

C₄³

C₂×D₈₉

yes

characteristic

#336

—

178

—

#336

Dic₈₉

yes

—

C₂²×C₈

—

178

—

C₂²×C₈

C₂×Dic₈₉

yes

—

C₂×C₄²

Dic₁₇₈

yes

—

C₂×C₄²

D₃₅₆

yes

—

C₂×C₄²

C₂×Dic₈₉

yes

—

C₂×C₄²

C₂²×D₈₉

yes

characteristic

C₄⋊C₈

—

178

—

C₄×C₈

—

C₂²×C₄

C₄⋊Dic₈₉

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

characteristic

C₂²×C₄

C₂×D₃₅₆

yes

characteristic

C₂²×C₄

C₂×Dic₁₇₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

356

—

C₂×C₈

—

356

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#13083574

yes

—

C₂³

C₂×C₄⋊Dic₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#15667971

yes

—

C₂×C₄

#15668000

yes

—

C₂×C₄

#15668004

yes

—

C₂×C₄

#15668019

yes

—

C₂×C₄

#15668034

yes

—

C₈

—

C₈

—

712

—

C₂²

#19151464

yes

characteristic

C₂²

#19151625

yes

—

C₂²

#19151636

yes

—

C₂²

#19151647

yes

—

C₄

—

100

C₄

—

101

C₄

#19151659

yes

—

102

C₄

#19151953

yes

—

103

C₂

#26019032

yes

—

104

C₂

#26020372

yes

characteristic

105

C₂

#26020383

yes

characteristic

106

C₂

#26020393

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1744	C₂
socle, prime radical	C₂³×C₈₉	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1744	D₈₉
hypocenter	C₈₉	#8788
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₃₅₆

commutator

C₂²×C₄×C₈₉

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₉

Fitting

C₈₉×#1744

C₂

socle, prime radical

C₂³×C₈₉

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₈₉

C₂³

D₈₉

C₈₉

G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
57. C₈₉	#8788	semidirect

H ≅ G/N

type

57. C₈₉

#8788

semidirect

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

subgroup	index
#26020369	2
#26018087	2
C₈₉×#1744	2
#8788	89

subgroup

index

#26020369

#26018087

C₈₉×#1744

#8788

Subgroups — #36448102

Conjugacy classes of subgroups

Complements

Maximal subgroups