Group #36385229 — subgroups

Subgroups — C₈₉×#7381 #36385229

Order 22784.

120 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 58 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	C₈₉×#2170	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1422	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#504×C₈₉	—	2	1	no	—
5	#443×C₈₉	C₄	1	1	yes	characteristic
6	M₆₄×C₈₉	—	2	1	no	—
7	C₂×C₃₂×C₈₉	—	2	1	no	—
8	#364×C₈₉	C₄	1	1	yes	characteristic
9	#350×C₈₉	—	2	1	no	—
10	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	#135×C₈₉	C₈	1	1	yes	characteristic
12	#131×C₈₉	—	2	1	no	—
13	M₃₂×C₈₉	—	2	1	no	—
14	M₃₂×C₈₉	—	4	1	no	—
15	C₂×C₁₆×C₈₉	—	4	1	no	—
16	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
17	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
18	#108×C₈₉	C₈	1	1	yes	characteristic
19	#104×C₈₉	—	2	1	no	—
20	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₈₉	—	2	2	no	—
22	C₃₂×C₈₉	—	4	1	no	—
23	#41×C₈₉	—	2	1	no	—
24	Q₁₆×C₈₉	C₁₆	1	1	yes	characteristic
25	SD₁₆×C₈₉	—	2	1	no	—
26	D₈×C₈₉	C₁₆	1	1	yes	characteristic
27	M₁₆×C₈₉	—	2	1	no	—
28	M₁₆×C₈₉	—	4	1	no	—
29	C₂×C₈×C₈₉	—	4	1	no	—
30	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
32	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
33	C₁₆×C₈₉	—	2	1	no	—
34	C₁₆×C₈₉	—	4	1	no	—
35	C₁₆×C₈₉	D₈	1	1	yes	characteristic
36	C₁₆×C₈₉	SD₁₆	1	1	yes	characteristic
37	Q₈×C₈₉	—	2	1	no	—
38	D₄×C₈₉	—	2	1	no	—
39	D₄×C₈₉	—	4	1	no	—
40	C₂×C₄×C₈₉	—	2	1	no	—
41	C₂×C₄×C₈₉	—	4	1	no	—
42	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
43	C₈×C₈₉	—	4	1	no	—
44	C₈×C₈₉	#102	1	1	yes	characteristic
45	C₈×C₈₉	#104	1	1	yes	characteristic
46	C₈×C₈₉	C₂×C₁₆	1	1	yes	characteristic
47	C₈×C₈₉	M₃₂	1	1	yes	characteristic
48	C₂²×C₈₉	—	4	1	no	—
49	C₂²×C₈₉	#350	1	1	yes	characteristic
50	C₄×C₈₉	—	2	2	no	—
51	C₄×C₈₉	—	4	1	no	—
52	C₄×C₈₉	D₄⋊C₈	1	1	yes	characteristic
53	C₄×C₈₉	C₂²⋊C₁₆	1	1	yes	characteristic
54	#7381	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₈₉	—	2	1	no	—
56	C₂×C₈₉	—	8	1	no	—
57	C₂×C₈₉	D₄⋊C₁₆	1	1	yes	characteristic
58	#2170	C₂×C₈₉	1	1	yes	characteristic
59	#1422	C₂×C₈₉	1	1	yes	characteristic, complemented
60	C₄×C₃₂	C₂×C₈₉	1	1	yes	characteristic, complemented
61	C₈₉	#7381	1	1	yes	characteristic, direct product factor, Sylow
62	#504	—	2	1	no	—
63	#443	C₄×C₈₉	1	1	yes	characteristic
64	M₆₄	—	2	1	no	—
65	C₂×C₃₂	—	2	1	no	—
66	#364	C₄×C₈₉	1	1	yes	characteristic
67	#350	—	2	1	no	—
68	C₄×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
69	#135	C₈×C₈₉	1	1	yes	characteristic
70	#131	—	2	1	no	—
71	M₃₂	—	2	1	no	—
72	M₃₂	—	4	1	no	—
73	C₂×C₁₆	—	4	1	no	—
74	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
75	C₂×C₁₆	D₄×C₈₉	1	1	yes	characteristic
76	#108	C₈×C₈₉	1	1	yes	characteristic
77	#104	—	2	1	no	—
78	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₃₂	—	4	1	no	—
81	#41	—	2	1	no	—
82	Q₁₆	C₁₆×C₈₉	1	1	yes	characteristic
83	SD₁₆	—	2	1	no	—
84	D₈	C₁₆×C₈₉	1	1	yes	characteristic
85	M₁₆	—	2	1	no	—
86	M₁₆	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
89	C₂×C₈	C₂×C₈×C₈₉	1	1	yes	characteristic
90	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	4	1	no	—
93	C₁₆	D₈×C₈₉	1	1	yes	characteristic
94	C₁₆	SD₁₆×C₈₉	1	1	yes	characteristic
95	Q₈	—	2	1	no	—
96	D₄	—	2	1	no	—
97	D₄	—	4	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
101	C₈	—	4	1	no	—
102	C₈	#102×C₈₉	1	1	yes	characteristic
103	C₈	#104×C₈₉	1	1	yes	characteristic
104	C₈	C₂×C₁₆×C₈₉	1	1	yes	characteristic
105	C₈	M₃₂×C₈₉	1	1	yes	characteristic
106	C₂²	—	4	1	no	—
107	C₂²	#350×C₈₉	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	4	1	no	—
110	C₄	D₄⋊C₈×C₈₉	1	1	yes	characteristic
111	C₄	C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
112	C₂	—	2	1	no	—
113	C₂	—	8	1	no	—
114	C₂	C₈₉×D₄⋊C₁₆	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈₉×#2170

C₂

yes

maximal, characteristic

C₈₉×#1422

C₂

yes

maximal, characteristic, complemented

C₄×C₃₂×C₈₉

C₂

yes

maximal, characteristic, complemented

#504×C₈₉

—

#443×C₈₉

C₄

yes

characteristic

M₆₄×C₈₉

—

C₂×C₃₂×C₈₉

—

#364×C₈₉

C₄

yes

characteristic

#350×C₈₉

—

C₄×C₁₆×C₈₉

C₂²

yes

characteristic

#135×C₈₉

C₈

yes

characteristic

#131×C₈₉

—

M₃₂×C₈₉

—

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

D₄

yes

characteristic

#108×C₈₉

C₈

yes

characteristic

#104×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

characteristic

C₃₂×C₈₉

—

C₃₂×C₈₉

—

#41×C₈₉

—

Q₁₆×C₈₉

C₁₆

yes

characteristic

SD₁₆×C₈₉

—

D₈×C₈₉

C₁₆

yes

characteristic

M₁₆×C₈₉

—

M₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₈

yes

characteristic

C₄²×C₈₉

M₁₆

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

D₈

yes

characteristic

C₁₆×C₈₉

SD₁₆

yes

characteristic

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂²⋊C₈

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

yes

C₈×C₈₉

yes

C₈×C₈₉

C₂×C₁₆

yes

characteristic

C₈×C₈₉

M₃₂

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

#350

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

D₄⋊C₈

yes

characteristic

C₄×C₈₉

C₂²⋊C₁₆

yes

characteristic

#7381

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

—

C₂×C₈₉

—

C₂×C₈₉

D₄⋊C₁₆

yes

characteristic

#2170

C₂×C₈₉

yes

characteristic

#1422

C₂×C₈₉

yes

characteristic, complemented

C₄×C₃₂

C₂×C₈₉

yes

characteristic, complemented

C₈₉

#7381

yes

characteristic, direct product factor, Sylow

#504

—

#443

C₄×C₈₉

yes

characteristic

M₆₄

—

C₂×C₃₂

—

#364

C₄×C₈₉

yes

characteristic

#350

—

C₄×C₁₆

C₂²×C₈₉

yes

characteristic

#135

C₈×C₈₉

yes

characteristic

#131

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

D₄×C₈₉

yes

characteristic

C₂×C₁₆

D₄×C₈₉

yes

characteristic

#108

C₈×C₈₉

yes

characteristic

#104

—

C₄×C₈

C₂×C₄×C₈₉

yes

characteristic

C₃₂

—

C₃₂

—

#41

—

Q₁₆

C₁₆×C₈₉

yes

characteristic

SD₁₆

—

D₈

C₁₆×C₈₉

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄×C₈₉

yes

characteristic

C₂×C₈

C₂×C₈×C₈₉

yes

characteristic

C₄²

M₁₆×C₈₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

D₈×C₈₉

yes

characteristic

C₁₆

SD₁₆×C₈₉

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊C₈×C₈₉

yes

characteristic

101

C₈

—

102

C₈

#102×C₈₉

yes

characteristic

103

C₈

#104×C₈₉

yes

characteristic

104

C₈

C₂×C₁₆×C₈₉

yes

characteristic

105

C₈

M₃₂×C₈₉

yes

characteristic

106

C₂²

—

107

C₂²

#350×C₈₉

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

D₄⋊C₈×C₈₉

yes

characteristic

111

C₄

C₂²⋊C₁₆×C₈₉

yes

characteristic

112

C₂

—

113

C₂

—

114

C₂

C₈₉×D₄⋊C₁₆

yes

characteristic

	subgroup	G/H
center	C₁₆×C₈₉	D₈
commutator	C₈	C₂×C₁₆×C₈₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₈₉
socle	C₂×C₈₉	D₄⋊C₁₆
prime radical	#135×C₈₉	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₈₉ ◁ C₂×C₁₆×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

subgroup

G/H

center

C₁₆×C₈₉

D₈

commutator

C₈

C₂×C₁₆×C₈₉

Frattini, cosocle

C₄×C₁₆

C₂²×C₈₉

socle

C₂×C₈₉

D₄⋊C₁₆

prime radical

#135×C₈₉

C₈

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆×C₈₉ ◁ C₂×C₁₆×C₈₉ ◁ C₄×C₁₆×C₈₉ ◁ G

N	H ≅ G/N	type
2. C₈₉×#1422	C₂	semidirect
3. C₄×C₃₂×C₈₉	C₂	semidirect
54. #7381	C₈₉	direct (N × H)
59. #1422	C₂×C₈₉	semidirect
60. C₄×C₃₂	C₂×C₈₉	semidirect
61. C₈₉	#7381	direct (N × H)

H ≅ G/N

type

2. C₈₉×#1422

C₂

semidirect

3. C₄×C₃₂×C₈₉

C₂

semidirect

54. #7381

C₈₉

direct (N × H)

59. #1422

C₂×C₈₉

semidirect

60. C₄×C₃₂

C₂×C₈₉

semidirect

61. C₈₉

#7381

direct (N × H)

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

subgroup	index
C₈₉×#2170	2
C₈₉×#1422	2
C₄×C₃₂×C₈₉	2
#7381	89

subgroup

index

C₈₉×#2170

C₈₉×#1422

C₄×C₃₂×C₈₉

#7381

Subgroups — C₈₉×#7381 #36385229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#7381 #36385229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#7381 #36385229