Group #36390737 — subgroups

Subgroups — C₈₉×#12889 #36390737

Order 22784.

310 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 174 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	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1846	C₂	1	2	yes	maximal
3	C₈₉×#1380	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
5	C₂×#107×C₈₉	C₂²	1	4	yes	—
6	C₂×#106×C₈₉	—	2	2	no	—
7	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
8	#382×C₈₉	—	2	2	no	—
9	C₄×C₁₆×C₈₉	C₄	1	4	yes	complemented
10	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
11	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
13	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₈₉	—	2	4	no	—
15	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
16	#107×C₈₉	—	2	8	no	—
17	#106×C₈₉	—	4	4	no	—
18	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₈₉	Q₈	1	2	yes	—
20	#95×C₈₉	—	4	4	no	—
21	C₂²×C₄×C₈₉	—	4	4	no	—
22	C₂²×C₄×C₈₉	D₈	1	2	yes	—
23	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	C₄⋊C₄	1	2	yes	—
25	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
26	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₈×C₈₉	#41	1	4	yes	—
29	C₄⋊C₄×C₈₉	—	4	4	no	—
30	C₄⋊C₄×C₈₉	—	4	4	no	—
31	C₄²×C₈₉	C₄⋊C₄	1	2	yes	—
32	C₄²×C₈₉	Q₁₆	1	2	yes	—
33	C₁₆×C₈₉	—	2	4	no	—
34	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
35	C₂×C₄×C₈₉	—	2	4	no	—
36	C₂×C₄×C₈₉	—	8	4	no	—
37	C₂×C₄×C₈₉	—	8	8	no	—
38	C₂×C₄×C₈₉	#107	1	4	yes	—
39	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₉	#124	1	2	yes	—
41	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
42	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
43	C₈×C₈₉	#106	1	4	yes	—
44	C₈×C₈₉	#117	1	2	yes	—
45	C₈×C₈₉	#125	1	2	yes	—
46	C₂²×C₈₉	C₂×#107	1	1	yes	characteristic
47	C₂²×C₈₉	#486	1	2	yes	—
48	C₂²×C₈₉	#508	1	4	yes	—
49	C₄×C₈₉	—	2	4	no	—
50	C₄×C₈₉	—	16	4	no	—
51	C₄×C₈₉	#382	1	1	yes	characteristic
52	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
53	C₄×C₈₉	#487	1	2	yes	—
54	#12889	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₈₉	#1834	1	1	yes	characteristic
56	C₂×C₈₉	#2158	1	2	yes	—
57	C₂×C₈₉	#2242	1	4	yes	—
58	C₂×C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic
59	#1846	C₂×C₈₉	1	2	yes	—
60	#1380	C₂×C₈₉	1	4	yes	—
61	C₈₉	#12889	1	1	yes	characteristic, direct product factor, Sylow
62	C₂²×C₁₆	C₂²×C₈₉	1	2	yes	—
63	C₂×#107	C₂²×C₈₉	1	4	yes	—
64	C₂×#106	—	2	2	no	—
65	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
66	#382	—	2	2	no	—
67	C₄×C₁₆	C₄×C₈₉	1	4	yes	complemented
68	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
69	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	2	4	no	—
71	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
72	C₂×C₁₆	—	2	4	no	—
73	C₂×C₁₆	C₂×C₄×C₈₉	1	4	yes	—
74	#107	—	2	8	no	—
75	#106	—	4	4	no	—
76	C₄×C₈	C₂×C₄×C₈₉	1	2	yes	—
77	C₄×C₈	Q₈×C₈₉	1	2	yes	—
78	#95	—	4	4	no	—
79	C₂²×C₄	—	4	4	no	—
80	C₂²×C₄	D₈×C₈₉	1	2	yes	—
81	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
82	C₂×C₈	C₄⋊C₄×C₈₉	1	2	yes	—
83	C₂×C₈	SD₁₆×C₈₉	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
86	C₂×C₈	#41×C₈₉	1	4	yes	—
87	C₄⋊C₄	—	4	4	no	—
88	C₄⋊C₄	—	4	4	no	—
89	C₄²	C₄⋊C₄×C₈₉	1	2	yes	—
90	C₄²	Q₁₆×C₈₉	1	2	yes	—
91	C₁₆	—	2	4	no	—
92	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	8	4	no	—
95	C₂×C₄	—	8	8	no	—
96	C₂×C₄	#107×C₈₉	1	4	yes	—
97	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
98	C₂×C₄	#124×C₈₉	1	2	yes	—
99	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
100	C₂×C₄	C₂×Q₁₆×C₈₉	1	1	yes	characteristic
101	C₈	#106×C₈₉	1	4	yes	—
102	C₈	#117×C₈₉	1	2	yes	—
103	C₈	#125×C₈₉	1	2	yes	—
104	C₂²	C₂×#107×C₈₉	1	1	yes	characteristic
105	C₂²	#486×C₈₉	1	2	yes	—
106	C₂²	#508×C₈₉	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	16	4	no	—
109	C₄	#382×C₈₉	1	1	yes	characteristic
110	C₄	C₂×#106×C₈₉	1	1	yes	characteristic
111	C₄	#487×C₈₉	1	2	yes	—
112	C₂	C₈₉×#1834	1	1	yes	characteristic
113	C₂	C₈₉×#2158	1	2	yes	—
114	C₂	C₈₉×#2242	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic

C₈₉×#1846

C₂

yes

maximal

C₈₉×#1380

C₂

yes

maximal

C₂²×C₁₆×C₈₉

C₂²

yes

—

C₂×#107×C₈₉

C₂²

yes

—

C₂×#106×C₈₉

—

C₂×C₄×C₈×C₈₉

C₂²

yes

characteristic

#382×C₈₉

—

C₄×C₁₆×C₈₉

C₄

yes

complemented

C₂²×C₈×C₈₉

D₄

yes

characteristic

C₂²×C₈×C₈₉

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₈₉

—

C₂×C₄²×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

—

C₂×C₁₆×C₈₉

C₂×C₄

yes

—

#107×C₈₉

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂×C₄

yes

—

C₄×C₈×C₈₉

Q₈

yes

—

#95×C₈₉

—

C₂²×C₄×C₈₉

—

C₂²×C₄×C₈₉

D₈

yes

—

C₂²×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₈×C₈₉

C₄⋊C₄

yes

—

C₂×C₈×C₈₉

SD₁₆

yes

—

C₂×C₈×C₈₉

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×Q₈

yes

characteristic

C₂×C₈×C₈₉

#41

yes

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄²×C₈₉

C₄⋊C₄

yes

—

C₄²×C₈₉

Q₁₆

yes

—

C₁₆×C₈₉

—

C₂³×C₈₉

C₂×D₈

yes

characteristic

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

#107

yes

—

C₂×C₄×C₈₉

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₉

#124

yes

—

C₂×C₄×C₈₉

C₂×SD₁₆

yes

—

C₂×C₄×C₈₉

C₂×Q₁₆

yes

characteristic

C₈×C₈₉

#106

yes

—

C₈×C₈₉

#117

yes

—

C₈×C₈₉

#125

yes

—

C₂²×C₈₉

C₂×#107

yes

characteristic

C₂²×C₈₉

#486

yes

—

C₂²×C₈₉

#508

yes

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

#12889

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₈₉

#1834

yes

characteristic

C₂×C₈₉

#2158

yes

—

C₂×C₈₉

#2242

yes

—

C₂×C₄×C₁₆

C₂×C₈₉

yes

characteristic

#1846

C₂×C₈₉

yes

—

#1380

C₂×C₈₉

yes

—

C₈₉

#12889

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₈₉

yes

—

C₂×#107

C₂²×C₈₉

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²×C₈₉

yes

characteristic

#382

—

C₄×C₁₆

C₄×C₈₉

yes

complemented

C₂²×C₈

D₄×C₈₉

yes

characteristic

C₂²×C₈

C₂³×C₈₉

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

D₄×C₈₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₈₉

yes

—

#107

—

#106

—

C₄×C₈

C₂×C₄×C₈₉

yes

—

C₄×C₈

Q₈×C₈₉

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

D₈×C₈₉

yes

—

C₂²×C₄

C₂×D₄×C₈₉

yes

characteristic

C₂×C₈

C₄⋊C₄×C₈₉

yes

—

C₂×C₈

SD₁₆×C₈₉

yes

—

C₂×C₈

C₂²×C₄×C₈₉

yes

characteristic

C₂×C₈

C₂×Q₈×C₈₉

yes

characteristic

C₂×C₈

#41×C₈₉

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄⋊C₄×C₈₉

yes

—

C₄²

Q₁₆×C₈₉

yes

—

C₁₆

—

C₂³

C₂×D₈×C₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#107×C₈₉

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₈₉

yes

characteristic

C₂×C₄

#124×C₈₉

yes

—

C₂×C₄

C₂×SD₁₆×C₈₉

yes

—

100

C₂×C₄

C₂×Q₁₆×C₈₉

yes

characteristic

101

C₈

#106×C₈₉

yes

—

102

C₈

#117×C₈₉

yes

—

103

C₈

#125×C₈₉

yes

—

104

C₂²

C₂×#107×C₈₉

yes

characteristic

105

C₂²

#486×C₈₉

yes

—

106

C₂²

#508×C₈₉

yes

—

107

C₄

—

108

C₄

—

109

C₄

#382×C₈₉

yes

characteristic

110

C₄

C₂×#106×C₈₉

yes

characteristic

111

C₄

#487×C₈₉

yes

—

112

C₂

C₈₉×#1834

yes

characteristic

113

C₂

C₈₉×#2158

yes

—

114

C₂

C₈₉×#2242

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₈₉

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₈₉

Frattini, cosocle

C₂²×C₈

C₂³×C₈₉

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
9. C₄×C₁₆×C₈₉	C₄	semidirect
54. #12889	C₈₉	direct (N × H)
61. C₈₉	#12889	direct (N × H)
67. C₄×C₁₆	C₄×C₈₉	semidirect

H ≅ G/N

type

9. C₄×C₁₆×C₈₉

C₄

semidirect

54. #12889

C₈₉

direct (N × H)

61. C₈₉

#12889

direct (N × H)

67. C₄×C₁₆

C₄×C₈₉

semidirect

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

subgroup	index
C₂×C₄×C₁₆×C₈₉	2
C₈₉×#1846	2
C₈₉×#1380	2
#12889	89

subgroup

index

C₂×C₄×C₁₆×C₈₉

C₈₉×#1846

C₈₉×#1380

#12889

Subgroups — C₈₉×#12889 #36390737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#12889 #36390737

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12889 #36390737