Group #26016164 — subgroups

Subgroups — C₈₉×#1690 #26016164

Order 11392.

152 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 68 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	#494×C₈₉	C₂	1	1	yes	maximal, characteristic
2	#475×C₈₉	C₂	1	2	yes	maximal
3	#446×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#334×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#332×C₈₉	C₂	1	2	yes	maximal
6	C₂×Q₁₆×C₈₉	—	2	1	no	—
7	C₂×Q₁₆×C₈₉	—	2	1	no	—
8	C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
9	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
10	#106×C₈₉	—	2	2	no	—
11	C₄⋊C₈×C₈₉	—	2	1	no	—
12	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
14	#103×C₈₉	—	2	2	no	—
15	#97×C₈₉	—	2	1	no	—
16	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₈₉	—	2	2	no	—
18	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
19	Q₁₆×C₈₉	—	4	2	no	—
20	Q₁₆×C₈₉	—	4	2	no	—
21	C₂×C₈×C₈₉	—	2	1	no	—
22	C₂×C₈×C₈₉	—	2	2	no	—
23	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
24	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
25	C₄⋊C₄×C₈₉	—	2	1	no	—
26	C₄⋊C₄×C₈₉	—	2	2	no	—
27	C₄⋊C₄×C₈₉	—	2	2	no	—
28	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₄²×C₈₉	—	2	1	no	—
30	C₄²×C₈₉	C₂³	1	1	yes	characteristic
31	Q₈×C₈₉	—	2	2	no	—
32	Q₈×C₈₉	—	4	2	no	—
33	Q₈×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	—	2	1	no	—
35	C₂×C₄×C₈₉	—	2	1	no	—
36	C₂×C₄×C₈₉	—	4	2	no	—
37	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₈₉	#41	1	1	yes	characteristic
40	C₈×C₈₉	—	2	1	no	—
41	C₈×C₈₉	—	4	1	no	—
42	C₈×C₈₉	—	4	2	no	—
43	C₈×C₈₉	SD₁₆	1	2	yes	—
44	C₂²×C₈₉	#121	1	1	yes	characteristic
45	C₄×C₈₉	—	2	1	no	—
46	C₄×C₈₉	—	4	1	no	—
47	C₄×C₈₉	—	4	1	no	—
48	C₄×C₈₉	—	8	2	no	—
49	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
50	C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
51	C₄×C₈₉	#137	1	2	yes	—
52	C₂×C₈₉	#460	1	1	yes	characteristic
53	C₂×C₈₉	#470	1	1	yes	characteristic
54	C₂×C₈₉	#473	1	1	yes	characteristic
55	#1690	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₈₉	#1690	1	1	yes	characteristic, direct product factor, Sylow
57	#494	C₂×C₈₉	1	1	yes	characteristic
58	#475	C₂×C₈₉	1	2	yes	—
59	#446	C₂×C₈₉	1	1	yes	characteristic
60	#334	C₂×C₈₉	1	1	yes	characteristic
61	#332	C₂×C₈₉	1	2	yes	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
65	C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
66	#106	—	2	2	no	—
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
69	C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
70	#103	—	2	2	no	—
71	#97	—	2	1	no	—
72	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	D₄×C₈₉	1	1	yes	characteristic
75	Q₁₆	—	4	2	no	—
76	Q₁₆	—	4	2	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	D₄×C₈₉	1	1	yes	characteristic
80	C₂×C₈	D₄×C₈₉	1	1	yes	characteristic
81	C₄⋊C₄	—	2	1	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
85	C₄²	—	2	1	no	—
86	C₄²	C₂³×C₈₉	1	1	yes	characteristic
87	Q₈	—	2	2	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
94	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
95	C₂×C₄	#41×C₈₉	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	4	1	no	—
98	C₈	—	4	2	no	—
99	C₈	SD₁₆×C₈₉	1	2	yes	—
100	C₂²	#121×C₈₉	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	4	1	no	—
103	C₄	—	4	1	no	—
104	C₄	—	8	2	no	—
105	C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
106	C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
107	C₄	#137×C₈₉	1	2	yes	—
108	C₂	#460×C₈₉	1	1	yes	characteristic
109	C₂	#470×C₈₉	1	1	yes	characteristic
110	C₂	#473×C₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#494×C₈₉

C₂

yes

maximal, characteristic

#475×C₈₉

C₂

yes

maximal

#446×C₈₉

C₂

yes

maximal, characteristic

#334×C₈₉

C₂

yes

maximal, characteristic

#332×C₈₉

C₂

yes

maximal

C₂×Q₁₆×C₈₉

—

C₂×Q₁₆×C₈₉

—

C₄⋊Q₈×C₈₉

C₂²

yes

—

C₄×Q₈×C₈₉

C₂²

yes

characteristic

#106×C₈₉

—

C₄⋊C₈×C₈₉

—

C₄⋊C₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₂²

yes

—

#103×C₈₉

—

#97×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×Q₈×C₈₉

D₄

yes

characteristic

Q₁₆×C₈₉

—

Q₁₆×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

D₄

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

D₄

yes

characteristic

C₄²×C₈₉

—

C₄²×C₈₉

C₂³

yes

characteristic

Q₈×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₂×C₄×C₈₉

#41

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

SD₁₆

yes

—

C₂²×C₈₉

#121

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

C₂×SD₁₆

yes

C₄×C₈₉

yes

C₄×C₈₉

yes

—

C₂×C₈₉

yes

C₂×C₈₉

yes

C₂×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#1690

yes

characteristic, direct product factor, Sylow

#494

C₂×C₈₉

yes

characteristic

#475

C₂×C₈₉

yes

—

#446

C₂×C₈₉

yes

characteristic

#334

C₂×C₈₉

yes

characteristic

#332

C₂×C₈₉

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₈₉

yes

—

C₄×Q₈

C₂²×C₈₉

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₈₉

yes

characteristic

C₄⋊C₈

C₂²×C₈₉

yes

—

#103

—

#97

—

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₈₉

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₈₉

yes

characteristic

C₂×C₈

D₄×C₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₈₉

yes

characteristic

C₄²

—

C₄²

C₂³×C₈₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

C₂×C₄

#41×C₈₉

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

SD₁₆×C₈₉

yes

—

100

C₂²

#121×C₈₉

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₂×SD₁₆×C₈₉

yes

characteristic

106

C₄

AGL(1,ℤ/8)×C₈₉

yes

characteristic

107

C₄

#137×C₈₉

yes

—

108

C₂

#460×C₈₉

yes

characteristic

109

C₂

#470×C₈₉

yes

characteristic

110

C₂

#473×C₈₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₈₉	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₈₉
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₈₉

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₈₉

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ G

N	H ≅ G/N	type
55. #1690	C₈₉	direct (N × H)
56. C₈₉	#1690	direct (N × H)

H ≅ G/N

type

55. #1690

C₈₉

direct (N × H)

56. C₈₉

#1690

direct (N × H)

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

subgroup	index
#494×C₈₉	2
#475×C₈₉	2
#446×C₈₉	2
#334×C₈₉	2
#332×C₈₉	2
#1690	89

subgroup

index

#494×C₈₉

#475×C₈₉

#446×C₈₉

#334×C₈₉

#332×C₈₉

#1690

Subgroups — C₈₉×#1690 #26016164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#1690 #26016164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#1690 #26016164