Group #26016716 — subgroups

Subgroups — C₈₉×#2242 #26016716

Order 11392.

138 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 64 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	#499×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×C₈₉	C₂	1	1	yes	maximal, characteristic
3	#358×C₈₉	C₂	1	2	yes	maximal, complemented
4	#357×C₈₉	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×SD₁₆×C₈₉	—	2	1	no	—
8	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
9	#125×C₈₉	—	2	1	no	—
10	#124×C₈₉	—	2	1	no	—
11	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
12	#107×C₈₉	C₂²	1	2	yes	—
13	#106×C₈₉	—	2	1	no	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₈₉	—	2	1	no	—
16	C₂×D₄×C₈₉	—	2	1	no	—
17	Q₁₆×C₈₉	—	2	2	no	—
18	SD₁₆×C₈₉	—	4	2	no	—
19	D₈×C₈₉	—	2	2	no	—
20	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₈₉	—	2	2	no	—
23	C₄⋊C₄×C₈₉	—	4	2	no	—
24	C₂²⋊C₄×C₈₉	—	4	2	no	—
25	C₄²×C₈₉	D₄	1	1	yes	characteristic
26	C₁₆×C₈₉	—	2	2	no	—
27	C₂³×C₈₉	—	4	1	no	—
28	Q₈×C₈₉	—	4	1	no	—
29	Q₈×C₈₉	—	4	1	no	—
30	D₄×C₈₉	—	4	1	no	—
31	D₄×C₈₉	—	4	1	no	—
32	C₂×C₄×C₈₉	—	4	1	no	—
33	C₂×C₄×C₈₉	—	4	2	no	—
34	C₂×C₄×C₈₉	D₈	1	2	yes	—
35	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
36	C₈×C₈₉	SD₁₆	1	2	yes	—
37	C₈×C₈₉	#41	1	2	yes	—
38	C₂²×C₈₉	—	8	1	no	—
39	C₂²×C₈₉	—	8	2	no	—
40	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
41	C₄×C₈₉	—	2	2	no	—
42	C₄×C₈₉	—	8	1	no	—
43	C₄×C₈₉	—	8	2	no	—
44	C₄×C₈₉	#124	1	1	yes	characteristic
45	C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
46	C₂×C₈₉	—	16	1	no	—
47	C₂×C₈₉	#486	1	1	yes	characteristic
48	C₂×C₈₉	#508	1	2	yes	—
49	#2242	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₈₉	#2242	1	1	yes	characteristic, direct product factor, Sylow
51	#499	C₂×C₈₉	1	1	yes	characteristic, complemented
52	#495	C₂×C₈₉	1	1	yes	characteristic
53	#358	C₂×C₈₉	1	2	yes	complemented
54	#357	C₂×C₈₉	1	2	yes	—
55	C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
56	C₂×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
57	C₂×SD₁₆	—	2	1	no	—
58	C₂×D₈	C₂²×C₈₉	1	1	yes	characteristic
59	#125	—	2	1	no	—
60	#124	—	2	1	no	—
61	C₂×C₁₆	C₂²×C₈₉	1	2	yes	—
62	#107	C₂²×C₈₉	1	2	yes	—
63	#106	—	2	1	no	—
64	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
65	C₂×Q₈	—	2	1	no	—
66	C₂×D₄	—	2	1	no	—
67	Q₁₆	—	2	2	no	—
68	SD₁₆	—	4	2	no	—
69	D₈	—	2	2	no	—
70	C₂×C₈	D₄×C₈₉	1	1	yes	characteristic
71	C₂×C₈	C₂³×C₈₉	1	1	yes	characteristic
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₂²⋊C₄	—	4	2	no	—
75	C₄²	D₄×C₈₉	1	1	yes	characteristic
76	C₁₆	—	2	2	no	—
77	C₂³	—	4	1	no	—
78	Q₈	—	4	1	no	—
79	Q₈	—	4	1	no	—
80	D₄	—	4	1	no	—
81	D₄	—	4	1	no	—
82	C₂×C₄	—	4	1	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	D₈×C₈₉	1	2	yes	—
85	C₂×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
86	C₈	SD₁₆×C₈₉	1	2	yes	—
87	C₈	#41×C₈₉	1	2	yes	—
88	C₂²	—	8	1	no	—
89	C₂²	—	8	2	no	—
90	C₂²	C₂×D₈×C₈₉	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	8	1	no	—
93	C₄	—	8	2	no	—
94	C₄	#124×C₈₉	1	1	yes	characteristic
95	C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
96	C₂	—	16	1	no	—
97	C₂	#486×C₈₉	1	1	yes	characteristic
98	C₂	#508×C₈₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#499×C₈₉

C₂

yes

maximal, characteristic, complemented

#495×C₈₉

C₂

yes

maximal, characteristic

#358×C₈₉

C₂

yes

maximal, complemented

#357×C₈₉

C₂

yes

maximal

C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂×Q₁₆×C₈₉

C₂²

yes

characteristic

C₂×SD₁₆×C₈₉

—

C₂×D₈×C₈₉

C₂²

yes

characteristic

#125×C₈₉

—

#124×C₈₉

—

C₂×C₁₆×C₈₉

C₂²

yes

—

#107×C₈₉

C₂²

yes

—

#106×C₈₉

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×Q₈×C₈₉

—

C₂×D₄×C₈₉

—

Q₁₆×C₈₉

—

SD₁₆×C₈₉

—

D₈×C₈₉

—

C₂×C₈×C₈₉

D₄

yes

characteristic

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

—

C₄⋊C₄×C₈₉

—

C₂²⋊C₄×C₈₉

—

C₄²×C₈₉

D₄

yes

characteristic

C₁₆×C₈₉

—

C₂³×C₈₉

—

Q₈×C₈₉

—

Q₈×C₈₉

—

D₄×C₈₉

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

D₈

yes

—

C₂×C₄×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

SD₁₆

yes

—

C₈×C₈₉

#41

yes

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

#124

yes

characteristic

C₄×C₈₉

C₂×SD₁₆

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

#486

yes

characteristic

C₂×C₈₉

#508

yes

—

#2242

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#2242

yes

characteristic, direct product factor, Sylow

#499

C₂×C₈₉

yes

characteristic, complemented

#495

C₂×C₈₉

yes

characteristic

#358

C₂×C₈₉

yes

complemented

#357

C₂×C₈₉

yes

—

C₄×C₁₆

C₂×C₈₉

yes

characteristic, complemented

C₂×Q₁₆

C₂²×C₈₉

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂²×C₈₉

yes

characteristic

#125

—

#124

—

C₂×C₁₆

C₂²×C₈₉

yes

—

#107

C₂²×C₈₉

yes

—

#106

—

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₈₉

yes

characteristic

C₂×C₈

C₂³×C₈₉

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₄×C₈₉

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₈×C₈₉

yes

—

C₂×C₄

C₂×D₄×C₈₉

yes

characteristic

C₈

SD₁₆×C₈₉

yes

—

C₈

#41×C₈₉

yes

—

C₂²

—

C₂²

—

C₂²

C₂×D₈×C₈₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#124×C₈₉

yes

characteristic

C₄

C₂×SD₁₆×C₈₉

yes

characteristic

C₂

—

C₂

#486×C₈₉

yes

characteristic

C₂

#508×C₈₉

yes

—

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

subgroup

G/H

center, socle

C₂²×C₈₉

C₂×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂³×C₈₉

prime radical

C₂×D₈×C₈₉

C₂²

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈₉ ◁ C₄²×C₈₉ ◁ C₄×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
1. #499×C₈₉	C₂	semidirect
3. #358×C₈₉	C₂	semidirect
5. C₄×C₁₆×C₈₉	C₂	semidirect
49. #2242	C₈₉	direct (N × H)
50. C₈₉	#2242	direct (N × H)
51. #499	C₂×C₈₉	semidirect
53. #358	C₂×C₈₉	semidirect
55. C₄×C₁₆	C₂×C₈₉	semidirect

H ≅ G/N

type

1. #499×C₈₉

C₂

semidirect

3. #358×C₈₉

C₂

semidirect

5. C₄×C₁₆×C₈₉

C₂

semidirect

49. #2242

C₈₉

direct (N × H)

50. C₈₉

#2242

direct (N × H)

51. #499

C₂×C₈₉

semidirect

53. #358

C₂×C₈₉

semidirect

55. C₄×C₁₆

C₂×C₈₉

semidirect

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

subgroup	index
#499×C₈₉	2
#495×C₈₉	2
#358×C₈₉	2
#357×C₈₉	2
C₄×C₁₆×C₈₉	2
#2242	89

subgroup

index

#499×C₈₉

#495×C₈₉

#358×C₈₉

#357×C₈₉

C₄×C₁₆×C₈₉

#2242

Subgroups — C₈₉×#2242 #26016716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#2242 #26016716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#2242 #26016716