Group #26016643 — subgroups

Subgroups — C₈₉×#2169 #26016643

Order 11392.

162 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 100 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×M₃₂×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×C₈₉	C₂²	1	2	yes	—
7	C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
8	M₃₂×C₈₉	—	2	4	no	—
9	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
10	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
13	C₂²⋊C₈×C₈₉	C₄	1	2	yes	—
14	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×D₄×C₈₉	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₈₉	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₈₉	—	2	2	no	—
18	C₂×C₈×C₈₉	—	2	2	no	—
19	C₂×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₈₉	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
23	C₂²⋊C₄×C₈₉	C₈	1	2	yes	—
24	C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₁₆×C₈₉	—	2	2	no	—
26	C₁₆×C₈₉	—	2	1	no	—
27	C₁₆×C₈₉	D₄	1	2	yes	complemented
28	C₂³×C₈₉	C₂×C₈	1	2	yes	—
29	D₄×C₈₉	—	2	2	no	—
30	C₂×C₄×C₈₉	—	2	2	no	—
31	C₂×C₄×C₈₉	—	2	2	no	—
32	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₈₉	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
36	C₈×C₈₉	—	2	2	no	—
37	C₈×C₈₉	—	2	1	no	—
38	C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₈₉	#41	1	1	yes	characteristic
40	C₂²×C₈₉	—	2	2	no	—
41	C₂²×C₈₉	—	2	2	no	—
42	C₂²×C₈₉	—	2	2	no	—
43	C₂²×C₈₉	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₈₉	—	2	2	no	—
45	C₄×C₈₉	—	2	1	no	—
46	C₄×C₈₉	M₃₂	1	2	yes	complemented
47	C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
48	C₄×C₈₉	#131	1	1	yes	characteristic
49	C₂×C₈₉	—	4	2	no	—
50	C₂×C₈₉	C₈×D₄	1	1	yes	characteristic
51	C₂×C₈₉	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₈₉	#504	1	1	yes	characteristic
53	#2169	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₈₉	#2169	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×M₃₂	C₂×C₈₉	1	2	yes	complemented
56	C₈×D₄	C₂×C₈₉	1	1	yes	characteristic
57	C₄⋊C₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
58	C₂²⋊C₁₆	C₂×C₈₉	1	2	yes	complemented
59	C₄×C₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
60	C₂²×C₈	C₂²×C₈₉	1	2	yes	—
61	C₄×D₄	C₄×C₈₉	1	1	yes	characteristic
62	M₃₂	—	2	4	no	—
63	C₂×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
64	C₂×C₁₆	C₂²×C₈₉	1	2	yes	—
65	C₂×C₁₆	C₂²×C₈₉	1	1	yes	characteristic
66	C₄⋊C₈	C₄×C₈₉	1	1	yes	characteristic
67	C₂²⋊C₈	C₄×C₈₉	1	2	yes	—
68	C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
69	C₂×D₄	C₈×C₈₉	1	1	yes	characteristic
70	C₂²×C₄	C₂×C₄×C₈₉	1	2	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
74	C₂×C₈	C₂×C₄×C₈₉	1	2	yes	—
75	C₂×C₈	C₂³×C₈₉	1	1	yes	characteristic
76	C₄⋊C₄	C₈×C₈₉	1	1	yes	characteristic
77	C₂²⋊C₄	C₈×C₈₉	1	2	yes	—
78	C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	C₁₆	—	2	1	no	—
81	C₁₆	D₄×C₈₉	1	2	yes	complemented
82	C₂³	C₂×C₈×C₈₉	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
87	C₂×C₄	C₂×C₈×C₈₉	1	2	yes	—
88	C₂×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
89	C₂×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
90	C₈	—	2	2	no	—
91	C₈	—	2	1	no	—
92	C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
93	C₈	#41×C₈₉	1	1	yes	characteristic
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×C₈×C₈₉	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	M₃₂×C₈₉	1	2	yes	complemented
101	C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
102	C₄	#131×C₈₉	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	C₈×D₄×C₈₉	1	1	yes	characteristic
105	C₂	C₂×M₃₂×C₈₉	1	1	yes	characteristic
106	C₂	#504×C₈₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₃₂×C₈₉

C₂

yes

maximal, complemented

C₈×D₄×C₈₉

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²⋊C₁₆×C₈₉

C₂

yes

maximal, complemented

C₄×C₁₆×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×C₈₉

C₂²

yes

—

C₄×D₄×C₈₉

C₄

yes

characteristic

M₃₂×C₈₉

—

C₂×C₁₆×C₈₉

C₂²

yes

—

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

C₂×C₁₆×C₈₉

C₂²

yes

characteristic

C₄⋊C₈×C₈₉

C₄

yes

characteristic

C₂²⋊C₈×C₈₉

C₄

yes

—

C₄×C₈×C₈₉

C₂²

yes

characteristic

C₂×D₄×C₈₉

C₈

yes

characteristic

C₂²×C₄×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

—

C₂×C₈×C₈₉

C₂×C₄

yes

characteristic

C₂×C₈×C₈₉

C₂×C₄

yes

—

C₂×C₈×C₈₉

C₂³

yes

characteristic

C₄⋊C₄×C₈₉

C₈

yes

characteristic

C₂²⋊C₄×C₈₉

C₈

yes

—

C₄²×C₈₉

C₂×C₄

yes

characteristic

C₁₆×C₈₉

—

C₁₆×C₈₉

—

C₁₆×C₈₉

D₄

yes

complemented

C₂³×C₈₉

C₂×C₈

yes

—

D₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

—

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂×C₈

yes

—

C₂×C₄×C₈₉

C₂×C₈

yes

characteristic

C₂×C₄×C₈₉

C₂²×C₄

yes

characteristic

C₈×C₈₉

—

C₈×C₈₉

—

C₈×C₈₉

C₂×D₄

yes

characteristic

C₈×C₈₉

#41

yes

characteristic

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

—

C₂²×C₈₉

C₂²×C₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

—

C₄×C₈₉

M₃₂

yes

complemented

C₄×C₈₉

C₄×D₄

yes

characteristic

C₄×C₈₉

#131

yes

characteristic

C₂×C₈₉

—

C₂×C₈₉

C₈×D₄

yes

characteristic

C₂×C₈₉

C₂×M₃₂

yes

C₂×C₈₉

yes

C₈₉

yes

maximal, characteristic, direct product factor, Sylow

C₈₉

#2169

yes

characteristic, direct product factor, Sylow

C₂×M₃₂

C₂×C₈₉

yes

complemented

C₈×D₄

C₂×C₈₉

yes

characteristic

C₄⋊C₁₆

C₂×C₈₉

yes

characteristic, complemented

C₂²⋊C₁₆

C₂×C₈₉

yes

complemented

C₄×C₁₆

C₂×C₈₉

yes

characteristic, complemented

C₂²×C₈

C₂²×C₈₉

yes

—

C₄×D₄

C₄×C₈₉

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂²×C₈₉

yes

characteristic

C₂×C₁₆

C₂²×C₈₉

yes

—

C₂×C₁₆

C₂²×C₈₉

yes

characteristic

C₄⋊C₈

C₄×C₈₉

yes

characteristic

C₂²⋊C₈

C₄×C₈₉

yes

—

C₄×C₈

C₂²×C₈₉

yes

characteristic

C₂×D₄

C₈×C₈₉

yes

characteristic

C₂²×C₄

C₂×C₄×C₈₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄×C₈₉

yes

characteristic

C₂×C₈

C₂×C₄×C₈₉

yes

—

C₂×C₈

C₂³×C₈₉

yes

characteristic

C₄⋊C₄

C₈×C₈₉

yes

characteristic

C₂²⋊C₄

C₈×C₈₉

yes

—

C₄²

C₂×C₄×C₈₉

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

D₄×C₈₉

yes

complemented

C₂³

C₂×C₈×C₈₉

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₈×C₈₉

yes

characteristic

C₂×C₄

C₂×C₈×C₈₉

yes

—

C₂×C₄

C₂×C₈×C₈₉

yes

characteristic

C₂×C₄

C₂²×C₄×C₈₉

yes

characteristic

C₈

—

C₈

—

C₈

C₂×D₄×C₈₉

yes

characteristic

C₈

#41×C₈₉

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×C₈×C₈₉

yes

characteristic

C₄

—

C₄

—

100

C₄

M₃₂×C₈₉

yes

complemented

101

C₄

C₄×D₄×C₈₉

yes

characteristic

102

C₄

#131×C₈₉

yes

characteristic

103

C₂

—

104

C₂

C₈×D₄×C₈₉

yes

characteristic

105

C₂

C₂×M₃₂×C₈₉

yes

characteristic

106

C₂

#504×C₈₉

yes

characteristic

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂³
commutator	C₂²	C₂²×C₈×C₈₉
Frattini, cosocle	C₂×C₈	C₂³×C₈₉
socle	C₂²×C₈₉	C₂²×C₈
prime radical	C₂×D₄×C₈₉	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ G

subgroup

G/H

center

C₂×C₈×C₈₉

C₂³

commutator

C₂²

C₂²×C₈×C₈₉

Frattini, cosocle

C₂×C₈

C₂³×C₈₉

socle

C₂²×C₈₉

C₂²×C₈

prime radical

C₂×D₄×C₈₉

C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈×C₈₉ ◁ G

N	H ≅ G/N	type
1. C₂×M₃₂×C₈₉	C₂	semidirect
3. C₄⋊C₁₆×C₈₉	C₂	semidirect
4. C₂²⋊C₁₆×C₈₉	C₂	semidirect
5. C₄×C₁₆×C₈₉	C₂	semidirect
27. C₁₆×C₈₉	D₄	semidirect
46. C₄×C₈₉	M₃₂	semidirect
53. #2169	C₈₉	direct (N × H)
54. C₈₉	#2169	direct (N × H)
55. C₂×M₃₂	C₂×C₈₉	semidirect
57. C₄⋊C₁₆	C₂×C₈₉	semidirect
58. C₂²⋊C₁₆	C₂×C₈₉	semidirect
59. C₄×C₁₆	C₂×C₈₉	semidirect
81. C₁₆	D₄×C₈₉	semidirect
100. C₄	M₃₂×C₈₉	semidirect

H ≅ G/N

type

1. C₂×M₃₂×C₈₉

C₂

semidirect

3. C₄⋊C₁₆×C₈₉

C₂

semidirect

4. C₂²⋊C₁₆×C₈₉

C₂

semidirect

5. C₄×C₁₆×C₈₉

C₂

semidirect

27. C₁₆×C₈₉

D₄

semidirect

46. C₄×C₈₉

M₃₂

semidirect

53. #2169

C₈₉

direct (N × H)

54. C₈₉

#2169

direct (N × H)

55. C₂×M₃₂

C₂×C₈₉

semidirect

57. C₄⋊C₁₆

C₂×C₈₉

semidirect

58. C₂²⋊C₁₆

C₂×C₈₉

semidirect

59. C₄×C₁₆

C₂×C₈₉

semidirect

81. C₁₆

D₄×C₈₉

semidirect

100. C₄

M₃₂×C₈₉

semidirect

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

subgroup	index
C₂×M₃₂×C₈₉	2
C₈×D₄×C₈₉	2
C₄⋊C₁₆×C₈₉	2
C₂²⋊C₁₆×C₈₉	2
C₄×C₁₆×C₈₉	2
#2169	89

subgroup

index

C₂×M₃₂×C₈₉

C₈×D₄×C₈₉

C₄⋊C₁₆×C₈₉

C₂²⋊C₁₆×C₈₉

C₄×C₁₆×C₈₉

#2169

Subgroups — C₈₉×#2169 #26016643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C89×#2169 #26016643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#2169 #26016643