Group #28586009 — subgroups

Subgroups — C₂×C₄²×C₅₉⋊C₈ #28586009

Order 15104.

2046 conjugacy classes of subgroups; only the 1659 normal subgroups are shown.

normal subgroups only

This group has 2046 conjugacy classes of subgroups — too many to draw. Only its 1659 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	C₂²×C₄×C₅₉⋊C₈	C₂	1	6	yes	maximal
2	C₄²×C₅₉⋊C₈	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄³×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₅₉⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₅₉⋊C₈	C₄	1	48	yes	direct product factor
6	C₂×C₄×C₅₉⋊C₈	C₂²	1	24	yes	—
7	C₂²×C₄²×C₅₉	C₄	1	4	yes	—
8	C₂²×C₄²×C₅₉	C₂²	1	3	yes	—
9	C₄³×C₅₉	C₄	1	4	yes	—
10	C₄³×C₅₉	C₂²	1	4	yes	—
11	C₂²×C₅₉⋊C₈	C₂×C₄	1	48	yes	—
12	C₂²×C₅₉⋊C₈	C₂³	1	8	yes	—
13	C₄×C₅₉⋊C₈	C₂×C₄	1	96	yes	direct product factor
14	C₂³×C₄×C₅₉	C₂×C₄	1	6	yes	—
15	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₉	C₈	1	32	yes	complemented
17	C₂×C₄²×C₅₉	C₂×C₄	1	24	yes	—
18	C₂×C₄²×C₅₉	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₅₉	C₂×C₄	1	16	yes	—
20	C₂×C₄²×C₅₉	C₂³	1	6	yes	—
21	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₉	C₄²	1	24	yes	—
23	C₂²×C₄×C₅₉	C₂×C₈	1	48	yes	—
24	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
26	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
27	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
28	C₄²×C₅₉	C₄²	1	24	yes	—
29	C₄²×C₅₉	C₂×C₈	1	64	yes	complemented
30	C₄²×C₅₉	C₂²×C₄	1	24	yes	—
31	C₂×C₅₉⋊C₈	C₄²	1	64	yes	direct product factor
32	C₂×C₅₉⋊C₈	C₂²×C₄	1	48	yes	—
33	C₂³×C₅₉	C₂×C₄²	1	6	yes	—
34	C₂³×C₅₉	C₂²×C₈	1	8	yes	—
35	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₉	C₄×C₈	1	96	yes	complemented
37	C₂×C₄×C₅₉	C₂×C₄²	1	16	yes	—
38	C₂×C₄×C₅₉	C₂×C₄²	1	6	yes	—
39	C₂×C₄×C₅₉	C₂×C₄²	1	24	yes	—
40	C₂×C₄×C₅₉	C₂²×C₈	1	48	yes	—
41	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
42	C₅₉⋊C₈	C₂×C₄²	1	32	yes	direct product factor
43	C₂×C₄²×C₈	—	59	1	no	maximal, Sylow
44	C₂²×C₅₉	C₄³	1	4	yes	—
45	C₂²×C₅₉	C₂×C₄×C₈	1	24	yes	—
46	C₂²×C₅₉	C₂²×C₄²	1	3	yes	—
47	C₂²×C₅₉	C₂³×C₈	1	4	yes	—
48	C₄×C₅₉	C₄³	1	4	yes	—
49	C₄×C₅₉	C₂×C₄×C₈	1	48	yes	complemented
50	C₄×C₅₉	C₂²×C₄²	1	4	yes	—
51	C₂²×C₄×C₈	—	59	6	no	—
52	C₂×C₄³	D₅₉	1	1	yes	characteristic
53	C₄²×C₈	—	59	8	no	—
54	C₂×C₅₉	C₄²×C₈	1	8	yes	complemented
55	C₂×C₅₉	C₂×C₄³	1	1	yes	characteristic
56	C₂×C₅₉	C₂²×C₄×C₈	1	6	yes	—
57	C₂³×C₈	—	59	4	no	—
58	C₂²×C₄²	Dic₅₉	1	4	yes	—
59	C₂²×C₄²	C₂×D₅₉	1	3	yes	—
60	C₂×C₄×C₈	—	59	48	no	—
61	C₂×C₄×C₈	—	59	24	no	—
62	C₄³	Dic₅₉	1	4	yes	—
63	C₄³	C₂×D₅₉	1	4	yes	—
64	C₅₉	C₂×C₄²×C₈	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₄	C₂×Dic₅₉	1	6	yes	—
66	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
67	C₂²×C₈	—	59	48	no	—
68	C₂²×C₈	—	59	8	no	—
69	C₂×C₄²	C₅₉⋊C₈	1	32	yes	direct product factor
70	C₂×C₄²	C₄×D₅₉	1	24	yes	—
71	C₂×C₄²	C₂×Dic₅₉	1	16	yes	—
72	C₂×C₄²	C₂×Dic₅₉	1	6	yes	—
73	C₂×C₄²	C₂²×D₅₉	1	6	yes	—
74	C₄×C₈	—	59	96	no	—
75	C₂⁴	C₂²×Dic₅₉	1	1	yes	characteristic
76	C₂²×C₄	C₂×C₅₉⋊C₈	1	48	yes	—
77	C₂²×C₄	C₄×Dic₅₉	1	24	yes	—
78	C₂²×C₄	C₂×C₄×D₅₉	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
80	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
81	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
82	C₂×C₈	—	59	64	no	—
83	C₂×C₈	—	59	48	no	—
84	C₄²	C₂×C₅₉⋊C₈	1	64	yes	direct product factor
85	C₄²	C₄×Dic₅₉	1	24	yes	—
86	C₄²	C₂×C₄×D₅₉	1	24	yes	—
87	C₂³	C₂×C₄×Dic₅₉	1	6	yes	—
88	C₂³	C₂²×C₅₉⋊C₈	1	8	yes	—
89	C₂³	C₂³×Dic₅₉	1	1	yes	characteristic
90	C₂×C₄	C₄×C₅₉⋊C₈	1	96	yes	direct product factor
91	C₂×C₄	C₄²×D₅₉	1	16	yes	—
92	C₂×C₄	C₂×C₄×Dic₅₉	1	6	yes	—
93	C₂×C₄	C₂×C₄×Dic₅₉	1	24	yes	—
94	C₂×C₄	C₂²×C₅₉⋊C₈	1	48	yes	—
95	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
96	C₈	—	59	32	no	—
97	C₂²	C₄²×Dic₅₉	1	4	yes	—
98	C₂²	C₂×C₄×C₅₉⋊C₈	1	24	yes	—
99	C₂²	C₂²×C₄×Dic₅₉	1	3	yes	—
100	C₂²	C₂³×C₅₉⋊C₈	1	4	yes	—
101	C₄	C₄²×Dic₅₉	1	4	yes	—
102	C₄	C₂×C₄×C₅₉⋊C₈	1	48	yes	direct product factor
103	C₄	C₂×C₄²×D₅₉	1	4	yes	—
104	C₂	C₄²×C₅₉⋊C₈	1	8	yes	direct product factor
105	C₂	C₂×C₄²×Dic₅₉	1	1	yes	characteristic
106	C₂	C₂²×C₄×C₅₉⋊C₈	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₅₉⋊C₈

C₂

yes

maximal

C₄²×C₅₉⋊C₈

C₂

yes

maximal, direct product factor

C₂×C₄³×C₅₉

C₂

yes

maximal, characteristic

C₂³×C₅₉⋊C₈

C₂²

yes

—

C₂×C₄×C₅₉⋊C₈

C₄

yes

direct product factor

C₂×C₄×C₅₉⋊C₈

C₂²

yes

—

C₂²×C₄²×C₅₉

C₄

yes

—

C₂²×C₄²×C₅₉

C₂²

yes

—

C₄³×C₅₉

C₄

yes

—

C₄³×C₅₉

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂×C₄

yes

—

C₂²×C₅₉⋊C₈

C₂³

yes

—

C₄×C₅₉⋊C₈

C₂×C₄

yes

direct product factor

C₂³×C₄×C₅₉

C₂×C₄

yes

—

C₂³×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₄²×C₅₉

C₈

yes

complemented

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂×C₄

yes

—

C₂×C₄²×C₅₉

C₂³

yes

—

C₂⁴×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₄²

yes

—

C₂²×C₄×C₅₉

C₂×C₈

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×C₄×C₅₉

C₂²×C₄

yes

—

C₂²×C₄×C₅₉

C₂⁴

yes

characteristic

C₄²×C₅₉

C₄²

yes

—

C₄²×C₅₉

C₂×C₈

yes

complemented

C₄²×C₅₉

C₂²×C₄

yes

—

C₂×C₅₉⋊C₈

C₄²

yes

direct product factor

C₂×C₅₉⋊C₈

C₂²×C₄

yes

—

C₂³×C₅₉

C₂×C₄²

yes

—

C₂³×C₅₉

C₂²×C₈

yes

—

C₂³×C₅₉

C₂³×C₄

yes

characteristic

C₂×C₄×C₅₉

C₄×C₈

yes

complemented

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂×C₄²

yes

—

C₂×C₄×C₅₉

C₂²×C₈

yes

—

C₂×C₄×C₅₉

C₂³×C₄

yes

—

C₅₉⋊C₈

C₂×C₄²

yes

direct product factor

C₂×C₄²×C₈

—

maximal, Sylow

C₂²×C₅₉

C₄³

yes

—

C₂²×C₅₉

C₂×C₄×C₈

yes

—

C₂²×C₅₉

C₂²×C₄²

yes

—

C₂²×C₅₉

C₂³×C₈

yes

—

C₄×C₅₉

C₄³

yes

—

C₄×C₅₉

C₂×C₄×C₈

yes

complemented

C₄×C₅₉

C₂²×C₄²

yes

—

C₂²×C₄×C₈

—

C₂×C₄³

D₅₉

yes

characteristic

C₄²×C₈

—

C₂×C₅₉

C₄²×C₈

yes

complemented

C₂×C₅₉

C₂×C₄³

yes

characteristic

C₂×C₅₉

C₂²×C₄×C₈

yes

—

C₂³×C₈

—

C₂²×C₄²

Dic₅₉

yes

—

C₂²×C₄²

C₂×D₅₉

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄³

Dic₅₉

yes

—

C₄³

C₂×D₅₉

yes

—

C₅₉

C₂×C₄²×C₈

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₉

yes

—

C₂³×C₄

C₂²×D₅₉

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₅₉⋊C₈

yes

direct product factor

C₂×C₄²

C₄×D₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂×Dic₅₉

yes

—

C₂×C₄²

C₂²×D₅₉

yes

—

C₄×C₈

—

C₂⁴

C₂²×Dic₅₉

yes

characteristic

C₂²×C₄

C₂×C₅₉⋊C₈

yes

—

C₂²×C₄

C₄×Dic₅₉

yes

—

C₂²×C₄

C₂×C₄×D₅₉

yes

—

C₂²×C₄

C₂²×Dic₅₉

yes

characteristic

C₂²×C₄

C₂²×Dic₅₉

yes

—

C₂²×C₄

C₂³×D₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

C₂×C₅₉⋊C₈

yes

direct product factor

C₄²

C₄×Dic₅₉

yes

—

C₄²

C₂×C₄×D₅₉

yes

—

C₂³

C₂×C₄×Dic₅₉

yes

—

C₂³

C₂²×C₅₉⋊C₈

yes

—

C₂³

C₂³×Dic₅₉

yes

characteristic

C₂×C₄

C₄×C₅₉⋊C₈

yes

direct product factor

C₂×C₄

C₄²×D₅₉

yes

—

C₂×C₄

C₂×C₄×Dic₅₉

yes

—

C₂×C₄

C₂×C₄×Dic₅₉

yes

—

C₂×C₄

C₂²×C₅₉⋊C₈

yes

—

C₂×C₄

C₂²×C₄×D₅₉

yes

—

C₈

—

C₂²

C₄²×Dic₅₉

yes

—

C₂²

C₂×C₄×C₅₉⋊C₈

yes

—

C₂²

C₂²×C₄×Dic₅₉

yes

—

100

C₂²

C₂³×C₅₉⋊C₈

yes

—

101

C₄

C₄²×Dic₅₉

yes

—

102

C₄

C₂×C₄×C₅₉⋊C₈

yes

direct product factor

103

C₄

C₂×C₄²×D₅₉

yes

—

104

C₂

C₄²×C₅₉⋊C₈

yes

direct product factor

105

C₂

C₂×C₄²×Dic₅₉

yes

characteristic

106

C₂

C₂²×C₄×C₅₉⋊C₈

yes

—

	subgroup	G/H
center, hypercenter	C₂×C₄³	D₅₉
commutator, hypocenter	C₅₉	C₂×C₄²×C₈
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₂×C₄³×C₅₉	C₂
socle, prime radical	C₂⁴×C₅₉	C₂²×C₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
LCS	G ▷ C₅₉
UCS	C₁ ◁ C₂×C₄³

subgroup

G/H

center, hypercenter

C₂×C₄³

D₅₉

commutator, hypocenter

C₅₉

C₂×C₄²×C₈

Frattini

C₂²×C₄

C₂³×D₅₉

Fitting

C₂×C₄³×C₅₉

C₂

socle, prime radical

C₂⁴×C₅₉

C₂²×C₄

cosocle

C₂²×C₄×C₅₉

C₂⁴

LCS

G ▷ C₅₉

UCS

C₁ ◁ C₂×C₄³

N	H ≅ G/N	type
2. C₄²×C₅₉⋊C₈	C₂	direct (N × H)
5. C₂×C₄×C₅₉⋊C₈	C₄	direct (N × H)
13. C₄×C₅₉⋊C₈	C₂×C₄	direct (N × H)
16. C₂×C₄²×C₅₉	C₈	semidirect
29. C₄²×C₅₉	C₂×C₈	semidirect
31. C₂×C₅₉⋊C₈	C₄²	direct (N × H)
36. C₂×C₄×C₅₉	C₄×C₈	semidirect
42. C₅₉⋊C₈	C₂×C₄²	direct (N × H)
49. C₄×C₅₉	C₂×C₄×C₈	semidirect
54. C₂×C₅₉	C₄²×C₈	semidirect
64. C₅₉	C₂×C₄²×C₈	semidirect
69. C₂×C₄²	C₅₉⋊C₈	direct (N × H)
84. C₄²	C₂×C₅₉⋊C₈	direct (N × H)
90. C₂×C₄	C₄×C₅₉⋊C₈	direct (N × H)
102. C₄	C₂×C₄×C₅₉⋊C₈	direct (N × H)
104. C₂	C₄²×C₅₉⋊C₈	direct (N × H)

H ≅ G/N

type

2. C₄²×C₅₉⋊C₈

C₂

direct (N × H)

5. C₂×C₄×C₅₉⋊C₈

C₄

direct (N × H)

13. C₄×C₅₉⋊C₈

C₂×C₄

direct (N × H)

16. C₂×C₄²×C₅₉

C₈

semidirect

29. C₄²×C₅₉

C₂×C₈

semidirect

31. C₂×C₅₉⋊C₈

C₄²

direct (N × H)

36. C₂×C₄×C₅₉

C₄×C₈

semidirect

42. C₅₉⋊C₈

C₂×C₄²

direct (N × H)

49. C₄×C₅₉

C₂×C₄×C₈

semidirect

54. C₂×C₅₉

C₄²×C₈

semidirect

64. C₅₉

C₂×C₄²×C₈

semidirect

69. C₂×C₄²

C₅₉⋊C₈

direct (N × H)

84. C₄²

C₂×C₅₉⋊C₈

direct (N × H)

90. C₂×C₄

C₄×C₅₉⋊C₈

direct (N × H)

102. C₄

C₂×C₄×C₅₉⋊C₈

direct (N × H)

104. C₂

C₄²×C₅₉⋊C₈

direct (N × H)

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

subgroup	index
C₂²×C₄×C₅₉⋊C₈	2
C₄²×C₅₉⋊C₈	2
C₂×C₄³×C₅₉	2
C₂×C₄²×C₈	59

subgroup

index

C₂²×C₄×C₅₉⋊C₈

C₄²×C₅₉⋊C₈

C₂×C₄³×C₅₉

C₂×C₄²×C₈

Subgroups — C₂×C₄²×C₅₉⋊C₈ #28586009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C42×C59⋊C8 #28586009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄²×C₅₉⋊C₈ #28586009