Group #21420350 — subgroups

Subgroups — C₅₉×#2032 #21420350

Order 7552.

278 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₅₉×#543	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#99×C₅₉	C₂	1	3	yes	maximal
3	C₅₉×#342	C₂	1	3	yes	maximal, complemented
4	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
5	C₂²⋊Q₈×C₅₉	—	2	6	no	—
6	C₂×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	—
7	C₂×C₂²⋊C₄×C₅₉	C₂²	1	3	yes	—
8	#99×C₅₉	—	2	6	no	—
9	C₂⁴×C₅₉	Q₈	1	1	yes	characteristic, complemented
10	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
11	C₂×Q₈×C₅₉	—	2	2	no	—
12	C₂×D₄×C₅₉	—	2	3	no	—
13	C₂×D₄×C₅₉	—	2	3	no	—
14	C₂²×C₄×C₅₉	—	2	3	no	—
15	C₂²×C₄×C₅₉	—	2	3	no	—
16	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic, complemented
17	C₄⋊C₄×C₅₉	—	2	6	no	—
18	C₂²⋊C₄×C₅₉	—	2	3	no	—
19	C₂²⋊C₄×C₅₉	—	2	3	no	—
20	C₂²⋊C₄×C₅₉	—	2	3	no	—
21	C₂²⋊C₄×C₅₉	D₄	1	6	yes	—
22	C₂³×C₅₉	—	2	3	no	—
23	C₂³×C₅₉	—	8	1	no	—
24	C₂³×C₅₉	—	8	1	no	—
25	C₂³×C₅₉	C₂×D₄	1	3	yes	—
26	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
27	C₂³×C₅₉	#41	1	3	yes	—
28	Q₈×C₅₉	—	8	2	no	—
29	D₄×C₅₉	—	4	1	no	—
30	D₄×C₅₉	—	4	3	no	—
31	C₂×C₄×C₅₉	—	2	6	no	—
32	C₂×C₄×C₅₉	—	2	3	no	—
33	C₂×C₄×C₅₉	—	4	3	no	—
34	C₂×C₄×C₅₉	—	8	3	no	—
35	C₂×C₄×C₅₉	—	8	3	no	—
36	C₂²×C₅₉	—	4	3	no	—
37	C₂²×C₅₉	—	4	3	no	—
38	C₂²×C₅₉	—	4	1	no	—
39	C₂²×C₅₉	—	4	1	no	—
40	C₂²×C₅₉	—	8	3	no	—
41	C₂²×C₅₉	#120	1	1	yes	characteristic
42	C₂²×C₅₉	C₂²⋊Q₈	1	3	yes	—
43	C₂²×C₅₉	#124	1	3	yes	—
44	C₄×C₅₉	—	4	6	no	—
45	C₄×C₅₉	—	4	1	no	—
46	C₄×C₅₉	—	8	3	no	—
47	#2032	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂×C₅₉	—	2	1	no	—
49	C₂×C₅₉	—	2	3	no	—
50	C₂×C₅₉	—	2	3	no	—
51	C₂×C₅₉	—	8	1	no	—
52	C₂×C₅₉	#393	1	1	yes	characteristic
53	#543	C₂×C₅₉	1	1	yes	characteristic, complemented
54	C₂×#99	C₂×C₅₉	1	3	yes	—
55	#342	C₂×C₅₉	1	3	yes	complemented
56	C₅₉	#2032	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×D₄	C₂²×C₅₉	1	1	yes	characteristic
58	C₂²⋊Q₈	—	2	6	no	—
59	C₂×C₂²⋊C₄	C₂²×C₅₉	1	3	yes	—
60	C₂×C₂²⋊C₄	C₂²×C₅₉	1	3	yes	—
61	#99	—	2	6	no	—
62	C₂⁴	Q₈×C₅₉	1	1	yes	characteristic, complemented
63	C₂⁴	C₂³×C₅₉	1	1	yes	characteristic
64	C₂×Q₈	—	2	2	no	—
65	C₂×D₄	—	2	3	no	—
66	C₂×D₄	—	2	3	no	—
67	C₂²×C₄	—	2	3	no	—
68	C₂²×C₄	—	2	3	no	—
69	C₂²×C₄	Q₈×C₅₉	1	1	yes	characteristic, complemented
70	C₄⋊C₄	—	2	6	no	—
71	C₂²⋊C₄	—	2	3	no	—
72	C₂²⋊C₄	—	2	3	no	—
73	C₂²⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	D₄×C₅₉	1	6	yes	—
75	C₂³	—	2	3	no	—
76	C₂³	—	8	1	no	—
77	C₂³	—	8	1	no	—
78	C₂³	C₂×D₄×C₅₉	1	3	yes	—
79	C₂³	C₂×Q₈×C₅₉	1	1	yes	characteristic
80	C₂³	#41×C₅₉	1	3	yes	—
81	Q₈	—	8	2	no	—
82	D₄	—	4	1	no	—
83	D₄	—	4	3	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	2	3	no	—
86	C₂×C₄	—	4	3	no	—
87	C₂×C₄	—	8	3	no	—
88	C₂×C₄	—	8	3	no	—
89	C₂²	—	4	3	no	—
90	C₂²	—	4	3	no	—
91	C₂²	—	4	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	—	8	3	no	—
94	C₂²	#120×C₅₉	1	1	yes	characteristic
95	C₂²	C₂²⋊Q₈×C₅₉	1	3	yes	—
96	C₂²	#124×C₅₉	1	3	yes	—
97	C₄	—	4	6	no	—
98	C₄	—	4	1	no	—
99	C₄	—	8	3	no	—
100	C₂	—	2	1	no	—
101	C₂	—	2	3	no	—
102	C₂	—	2	3	no	—
103	C₂	—	8	1	no	—
104	C₂	C₅₉×#393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₉×#543

C₂

yes

maximal, characteristic, complemented

C₂×#99×C₅₉

C₂

yes

maximal

C₅₉×#342

C₂

yes

maximal, complemented

C₂²×D₄×C₅₉

C₂²

yes

characteristic

C₂²⋊Q₈×C₅₉

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

C₂×C₂²⋊C₄×C₅₉

C₂²

yes

—

#99×C₅₉

—

C₂⁴×C₅₉

Q₈

yes

characteristic, complemented

C₂⁴×C₅₉

C₂³

yes

characteristic

C₂×Q₈×C₅₉

—

C₂×D₄×C₅₉

—

C₂×D₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

Q₈

yes

characteristic, complemented

C₄⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

—

C₂²⋊C₄×C₅₉

D₄

yes

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂×D₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂³×C₅₉

#41

yes

—

Q₈×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#120

yes

characteristic

C₂²×C₅₉

C₂²⋊Q₈

yes

—

C₂²×C₅₉

#124

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

—

#2032

C₅₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#393

yes

characteristic

#543

C₂×C₅₉

yes

characteristic, complemented

C₂×#99

C₂×C₅₉

yes

—

#342

C₂×C₅₉

yes

complemented

C₅₉

#2032

yes

characteristic, direct product factor, Sylow

C₂²×D₄

C₂²×C₅₉

yes

characteristic

C₂²⋊Q₈

—

C₂×C₂²⋊C₄

C₂²×C₅₉

yes

—

C₂×C₂²⋊C₄

C₂²×C₅₉

yes

—

#99

—

C₂⁴

Q₈×C₅₉

yes

characteristic, complemented

C₂⁴

C₂³×C₅₉

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Q₈×C₅₉

yes

characteristic, complemented

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

D₄×C₅₉

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×D₄×C₅₉

yes

—

C₂³

C₂×Q₈×C₅₉

yes

characteristic

C₂³

#41×C₅₉

yes

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#120×C₅₉

yes

characteristic

C₂²

C₂²⋊Q₈×C₅₉

yes

—

C₂²

#124×C₅₉

yes

—

C₄

—

C₄

—

C₄

—

100

C₂

—

101

C₂

—

102

C₂

—

103

C₂

—

104

C₂

C₅₉×#393

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₅₉	#393
commutator, Frattini, cosocle	C₂⁴	C₂³×C₅₉
prime radical	C₂²×D₄×C₅₉	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₉ ◁ C₂⁴×C₅₉ ◁ G

subgroup

G/H

center, socle

C₂×C₅₉

#393

commutator, Frattini, cosocle

C₂⁴

C₂³×C₅₉

prime radical

C₂²×D₄×C₅₉

C₂²

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₅₉ ◁ C₂⁴×C₅₉ ◁ G

N	H ≅ G/N	type
1. C₅₉×#543	C₂	semidirect
3. C₅₉×#342	C₂	semidirect
9. C₂⁴×C₅₉	Q₈	semidirect
16. C₂²×C₄×C₅₉	Q₈	semidirect
47. #2032	C₅₉	direct (N × H)
53. #543	C₂×C₅₉	semidirect
55. #342	C₂×C₅₉	semidirect
56. C₅₉	#2032	direct (N × H)
62. C₂⁴	Q₈×C₅₉	semidirect
69. C₂²×C₄	Q₈×C₅₉	semidirect

H ≅ G/N

type

1. C₅₉×#543

C₂

semidirect

3. C₅₉×#342

C₂

semidirect

9. C₂⁴×C₅₉

Q₈

semidirect

16. C₂²×C₄×C₅₉

Q₈

semidirect

47. #2032

C₅₉

direct (N × H)

53. #543

C₂×C₅₉

semidirect

55. #342

C₂×C₅₉

semidirect

56. C₅₉

#2032

direct (N × H)

62. C₂⁴

Q₈×C₅₉

semidirect

69. C₂²×C₄

Q₈×C₅₉

semidirect

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

subgroup	index
C₅₉×#543	2
C₂×#99×C₅₉	2
C₅₉×#342	2
#2032	59

subgroup

index

C₅₉×#543

C₂×#99×C₅₉

C₅₉×#342

#2032

Subgroups — C₅₉×#2032 #21420350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C59×#2032 #21420350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#2032 #21420350