Group #21424006 — subgroups

Subgroups — #21424006

Order 7552.

852 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 852 conjugacy classes of subgroups — too many to draw. Only its 197 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×C₂²⋊(C₅₉⋊C₈)	C₂	1	6	yes	maximal, complemented
2	C₂⁴×C₄×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅₉⋊C₈	C₂²	1	4	yes	complemented
4	C₂²⋊(C₅₉⋊C₈)	—	2	12	no	—
5	C₂⁵×C₅₉	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₅₉	—	2	12	no	—
7	C₂³×C₄×C₅₉	C₄	1	3	yes	—
8	C₂³×C₄×C₅₉	C₂²	1	3	yes	—
9	C₂⁴×C₅₉	—	2	6	no	—
10	C₂⁴×C₅₉	—	2	6	no	—
11	C₂⁴×C₅₉	C₈	1	4	yes	complemented
12	C₂⁴×C₅₉	C₂×C₄	1	3	yes	—
13	C₂²×C₄×C₅₉	—	2	32	no	—
14	C₂²×C₄×C₅₉	—	2	6	no	—
15	C₂²×C₄×C₅₉	—	2	12	no	—
16	C₂²×C₄×C₅₉	—	2	12	no	—
17	C₂²×C₄×C₅₉	C₂×C₄	1	3	yes	—
18	C₂²×C₄×C₅₉	D₄	1	12	yes	—
19	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
20	C₂×C₅₉⋊C₈	—	2	12	no	—
21	C₂³×C₅₉	—	2	8	no	—
22	C₂³×C₅₉	—	2	24	no	—
23	C₂³×C₅₉	—	2	24	no	—
24	C₂³×C₅₉	—	2	6	no	—
25	C₂³×C₅₉	—	2	6	no	—
26	C₂³×C₅₉	C₂²⋊C₄	1	6	yes	—
27	C₂³×C₅₉	C₂×C₈	1	6	yes	—
28	C₂³×C₅₉	M₁₆	1	6	yes	—
29	C₂³×C₅₉	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	—	2	24	no	—
31	C₂×C₄×C₅₉	—	2	6	no	—
32	C₂×C₄×C₅₉	—	2	24	no	—
33	C₂×C₄×C₅₉	C₂²⋊C₄	1	6	yes	—
34	C₂×C₄×C₅₉	C₂×D₄	1	6	yes	—
35	C₅₉⋊C₈	—	4	4	no	—
36	C₂²×C₅₉	—	2	32	no	—
37	C₂²×C₅₉	—	2	6	no	—
38	C₂²×C₅₉	—	2	12	no	—
39	C₂²×C₅₉	—	2	6	no	—
40	C₂²×C₅₉	—	2	12	no	—
41	C₂²×C₅₉	C₂²⋊C₈	1	12	yes	complemented
42	C₂²×C₅₉	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂²×C₅₉	C₂²×C₈	1	1	yes	characteristic
44	C₂²×C₅₉	C₂×M₁₆	1	3	yes	—
45	C₄×C₅₉	—	2	6	no	—
46	C₄×C₅₉	#120	1	4	yes	—
47	#1779	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	—	2	12	no	—
49	C₂×C₅₉	#379	1	1	yes	characteristic
50	C₂×C₅₉	C₂×C₂²⋊C₈	1	3	yes	—
51	C₂×C₅₉	#407	1	3	yes	—
52	C₂⁴×C₄	D₅₉	1	1	yes	characteristic
53	C₂×C₂²⋊C₈	—	59	6	no	—
54	C₅₉	#1779	1	1	yes	characteristic, complemented, Sylow
55	C₂⁵	Dic₅₉	1	1	yes	characteristic
56	C₂³×C₄	—	2	12	no	—
57	C₂³×C₄	Dic₅₉	1	3	yes	—
58	C₂³×C₄	C₂×D₅₉	1	3	yes	—
59	C₂²×C₈	—	59	4	no	—
60	C₂²⋊C₈	—	118	12	no	—
61	C₂⁴	—	2	6	no	—
62	C₂⁴	—	2	6	no	—
63	C₂⁴	C₅₉⋊C₈	1	4	yes	complemented
64	C₂⁴	C₂×Dic₅₉	1	3	yes	—
65	C₂²×C₄	—	2	32	no	—
66	C₂²×C₄	—	2	6	no	—
67	C₂²×C₄	—	2	12	no	—
68	C₂²×C₄	—	2	12	no	—
69	C₂²×C₄	C₂×Dic₅₉	1	3	yes	—
70	C₂²×C₄	C₂²⋊D₅₉	1	12	yes	—
71	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
72	C₂×C₈	—	118	12	no	—
73	C₂³	—	2	8	no	—
74	C₂³	—	2	24	no	—
75	C₂³	—	2	24	no	—
76	C₂³	—	2	6	no	—
77	C₂³	—	2	6	no	—
78	C₂³	C₂×C₅₉⋊C₈	1	6	yes	—
79	C₂³	#11745664	1	6	yes	—
80	C₂³	C₂²⋊Dic₅₉	1	6	yes	—
81	C₂³	C₂²×Dic₅₉	1	1	yes	characteristic
82	C₂×C₄	—	2	24	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	2	24	no	—
85	C₂×C₄	C₂²⋊Dic₅₉	1	6	yes	—
86	C₂×C₄	C₂×C₂²⋊D₅₉	1	6	yes	—
87	C₈	—	236	4	no	—
88	C₂²	—	2	32	no	—
89	C₂²	—	2	6	no	—
90	C₂²	—	2	12	no	—
91	C₂²	—	2	12	no	—
92	C₂²	—	2	6	no	—
93	C₂²	C₂²⋊(C₅₉⋊C₈)	1	12	yes	complemented
94	C₂²	C₂×C₂²⋊Dic₅₉	1	3	yes	—
95	C₂²	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
96	C₂²	C₂×#11745664	1	3	yes	—
97	C₄	—	2	6	no	—
98	C₄	#14273696	1	4	yes	—
99	C₂	—	2	12	no	—
100	C₂	#16837258	1	1	yes	characteristic
101	C₂	C₂×C₂²⋊(C₅₉⋊C₈)	1	3	yes	—
102	C₂	#16837366	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₅₉⋊C₈)

C₂

yes

maximal, complemented

C₂⁴×C₄×C₅₉

C₂

yes

maximal, characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

complemented

C₂²⋊(C₅₉⋊C₈)

—

C₂⁵×C₅₉

C₄

yes

characteristic

C₂³×C₄×C₅₉

—

C₂³×C₄×C₅₉

C₄

yes

—

C₂³×C₄×C₅₉

C₂²

yes

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

—

C₂⁴×C₅₉

C₈

yes

complemented

C₂⁴×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

—

C₂²×C₄×C₅₉

D₄

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

—

C₂³×C₅₉

C₂²⋊C₄

yes

—

C₂³×C₅₉

C₂×C₈

yes

—

C₂³×C₅₉

M₁₆

yes

—

C₂³×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₂²⋊C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂²⋊C₈

yes

complemented

C₂²×C₅₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₅₉

C₂²×C₈

yes

characteristic

C₂²×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

—

C₄×C₅₉

#120

yes

—

#1779

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#379

yes

characteristic

C₂×C₅₉

C₂×C₂²⋊C₈

yes

—

C₂×C₅₉

#407

yes

—

C₂⁴×C₄

D₅₉

yes

characteristic

C₂×C₂²⋊C₈

—

C₅₉

#1779

yes

characteristic, complemented, Sylow

C₂⁵

Dic₅₉

yes

characteristic

C₂³×C₄

—

C₂³×C₄

Dic₅₉

yes

—

C₂³×C₄

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²⋊C₈

—

118

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₅₉⋊C₈

yes

complemented

C₂⁴

C₂×Dic₅₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₉

yes

—

C₂²×C₄

C₂²⋊D₅₉

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

C₂×C₈

—

118

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂×C₅₉⋊C₈

yes

—

C₂³

#11745664

yes

—

C₂³

C₂²⋊Dic₅₉

yes

—

C₂³

C₂²×Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₅₉

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

—

C₈

—

236

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊(C₅₉⋊C₈)

yes

complemented

C₂²

C₂×C₂²⋊Dic₅₉

yes

—

C₂²

C₂²×C₅₉⋊C₈

yes

characteristic

C₂²

C₂×#11745664

yes

—

C₄

—

C₄

#14273696

yes

—

C₂

—

100

C₂

#16837258

yes

characteristic

101

C₂

C₂×C₂²⋊(C₅₉⋊C₈)

yes

—

102

C₂

#16837366

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂²×C₈
Fitting	C₂⁴×C₄×C₅₉	C₂
socle	C₂³×C₅₉	C₂²×C₄
hypercenter	C₂⁴×C₄	D₅₉
hypocenter	C₅₉	#1779
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₉

commutator

C₂²×C₅₉

C₂²×C₈

Fitting

C₂⁴×C₄×C₅₉

C₂

socle

C₂³×C₅₉

C₂²×C₄

hypercenter

C₂⁴×C₄

D₅₉

C₅₉

G ▷ C₂²×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₅₉⋊C₈)	C₂	semidirect
3. C₂²×C₅₉⋊C₈	C₂²	semidirect
11. C₂⁴×C₅₉	C₈	semidirect
41. C₂²×C₅₉	C₂²⋊C₈	semidirect
54. C₅₉	#1779	semidirect
63. C₂⁴	C₅₉⋊C₈	semidirect
93. C₂²	C₂²⋊(C₅₉⋊C₈)	semidirect

H ≅ G/N

type

1. C₂×C₂²⋊(C₅₉⋊C₈)

C₂

semidirect

3. C₂²×C₅₉⋊C₈

C₂²

semidirect

11. C₂⁴×C₅₉

C₈

semidirect

41. C₂²×C₅₉

C₂²⋊C₈

semidirect

54. C₅₉

#1779

semidirect

63. C₂⁴

C₅₉⋊C₈

semidirect

93. C₂²

C₂²⋊(C₅₉⋊C₈)

semidirect

Subgroups — #21424006

Conjugacy classes of subgroups

Complements