Group #45688173 — subgroups

Subgroups — #45688173

Order 33344.

252 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 91 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	#30659705	C₂	1	1	yes	maximal, characteristic, complemented
2	#30659702	C₂	1	1	yes	maximal, characteristic, complemented
3	#30659693	C₂	1	3	yes	maximal, complemented
4	#30659683	C₂	1	3	yes	maximal, complemented
5	#30659681	C₂	1	3	yes	maximal, complemented
6	#30659678	C₂	1	3	yes	maximal, complemented
7	#126×C₅₂₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₅₂₁	C₂²	1	1	yes	characteristic
9	C₂×D₂₀₈₄	C₂²	1	3	yes	—
10	C₂×D₂₀₈₄	C₂²	1	3	yes	—
11	C₂×D₂₀₈₄	C₂²	1	3	yes	complemented
12	C₂×C₂²⋊D₅₂₁	C₂²	1	3	yes	—
13	C₂×C₄×D₅₂₁	C₂²	1	3	yes	complemented
14	#22552343	C₂²	1	3	yes	complemented
15	#22552341	C₂²	1	3	yes	—
16	#22552341	C₂²	1	3	yes	—
17	#22552341	C₂²	1	3	yes	complemented
18	C₄⋊C₄×C₅₂₁	C₂²	1	3	yes	complemented
19	C₂²⋊C₄×C₅₂₁	C₂²	1	3	yes	—
20	C₄²×C₅₂₁	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₅₂₁	—	4	3	no	—
22	C₂²×D₅₂₁	C₂³	1	1	yes	characteristic
23	C₂²×D₅₂₁	C₂³	1	1	yes	characteristic
24	C₂²×D₅₂₁	C₂³	1	3	yes	—
25	D₂₀₈₄	—	4	3	no	—
26	D₂₀₈₄	—	4	3	no	—
27	D₂₀₈₄	—	4	3	no	—
28	C₂²⋊D₅₂₁	—	4	3	no	—
29	C₂×Dic₅₂₁	C₂³	1	3	yes	—
30	C₄×D₅₂₁	—	4	3	no	—
31	C₂³×C₅₂₁	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
33	C₂×C₄×C₅₂₁	C₂³	1	3	yes	—
34	C₂²×C₅₂₁	—	2	3	no	—
35	C₂²×C₅₂₁	C₂⁴	1	1	yes	characteristic
36	C₂×D₅₂₁	—	2	3	no	—
37	C₂×D₅₂₁	—	2	3	no	—
38	C₂×D₅₂₁	—	2	3	no	—
39	C₂×D₅₂₁	—	2	3	no	—
40	C₂×D₅₂₁	—	2	3	no	—
41	C₂×D₅₂₁	—	4	4	no	—
42	C₄×C₅₂₁	—	2	3	no	—
43	C₄×C₅₂₁	—	2	3	no	—
44	Dic₅₂₁	—	2	3	no	—
45	C₂×C₅₂₁	—	4	1	no	—
46	C₂×C₅₂₁	H(5,𝔽₂)	1	3	yes	—
47	D₅₂₁	—	4	3	no	—
48	D₅₂₁	—	4	1	no	—
49	D₅₂₁	—	4	1	no	—
50	C₅₂₁	#560	1	1	yes	characteristic, complemented, Sylow
51	#560	—	521	1	no	maximal, Sylow
52	#127	—	521	1	no	—
53	#126	—	521	1	no	—
54	#126	D₅₂₁	1	1	yes	characteristic, complemented
55	#123	—	521	3	no	—
56	#121	—	521	3	no	—
57	#121	—	521	3	no	—
58	#120	—	521	3	no	—
59	C₂⁴	—	521	1	no	—
60	C₂×D₄	—	521	3	no	—
61	C₂×D₄	—	521	3	no	—
62	C₂×D₄	—	521	3	no	—
63	C₂×D₄	—	521	3	no	—
64	C₂²×C₄	—	521	3	no	—
65	C₄⋊C₄	—	521	3	no	—
66	C₄⋊C₄	C₂×D₅₂₁	1	3	yes	complemented
67	C₂²⋊C₄	—	521	3	no	—
68	C₂²⋊C₄	—	521	3	no	—
69	C₂²⋊C₄	—	521	3	no	—
70	C₂²⋊C₄	C₂×D₅₂₁	1	3	yes	—
71	C₄²	C₂×D₅₂₁	1	1	yes	characteristic, complemented
72	C₂³	—	521	1	no	—
73	C₂³	—	521	1	no	—
74	C₂³	—	521	3	no	—
75	C₂³	—	2084	3	no	—
76	C₂³	C₂²×D₅₂₁	1	1	yes	characteristic
77	D₄	—	2084	3	no	—
78	D₄	—	2084	3	no	—
79	D₄	—	2084	3	no	—
80	D₄	—	2084	3	no	—
81	C₂×C₄	—	521	3	no	—
82	C₂×C₄	—	2084	3	no	—
83	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
84	C₂×C₄	C₂²×D₅₂₁	1	3	yes	—
85	C₂²	—	2	3	no	—
86	C₂²	—	1042	3	no	—
87	C₂²	—	1042	3	no	—
88	C₂²	—	1042	3	no	—
89	C₂²	—	1042	3	no	—
90	C₂²	—	1042	3	no	—
91	C₂²	—	2084	4	no	—
92	C₂²	C₂³×D₅₂₁	1	1	yes	characteristic
93	C₄	—	2	3	no	—
94	C₄	—	2	3	no	—
95	C₄	—	1042	3	no	—
96	C₂	—	4	1	no	—
97	C₂	—	2084	1	no	—
98	C₂	—	2084	1	no	—
99	C₂	—	2084	3	no	—
100	C₂	#30659754	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#30659705

C₂

yes

maximal, characteristic, complemented

#30659702

C₂

yes

maximal, characteristic, complemented

#30659693

C₂

yes

maximal, complemented

#30659683

C₂

yes

maximal, complemented

#30659681

C₂

yes

maximal, complemented

#30659678

C₂

yes

maximal, complemented

#126×C₅₂₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₅₂₁

C₂²

yes

characteristic

C₂×D₂₀₈₄

C₂²

yes

—

C₂×D₂₀₈₄

C₂²

yes

—

C₂×D₂₀₈₄

C₂²

yes

complemented

C₂×C₂²⋊D₅₂₁

C₂²

yes

—

C₂×C₄×D₅₂₁

C₂²

yes

complemented

#22552343

C₂²

yes

complemented

#22552341

C₂²

yes

—

#22552341

C₂²

yes

—

#22552341

C₂²

yes

complemented

C₄⋊C₄×C₅₂₁

C₂²

yes

complemented

C₂²⋊C₄×C₅₂₁

C₂²

yes

—

C₄²×C₅₂₁

C₂²

yes

characteristic, complemented

C₂²×D₅₂₁

—

C₂²×D₅₂₁

C₂³

yes

characteristic

C₂²×D₅₂₁

C₂³

yes

characteristic

C₂²×D₅₂₁

C₂³

yes

—

D₂₀₈₄

—

D₂₀₈₄

—

D₂₀₈₄

—

C₂²⋊D₅₂₁

—

C₂×Dic₅₂₁

C₂³

yes

—

C₄×D₅₂₁

—

C₂³×C₅₂₁

C₂³

yes

characteristic

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂×C₄×C₅₂₁

C₂³

yes

—

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂⁴

yes

characteristic

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₂×D₅₂₁

—

C₄×C₅₂₁

—

C₄×C₅₂₁

—

Dic₅₂₁

—

C₂×C₅₂₁

—

C₂×C₅₂₁

H(5,𝔽₂)

yes

—

D₅₂₁

—

D₅₂₁

—

D₅₂₁

—

C₅₂₁

#560

yes

characteristic, complemented, Sylow

#560

—

521

maximal, Sylow

#127

—

521

—

#126

—

521

—

#126

D₅₂₁

yes

characteristic, complemented

#123

—

521

—

#121

—

521

—

#121

—

521

—

#120

—

521

—

C₂⁴

—

521

—

C₂×D₄

—

521

—

C₂×D₄

—

521

—

C₂×D₄

—

521

—

C₂×D₄

—

521

—

C₂²×C₄

—

521

—

C₄⋊C₄

—

521

—

C₄⋊C₄

C₂×D₅₂₁

yes

complemented

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

—

521

—

C₂²⋊C₄

C₂×D₅₂₁

yes

—

C₄²

C₂×D₅₂₁

yes

characteristic, complemented

C₂³

—

521

—

C₂³

—

521

—

C₂³

—

521

—

C₂³

—

2084

—

C₂³

C₂²×D₅₂₁

yes

characteristic

D₄

—

2084

—

D₄

—

2084

—

D₄

—

2084

—

D₄

—

2084

—

C₂×C₄

—

521

—

C₂×C₄

—

2084

—

C₂×C₄

C₂²×D₅₂₁

yes

—

C₂×C₄

C₂²×D₅₂₁

yes

—

C₂²

—

C₂²

—

1042

—

C₂²

—

1042

—

C₂²

—

1042

—

C₂²

—

1042

—

C₂²

—

1042

—

C₂²

—

2084

—

C₂²

C₂³×D₅₂₁

yes

characteristic

C₄

—

C₄

—

C₄

—

1042

—

C₂

—

C₂

—

2084

—

C₂

—

2084

—

C₂

—

2084

—

100

C₂

#30659754

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₂₁
commutator, socle, cosocle	C₂²×C₅₂₁	C₂⁴
Fitting	#126×C₅₂₁	C₂
hypercenter	#126	D₅₂₁
hypocenter	C₅₂₁	#560
LCS	G ▷ C₂²×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ #126

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₂₁

commutator, socle, cosocle

C₂²×C₅₂₁

C₂⁴

Fitting

#126×C₅₂₁

C₂

D₅₂₁

C₅₂₁

G ▷ C₂²×C₅₂₁ ▷ C₅₂₁

C₁ ◁ C₂² ◁ #126

N	H ≅ G/N	type
1. #30659705	C₂	semidirect
2. #30659702	C₂	semidirect
3. #30659693	C₂	semidirect
4. #30659683	C₂	semidirect
5. #30659681	C₂	semidirect
6. #30659678	C₂	semidirect
7. #126×C₅₂₁	C₂	semidirect
11. C₂×D₂₀₈₄	C₂²	semidirect
13. C₂×C₄×D₅₂₁	C₂²	semidirect
14. #22552343	C₂²	semidirect
17. #22552341	C₂²	semidirect
18. C₄⋊C₄×C₅₂₁	C₂²	semidirect
20. C₄²×C₅₂₁	C₂²	semidirect
50. C₅₂₁	#560	semidirect
54. #126	D₅₂₁	semidirect
66. C₄⋊C₄	C₂×D₅₂₁	semidirect
71. C₄²	C₂×D₅₂₁	semidirect

H ≅ G/N

type

1. #30659705

C₂

semidirect

2. #30659702

C₂

semidirect

3. #30659693

C₂

semidirect

4. #30659683

C₂

semidirect

5. #30659681

C₂

semidirect

6. #30659678

C₂

semidirect

7. #126×C₅₂₁

C₂

semidirect

11. C₂×D₂₀₈₄

C₂²

semidirect

13. C₂×C₄×D₅₂₁

C₂²

semidirect

14. #22552343

C₂²

semidirect

17. #22552341

C₂²

semidirect

18. C₄⋊C₄×C₅₂₁

C₂²

semidirect

20. C₄²×C₅₂₁

C₂²

semidirect

50. C₅₂₁

#560

semidirect

54. #126

D₅₂₁

semidirect

66. C₄⋊C₄

C₂×D₅₂₁

semidirect

71. C₄²

C₂×D₅₂₁

semidirect

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

subgroup	index
#30659705	2
#30659702	2
#30659693	2
#30659683	2
#30659681	2
#30659678	2
#126×C₅₂₁	2
#560	521

subgroup

index

#126×C₅₂₁

521

Subgroups — #45688173

Conjugacy classes of subgroups

Complements

Maximal subgroups