Group #23753023 — subgroups

Subgroups — #23753023

Order 9664.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#18012786	C₂	1	1	yes	maximal, characteristic, complemented
2	#18012769	C₂	1	3	yes	maximal, complemented
3	#18012765	C₂	1	3	yes	maximal, complemented
4	#18012759	C₂	1	3	yes	maximal, complemented
5	C₂×#14558992	C₂	1	3	yes	maximal, complemented
6	#18012746	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#14559004	C₂²	1	3	yes	complemented
9	#14559003	—	2	3	no	—
10	C₂×Dic₃₀₂	C₂²	1	3	yes	complemented
11	Q₈×D₁₅₁	—	2	1	no	—
12	#14558994	C₂²	1	3	yes	complemented
13	#14558992	C₂²	1	3	yes	complemented
14	#14558992	C₂²	1	6	yes	complemented
15	#14558991	C₂²	1	1	yes	characteristic, complemented
16	#14558989	C₂²	1	3	yes	complemented
17	#14558988	C₂²	1	3	yes	complemented
18	C₂×C₁₅₁⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₅₁	—	2	1	no	—
20	#41×C₁₅₁	C₂²	1	3	yes	—
21	#41×C₁₅₁	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₅₁	—	2	3	no	—
23	C₂×D₄×C₁₅₁	C₂²	1	3	yes	—
24	Dic₃₀₂	—	2	3	no	—
25	Dic₃₀₂	C₂³	1	3	yes	—
26	D₆₀₄	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₅₁	—	2	3	no	—
28	C₂×Dic₁₅₁	—	2	3	no	—
29	C₄×D₁₅₁	—	2	3	no	—
30	C₁₅₁⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₅₁⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₅₁	—	2	3	no	—
33	Q₈×C₁₅₁	D₄	1	1	yes	characteristic
34	Q₈×C₁₅₁	C₂³	1	1	yes	characteristic
35	D₄×C₁₅₁	—	2	3	no	—
36	D₄×C₁₅₁	—	2	3	no	—
37	D₄×C₁₅₁	D₄	1	3	yes	—
38	D₄×C₁₅₁	C₂³	1	3	yes	—
39	C₂×C₄×C₁₅₁	—	2	3	no	—
40	C₂×C₄×C₁₅₁	C₂³	1	3	yes	—
41	C₂²×C₁₅₁	—	2	3	no	—
42	C₂²×C₁₅₁	—	8	3	no	—
43	C₂²×C₁₅₁	C₂×D₄	1	3	yes	—
44	C₂×D₁₅₁	—	2	1	no	—
45	C₄×C₁₅₁	—	2	1	no	—
46	C₄×C₁₅₁	C₂×D₄	1	3	yes	—
47	C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
48	Dic₁₅₁	—	2	3	no	—
49	C₂×C₁₅₁	—	2	3	no	—
50	C₂×C₁₅₁	—	4	3	no	—
51	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
52	D₁₅₁	—	4	1	no	—
53	C₁₅₁	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	151	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	151	1	no	—
56	H(5,𝔽₂)	D₁₅₁	1	1	yes	characteristic, complemented
57	#137	—	151	3	no	—
58	AGL(1,ℤ/8)	—	151	3	no	—
59	#135	—	151	3	no	—
60	C₂×SD₁₆	—	151	3	no	—
61	#131	—	151	1	no	—
62	#41	—	2	1	no	—
63	#41	—	302	3	no	—
64	#41	—	151	3	no	—
65	#41	C₂×D₁₅₁	1	3	yes	—
66	#41	C₂×D₁₅₁	1	1	yes	characteristic
67	C₂×Q₈	—	302	1	no	—
68	C₂×Q₈	—	151	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₅₁	1	3	yes	—
71	Q₁₆	—	151	3	no	—
72	SD₁₆	—	151	1	no	—
73	SD₁₆	—	151	3	no	—
74	SD₁₆	—	151	6	no	—
75	D₈	—	151	3	no	—
76	M₁₆	—	151	3	no	—
77	C₂×C₈	—	151	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	302	3	no	—
80	Q₈	—	151	3	no	—
81	Q₈	C₂²⋊D₁₅₁	1	1	yes	characteristic
82	Q₈	C₂²×D₁₅₁	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	302	3	no	—
86	D₄	—	151	1	no	—
87	D₄	C₂²⋊D₁₅₁	1	3	yes	—
88	D₄	C₂²×D₁₅₁	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	302	3	no	—
91	C₂×C₄	—	302	3	no	—
92	C₂×C₄	C₂²×D₁₅₁	1	3	yes	—
93	C₈	—	151	1	no	—
94	C₈	—	151	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	302	1	no	—
98	C₂²	C₂×C₂²⋊D₁₅₁	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	302	3	no	—
101	C₄	C₂×C₂²⋊D₁₅₁	1	3	yes	—
102	C₄	C₂³×D₁₅₁	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	604	1	no	—
106	C₂	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#18012786

C₂

yes

maximal, characteristic, complemented

#18012769

C₂

yes

maximal, complemented

#18012765

C₂

yes

maximal, complemented

#18012759

C₂

yes

maximal, complemented

C₂×#14558992

C₂

yes

maximal, complemented

#18012746

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₅₁

C₂

yes

maximal, characteristic, complemented

#14559004

C₂²

yes

complemented

#14559003

—

C₂×Dic₃₀₂

C₂²

yes

complemented

Q₈×D₁₅₁

—

#14558994

C₂²

yes

complemented

#14558992

C₂²

yes

complemented

#14558992

C₂²

yes

complemented

#14558991

C₂²

yes

characteristic, complemented

#14558989

C₂²

yes

complemented

#14558988

C₂²

yes

complemented

C₂×C₁₅₁⋊C₈

C₂²

yes

complemented

#41×C₁₅₁

—

#41×C₁₅₁

C₂²

yes

—

#41×C₁₅₁

C₂²

yes

characteristic

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

C₂²

yes

—

Dic₃₀₂

—

Dic₃₀₂

C₂³

yes

—

D₆₀₄

C₂³

yes

characteristic

C₂²⋊D₁₅₁

—

C₂×Dic₁₅₁

—

C₄×D₁₅₁

—

C₁₅₁⋊C₈

C₂³

yes

characteristic

C₁₅₁⋊C₈

C₂³

yes

—

C₂³×C₁₅₁

—

Q₈×C₁₅₁

D₄

yes

characteristic

Q₈×C₁₅₁

C₂³

yes

characteristic

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

D₄

yes

—

D₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂×D₄

yes

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

C₄×C₁₅₁

C₂⁴

yes

characteristic

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

D₁₅₁

—

C₁₅₁

#577

yes

characteristic, complemented, Sylow

#577

—

151

maximal, Sylow

2₋¹⁺⁴

—

151

—

H(5,𝔽₂)

D₁₅₁

yes

characteristic, complemented

#137

—

151

—

AGL(1,ℤ/8)

—

151

—

#135

—

151

—

C₂×SD₁₆

—

151

—

#131

—

151

—

#41

—

#41

—

302

—

#41

—

151

—

#41

C₂×D₁₅₁

yes

—

#41

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂×D₄

—

C₂×D₄

C₂×D₁₅₁

yes

—

Q₁₆

—

151

—

SD₁₆

—

151

—

SD₁₆

—

151

—

SD₁₆

—

151

—

D₈

—

151

—

M₁₆

—

151

—

C₂×C₈

—

151

—

C₂³

—

Q₈

—

302

—

Q₈

—

151

—

Q₈

C₂²⋊D₁₅₁

yes

characteristic

Q₈

C₂²×D₁₅₁

yes

characteristic

D₄

—

D₄

—

D₄

—

302

—

D₄

—

151

—

D₄

C₂²⋊D₁₅₁

yes

—

D₄

C₂²×D₁₅₁

yes

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

C₂²×D₁₅₁

yes

—

C₈

—

151

—

C₈

—

151

—

C₂²

—

C₂²

—

C₂²

—

302

—

C₂²

C₂×C₂²⋊D₁₅₁

yes

—

C₄

—

100

C₄

—

302

—

101

C₄

C₂×C₂²⋊D₁₅₁

yes

—

102

C₄

C₂³×D₁₅₁

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

604

—

106

C₂

C₂²×C₂²⋊D₁₅₁

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₅₁
commutator, cosocle	C₄×C₁₅₁	C₂⁴
Frattini	C₄	C₂³×D₁₅₁
Fitting	H(5,𝔽₂)×C₁₅₁	C₂
socle	C₂×C₁₅₁	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₅₁
hypocenter	C₁₅₁	#577
LCS	G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₅₁

commutator, cosocle

C₄×C₁₅₁

C₂⁴

Frattini

C₄

C₂³×D₁₅₁

Fitting

H(5,𝔽₂)×C₁₅₁

C₂

socle

C₂×C₁₅₁

C₂²×D₄

D₁₅₁

C₁₅₁

G ▷ C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #18012786	C₂	semidirect
2. #18012769	C₂	semidirect
3. #18012765	C₂	semidirect
4. #18012759	C₂	semidirect
5. C₂×#14558992	C₂	semidirect
6. #18012746	C₂	semidirect
7. H(5,𝔽₂)×C₁₅₁	C₂	semidirect
8. #14559004	C₂²	semidirect
10. C₂×Dic₃₀₂	C₂²	semidirect
12. #14558994	C₂²	semidirect
13. #14558992	C₂²	semidirect
14. #14558992	C₂²	semidirect
15. #14558991	C₂²	semidirect
16. #14558989	C₂²	semidirect
17. #14558988	C₂²	semidirect
18. C₂×C₁₅₁⋊C₈	C₂²	semidirect
53. C₁₅₁	#577	semidirect
56. H(5,𝔽₂)	D₁₅₁	semidirect

H ≅ G/N

type

1. #18012786

C₂

semidirect

2. #18012769

C₂

semidirect

3. #18012765

C₂

semidirect

4. #18012759

C₂

semidirect

5. C₂×#14558992

C₂

semidirect

6. #18012746

C₂

semidirect

7. H(5,𝔽₂)×C₁₅₁

C₂

semidirect

8. #14559004

C₂²

semidirect

10. C₂×Dic₃₀₂

C₂²

semidirect

12. #14558994

C₂²

semidirect

13. #14558992

C₂²

semidirect

14. #14558992

C₂²

semidirect

15. #14558991

C₂²

semidirect

16. #14558989

C₂²

semidirect

17. #14558988

C₂²

semidirect

18. C₂×C₁₅₁⋊C₈

C₂²

semidirect

53. C₁₅₁

#577

semidirect

56. H(5,𝔽₂)

D₁₅₁

semidirect

Subgroups — #23753023

Conjugacy classes of subgroups

Complements