Group #23758084 — subgroups

Subgroups — #23758084

Order 10048.

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	#19097818	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097801	C₂	1	3	yes	maximal, complemented
3	#19097797	C₂	1	3	yes	maximal, complemented
4	#19097791	C₂	1	3	yes	maximal, complemented
5	C₂×#14578942	C₂	1	3	yes	maximal, complemented
6	#19097778	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14578954	C₂²	1	3	yes	complemented
9	#14578953	—	2	3	no	—
10	C₂×Dic₃₁₄	C₂²	1	3	yes	complemented
11	Q₈×D₁₅₇	—	2	1	no	—
12	#14578944	C₂²	1	3	yes	complemented
13	#14578942	C₂²	1	3	yes	complemented
14	#14578942	C₂²	1	6	yes	complemented
15	#14578941	C₂²	1	1	yes	characteristic, complemented
16	#14578939	C₂²	1	3	yes	complemented
17	#14578938	C₂²	1	3	yes	complemented
18	C₂×#11929434	C₂²	1	3	yes	complemented
19	#41×C₁₅₇	—	2	1	no	—
20	#41×C₁₅₇	C₂²	1	3	yes	—
21	#41×C₁₅₇	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₅₇	—	2	3	no	—
23	C₂×D₄×C₁₅₇	C₂²	1	3	yes	—
24	Dic₃₁₄	—	2	3	no	—
25	Dic₃₁₄	C₂³	1	3	yes	—
26	D₆₂₈	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₅₇	—	2	3	no	—
28	C₂×Dic₁₅₇	—	2	3	no	—
29	C₄×D₁₅₇	—	2	3	no	—
30	#11929434	C₂³	1	3	yes	—
31	#11929434	C₂³	1	1	yes	characteristic
32	C₂³×C₁₅₇	—	2	3	no	—
33	Q₈×C₁₅₇	D₄	1	1	yes	characteristic
34	Q₈×C₁₅₇	C₂³	1	1	yes	characteristic
35	D₄×C₁₅₇	—	2	3	no	—
36	D₄×C₁₅₇	—	2	3	no	—
37	D₄×C₁₅₇	D₄	1	3	yes	—
38	D₄×C₁₅₇	C₂³	1	3	yes	—
39	C₂×C₄×C₁₅₇	—	2	3	no	—
40	C₂×C₄×C₁₅₇	C₂³	1	3	yes	—
41	C₂²×C₁₅₇	—	2	3	no	—
42	C₂²×C₁₅₇	—	8	3	no	—
43	C₂²×C₁₅₇	C₂×D₄	1	3	yes	—
44	C₂×D₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	2	1	no	—
46	C₄×C₁₅₇	C₂×D₄	1	3	yes	—
47	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
48	Dic₁₅₇	—	2	3	no	—
49	C₂×C₁₅₇	—	2	3	no	—
50	C₂×C₁₅₇	—	4	3	no	—
51	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
52	D₁₅₇	—	4	1	no	—
53	C₁₅₇	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	157	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	157	1	no	—
56	H(5,𝔽₂)	D₁₅₇	1	1	yes	characteristic, complemented
57	#137	—	157	3	no	—
58	AGL(1,ℤ/8)	—	157	3	no	—
59	#135	—	157	3	no	—
60	C₂×SD₁₆	—	157	3	no	—
61	#131	—	157	1	no	—
62	#41	—	2	1	no	—
63	#41	—	314	3	no	—
64	#41	—	157	3	no	—
65	#41	C₂×D₁₅₇	1	3	yes	—
66	#41	C₂×D₁₅₇	1	1	yes	characteristic
67	C₂×Q₈	—	314	1	no	—
68	C₂×Q₈	—	157	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₅₇	1	3	yes	—
71	Q₁₆	—	157	3	no	—
72	SD₁₆	—	157	1	no	—
73	SD₁₆	—	157	3	no	—
74	SD₁₆	—	157	6	no	—
75	D₈	—	157	3	no	—
76	M₁₆	—	157	3	no	—
77	C₂×C₈	—	157	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	314	3	no	—
80	Q₈	—	157	3	no	—
81	Q₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
82	Q₈	C₂²×D₁₅₇	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	314	3	no	—
86	D₄	—	157	1	no	—
87	D₄	C₂²⋊D₁₅₇	1	3	yes	—
88	D₄	C₂²×D₁₅₇	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	314	3	no	—
91	C₂×C₄	—	314	3	no	—
92	C₂×C₄	C₂²×D₁₅₇	1	3	yes	—
93	C₈	—	157	1	no	—
94	C₈	—	157	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	314	1	no	—
98	C₂²	C₂×C₂²⋊D₁₅₇	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	314	3	no	—
101	C₄	C₂×C₂²⋊D₁₅₇	1	3	yes	—
102	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	628	1	no	—
106	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097818

C₂

yes

maximal, characteristic, complemented

#19097801

C₂

yes

maximal, complemented

#19097797

C₂

yes

maximal, complemented

#19097791

C₂

yes

maximal, complemented

C₂×#14578942

C₂

yes

maximal, complemented

#19097778

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

C₂²

yes

complemented

#14578953

—

C₂×Dic₃₁₄

C₂²

yes

complemented

Q₈×D₁₅₇

—

#14578944

C₂²

yes

complemented

#14578942

C₂²

yes

complemented

#14578942

C₂²

yes

complemented

#14578941

C₂²

yes

characteristic, complemented

#14578939

C₂²

yes

complemented

#14578938

C₂²

yes

complemented

C₂×#11929434

C₂²

yes

complemented

#41×C₁₅₇

—

#41×C₁₅₇

C₂²

yes

—

#41×C₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂²

yes

—

Dic₃₁₄

—

Dic₃₁₄

C₂³

yes

—

D₆₂₈

C₂³

yes

characteristic

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₄×D₁₅₇

—

#11929434

C₂³

yes

—

#11929434

C₂³

yes

characteristic

C₂³×C₁₅₇

—

Q₈×C₁₅₇

D₄

yes

characteristic

Q₈×C₁₅₇

C₂³

yes

characteristic

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

D₄

yes

—

D₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂⁴

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

D₁₅₇

—

C₁₅₇

#577

yes

characteristic, complemented, Sylow

#577

—

157

maximal, Sylow

2₋¹⁺⁴

—

157

—

H(5,𝔽₂)

D₁₅₇

yes

characteristic, complemented

#137

—

157

—

AGL(1,ℤ/8)

—

157

—

#135

—

157

—

C₂×SD₁₆

—

157

—

#131

—

157

—

#41

—

#41

—

314

—

#41

—

157

—

#41

C₂×D₁₅₇

yes

—

#41

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

C₂×D₄

C₂×D₁₅₇

yes

—

Q₁₆

—

157

—

SD₁₆

—

157

—

SD₁₆

—

157

—

SD₁₆

—

157

—

D₈

—

157

—

M₁₆

—

157

—

C₂×C₈

—

157

—

C₂³

—

Q₈

—

314

—

Q₈

—

157

—

Q₈

C₂²⋊D₁₅₇

yes

characteristic

Q₈

C₂²×D₁₅₇

yes

characteristic

D₄

—

D₄

—

D₄

—

314

—

D₄

—

157

—

D₄

C₂²⋊D₁₅₇

yes

—

D₄

C₂²×D₁₅₇

yes

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₈

—

157

—

C₈

—

157

—

C₂²

—

C₂²

—

C₂²

—

314

—

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₄

—

100

C₄

—

314

—

101

C₄

C₂×C₂²⋊D₁₅₇

yes

—

102

C₄

C₂³×D₁₅₇

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

628

—

106

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₅₇
commutator, cosocle	C₄×C₁₅₇	C₂⁴
Frattini	C₄	C₂³×D₁₅₇
Fitting	H(5,𝔽₂)×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₅₇
hypocenter	C₁₅₇	#577
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₅₇

commutator, cosocle

C₄×C₁₅₇

C₂⁴

Frattini

C₄

C₂³×D₁₅₇

Fitting

H(5,𝔽₂)×C₁₅₇

C₂

socle

C₂×C₁₅₇

C₂²×D₄

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #19097818	C₂	semidirect
2. #19097801	C₂	semidirect
3. #19097797	C₂	semidirect
4. #19097791	C₂	semidirect
5. C₂×#14578942	C₂	semidirect
6. #19097778	C₂	semidirect
7. H(5,𝔽₂)×C₁₅₇	C₂	semidirect
8. #14578954	C₂²	semidirect
10. C₂×Dic₃₁₄	C₂²	semidirect
12. #14578944	C₂²	semidirect
13. #14578942	C₂²	semidirect
14. #14578942	C₂²	semidirect
15. #14578941	C₂²	semidirect
16. #14578939	C₂²	semidirect
17. #14578938	C₂²	semidirect
18. C₂×#11929434	C₂²	semidirect
53. C₁₅₇	#577	semidirect
56. H(5,𝔽₂)	D₁₅₇	semidirect

H ≅ G/N

type

1. #19097818

C₂

semidirect

2. #19097801

C₂

semidirect

3. #19097797

C₂

semidirect

4. #19097791

C₂

semidirect

5. C₂×#14578942

C₂

semidirect

6. #19097778

C₂

semidirect

7. H(5,𝔽₂)×C₁₅₇

C₂

semidirect

8. #14578954

C₂²

semidirect

10. C₂×Dic₃₁₄

C₂²

semidirect

12. #14578944

C₂²

semidirect

13. #14578942

C₂²

semidirect

14. #14578942

C₂²

semidirect

15. #14578941

C₂²

semidirect

16. #14578939

C₂²

semidirect

17. #14578938

C₂²

semidirect

18. C₂×#11929434

C₂²

semidirect

53. C₁₅₇

#577

semidirect

56. H(5,𝔽₂)

D₁₅₇

semidirect

Subgroups — #23758084

Conjugacy classes of subgroups

Complements