Group #23758073 — subgroups

Subgroups — #23758073

Order 10048.

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

normal subgroups only

This group has 262 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	C₂×#14578954	C₂	1	2	yes	maximal, complemented
2	#19097804	C₂	1	4	yes	maximal, complemented
3	#19097799	C₂	1	4	yes	maximal, complemented
4	#19097793	C₂	1	4	yes	maximal, complemented
5	C₂×M₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#14578954	—	2	2	no	—
7	#14578954	—	2	2	no	—
8	#14578954	C₂²	1	4	yes	—
9	#14578954	C₂²	1	4	yes	complemented
10	C₂×Dic₃₁₄	C₂²	1	2	yes	complemented
11	C₂×D₆₂₈	C₂²	1	2	yes	complemented
12	C₂×C₂²⋊D₁₅₇	—	2	2	no	—
13	C₂×C₄×D₁₅₇	—	2	2	no	—
14	Dic₆₂₈	C₂²	1	4	yes	complemented
15	#14578943	C₂²	1	8	yes	complemented
16	D₁₂₅₆	C₂²	1	4	yes	complemented
17	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
18	M₁₆×C₁₅₇	C₂²	1	4	yes	complemented
19	C₂×C₈×C₁₅₇	C₂²	1	2	yes	complemented
20	C₂²×D₁₅₇	—	2	2	no	—
21	Dic₃₁₄	—	2	2	no	—
22	Dic₃₁₄	C₂³	1	4	yes	—
23	D₆₂₈	—	2	2	no	—
24	D₆₂₈	C₂³	1	4	yes	—
25	C₂²⋊D₁₅₇	—	2	4	no	—
26	C₂²⋊D₁₅₇	—	2	4	no	—
27	C₂×Dic₁₅₇	—	2	2	no	—
28	C₄×D₁₅₇	—	2	4	no	—
29	C₄×D₁₅₇	—	2	4	no	—
30	C₂³×C₁₅₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	D₄	1	2	yes	—
32	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₅₇	C₂³	1	4	yes	—
36	C₂²×C₁₅₇	—	4	1	no	—
37	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
38	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
39	C₂×D₁₅₇	—	2	4	no	—
40	C₂×D₁₅₇	—	4	4	no	—
41	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
42	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
44	Dic₁₅₇	—	2	4	no	—
45	C₂×C₁₅₇	—	2	2	no	—
46	C₂×C₁₅₇	—	2	1	no	—
47	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
48	D₁₅₇	—	4	4	no	—
49	C₁₅₇	#575	1	1	yes	characteristic, complemented, Sylow
50	#575	—	157	1	no	maximal, Sylow
51	C₂×#41	—	157	2	no	—
52	#137	—	157	4	no	—
53	AGL(1,ℤ/8)	—	157	4	no	—
54	#135	—	157	4	no	—
55	C₂×M₁₆	D₁₅₇	1	1	yes	characteristic, complemented
56	#41	—	314	2	no	—
57	#41	—	314	2	no	—
58	#41	—	157	4	no	—
59	#41	—	157	4	no	—
60	C₂×Q₈	—	157	2	no	—
61	C₂×D₄	—	314	2	no	—
62	C₂×D₄	—	157	2	no	—
63	C₂²×C₄	—	314	2	no	—
64	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
65	Q₁₆	—	157	4	no	—
66	SD₁₆	—	157	8	no	—
67	D₈	—	157	4	no	—
68	M₁₆	C₂×D₁₅₇	1	4	yes	complemented
69	C₂×C₈	C₂×D₁₅₇	1	2	yes	complemented
70	C₂³	—	314	2	no	—
71	C₂³	D₆₂₈	1	1	yes	characteristic
72	Q₈	—	314	2	no	—
73	Q₈	—	157	4	no	—
74	D₄	—	314	4	no	—
75	D₄	—	314	4	no	—
76	D₄	—	314	2	no	—
77	D₄	—	157	4	no	—
78	C₂×C₄	—	314	2	no	—
79	C₂×C₄	—	314	4	no	—
80	C₂×C₄	—	314	4	no	—
81	C₂×C₄	D₆₂₈	1	1	yes	characteristic
82	C₂×C₄	D₆₂₈	1	2	yes	—
83	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
85	C₈	C₂²×D₁₅₇	1	4	yes	—
86	C₂²	—	4	1	no	—
87	C₂²	—	314	4	no	—
88	C₂²	—	628	4	no	—
89	C₂²	C₂×D₆₂₈	1	2	yes	—
90	C₂²	C₂×D₆₂₈	1	1	yes	characteristic
91	C₄	—	314	4	no	—
92	C₄	C₂×D₆₂₈	1	2	yes	—
93	C₄	C₂×D₆₂₈	1	1	yes	characteristic
94	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	2	1	no	—
97	C₂	—	628	4	no	—
98	C₂	C₂²×D₆₂₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578954

C₂

yes

maximal, complemented

#19097804

C₂

yes

maximal, complemented

#19097799

C₂

yes

maximal, complemented

#19097793

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

—

#14578954

—

#14578954

C₂²

yes

—

#14578954

C₂²

yes

complemented

C₂×Dic₃₁₄

C₂²

yes

complemented

C₂×D₆₂₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₅₇

—

C₂×C₄×D₁₅₇

—

Dic₆₂₈

C₂²

yes

complemented

#14578943

C₂²

yes

complemented

D₁₂₅₆

C₂²

yes

complemented

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

M₁₆×C₁₅₇

C₂²

yes

complemented

C₂×C₈×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

C₂³

yes

—

D₆₂₈

—

D₆₂₈

C₂³

yes

—

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

D₄

yes

characteristic

C₂×C₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

C₂⁴

yes

characteristic

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

D₁₅₇

—

C₁₅₇

#575

yes

characteristic, complemented, Sylow

#575

—

157

maximal, Sylow

C₂×#41

—

157

—

#137

—

157

—

AGL(1,ℤ/8)

—

157

—

#135

—

157

—

C₂×M₁₆

D₁₅₇

yes

characteristic, complemented

#41

—

314

—

#41

—

314

—

#41

—

157

—

#41

—

157

—

C₂×Q₈

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

Q₁₆

—

157

—

SD₁₆

—

157

—

D₈

—

157

—

M₁₆

C₂×D₁₅₇

yes

complemented

C₂×C₈

C₂×D₁₅₇

yes

complemented

C₂³

—

314

—

C₂³

D₆₂₈

yes

characteristic

Q₈

—

314

—

Q₈

—

157

—

D₄

—

314

—

D₄

—

314

—

D₄

—

314

—

D₄

—

157

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

D₆₂₈

yes

characteristic

C₂×C₄

D₆₂₈

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

C₂²×D₁₅₇

yes

—

C₂²

—

C₂²

—

314

—

C₂²

—

628

—

C₂²

C₂×D₆₂₈

yes

—

C₂²

C₂×D₆₂₈

yes

characteristic

C₄

—

314

—

C₄

C₂×D₆₂₈

yes

—

C₄

C₂×D₆₂₈

yes

characteristic

C₄

C₂³×D₁₅₇

yes

characteristic

C₂

—

C₂

—

C₂

—

628

—

C₂

C₂²×D₆₂₈

yes

characteristic

	subgroup	G/H
center	C₄	C₂×D₆₂₈
commutator, cosocle	C₄×C₁₅₇	C₂⁴
Frattini	C₄	C₂³×D₁₅₇
Fitting	C₂×M₁₆×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×D₄
hypercenter	C₂×M₁₆	D₁₅₇
hypocenter	C₁₅₇	#575
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₂×D₆₂₈

commutator, cosocle

C₄×C₁₅₇

C₂⁴

Frattini

C₄

C₂³×D₁₅₇

Fitting

C₂×M₁₆×C₁₅₇

C₂

socle

C₂×C₁₅₇

C₂²×D₄

hypercenter

C₂×M₁₆

D₁₅₇

hypocenter

C₁₅₇

#575

LCS

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
1. C₂×#14578954	C₂	semidirect
2. #19097804	C₂	semidirect
3. #19097799	C₂	semidirect
4. #19097793	C₂	semidirect
5. C₂×M₁₆×C₁₅₇	C₂	semidirect
9. #14578954	C₂²	semidirect
10. C₂×Dic₃₁₄	C₂²	semidirect
11. C₂×D₆₂₈	C₂²	semidirect
14. Dic₆₂₈	C₂²	semidirect
15. #14578943	C₂²	semidirect
16. D₁₂₅₆	C₂²	semidirect
18. M₁₆×C₁₅₇	C₂²	semidirect
19. C₂×C₈×C₁₅₇	C₂²	semidirect
49. C₁₅₇	#575	semidirect
55. C₂×M₁₆	D₁₅₇	semidirect
68. M₁₆	C₂×D₁₅₇	semidirect
69. C₂×C₈	C₂×D₁₅₇	semidirect

H ≅ G/N

type

1. C₂×#14578954

C₂

semidirect

2. #19097804

C₂

semidirect

3. #19097799

C₂

semidirect

4. #19097793

C₂

semidirect

5. C₂×M₁₆×C₁₅₇

C₂

semidirect

9. #14578954

C₂²

semidirect

10. C₂×Dic₃₁₄

C₂²

semidirect

11. C₂×D₆₂₈

C₂²

semidirect

14. Dic₆₂₈

C₂²

semidirect

15. #14578943

C₂²

semidirect

16. D₁₂₅₆

C₂²

semidirect

18. M₁₆×C₁₅₇

C₂²

semidirect

19. C₂×C₈×C₁₅₇

C₂²

semidirect

49. C₁₅₇

#575

semidirect

55. C₂×M₁₆

D₁₅₇

semidirect

68. M₁₆

C₂×D₁₅₇

semidirect

69. C₂×C₈

C₂×D₁₅₇

semidirect

Subgroups — #23758073

Conjugacy classes of subgroups

Complements