Group #23758085 — subgroups

Subgroups — #23758085

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	#19097817	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097804	C₂	1	3	yes	maximal, complemented
4	#19097799	C₂	1	3	yes	maximal, complemented
5	#19097793	C₂	1	3	yes	maximal, complemented
6	C₂×#14578943	C₂	1	3	yes	maximal, complemented
7	#131×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14578955	—	2	1	no	—
9	#14578954	C₂²	1	3	yes	—
10	#14578954	C₂²	1	3	yes	complemented
11	#14578953	—	2	3	no	—
12	C₂×Dic₃₁₄	C₂²	1	3	yes	complemented
13	Q₈×D₁₅₇	—	2	1	no	—
14	C₂×D₆₂₈	C₂²	1	3	yes	—
15	D₄×D₁₅₇	—	2	3	no	—
16	Dic₆₂₈	C₂²	1	3	yes	complemented
17	#14578943	C₂²	1	3	yes	complemented
18	#14578943	C₂²	1	6	yes	complemented
19	#14578943	C₂²	1	1	yes	characteristic, complemented
20	D₁₂₅₆	C₂²	1	3	yes	complemented
21	#41×C₁₅₇	C₂²	1	1	yes	characteristic
22	M₁₆×C₁₅₇	C₂²	1	3	yes	complemented
23	C₂×C₈×C₁₅₇	C₂²	1	3	yes	complemented
24	C₂²×D₁₅₇	—	2	3	no	—
25	Dic₃₁₄	—	2	3	no	—
26	Dic₃₁₄	C₂³	1	1	yes	characteristic
27	Dic₃₁₄	C₂³	1	3	yes	—
28	D₆₂₈	—	2	3	no	—
29	D₆₂₈	C₂³	1	3	yes	—
30	D₆₂₈	C₂³	1	1	yes	characteristic
31	C₂²⋊D₁₅₇	—	2	3	no	—
32	C₂²⋊D₁₅₇	—	2	3	no	—
33	C₂×Dic₁₅₇	—	2	3	no	—
34	C₄×D₁₅₇	—	2	3	no	—
35	C₄×D₁₅₇	—	2	3	no	—
36	Q₈×C₁₅₇	D₄	1	1	yes	characteristic
37	D₄×C₁₅₇	D₄	1	3	yes	—
38	C₂×C₄×C₁₅₇	C₂³	1	3	yes	—
39	C₈×C₁₅₇	C₂³	1	3	yes	—
40	C₈×C₁₅₇	C₂³	1	1	yes	characteristic
41	C₂²×C₁₅₇	C₂×D₄	1	3	yes	—
42	C₂×D₁₅₇	—	2	1	no	—
43	C₂×D₁₅₇	—	2	3	no	—
44	C₂×D₁₅₇	—	8	3	no	—
45	C₄×C₁₅₇	C₂×D₄	1	3	yes	—
46	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
47	Dic₁₅₇	—	2	1	no	—
48	Dic₁₅₇	—	2	3	no	—
49	C₂×C₁₅₇	—	2	3	no	—
50	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
51	D₁₅₇	—	4	1	no	—
52	D₁₅₇	—	4	3	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)	—	157	1	no	—
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	D₁₅₇	1	1	yes	characteristic, complemented
62	#41	—	314	1	no	—
63	#41	—	314	3	no	—
64	#41	—	157	3	no	—
65	#41	—	157	3	no	—
66	#41	C₂×D₁₅₇	1	1	yes	characteristic
67	C₂×Q₈	—	314	1	no	—
68	C₂×Q₈	—	157	3	no	—
69	C₂×D₄	—	314	3	no	—
70	C₂×D₄	—	157	3	no	—
71	Q₁₆	—	157	3	no	—
72	SD₁₆	—	157	3	no	—
73	SD₁₆	—	157	6	no	—
74	SD₁₆	—	157	1	no	—
75	D₈	—	157	3	no	—
76	M₁₆	C₂×D₁₅₇	1	3	yes	complemented
77	C₂×C₈	C₂×D₁₅₇	1	3	yes	complemented
78	C₂³	—	314	3	no	—
79	Q₈	—	314	3	no	—
80	Q₈	—	157	3	no	—
81	Q₈	—	157	1	no	—
82	Q₈	D₆₂₈	1	1	yes	characteristic
83	D₄	—	314	3	no	—
84	D₄	—	314	3	no	—
85	D₄	—	314	3	no	—
86	D₄	—	157	1	no	—
87	D₄	—	157	3	no	—
88	D₄	D₆₂₈	1	3	yes	—
89	C₂×C₄	—	314	3	no	—
90	C₂×C₄	—	314	3	no	—
91	C₂×C₄	—	314	3	no	—
92	C₂×C₄	C₂²×D₁₅₇	1	3	yes	—
93	C₈	C₂²×D₁₅₇	1	1	yes	characteristic
94	C₈	C₂²×D₁₅₇	1	3	yes	—
95	C₂²	—	314	1	no	—
96	C₂²	—	314	3	no	—
97	C₂²	—	1256	3	no	—
98	C₂²	C₂×D₆₂₈	1	3	yes	—
99	C₄	—	314	1	no	—
100	C₄	—	314	3	no	—
101	C₄	C₂×D₆₂₈	1	3	yes	—
102	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	628	3	no	—
105	C₂	—	628	1	no	—
106	C₂	C₂²×D₆₂₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19097818

C₂

yes

maximal, characteristic, complemented

#19097817

C₂

yes

maximal, characteristic, complemented

#19097804

C₂

yes

maximal, complemented

#19097799

C₂

yes

maximal, complemented

#19097793

C₂

yes

maximal, complemented

C₂×#14578943

C₂

yes

maximal, complemented

#131×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578955

—

#14578954

C₂²

yes

—

#14578954

C₂²

yes

complemented

#14578953

—

C₂×Dic₃₁₄

C₂²

yes

complemented

Q₈×D₁₅₇

—

C₂×D₆₂₈

C₂²

yes

—

D₄×D₁₅₇

—

Dic₆₂₈

C₂²

yes

complemented

#14578943

C₂²

yes

complemented

#14578943

C₂²

yes

complemented

#14578943

C₂²

yes

characteristic, complemented

D₁₂₅₆

C₂²

yes

complemented

#41×C₁₅₇

C₂²

yes

characteristic

M₁₆×C₁₅₇

C₂²

yes

complemented

C₂×C₈×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

C₂³

yes

characteristic

Dic₃₁₄

C₂³

yes

—

D₆₂₈

—

D₆₂₈

C₂³

yes

—

D₆₂₈

C₂³

yes

characteristic

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

Q₈×C₁₅₇

D₄

yes

characteristic

D₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₈×C₁₅₇

C₂³

yes

—

C₈×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂⁴

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#577

yes

characteristic, complemented, Sylow

#577

—

157

maximal, Sylow

2₋¹⁺⁴

—

157

—

H(5,𝔽₂)

—

157

—

#137

—

157

—

AGL(1,ℤ/8)

—

157

—

#135

—

157

—

C₂×SD₁₆

—

157

—

#131

D₁₅₇

yes

characteristic, complemented

#41

—

314

—

#41

—

314

—

#41

—

157

—

#41

—

157

—

#41

C₂×D₁₅₇

yes

characteristic

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

Q₁₆

—

157

—

SD₁₆

—

157

—

SD₁₆

—

157

—

SD₁₆

—

157

—

D₈

—

157

—

M₁₆

C₂×D₁₅₇

yes

complemented

C₂×C₈

C₂×D₁₅₇

yes

complemented

C₂³

—

314

—

Q₈

—

314

—

Q₈

—

157

—

Q₈

—

157

—

Q₈

D₆₂₈

yes

characteristic

D₄

—

314

—

D₄

—

314

—

D₄

—

314

—

D₄

—

157

—

D₄

—

157

—

D₄

D₆₂₈

yes

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₈

C₂²×D₁₅₇

yes

characteristic

C₈

C₂²×D₁₅₇

yes

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

1256

—

C₂²

C₂×D₆₂₈

yes

—

C₄

—

314

—

100

C₄

—

314

—

101

C₄

C₂×D₆₂₈

yes

—

102

C₄

C₂³×D₁₅₇

yes

characteristic

103

C₂

—

104

C₂

—

628

—

105

C₂

—

628

—

106

C₂

C₂²×D₆₂₈

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×D₆₂₈
commutator, cosocle	C₄×C₁₅₇	C₂⁴
Frattini	C₄	C₂³×D₁₅₇
Fitting	#131×C₁₅₇	C₂
socle	C₂×C₁₅₇	C₂²×D₄
hypercenter	#131	D₁₅₇
hypocenter	C₁₅₇	#577
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

subgroup

G/H

center

C₂

C₂²×D₆₂₈

commutator, cosocle

C₄×C₁₅₇

C₂⁴

Frattini

C₄

C₂³×D₁₅₇

Fitting

#131×C₁₅₇

C₂

socle

C₂×C₁₅₇

C₂²×D₄

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131

N	H ≅ G/N	type
1. #19097818	C₂	semidirect
2. #19097817	C₂	semidirect
3. #19097804	C₂	semidirect
4. #19097799	C₂	semidirect
5. #19097793	C₂	semidirect
6. C₂×#14578943	C₂	semidirect
7. #131×C₁₅₇	C₂	semidirect
10. #14578954	C₂²	semidirect
12. C₂×Dic₃₁₄	C₂²	semidirect
16. Dic₆₂₈	C₂²	semidirect
17. #14578943	C₂²	semidirect
18. #14578943	C₂²	semidirect
19. #14578943	C₂²	semidirect
20. D₁₂₅₆	C₂²	semidirect
22. M₁₆×C₁₅₇	C₂²	semidirect
23. C₂×C₈×C₁₅₇	C₂²	semidirect
53. C₁₅₇	#577	semidirect
61. #131	D₁₅₇	semidirect
76. M₁₆	C₂×D₁₅₇	semidirect
77. C₂×C₈	C₂×D₁₅₇	semidirect

H ≅ G/N

type

1. #19097818

C₂

semidirect

2. #19097817

C₂

semidirect

3. #19097804

C₂

semidirect

4. #19097799

C₂

semidirect

5. #19097793

C₂

semidirect

6. C₂×#14578943

C₂

semidirect

7. #131×C₁₅₇

C₂

semidirect

10. #14578954

C₂²

semidirect

12. C₂×Dic₃₁₄

C₂²

semidirect

16. Dic₆₂₈

C₂²

semidirect

17. #14578943

C₂²

semidirect

18. #14578943

C₂²

semidirect

19. #14578943

C₂²

semidirect

20. D₁₂₅₆

C₂²

semidirect

22. M₁₆×C₁₅₇

C₂²

semidirect

23. C₂×C₈×C₁₅₇

C₂²

semidirect

53. C₁₅₇

#577

semidirect

61. #131

D₁₅₇

semidirect

76. M₁₆

C₂×D₁₅₇

semidirect

77. C₂×C₈

C₂×D₁₅₇

semidirect

Subgroups — #23758085

Conjugacy classes of subgroups

Complements