Group #23758015 — subgroups

Subgroups — #23758015

Order 10048.

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

normal subgroups only

This group has 260 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	#19097760	C₂	1	2	yes	maximal, complemented
2	#19097759	C₂	1	2	yes	maximal, complemented
3	#19097757	C₂	1	4	yes	maximal, complemented
4	#19097741	C₂	1	4	yes	maximal, complemented
5	#19097739	C₂	1	2	yes	maximal, complemented
6	#124×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₅₇	C₂²	1	2	yes	complemented
8	C₂×Dic₃₁₄	C₂²	1	2	yes	complemented
9	C₂×D₆₂₈	C₂²	1	2	yes	—
10	C₂×D₆₂₈	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₅₇	C₂²	1	4	yes	—
12	C₂²⋊Dic₁₅₇	C₂²	1	2	yes	complemented
13	#14578931	C₂²	1	4	yes	complemented
14	#14578931	C₂²	1	4	yes	complemented
15	#14578931	C₂²	1	4	yes	complemented
16	C₄×Dic₁₅₇	C₂²	1	2	yes	complemented
17	C₂×Q₈×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
18	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₁₅₇	C₂²	1	4	yes	complemented
20	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic, complemented
21	C₂²×D₁₅₇	—	4	4	no	—
22	C₂²×D₁₅₇	—	4	2	no	—
23	C₂²×D₁₅₇	C₂³	1	4	yes	—
24	Dic₃₁₄	—	4	2	no	—
25	D₆₂₈	—	4	2	no	—
26	D₆₂₈	—	4	2	no	—
27	C₂²⋊D₁₅₇	—	4	4	no	—
28	C₂×Dic₁₅₇	C₂³	1	4	yes	—
29	C₂³×C₁₅₇	C₂³	1	2	yes	—
30	Q₈×C₁₅₇	—	4	1	no	—
31	D₄×C₁₅₇	—	4	1	no	—
32	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₂²×C₁₅₇	—	2	2	no	—
36	C₂²×C₁₅₇	—	2	4	no	—
37	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₅₇	—	2	4	no	—
39	C₂×D₁₅₇	—	2	4	no	—
40	C₂×D₁₅₇	—	2	4	no	—
41	C₂×D₁₅₇	—	4	8	no	—
42	C₄×C₁₅₇	—	2	1	no	—
43	C₄×C₁₅₇	—	2	2	no	—
44	C₄×C₁₅₇	—	2	2	no	—
45	Dic₁₅₇	—	2	4	no	—
46	C₂×C₁₅₇	—	4	2	no	—
47	C₂×C₁₅₇	H(5,𝔽₂)	1	1	yes	characteristic
48	C₂×C₁₅₇	H(5,𝔽₂)	1	2	yes	—
49	D₁₅₇	—	4	4	no	—
50	C₁₅₇	H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
51	H(3,𝔽₄)	—	157	1	no	maximal, Sylow
52	#124	—	157	2	no	—
53	#124	—	157	4	no	—
54	#124	—	157	2	no	—
55	#124	D₁₅₇	1	1	yes	characteristic, complemented
56	#120	—	157	2	no	—
57	#120	—	157	4	no	—
58	C₂⁴	—	157	2	no	—
59	C₂×Q₈	—	157	2	no	—
60	C₂×Q₈	C₂×D₁₅₇	1	1	yes	characteristic, complemented
61	C₂×D₄	—	157	4	no	—
62	C₂×D₄	—	157	2	no	—
63	C₂×D₄	—	157	2	no	—
64	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic
65	C₂²⋊C₄	—	157	2	no	—
66	C₂²⋊C₄	—	157	4	no	—
67	C₂²⋊C₄	—	157	4	no	—
68	C₂²⋊C₄	—	157	4	no	—
69	C₂²⋊C₄	C₂×D₁₅₇	1	4	yes	complemented
70	C₄²	—	157	2	no	—
71	C₄²	C₂×D₁₅₇	1	1	yes	characteristic, complemented
72	C₂³	—	628	2	no	—
73	C₂³	—	628	4	no	—
74	C₂³	—	157	4	no	—
75	C₂³	C₂²×D₁₅₇	1	2	yes	—
76	Q₈	—	4	1	no	—
77	Q₈	—	628	2	no	—
78	D₄	—	4	1	no	—
79	D₄	—	628	2	no	—
80	D₄	—	628	2	no	—
81	D₄	—	628	4	no	—
82	C₂×C₄	—	157	4	no	—
83	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
84	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
85	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	314	4	no	—
89	C₂²	—	314	4	no	—
90	C₂²	—	314	4	no	—
91	C₂²	—	628	8	no	—
92	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	314	4	no	—
97	C₂	—	4	2	no	—
98	C₂	—	628	4	no	—
99	C₂	#19097816	1	1	yes	characteristic
100	C₂	#19097817	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19097760

C₂

yes

maximal, complemented

#19097759

C₂

yes

maximal, complemented

#19097757

C₂

yes

maximal, complemented

#19097741

C₂

yes

maximal, complemented

#19097739

C₂

yes

maximal, complemented

#124×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

complemented

C₂×Dic₃₁₄

C₂²

yes

complemented

C₂×D₆₂₈

C₂²

yes

—

C₂×D₆₂₈

C₂²

yes

—

C₂×C₂²⋊D₁₅₇

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

complemented

#14578931

C₂²

yes

complemented

#14578931

C₂²

yes

complemented

#14578931

C₂²

yes

complemented

C₄×Dic₁₅₇

C₂²

yes

complemented

C₂×Q₈×C₁₅₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₁₅₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₅₇

C₂²

yes

complemented

C₄²×C₁₅₇

C₂²

yes

characteristic, complemented

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

C₂³

yes

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

H(5,𝔽₂)

yes

characteristic

C₂×C₁₅₇

H(5,𝔽₂)

yes

—

D₁₅₇

—

C₁₅₇

H(3,𝔽₄)

yes

characteristic, complemented, Sylow

H(3,𝔽₄)

—

157

maximal, Sylow

#124

—

157

—

#124

—

157

—

#124

—

157

—

#124

D₁₅₇

yes

characteristic, complemented

#120

—

157

—

#120

—

157

—

C₂⁴

—

157

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

characteristic, complemented

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

C₂×D₁₅₇

yes

complemented

C₄²

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic, complemented

C₂³

—

628

—

C₂³

—

628

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

—

Q₈

—

Q₈

—

628

—

D₄

—

D₄

—

628

—

D₄

—

628

—

D₄

—

628

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂²

—

C₂²

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

628

—

C₂²

C₂³×D₁₅₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

314

—

C₂

—

C₂

—

628

—

C₂

#19097816

yes

characteristic

100

C₂

#19097817

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	#124×C₁₅₇	C₂
hypercenter	#124	D₁₅₇
hypocenter	C₁₅₇	H(3,𝔽₄)
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

#124×C₁₅₇

C₂

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #19097760	C₂	semidirect
2. #19097759	C₂	semidirect
3. #19097757	C₂	semidirect
4. #19097741	C₂	semidirect
5. #19097739	C₂	semidirect
6. #124×C₁₅₇	C₂	semidirect
7. C₂³×D₁₅₇	C₂²	semidirect
8. C₂×Dic₃₁₄	C₂²	semidirect
12. C₂²⋊Dic₁₅₇	C₂²	semidirect
13. #14578931	C₂²	semidirect
14. #14578931	C₂²	semidirect
15. #14578931	C₂²	semidirect
16. C₄×Dic₁₅₇	C₂²	semidirect
17. C₂×Q₈×C₁₅₇	C₂²	semidirect
19. C₂²⋊C₄×C₁₅₇	C₂²	semidirect
20. C₄²×C₁₅₇	C₂²	semidirect
50. C₁₅₇	H(3,𝔽₄)	semidirect
55. #124	D₁₅₇	semidirect
60. C₂×Q₈	C₂×D₁₅₇	semidirect
69. C₂²⋊C₄	C₂×D₁₅₇	semidirect
71. C₄²	C₂×D₁₅₇	semidirect

H ≅ G/N

type

1. #19097760

C₂

semidirect

2. #19097759

C₂

semidirect

3. #19097757

C₂

semidirect

4. #19097741

C₂

semidirect

5. #19097739

C₂

semidirect

6. #124×C₁₅₇

C₂

semidirect

7. C₂³×D₁₅₇

C₂²

semidirect

8. C₂×Dic₃₁₄

C₂²

semidirect

12. C₂²⋊Dic₁₅₇

C₂²

semidirect

13. #14578931

C₂²

semidirect

14. #14578931

C₂²

semidirect

15. #14578931

C₂²

semidirect

16. C₄×Dic₁₅₇

C₂²

semidirect

17. C₂×Q₈×C₁₅₇

C₂²

semidirect

19. C₂²⋊C₄×C₁₅₇

C₂²

semidirect

20. C₄²×C₁₅₇

C₂²

semidirect

50. C₁₅₇

H(3,𝔽₄)

semidirect

55. #124

D₁₅₇

semidirect

60. C₂×Q₈

C₂×D₁₅₇

semidirect

69. C₂²⋊C₄

C₂×D₁₅₇

semidirect

71. C₄²

C₂×D₁₅₇

semidirect

Subgroups — #23758015

Conjugacy classes of subgroups

Complements