Group #23757424 — subgroups

Subgroups — #23757424

Order 10048.

188 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14578942	C₂	1	2	yes	maximal, complemented
2	#19097750	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097692	C₂	1	2	yes	maximal, complemented
4	C₂²⋊#11929434	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic, complemented
7	#14578942	—	2	4	no	—
8	C₂×#11929434	C₂²	1	2	yes	complemented
9	C₄⋊Dic₁₅₇	C₂²	1	1	yes	characteristic, complemented
10	#14578933	—	2	1	no	—
11	C₂²⋊Dic₁₅₇	—	2	1	no	—
12	C₂⁴×C₁₅₇	—	2	1	no	—
13	C₂×D₄×C₁₅₇	—	2	4	no	—
14	C₂×D₄×C₁₅₇	—	2	1	no	—
15	C₂×D₄×C₁₅₇	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
17	Dic₃₁₄	—	2	2	no	—
18	C₂×Dic₁₅₇	—	2	1	no	—
19	C₂×Dic₁₅₇	—	2	1	no	—
20	#11929434	—	2	2	no	—
21	C₂³×C₁₅₇	—	2	4	no	—
22	C₂³×C₁₅₇	—	2	2	no	—
23	C₂³×C₁₅₇	—	4	2	no	—
24	C₂³×C₁₅₇	—	4	2	no	—
25	C₂³×C₁₅₇	D₄	1	1	yes	characteristic
26	D₄×C₁₅₇	—	2	4	no	—
27	D₄×C₁₅₇	—	2	2	no	—
28	D₄×C₁₅₇	D₄	1	4	yes	—
29	C₂×C₄×C₁₅₇	—	2	2	no	—
30	C₂×C₄×C₁₅₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₂²×C₁₅₇	—	2	1	no	—
33	C₂²×C₁₅₇	—	2	1	no	—
34	C₂²×C₁₅₇	—	2	4	no	—
35	C₂²×C₁₅₇	—	4	8	no	—
36	C₂²×C₁₅₇	—	4	2	no	—
37	C₂²×C₁₅₇	—	4	2	no	—
38	C₂²×C₁₅₇	SD₁₆	1	2	yes	complemented
39	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₁₅₇	—	2	1	no	—
41	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
42	Dic₁₅₇	—	4	1	no	—
43	Dic₁₅₇	—	4	1	no	—
44	C₂×C₁₅₇	—	2	2	no	—
45	C₂×C₁₅₇	—	4	4	no	—
46	C₂×C₁₅₇	#120	1	1	yes	characteristic
47	C₂×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
48	C₂×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₁₅₇	#450	1	1	yes	characteristic, complemented, Sylow
50	#450	—	157	1	no	maximal, Sylow
51	C₂²×D₄	D₁₅₇	1	1	yes	characteristic
52	C₂×SD₁₆	—	157	2	no	—
53	C₂²⋊Q₈	—	157	1	no	—
54	#102	—	157	2	no	—
55	C₂²⋊C₈	—	157	1	no	—
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	—	157	1	no	—
58	C₂×D₄	—	2	4	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	C₂×D₁₅₇	1	2	yes	—
61	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
62	SD₁₆	—	314	4	no	—
63	C₂×C₈	—	157	2	no	—
64	C₄⋊C₄	—	314	1	no	—
65	C₄⋊C₄	—	157	1	no	—
66	C₂²⋊C₄	—	314	1	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	4	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	C₂²⋊D₁₅₇	1	1	yes	characteristic
72	Q₈	—	314	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	2	2	no	—
75	D₄	C₂²⋊D₁₅₇	1	4	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	314	1	no	—
78	C₂×C₄	—	314	1	no	—
79	C₂×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
81	C₈	—	314	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	8	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	#14578942	1	2	yes	complemented
89	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	628	1	no	—
92	C₄	—	628	1	no	—
93	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	4	no	—
96	C₂	#19097740	1	1	yes	characteristic
97	C₂	C₂×#14578942	1	1	yes	characteristic
98	C₂	#19097797	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578942

C₂

yes

maximal, complemented

#19097750

C₂

yes

maximal, characteristic, complemented

#19097692

C₂

yes

maximal, complemented

C₂²⋊#11929434

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂×Dic₃₁₄

C₂²

yes

characteristic, complemented

#14578942

—

C₂×#11929434

C₂²

yes

complemented

C₄⋊Dic₁₅₇

C₂²

yes

characteristic, complemented

#14578933

—

C₂²⋊Dic₁₅₇

—

C₂⁴×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

D₄

yes

characteristic

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

SD₁₆

yes

complemented

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#120

yes

characteristic

C₂×C₁₅₇

C₂×SD₁₆

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#450

—

157

maximal, Sylow

C₂²×D₄

D₁₅₇

yes

characteristic

C₂×SD₁₆

—

157

—

C₂²⋊Q₈

—

157

—

#102

—

157

—

C₂²⋊C₈

—

157

—

C₂⁴

—

C₂×Q₈

—

157

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

SD₁₆

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

314

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₅₇

yes

characteristic

Q₈

—

314

—

D₄

—

D₄

—

D₄

C₂²⋊D₁₅₇

yes

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

—

314

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14578942

yes

complemented

C₂²

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₄

—

C₄

—

628

—

C₄

—

628

—

C₄

C₂×C₂²⋊D₁₅₇

yes

—

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting, prime radical	C₂²×D₄×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₄
hypercenter	C₂²×D₄	D₁₅₇
hypocenter	C₁₅₇	#450
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₇

Fitting, prime radical

C₂²×D₄×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×D₄

hypercenter

C₂²×D₄

D₁₅₇

hypocenter

C₁₅₇

#450

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#14578942	C₂	semidirect
2. #19097750	C₂	semidirect
3. #19097692	C₂	semidirect
4. C₂²⋊#11929434	C₂	semidirect
6. C₂×Dic₃₁₄	C₂²	semidirect
8. C₂×#11929434	C₂²	semidirect
9. C₄⋊Dic₁₅₇	C₂²	semidirect
38. C₂²×C₁₅₇	SD₁₆	semidirect
49. C₁₅₇	#450	semidirect
88. C₂²	#14578942	semidirect

H ≅ G/N

type

1. C₂×#14578942

C₂

semidirect

2. #19097750

C₂

semidirect

3. #19097692

C₂

semidirect

4. C₂²⋊#11929434

C₂

semidirect

6. C₂×Dic₃₁₄

C₂²

semidirect

8. C₂×#11929434

C₂²

semidirect

9. C₄⋊Dic₁₅₇

C₂²

semidirect

38. C₂²×C₁₅₇

SD₁₆

semidirect

49. C₁₅₇

#450

semidirect

88. C₂²

#14578942

semidirect

Subgroups — #23757424

Conjugacy classes of subgroups

Complements