Group #23757146 — subgroups

Subgroups — #23757146

Order 10048.

234 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14578933	C₂	1	2	yes	maximal, complemented
2	C₂×C₂²⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097678	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₅₇	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
9	#14578933	—	2	4	no	—
10	C₂²⋊Dic₁₅₇	—	2	2	no	—
11	C₂²⋊Dic₁₅₇	C₄	1	4	yes	complemented
12	C₂⁴×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₁₅₇	—	2	2	no	—
14	C₂²×C₄×C₁₅₇	—	2	4	no	—
15	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
16	C₂×Dic₁₅₇	—	2	2	no	—
17	C₂×Dic₁₅₇	—	2	2	no	—
18	C₂×Dic₁₅₇	—	2	1	no	—
19	C₂×Dic₁₅₇	—	2	1	no	—
20	C₂×Dic₁₅₇	—	2	2	no	—
21	C₂×Dic₁₅₇	—	2	2	no	—
22	C₂×Dic₁₅₇	C₂×C₄	1	4	yes	complemented
23	C₂³×C₁₅₇	—	2	2	no	—
24	C₂³×C₁₅₇	—	2	2	no	—
25	C₂³×C₁₅₇	C₂×C₄	1	2	yes	—
26	C₂³×C₁₅₇	D₄	1	2	yes	—
27	C₂³×C₁₅₇	Q₈	1	2	yes	—
28	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	—	2	8	no	—
30	C₂×C₄×C₁₅₇	—	2	2	no	—
31	C₂×C₄×C₁₅₇	—	2	2	no	—
32	C₂×C₄×C₁₅₇	D₄	1	4	yes	—
33	C₂²×C₁₅₇	—	2	4	no	—
34	C₂²×C₁₅₇	—	2	2	no	—
35	C₂²×C₁₅₇	—	2	4	no	—
36	C₂²×C₁₅₇	—	2	2	no	—
37	C₂²×C₁₅₇	C₄⋊C₄	1	4	yes	complemented
38	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
41	C₂²×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₁₅₇	#41	1	2	yes	—
43	C₄×C₁₅₇	—	2	4	no	—
44	Dic₁₅₇	—	2	4	no	—
45	Dic₁₅₇	—	4	1	no	—
46	Dic₁₅₇	—	4	1	no	—
47	C₂×C₁₅₇	—	2	4	no	—
48	C₂×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₁₅₇	C₄×D₄	1	2	yes	—
50	C₂×C₁₅₇	#120	1	1	yes	characteristic
51	C₂×C₁₅₇	C₂²⋊Q₈	1	2	yes	—
52	C₂×C₁₅₇	#123	1	1	yes	characteristic
53	C₁₅₇	#385	1	1	yes	characteristic, complemented, Sylow
54	#385	—	157	1	no	maximal, Sylow
55	C₂³×C₄	D₁₅₇	1	1	yes	characteristic
56	C₂×C₄⋊C₄	—	157	2	no	—
57	C₂×C₂²⋊C₄	—	157	1	no	—
58	C₂×C₂²⋊C₄	—	157	1	no	—
59	#95	—	157	2	no	—
60	C₂⁴	C₂×D₁₅₇	1	1	yes	characteristic
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	2	4	no	—
63	C₂²×C₄	—	157	1	no	—
64	C₂²×C₄	—	157	1	no	—
65	C₂²×C₄	—	157	2	no	—
66	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
67	C₄⋊C₄	—	314	4	no	—
68	C₂²⋊C₄	—	314	2	no	—
69	C₂²⋊C₄	—	157	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	2	no	—
72	C₂³	C₄×D₁₅₇	1	2	yes	—
73	C₂³	C₂²⋊D₁₅₇	1	2	yes	—
74	C₂³	Dic₃₁₄	1	2	yes	—
75	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
76	C₂×C₄	—	2	8	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	314	2	no	—
80	C₂×C₄	—	314	2	no	—
81	C₂×C₄	—	314	1	no	—
82	C₂×C₄	—	314	1	no	—
83	C₂×C₄	—	314	2	no	—
84	C₂×C₄	—	314	2	no	—
85	C₂×C₄	—	157	4	no	—
86	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	—
87	C₂²	—	2	4	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	2	2	no	—
91	C₂²	#14578933	1	4	yes	complemented
92	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
93	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
94	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
95	C₂²	C₂×Dic₃₁₄	1	1	yes	characteristic
96	C₂²	#14578954	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	314	4	no	—
99	C₄	—	628	1	no	—
100	C₄	—	628	1	no	—
101	C₂	—	2	4	no	—
102	C₂	C₂×#14578933	1	1	yes	characteristic
103	C₂	C₄×C₂²⋊D₁₅₇	1	2	yes	—
104	C₂	#19097740	1	1	yes	characteristic
105	C₂	#19097750	1	2	yes	—
106	C₂	#19097754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578933

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₅₇

C₂

yes

maximal, characteristic

C₂×C₂²⋊Dic₁₅₇

C₂

yes

maximal, characteristic

#19097678

C₂

yes

maximal, complemented

C₂³×C₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

C₂²

yes

—

#14578933

—

C₂²⋊Dic₁₅₇

—

C₂²⋊Dic₁₅₇

C₄

yes

complemented

C₂⁴×C₁₅₇

C₂²

yes

characteristic

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×C₄

yes

complemented

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

C₂×C₄

yes

—

C₂³×C₁₅₇

D₄

yes

—

C₂³×C₁₅₇

Q₈

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₄⋊C₄

yes

complemented

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂×Q₈

yes

characteristic

C₂²×C₁₅₇

#41

yes

—

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₅₇

C₄×D₄

yes

—

C₂×C₁₅₇

#120

yes

characteristic

C₂×C₁₅₇

C₂²⋊Q₈

yes

—

C₂×C₁₅₇

#123

yes

characteristic

C₁₅₇

#385

yes

characteristic, complemented, Sylow

#385

—

157

maximal, Sylow

C₂³×C₄

D₁₅₇

yes

characteristic

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

#95

—

157

—

C₂⁴

C₂×D₁₅₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₄⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂²⋊C₄

—

157

—

C₂³

—

C₂³

—

C₂³

C₄×D₁₅₇

yes

—

C₂³

C₂²⋊D₁₅₇

yes

—

C₂³

Dic₃₁₄

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

C₂²⋊D₁₅₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14578933

yes

complemented

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₂²

C₂×Dic₃₁₄

yes

characteristic

C₂²

#14578954

yes

—

C₄

—

C₄

—

314

—

C₄

—

628

—

100

C₄

—

628

—

101

C₂

—

102

C₂

C₂×#14578933

yes

characteristic

103

C₂

C₄×C₂²⋊D₁₅₇

yes

—

104

C₂

#19097740

yes

characteristic

105

C₂

#19097750

yes

—

106

C₂

#19097754

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂²×C₄
Fitting	C₂³×C₄×C₁₅₇	C₂
socle, cosocle	C₂³×C₁₅₇	C₂³
prime radical	C₂⁴×C₁₅₇	C₂²
hypercenter	C₂³×C₄	D₁₅₇
hypocenter	C₁₅₇	#385
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂²×C₄

Fitting

C₂³×C₄×C₁₅₇

C₂

socle, cosocle

C₂³×C₁₅₇

C₂³

prime radical

C₂⁴×C₁₅₇

C₂²

hypercenter

C₂³×C₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#14578933	C₂	semidirect
4. #19097678	C₂	semidirect
11. C₂²⋊Dic₁₅₇	C₄	semidirect
22. C₂×Dic₁₅₇	C₂×C₄	semidirect
37. C₂²×C₁₅₇	C₄⋊C₄	semidirect
53. C₁₅₇	#385	semidirect
91. C₂²	#14578933	semidirect

H ≅ G/N

type

1. C₂×#14578933

C₂

semidirect

4. #19097678

C₂

semidirect

11. C₂²⋊Dic₁₅₇

C₄

semidirect

22. C₂×Dic₁₅₇

C₂×C₄

semidirect

37. C₂²×C₁₅₇

C₄⋊C₄

semidirect

53. C₁₅₇

#385

semidirect

91. C₂²

#14578933

semidirect

Subgroups — #23757146

Conjugacy classes of subgroups

Complements