Group #23757774 — subgroups

Subgroups — C₂×#19097752 #23757774

Order 10048.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂³×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#19097752	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14578933	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊Dic₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂²×D₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₅₇	—	2	4	no	—
8	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
9	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
10	C₂²×Dic₁₅₇	C₂²	1	4	yes	complemented
11	#14578933	C₂²	1	8	yes	complemented
12	C₂²⋊Dic₁₅₇	C₂²	1	8	yes	complemented
13	C₂²⋊Dic₁₅₇	C₂²	1	4	yes	complemented
14	C₂⁴×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂⁴×C₁₅₇	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₅₇	—	2	4	no	—
17	C₂×D₄×C₁₅₇	C₂²	1	4	yes	—
18	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₅₇	—	2	4	no	—
20	C₂×Dic₁₅₇	—	2	8	no	—
21	C₂×Dic₁₅₇	—	2	4	no	—
22	C₂×Dic₁₅₇	C₂³	1	4	yes	—
23	C₂×Dic₁₅₇	C₂³	1	4	yes	—
24	C₂³×C₁₅₇	—	2	4	no	—
25	C₂³×C₁₅₇	—	2	2	no	—
26	C₂³×C₁₅₇	—	2	4	no	—
27	C₂³×C₁₅₇	D₄	1	4	yes	—
28	C₂³×C₁₅₇	C₂³	1	2	yes	—
29	C₂³×C₁₅₇	C₂³	1	2	yes	—
30	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
31	D₄×C₁₅₇	—	2	8	no	—
32	C₂×C₄×C₁₅₇	—	2	2	no	—
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₂²×C₁₅₇	—	2	8	no	—
35	C₂²×C₁₅₇	—	2	4	no	—
36	C₂²×C₁₅₇	—	2	2	no	—
37	C₂²×C₁₅₇	—	2	4	no	—
38	C₂²×C₁₅₇	—	4	4	no	—
39	C₂²×C₁₅₇	C₂×D₄	1	4	yes	—
40	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
41	C₂²×C₁₅₇	#41	1	4	yes	—
42	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
43	C₄×C₁₅₇	—	2	2	no	—
44	Dic₁₅₇	—	2	4	no	—
45	Dic₁₅₇	—	2	4	no	—
46	C₂×C₁₅₇	—	2	4	no	—
47	C₂×C₁₅₇	—	4	2	no	—
48	C₂×C₁₅₇	#123	1	4	yes	complemented
49	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₅₇	C₂×#41	1	2	yes	—
51	C₁₅₇	C₂×#123	1	1	yes	characteristic, complemented, Sylow
52	C₂×#123	—	157	1	no	maximal, Sylow
53	C₂²×D₄	D₁₅₇	1	1	yes	characteristic
54	C₂³×C₄	—	157	1	no	—
55	#123	—	157	8	no	—
56	C₂×C₄⋊C₄	—	157	2	no	—
57	C₂×C₂²⋊C₄	—	157	1	no	—
58	C₂×C₂²⋊C₄	—	157	2	no	—
59	C₂⁴	C₂×D₁₅₇	1	1	yes	characteristic
60	C₂⁴	C₂×D₁₅₇	1	1	yes	characteristic
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	C₂×D₁₅₇	1	4	yes	—
63	C₂²×C₄	—	314	4	no	—
64	C₂²×C₄	—	157	4	no	—
65	C₂²×C₄	—	157	2	no	—
66	C₂²×C₄	—	157	2	no	—
67	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
68	C₄⋊C₄	—	157	8	no	—
69	C₂²⋊C₄	—	157	4	no	—
70	C₂²⋊C₄	—	157	8	no	—
71	C₂³	—	2	4	no	—
72	C₂³	—	2	2	no	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂²⋊D₁₅₇	1	4	yes	—
75	C₂³	C₂²×D₁₅₇	1	2	yes	—
76	C₂³	C₂²×D₁₅₇	1	2	yes	—
77	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
78	D₄	—	2	8	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	314	8	no	—
81	C₂×C₄	—	314	4	no	—
82	C₂×C₄	—	314	4	no	—
83	C₂×C₄	—	157	4	no	—
84	C₂×C₄	—	157	4	no	—
85	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
86	C₂²	—	2	8	no	—
87	C₂²	—	2	4	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	4	4	no	—
91	C₂²	C₂×C₂²⋊D₁₅₇	1	4	yes	—
92	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
93	C₂²	#14578953	1	4	yes	—
94	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	314	4	no	—
97	C₄	—	314	4	no	—
98	C₂	—	2	4	no	—
99	C₂	—	4	2	no	—
100	C₂	#19097752	1	4	yes	direct product factor
101	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
102	C₂	C₂×#14578953	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

#19097752

C₂

yes

maximal, direct product factor

C₂×#14578933

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×C₂²⋊Dic₁₅₇

C₂

yes

maximal, complemented

C₂²×D₄×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

C₂²

yes

complemented

#14578933

C₂²

yes

complemented

C₂²⋊Dic₁₅₇

C₂²

yes

complemented

C₂²⋊Dic₁₅₇

C₂²

yes

complemented

C₂⁴×C₁₅₇

C₂²

yes

characteristic

C₂⁴×C₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

—

C₂³×C₁₅₇

D₄

yes

—

C₂³×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

—

C₂³×C₁₅₇

C₂³

yes

characteristic

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₄×C₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#123

yes

complemented

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

C₂×#41

yes

—

C₁₅₇

C₂×#123

yes

characteristic, complemented, Sylow

C₂×#123

—

157

maximal, Sylow

C₂²×D₄

D₁₅₇

yes

characteristic

C₂³×C₄

—

157

—

#123

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂⁴

C₂×D₁₅₇

yes

characteristic

C₂⁴

C₂×D₁₅₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₁₅₇

yes

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

—

C₂³

C₂²×D₁₅₇

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₂²

#14578953

yes

—

C₂²

C₂³×D₁₅₇

yes

characteristic

C₄

—

C₄

—

314

—

C₄

—

314

—

C₂

—

C₂

—

100

C₂

#19097752

yes

direct product factor

101

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

102

C₂

C₂×#14578953

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₅₇
commutator, cosocle	C₂²×C₁₅₇	C₂⁴
Frattini	C₂²	C₂³×D₁₅₇
Fitting, prime radical	C₂²×D₄×C₁₅₇	C₂
socle	C₂³×C₁₅₇	C₂³
hypercenter	C₂²×D₄	D₁₅₇
hypocenter	C₁₅₇	C₂×#123
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₅₇

commutator, cosocle

C₂²×C₁₅₇

C₂⁴

Frattini

C₂²

C₂³×D₁₅₇

Fitting, prime radical

C₂²×D₄×C₁₅₇

C₂

socle

C₂³×C₁₅₇

C₂³

hypercenter

C₂²×D₄

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂³×Dic₁₅₇	C₂	semidirect
2. #19097752	C₂	direct (N × H)
3. C₂×#14578933	C₂	semidirect
4. C₂×C₂²⋊Dic₁₅₇	C₂	semidirect
5. C₂×C₂²⋊Dic₁₅₇	C₂	semidirect
10. C₂²×Dic₁₅₇	C₂²	semidirect
11. #14578933	C₂²	semidirect
12. C₂²⋊Dic₁₅₇	C₂²	semidirect
13. C₂²⋊Dic₁₅₇	C₂²	semidirect
48. C₂×C₁₅₇	#123	semidirect
51. C₁₅₇	C₂×#123	semidirect
100. C₂	#19097752	direct (N × H)

H ≅ G/N

type

1. C₂³×Dic₁₅₇

C₂

semidirect

2. #19097752

C₂

direct (N × H)

3. C₂×#14578933

C₂

semidirect

4. C₂×C₂²⋊Dic₁₅₇

C₂

semidirect

5. C₂×C₂²⋊Dic₁₅₇

C₂

semidirect

10. C₂²×Dic₁₅₇

C₂²

semidirect

11. #14578933

C₂²

semidirect

12. C₂²⋊Dic₁₅₇

C₂²

semidirect

13. C₂²⋊Dic₁₅₇

C₂²

semidirect

48. C₂×C₁₅₇

#123

semidirect

51. C₁₅₇

C₂×#123

semidirect

100. C₂

#19097752

direct (N × H)

Subgroups — C₂×#19097752 #23757774

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19097752 #23757774

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19097752 #23757774