Group #23758035 — subgroups

Subgroups — C₂×#19097777 #23758035

Order 10048.

266 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 151 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×#14578954	C₂	1	1	yes	maximal, characteristic
2	#19097777	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14578938	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14578937	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₁₅₇	C₂	1	2	yes	maximal, complemented
6	C₂²×C₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#14578954	C₄	1	4	yes	—
8	#14578954	C₂²	1	4	yes	—
9	C₂×Dic₃₁₄	C₄	1	1	yes	characteristic
10	C₂×D₆₂₈	C₄	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₅₇	C₄	1	2	yes	—
12	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
13	#14578938	C₂²	1	4	yes	complemented
14	#14578937	C₂²	1	8	yes	complemented
15	C₂×#11929434	C₂²	1	2	yes	—
16	C₈×D₁₅₇	C₂²	1	8	yes	complemented
17	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×C₈×C₁₅₇	C₂²	1	4	yes	complemented
21	C₂²×D₁₅₇	C₂×C₄	1	2	yes	—
22	Dic₃₁₄	C₂×C₄	1	4	yes	—
23	D₆₂₈	C₂×C₄	1	4	yes	—
24	C₂²⋊D₁₅₇	C₂×C₄	1	8	yes	—
25	C₂×Dic₁₅₇	C₂×C₄	1	2	yes	—
26	C₄×D₁₅₇	C₂×C₄	1	4	yes	—
27	C₄×D₁₅₇	C₂³	1	4	yes	—
28	#11929434	C₂³	1	4	yes	—
29	C₂³×C₁₅₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂³	1	2	yes	—
34	C₈×C₁₅₇	C₂³	1	2	yes	—
35	C₈×C₁₅₇	C₂³	1	2	yes	—
36	C₂²×C₁₅₇	—	2	2	no	—
37	C₂²×C₁₅₇	C₂²×C₄	1	2	yes	—
38	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₁₅₇	—	2	4	no	—
40	C₂×D₁₅₇	C₂²×C₄	1	4	yes	—
41	C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
43	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
44	Dic₁₅₇	C₂²×C₄	1	4	yes	—
45	C₂×C₁₅₇	—	2	2	no	—
46	C₂×C₁₅₇	#131	1	2	yes	complemented
47	C₂×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
48	D₁₅₇	—	2	4	no	—
49	C₁₅₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	157	1	no	maximal, Sylow
51	C₂×#41	—	157	1	no	—
52	#131	—	157	8	no	—
53	C₂×M₁₆	—	157	1	no	—
54	C₂×M₁₆	—	157	2	no	—
55	C₂²×C₈	—	157	2	no	—
56	C₂²×C₈	D₁₅₇	1	1	yes	characteristic, complemented
57	#41	—	157	4	no	—
58	#41	—	157	4	no	—
59	C₂×Q₈	—	157	1	no	—
60	C₂×D₄	—	157	2	no	—
61	C₂×D₄	—	157	1	no	—
62	C₂²×C₄	—	157	2	no	—
63	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
64	M₁₆	—	157	4	no	—
65	M₁₆	—	157	8	no	—
66	C₂×C₈	—	157	2	no	—
67	C₂×C₈	—	157	8	no	—
68	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
69	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
70	C₂×C₈	C₂×D₁₅₇	1	4	yes	complemented
71	C₂³	—	157	2	no	—
72	C₂³	C₄×D₁₅₇	1	1	yes	characteristic
73	Q₈	—	157	4	no	—
74	D₄	—	157	8	no	—
75	D₄	—	157	4	no	—
76	C₂×C₄	—	157	2	no	—
77	C₂×C₄	—	157	4	no	—
78	C₂×C₄	—	157	4	no	—
79	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
80	C₂×C₄	C₄×D₁₅₇	1	2	yes	—
81	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
82	C₂×C₄	C₂²×D₁₅₇	1	2	yes	—
83	C₈	—	157	4	no	—
84	C₈	C₂²×D₁₅₇	1	2	yes	—
85	C₈	C₂²×D₁₅₇	1	2	yes	—
86	C₂²	—	2	2	no	—
87	C₂²	—	314	4	no	—
88	C₂²	—	157	4	no	—
89	C₂²	C₂×C₄×D₁₅₇	1	2	yes	—
90	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
91	C₄	—	157	4	no	—
92	C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
93	C₄	C₂×C₄×D₁₅₇	1	2	yes	—
94	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
95	C₂	—	2	2	no	—
96	C₂	—	314	4	no	—
97	C₂	#19097777	1	2	yes	direct product factor
98	C₂	C₂²×C₄×D₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578954

C₂

yes

maximal, characteristic

#19097777

C₂

yes

maximal, direct product factor

C₂×#14578938

C₂

yes

maximal, characteristic, complemented

C₂×#14578937

C₂

yes

maximal, complemented

C₂×C₈×D₁₅₇

C₂

yes

maximal, complemented

C₂²×C₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

C₄

yes

—

#14578954

C₂²

yes

—

C₂×Dic₃₁₄

C₄

yes

characteristic

C₂×D₆₂₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₄

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

#14578938

C₂²

yes

complemented

#14578937

C₂²

yes

complemented

C₂×#11929434

C₂²

yes

—

C₈×D₁₅₇

C₂²

yes

complemented

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

C₂×C₄

yes

—

Dic₃₁₄

C₂×C₄

yes

—

D₆₂₈

C₂×C₄

yes

—

C₂²⋊D₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

C₂×C₄

yes

—

C₄×D₁₅₇

C₂×C₄

yes

—

C₄×D₁₅₇

C₂³

yes

—

#11929434

C₂³

yes

—

C₂³×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

—

C₈×C₁₅₇

C₂³

yes

—

C₈×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂²×C₄

yes

—

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂²×C₄

yes

—

C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₄×C₁₅₇

C₂²×C₄

yes

—

C₄×C₁₅₇

C₂⁴

yes

characteristic

Dic₁₅₇

C₂²×C₄

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#131

yes

complemented

C₂×C₁₅₇

C₂³×C₄

yes

characteristic

D₁₅₇

—

C₁₅₇

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

157

maximal, Sylow

C₂×#41

—

157

—

#131

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

D₁₅₇

yes

characteristic, complemented

#41

—

157

—

#41

—

157

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

M₁₆

—

157

—

M₁₆

—

157

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₈

C₂×D₁₅₇

yes

complemented

C₂³

—

157

—

C₂³

C₄×D₁₅₇

yes

characteristic

Q₈

—

157

—

D₄

—

157

—

D₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₄×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

—

C₈

—

157

—

C₈

C₂²×D₁₅₇

yes

—

C₈

C₂²×D₁₅₇

yes

—

C₂²

—

C₂²

—

314

—

C₂²

—

157

—

C₂²

C₂×C₄×D₁₅₇

yes

—

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

C₄

—

157

—

C₄

C₂×C₄×D₁₅₇

yes

characteristic

C₄

C₂×C₄×D₁₅₇

yes

—

C₄

C₂³×D₁₅₇

yes

characteristic

C₂

—

C₂

—

314

—

C₂

#19097777

yes

direct product factor

C₂

C₂²×C₄×D₁₅₇

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₁₅₇
commutator	C₂×C₁₅₇	C₂³×C₄
Frattini	C₄	C₂³×D₁₅₇
Fitting	C₂²×C₈×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂²×C₄
prime radical	C₂×#14578954	C₂
cosocle	C₄×C₁₅₇	C₂⁴
hypercenter	C₂²×C₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#131
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

subgroup

G/H

center

C₂×C₈

C₂×D₁₅₇

commutator

C₂×C₁₅₇

C₂³×C₄

Frattini

C₄

C₂³×D₁₅₇

Fitting

C₂²×C₈×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂²×C₄

prime radical

C₂×#14578954

C₂

cosocle

C₄×C₁₅₇

C₂⁴

hypercenter

C₂²×C₈

D₁₅₇

C₁₅₇

G ▷ C₂×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂×C₈ ◁ C₂²×C₈

N	H ≅ G/N	type
2. #19097777	C₂	direct (N × H)
3. C₂×#14578938	C₂	semidirect
4. C₂×#14578937	C₂	semidirect
5. C₂×C₈×D₁₅₇	C₂	semidirect
6. C₂²×C₈×C₁₅₇	C₂	semidirect
13. #14578938	C₂²	semidirect
14. #14578937	C₂²	semidirect
16. C₈×D₁₅₇	C₂²	semidirect
20. C₂×C₈×C₁₅₇	C₂²	semidirect
46. C₂×C₁₅₇	#131	semidirect
49. C₁₅₇	C₂×#131	semidirect
56. C₂²×C₈	D₁₅₇	semidirect
70. C₂×C₈	C₂×D₁₅₇	semidirect
97. C₂	#19097777	direct (N × H)

H ≅ G/N

type

2. #19097777

C₂

direct (N × H)

3. C₂×#14578938

C₂

semidirect

4. C₂×#14578937

C₂

semidirect

5. C₂×C₈×D₁₅₇

C₂

semidirect

6. C₂²×C₈×C₁₅₇

C₂

semidirect

13. #14578938

C₂²

semidirect

14. #14578937

C₂²

semidirect

16. C₈×D₁₅₇

C₂²

semidirect

20. C₂×C₈×C₁₅₇

C₂²

semidirect

46. C₂×C₁₅₇

#131

semidirect

49. C₁₅₇

C₂×#131

semidirect

56. C₂²×C₈

D₁₅₇

semidirect

70. C₂×C₈

C₂×D₁₅₇

semidirect

97. C₂

#19097777

direct (N × H)

Subgroups — C₂×#19097777 #23758035

Conjugacy classes of subgroups

Complements

Subgroups — C2×#19097777 #23758035

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#19097777 #23758035