Group #23757175 — subgroups

Subgroups — #23757175

Order 10048.

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

normal subgroups only

This group has 258 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	C₂²×D₆₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊Dic₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14578931	C₂	1	4	yes	maximal, complemented
4	C₂×C₄²×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₁₅₇	C₂²	1	2	yes	—
6	C₂×D₆₂₈	—	2	2	no	—
7	C₂×D₆₂₈	—	2	2	no	—
8	C₂×D₆₂₈	C₄	1	4	yes	complemented
9	C₂²×Dic₁₅₇	C₂²	1	2	yes	—
10	C₄⋊Dic₁₅₇	—	2	2	no	—
11	#14578931	—	2	8	no	—
12	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₄²×C₁₅₇	—	2	2	no	—
15	C₂²×D₁₅₇	—	2	2	no	—
16	C₂²×D₁₅₇	—	2	2	no	—
17	C₂²×D₁₅₇	—	2	4	no	—
18	C₂²×D₁₅₇	—	2	4	no	—
19	C₂²×D₁₅₇	C₂×C₄	1	4	yes	—
20	D₆₂₈	—	2	8	no	—
21	C₂×Dic₁₅₇	—	2	4	no	—
22	C₂×Dic₁₅₇	—	2	2	no	—
23	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₁₅₇	—	2	2	no	—
25	C₂×C₄×C₁₅₇	—	2	2	no	—
26	C₂×C₄×C₁₅₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₁₅₇	D₄	1	2	yes	—
28	C₂×C₄×C₁₅₇	D₄	1	2	yes	—
29	C₂×C₄×C₁₅₇	D₄	1	4	yes	complemented
30	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
31	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
34	C₂²×C₁₅₇	#41	1	2	yes	—
35	C₂×D₁₅₇	—	2	4	no	—
36	C₂×D₁₅₇	—	2	8	no	—
37	C₂×D₁₅₇	—	4	4	no	—
38	C₂×D₁₅₇	—	4	2	no	—
39	C₂×D₁₅₇	—	4	2	no	—
40	C₄×C₁₅₇	—	2	4	no	—
41	C₄×C₁₅₇	C₂²⋊C₄	1	4	yes	complemented
42	Dic₁₅₇	—	4	2	no	—
43	C₂×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₁₅₇	C₄×D₄	1	2	yes	—
45	C₂×C₁₅₇	#121	1	2	yes	—
46	C₂×C₁₅₇	#124	1	1	yes	characteristic
47	C₂×C₁₅₇	#127	1	1	yes	characteristic
48	D₁₅₇	—	4	4	no	—
49	C₁₅₇	#390	1	1	yes	characteristic, complemented, Sylow
50	#390	—	157	1	no	maximal, Sylow
51	C₂²×D₄	—	157	1	no	—
52	C₂×C₄⋊C₄	—	157	1	no	—
53	C₂×C₂²⋊C₄	—	157	4	no	—
54	C₂×C₄²	D₁₅₇	1	1	yes	characteristic, complemented
55	C₂⁴	—	157	2	no	—
56	C₂×D₄	—	314	2	no	—
57	C₂×D₄	—	314	2	no	—
58	C₂×D₄	—	157	4	no	—
59	C₂²×C₄	—	157	2	no	—
60	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
61	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
62	C₄⋊C₄	—	314	2	no	—
63	C₂²⋊C₄	—	314	8	no	—
64	C₄²	—	2	2	no	—
65	C₂³	—	314	2	no	—
66	C₂³	—	314	4	no	—
67	C₂³	—	314	2	no	—
68	C₂³	—	314	4	no	—
69	C₂³	—	157	4	no	—
70	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
71	D₄	—	314	8	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	314	4	no	—
75	C₂×C₄	—	314	2	no	—
76	C₂×C₄	C₄×D₁₅₇	1	2	yes	—
77	C₂×C₄	C₂²⋊D₁₅₇	1	2	yes	—
78	C₂×C₄	D₆₂₈	1	2	yes	—
79	C₂×C₄	D₆₂₈	1	4	yes	complemented
80	C₂²	—	314	4	no	—
81	C₂²	—	314	8	no	—
82	C₂²	—	628	4	no	—
83	C₂²	—	628	2	no	—
84	C₂²	—	628	2	no	—
85	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
86	C₂²	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
87	C₂²	C₂×D₆₂₈	1	1	yes	characteristic
88	C₂²	C₂×D₆₂₈	1	2	yes	—
89	C₂²	#14578954	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	628	2	no	—
92	C₄	#14578931	1	4	yes	complemented
93	C₂	—	628	4	no	—
94	C₂	C₂×#14578931	1	1	yes	characteristic
95	C₂	C₄×D₆₂₈	1	2	yes	—
96	C₂	#19097745	1	2	yes	—
97	C₂	#19097759	1	1	yes	characteristic
98	C₂	#19097768	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₆₂₈

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×#14578931

C₂

yes

maximal, complemented

C₂×C₄²×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₇

C₂²

yes

—

C₂×D₆₂₈

—

C₂×D₆₂₈

—

C₂×D₆₂₈

C₄

yes

complemented

C₂²×Dic₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

—

#14578931

—

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₄²×C₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

—

C₂²×D₁₅₇

C₂×C₄

yes

—

D₆₂₈

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×C₄

yes

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

D₄

yes

complemented

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

#41

yes

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂²⋊C₄

yes

complemented

Dic₁₅₇

—

C₂×C₁₅₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₁₅₇

C₄×D₄

yes

—

C₂×C₁₅₇

#121

yes

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

D₁₅₇

—

C₁₅₇

#390

yes

characteristic, complemented, Sylow

#390

—

157

maximal, Sylow

C₂²×D₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₂²⋊C₄

—

157

—

C₂×C₄²

D₁₅₇

yes

characteristic, complemented

C₂⁴

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₄²

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

314

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

characteristic

D₄

—

314

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

C₄×D₁₅₇

yes

—

C₂×C₄

C₂²⋊D₁₅₇

yes

—

C₂×C₄

D₆₂₈

yes

—

C₂×C₄

D₆₂₈

yes

complemented

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

628

—

C₂²

—

628

—

C₂²

—

628

—

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

C₂²

C₂×C₂²⋊D₁₅₇

yes

characteristic

C₂²

C₂×D₆₂₈

yes

characteristic

C₂²

C₂×D₆₂₈

yes

—

C₂²

#14578954

yes

—

C₄

—

C₄

—

628

—

C₄

#14578931

yes

complemented

C₂

—

628

—

C₂

C₂×#14578931

yes

characteristic

C₂

C₄×D₆₂₈

yes

—

C₂

#19097745

yes

—

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₇
commutator	C₂²×C₁₅₇	C₂²×C₄
Fitting	C₂×C₄²×C₁₅₇	C₂
socle, cosocle	C₂³×C₁₅₇	C₂³
prime radical	C₂²×D₆₂₈	C₂
hypercenter	C₂×C₄²	D₁₅₇
hypocenter	C₁₅₇	#390
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₇

commutator

C₂²×C₁₅₇

C₂²×C₄

Fitting

C₂×C₄²×C₁₅₇

C₂

socle, cosocle

C₂³×C₁₅₇

C₂³

prime radical

C₂²×D₆₂₈

C₂

hypercenter

C₂×C₄²

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
2. C₂×C₄⋊Dic₁₅₇	C₂	semidirect
3. C₂×#14578931	C₂	semidirect
4. C₂×C₄²×C₁₅₇	C₂	semidirect
8. C₂×D₆₂₈	C₄	semidirect
29. C₂×C₄×C₁₅₇	D₄	semidirect
41. C₄×C₁₅₇	C₂²⋊C₄	semidirect
49. C₁₅₇	#390	semidirect
54. C₂×C₄²	D₁₅₇	semidirect
79. C₂×C₄	D₆₂₈	semidirect
92. C₄	#14578931	semidirect

H ≅ G/N

type

2. C₂×C₄⋊Dic₁₅₇

C₂

semidirect

3. C₂×#14578931

C₂

semidirect

4. C₂×C₄²×C₁₅₇

C₂

semidirect

8. C₂×D₆₂₈

C₄

semidirect

29. C₂×C₄×C₁₅₇

D₄

semidirect

41. C₄×C₁₅₇

C₂²⋊C₄

semidirect

49. C₁₅₇

#390

semidirect

54. C₂×C₄²

D₁₅₇

semidirect

79. C₂×C₄

D₆₂₈

semidirect

92. C₄

#14578931

semidirect

Subgroups — #23757175

Conjugacy classes of subgroups

Complements