Group #23757847 — subgroups

Subgroups — #23757847

Order 10048.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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	#19097769	C₂	1	2	yes	maximal, complemented
3	#19097760	C₂	1	2	yes	maximal, complemented
4	#19097754	C₂	1	4	yes	maximal, complemented
5	#19097747	C₂	1	4	yes	maximal, complemented
6	#19097730	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#14578954	—	2	4	no	—
9	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
10	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
13	C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
14	#14578933	C₂²	1	8	yes	—
15	C₂²⋊Dic₁₅₇	C₂²	1	2	yes	—
16	#14578931	C₂²	1	8	yes	—
17	C₄×Dic₁₅₇	C₂²	1	2	yes	—
18	C₂×Q₈×C₁₅₇	—	2	4	no	—
19	C₂×Q₈×C₁₅₇	C₂²	1	2	yes	—
20	C₂×Q₈×C₁₅₇	C₂²	1	2	yes	—
21	C₂²×C₄×C₁₅₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
23	C₂²×D₁₅₇	C₂³	1	2	yes	—
24	Dic₃₁₄	—	2	2	no	—
25	D₆₂₈	—	2	2	no	—
26	C₂²⋊D₁₅₇	—	2	4	no	—
27	C₂×Dic₁₅₇	C₂³	1	2	yes	—
28	C₂×Dic₁₅₇	C₂³	1	4	yes	—
29	C₄×D₁₅₇	—	2	4	no	—
30	C₂³×C₁₅₇	C₂³	1	1	yes	characteristic
31	Q₈×C₁₅₇	—	2	4	no	—
32	Q₈×C₁₅₇	—	2	4	no	—
33	C₂×C₄×C₁₅₇	—	2	4	no	—
34	C₂×C₄×C₁₅₇	D₄	1	4	yes	—
35	C₂×C₄×C₁₅₇	C₂³	1	4	yes	—
36	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
38	C₂²×C₁₅₇	—	2	2	no	—
39	C₂²×C₁₅₇	C₂×D₄	1	2	yes	—
40	C₂²×C₁₅₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₁₅₇	—	2	2	no	—
42	C₂×D₁₅₇	—	2	4	no	—
43	C₄×C₁₅₇	—	2	4	no	—
44	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
45	C₄×C₁₅₇	C₂×D₄	1	2	yes	—
46	Dic₁₅₇	—	2	2	no	—
47	Dic₁₅₇	—	2	4	no	—
48	C₂×C₁₅₇	—	2	2	no	—
49	C₂×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₅₇	2₋¹⁺⁴	1	2	yes	—
51	D₁₅₇	—	4	2	no	—
52	C₁₅₇	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	157	1	no	maximal, Sylow
54	C₂×#41	—	157	1	no	—
55	C₂²×Q₈	D₁₅₇	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	157	2	no	—
57	#124	—	157	2	no	—
58	#123	—	157	4	no	—
59	C₂²⋊Q₈	—	157	4	no	—
60	#117	—	157	1	no	—
61	#41	—	314	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	157	1	no	—
64	C₂×Q₈	C₂×D₁₅₇	1	2	yes	—
65	C₂×Q₈	C₂×D₁₅₇	1	2	yes	—
66	C₂×D₄	—	157	2	no	—
67	C₂×D₄	—	157	1	no	—
68	C₂²×C₄	—	157	2	no	—
69	C₂²×C₄	C₂×D₁₅₇	1	2	yes	—
70	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
71	C₄⋊C₄	—	157	2	no	—
72	C₄⋊C₄	—	157	8	no	—
73	C₂²⋊C₄	—	157	2	no	—
74	C₂²⋊C₄	—	157	8	no	—
75	C₄²	—	157	2	no	—
76	C₂³	—	157	2	no	—
77	C₂³	C₂²×D₁₅₇	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	314	2	no	—
81	D₄	—	314	4	no	—
82	D₄	—	314	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	314	4	no	—
85	C₂×C₄	—	157	4	no	—
86	C₂×C₄	—	157	2	no	—
87	C₂×C₄	C₂²⋊D₁₅₇	1	4	yes	—
88	C₂×C₄	C₂²×D₁₅₇	1	4	yes	—
89	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	314	2	no	—
93	C₂²	—	314	4	no	—
94	C₂²	C₂×C₂²⋊D₁₅₇	1	2	yes	—
95	C₂²	C₂³×D₁₅₇	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	314	2	no	—
98	C₄	—	314	4	no	—
99	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
100	C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	628	2	no	—
103	C₂	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
104	C₂	#19097819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578954

C₂

yes

maximal, characteristic

#19097769

C₂

yes

maximal, complemented

#19097760

C₂

yes

maximal, complemented

#19097754

C₂

yes

maximal, complemented

#19097747

C₂

yes

maximal, complemented

#19097730

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

—

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₂²

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

C₄⋊Dic₁₅₇

C₂²

yes

—

#14578933

C₂²

yes

—

C₂²⋊Dic₁₅₇

C₂²

yes

—

#14578931

C₂²

yes

—

C₄×Dic₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

C₂²

yes

—

C₂×Q₈×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

—

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂³

yes

—

Dic₃₁₄

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

C₂³

yes

—

C₄×D₁₅₇

—

C₂³×C₁₅₇

C₂³

yes

characteristic

Q₈×C₁₅₇

—

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₄

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

C₂⁴

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

—

C₄×C₁₅₇

C₂×D₄

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂²×D₄

yes

characteristic

C₂×C₁₅₇

2₋¹⁺⁴

yes

—

D₁₅₇

—

C₁₅₇

#536

yes

characteristic, complemented, Sylow

#536

—

157

maximal, Sylow

C₂×#41

—

157

—

C₂²×Q₈

D₁₅₇

yes

characteristic, complemented

C₄⋊Q₈

—

157

—

#124

—

157

—

#123

—

157

—

C₂²⋊Q₈

—

157

—

#117

—

157

—

#41

—

314

—

C₂×Q₈

—

C₂×Q₈

—

157

—

C₂×Q₈

C₂×D₁₅₇

yes

—

C₂×Q₈

C₂×D₁₅₇

yes

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₄²

—

157

—

C₂³

—

157

—

C₂³

C₂²×D₁₅₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

314

—

D₄

—

314

—

D₄

—

314

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²⋊D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

C₂×C₂²⋊D₁₅₇

yes

—

C₂²

C₂³×D₁₅₇

yes

characteristic

C₄

—

C₄

—

314

—

C₄

—

314

—

C₄

C₂×C₂²⋊D₁₅₇

yes

—

100

C₄

C₂×C₂²⋊D₁₅₇

yes

—

101

C₂

—

102

C₂

—

628

—

103

C₂

C₂²×C₂²⋊D₁₅₇

yes

characteristic

104

C₂

#19097819

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₅₇
commutator, socle, cosocle	C₂²×C₁₅₇	C₂⁴
Fitting	C₂²×Q₈×C₁₅₇	C₂
prime radical	C₂×#14578954	C₂
hypercenter	C₂²×Q₈	D₁₅₇
hypocenter	C₁₅₇	#536
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₅₇

commutator, socle, cosocle

C₂²×C₁₅₇

C₂⁴

Fitting

C₂²×Q₈×C₁₅₇

C₂

prime radical

C₂×#14578954

C₂

hypercenter

C₂²×Q₈

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #19097769	C₂	semidirect
3. #19097760	C₂	semidirect
4. #19097754	C₂	semidirect
5. #19097747	C₂	semidirect
6. #19097730	C₂	semidirect
7. C₂²×Q₈×C₁₅₇	C₂	semidirect
52. C₁₅₇	#536	semidirect
55. C₂²×Q₈	D₁₅₇	semidirect

H ≅ G/N

type

2. #19097769

C₂

semidirect

3. #19097760

C₂

semidirect

4. #19097754

C₂

semidirect

5. #19097747

C₂

semidirect

6. #19097730

C₂

semidirect

7. C₂²×Q₈×C₁₅₇

C₂

semidirect

52. C₁₅₇

#536

semidirect

55. C₂²×Q₈

D₁₅₇

semidirect

Subgroups — #23757847

Conjugacy classes of subgroups

Complements