Group #23757520 — subgroups

Subgroups — #23757520

Order 10048.

120 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 41 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×#14578943	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14578937	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097751	C₂	1	1	yes	maximal, characteristic, complemented
4	#19097746	C₂	1	1	yes	maximal, characteristic
5	#19097694	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097691	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
9	C₂×D₆₂₈	—	2	1	no	—
10	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
12	#14578943	—	2	2	no	—
13	#14578937	—	2	2	no	—
14	C₂×#11929434	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₅₇	—	2	1	no	—
16	#14578931	—	2	1	no	—
17	#14578931	—	2	1	no	—
18	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₅₇	—	2	1	no	—
22	C₂²×D₁₅₇	D₄	1	1	yes	characteristic
23	Dic₃₁₄	—	2	1	no	—
24	Dic₃₁₄	—	2	1	no	—
25	D₆₂₈	—	2	1	no	—
26	D₆₂₈	—	2	1	no	—
27	D₆₂₈	—	4	2	no	—
28	C₂×Dic₁₅₇	—	2	1	no	—
29	C₂×Dic₁₅₇	D₄	1	1	yes	characteristic
30	C₄×D₁₅₇	—	2	1	no	—
31	C₄×D₁₅₇	—	2	1	no	—
32	#11929434	—	2	1	no	—
33	C₂×C₄×C₁₅₇	—	2	1	no	—
34	C₂×C₄×C₁₅₇	—	2	1	no	—
35	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
36	C₈×C₁₅₇	D₄	1	2	yes	complemented
37	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
38	C₂×D₁₅₇	—	2	1	no	—
39	C₂×D₁₅₇	—	2	1	no	—
40	C₂×D₁₅₇	—	2	1	no	—
41	C₂×D₁₅₇	—	4	1	no	—
42	C₂×D₁₅₇	—	4	1	no	—
43	C₂×D₁₅₇	—	4	1	no	—
44	C₄×C₁₅₇	—	4	1	no	—
45	C₄×C₁₅₇	—	4	1	no	—
46	C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₅₇	#41	1	1	yes	characteristic
48	Dic₁₅₇	—	2	1	no	—
49	Dic₁₅₇	—	4	1	no	—
50	C₂×C₁₅₇	#121	1	1	yes	characteristic
51	C₂×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
52	C₂×C₁₅₇	#137	1	1	yes	characteristic
53	D₁₅₇	—	4	1	no	—
54	D₁₅₇	—	8	1	no	—
55	C₁₅₇	#468	1	1	yes	characteristic, complemented, Sylow
56	#468	—	157	1	no	maximal, Sylow
57	C₂×SD₁₆	—	157	1	no	—
58	C₂×M₁₆	—	157	1	no	—
59	C₂²⋊Q₈	—	157	1	no	—
60	#121	—	157	1	no	—
61	#107	D₁₅₇	1	1	yes	characteristic, complemented
62	#103	—	157	1	no	—
63	#102	—	157	1	no	—
64	C₂×Q₈	—	157	1	no	—
65	C₂×D₄	—	314	1	no	—
66	C₂×D₄	—	157	1	no	—
67	C₂²×C₄	—	157	1	no	—
68	SD₁₆	—	314	2	no	—
69	M₁₆	—	314	2	no	—
70	C₂×C₈	—	157	1	no	—
71	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
72	C₄⋊C₄	—	314	1	no	—
73	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
74	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	314	1	no	—
76	C₂²⋊C₄	—	314	1	no	—
77	C₂³	—	314	1	no	—
78	C₂³	—	157	1	no	—
79	Q₈	—	314	1	no	—
80	Q₈	—	314	1	no	—
81	D₄	—	314	1	no	—
82	D₄	—	314	1	no	—
83	D₄	—	628	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	314	1	no	—
87	C₂×C₄	—	314	1	no	—
88	C₂×C₄	—	314	1	no	—
89	C₂×C₄	—	157	1	no	—
90	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
91	C₈	—	314	1	no	—
92	C₈	D₆₂₈	1	2	yes	complemented
93	C₂²	—	314	1	no	—
94	C₂²	—	314	1	no	—
95	C₂²	—	314	1	no	—
96	C₂²	—	628	1	no	—
97	C₂²	—	628	1	no	—
98	C₂²	—	628	1	no	—
99	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
100	C₄	—	4	1	no	—
101	C₄	—	4	1	no	—
102	C₄	—	314	1	no	—
103	C₄	—	628	1	no	—
104	C₄	C₂×D₆₂₈	1	1	yes	characteristic
105	C₄	#14578955	1	1	yes	characteristic
106	C₂	—	628	1	no	—
107	C₂	—	1256	1	no	—
108	C₂	#19097746	1	1	yes	characteristic
109	C₂	#19097795	1	1	yes	characteristic
110	C₂	#19097803	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578943

C₂

yes

maximal, characteristic, complemented

C₂×#14578937

C₂

yes

maximal, characteristic, complemented

#19097751

C₂

yes

maximal, characteristic, complemented

#19097746

C₂

yes

maximal, characteristic

#19097694

C₂

yes

maximal, characteristic, complemented

#19097691

C₂

yes

maximal, characteristic, complemented

#107×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×D₆₂₈

—

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Dic₁₅₇

—

#14578931

—

#14578931

—

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

C₂²×D₁₅₇

D₄

yes

characteristic

Dic₃₁₄

—

Dic₃₁₄

—

D₆₂₈

—

D₆₂₈

—

D₆₂₈

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

D₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

D₄

yes

complemented

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

#41

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#468

yes

characteristic, complemented, Sylow

#468

—

157

maximal, Sylow

C₂×SD₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂²⋊Q₈

—

157

—

#121

—

157

—

#107

D₁₅₇

yes

characteristic, complemented

#103

—

157

—

#102

—

157

—

C₂×Q₈

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

SD₁₆

—

314

—

M₁₆

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

314

—

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₂²⋊C₄

—

314

—

C₂²⋊C₄

—

314

—

C₂³

—

314

—

C₂³

—

157

—

Q₈

—

314

—

Q₈

—

314

—

D₄

—

314

—

D₄

—

314

—

D₄

—

628

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

—

314

—

C₈

D₆₂₈

yes

complemented

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

628

—

C₂²

—

628

—

C₂²

—

628

—

C₂²

D₄×D₁₅₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

314

—

103

C₄

—

628

—

104

C₄

C₂×D₆₂₈

yes

characteristic

105

C₄

#14578955

yes

characteristic

106

C₂

—

628

—

107

C₂

—

1256

—

108

C₂

#19097746

yes

characteristic

109

C₂

#19097795

yes

characteristic

110

C₂

#19097803

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting	#107×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₄
prime radical	#19097746	C₂
hypercenter	#107	D₁₅₇
hypocenter	C₁₅₇	#468
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

subgroup

G/H

center

C₂²

D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₇

Fitting

#107×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×D₄

C₂

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

N	H ≅ G/N	type
1. C₂×#14578943	C₂	semidirect
2. C₂×#14578937	C₂	semidirect
3. #19097751	C₂	semidirect
5. #19097694	C₂	semidirect
6. #19097691	C₂	semidirect
7. #107×C₁₅₇	C₂	semidirect
36. C₈×C₁₅₇	D₄	semidirect
55. C₁₅₇	#468	semidirect
61. #107	D₁₅₇	semidirect
92. C₈	D₆₂₈	semidirect

H ≅ G/N

type

1. C₂×#14578943

C₂

semidirect

2. C₂×#14578937

C₂

semidirect

3. #19097751

C₂

semidirect

5. #19097694

C₂

semidirect

6. #19097691

C₂

semidirect

7. #107×C₁₅₇

C₂

semidirect

36. C₈×C₁₅₇

D₄

semidirect

55. C₁₅₇

#468

semidirect

61. #107

D₁₅₇

semidirect

92. C₈

D₆₂₈

semidirect

Subgroups — #23757520

Conjugacy classes of subgroups

Complements