Group #23757472 — subgroups

Subgroups — #23757472

Order 10048.

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

normal subgroups only

This group has 128 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×#14578942	C₂	1	1	yes	maximal, characteristic, complemented
3	#19097767	C₂	1	1	yes	maximal, characteristic
4	#19097736	C₂	1	1	yes	maximal, characteristic, complemented
5	#19097700	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097691	C₂	1	1	yes	maximal, characteristic, complemented
7	#102×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₃₁₄	C₂²	1	1	yes	characteristic
9	C₂×D₆₂₈	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₁₅₇	—	2	1	no	—
11	C₂×C₂²⋊D₁₅₇	—	2	1	no	—
12	#14578943	—	2	2	no	—
13	#14578942	—	2	2	no	—
14	C₂×#11929434	C₂²	1	1	yes	characteristic
15	#14578933	—	2	1	no	—
16	C₄×Dic₁₅₇	—	2	1	no	—
17	C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₅₇	—	2	1	no	—
22	Dic₃₁₄	—	2	1	no	—
23	Dic₃₁₄	D₄	1	2	yes	complemented
24	D₆₂₈	—	2	1	no	—
25	D₆₂₈	—	2	1	no	—
26	C₂²⋊D₁₅₇	—	4	2	no	—
27	C₂²⋊D₁₅₇	—	4	2	no	—
28	C₂×Dic₁₅₇	—	2	1	no	—
29	C₂×Dic₁₅₇	D₄	1	1	yes	characteristic
30	C₂×Dic₁₅₇	D₄	1	1	yes	characteristic
31	#11929434	—	2	1	no	—
32	C₂³×C₁₅₇	—	2	1	no	—
33	D₄×C₁₅₇	—	2	1	no	—
34	D₄×C₁₅₇	—	2	1	no	—
35	C₂×C₄×C₁₅₇	—	2	1	no	—
36	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
37	C₈×C₁₅₇	—	2	1	no	—
38	C₂²×C₁₅₇	—	4	1	no	—
39	C₂²×C₁₅₇	—	4	1	no	—
40	C₂²×C₁₅₇	—	4	1	no	—
41	C₂²×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₅₇	—	4	1	no	—
43	C₂×D₁₅₇	—	4	1	no	—
44	C₂×D₁₅₇	—	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	2	no	—
49	Dic₁₅₇	—	2	1	no	—
50	Dic₁₅₇	SD₁₆	1	2	yes	complemented
51	C₂×C₁₅₇	—	8	1	no	—
52	C₂×C₁₅₇	#121	1	1	yes	characteristic
53	C₂×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₅₇	AGL(1,ℤ/8)	1	1	yes	characteristic
55	D₁₅₇	—	8	1	no	—
56	C₁₅₇	#460	1	1	yes	characteristic, complemented, Sylow
57	#460	—	157	1	no	maximal, Sylow
58	C₂×SD₁₆	—	157	1	no	—
59	C₂×SD₁₆	—	157	1	no	—
60	#127	—	157	1	no	—
61	C₄×Q₈	—	157	1	no	—
62	C₄⋊C₈	—	157	1	no	—
63	#102	—	157	1	no	—
64	#102	D₁₅₇	1	1	yes	characteristic, complemented
65	C₂×Q₈	—	157	1	no	—
66	C₂×D₄	—	314	1	no	—
67	C₂×D₄	—	314	1	no	—
68	C₂×D₄	—	157	1	no	—
69	C₂×D₄	C₂×D₁₅₇	1	1	yes	characteristic
70	SD₁₆	—	314	2	no	—
71	SD₁₆	—	314	2	no	—
72	C₂×C₈	—	157	1	no	—
73	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
74	C₄⋊C₄	—	314	1	no	—
75	C₄⋊C₄	C₂×D₁₅₇	1	1	yes	characteristic
76	C₄²	—	314	1	no	—
77	C₄²	—	157	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	314	1	no	—
80	Q₈	—	314	1	no	—
81	Q₈	—	157	2	no	—
82	D₄	—	2	1	no	—
83	D₄	—	2	1	no	—
84	D₄	—	314	1	no	—
85	D₄	—	314	1	no	—
86	D₄	—	628	2	no	—
87	D₄	—	628	2	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	314	1	no	—
90	C₂×C₄	—	157	1	no	—
91	C₂×C₄	—	157	1	no	—
92	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	314	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	628	1	no	—
99	C₂²	—	628	1	no	—
100	C₂²	—	628	1	no	—
101	C₂²	D₄×D₁₅₇	1	1	yes	characteristic
102	C₄	—	4	1	no	—
103	C₄	—	314	2	no	—
104	C₄	—	314	1	no	—
105	C₄	—	157	2	no	—
106	C₄	D₄×D₁₅₇	1	1	yes	characteristic
107	C₄	#14578954	1	1	yes	characteristic
108	C₂	—	8	1	no	—
109	C₂	—	1256	1	no	—
110	C₂	#19097744	1	1	yes	characteristic
111	C₂	SD₁₆×D₁₅₇	1	1	yes	characteristic
112	C₂	#19097795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578943

C₂

yes

maximal, characteristic, complemented

C₂×#14578942

C₂

yes

maximal, characteristic, complemented

#19097767

C₂

yes

maximal, characteristic

#19097736

C₂

yes

maximal, characteristic, complemented

#19097700

C₂

yes

maximal, characteristic, complemented

#19097691

C₂

yes

maximal, characteristic, complemented

#102×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₁₄

C₂²

yes

characteristic

C₂×D₆₂₈

C₂²

yes

characteristic

C₂×C₂²⋊D₁₅₇

—

C₂×C₂²⋊D₁₅₇

—

#14578943

—

#14578942

—

C₂×#11929434

C₂²

yes

characteristic

#14578933

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

C₂²

yes

characteristic

C₂×D₄×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

—

Dic₃₁₄

—

Dic₃₁₄

D₄

yes

complemented

D₆₂₈

—

D₆₂₈

—

C₂²⋊D₁₅₇

—

C₂²⋊D₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

D₄

yes

characteristic

C₂×Dic₁₅₇

D₄

yes

characteristic

#11929434

—

C₂³×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂×D₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

#41

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

Dic₁₅₇

SD₁₆

yes

complemented

C₂×C₁₅₇

—

C₂×C₁₅₇

#121

yes

characteristic

C₂×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂×C₁₅₇

AGL(1,ℤ/8)

yes

characteristic

D₁₅₇

—

C₁₅₇

#460

yes

characteristic, complemented, Sylow

#460

—

157

maximal, Sylow

C₂×SD₁₆

—

157

—

C₂×SD₁₆

—

157

—

#127

—

157

—

C₄×Q₈

—

157

—

C₄⋊C₈

—

157

—

#102

—

157

—

#102

D₁₅₇

yes

characteristic, complemented

C₂×Q₈

—

157

—

C₂×D₄

—

314

—

C₂×D₄

—

314

—

C₂×D₄

—

157

—

C₂×D₄

C₂×D₁₅₇

yes

characteristic

SD₁₆

—

314

—

SD₁₆

—

314

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

314

—

C₄⋊C₄

C₂×D₁₅₇

yes

characteristic

C₄²

—

314

—

C₄²

—

157

—

C₂³

—

C₂³

—

314

—

Q₈

—

314

—

Q₈

—

157

—

D₄

—

D₄

—

D₄

—

314

—

D₄

—

314

—

D₄

—

628

—

D₄

—

628

—

C₂×C₄

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

—

C₈

—

314

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

628

—

C₂²

—

628

—

100

C₂²

—

628

—

101

C₂²

D₄×D₁₅₇

yes

characteristic

102

C₄

—

103

C₄

—

314

—

104

C₄

—

314

—

105

C₄

—

157

—

106

C₄

D₄×D₁₅₇

yes

characteristic

107

C₄

#14578954

yes

characteristic

108

C₂

—

109

C₂

—

1256

—

110

C₂

#19097744

yes

characteristic

111

C₂

SD₁₆×D₁₅₇

yes

characteristic

112

C₂

#19097795

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₅₇
Fitting	#102×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×D₄
prime radical	#19097767	C₂
hypercenter	#102	D₁₅₇
hypocenter	C₁₅₇	#460
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

subgroup

G/H

center

C₂²

D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂³

Frattini

C₂×C₄

C₂²×D₁₅₇

Fitting

#102×C₁₅₇

C₂

socle

C₂²×C₁₅₇

C₂×D₄

C₂

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ #102

N	H ≅ G/N	type
1. C₂×#14578943	C₂	semidirect
2. C₂×#14578942	C₂	semidirect
4. #19097736	C₂	semidirect
5. #19097700	C₂	semidirect
6. #19097691	C₂	semidirect
7. #102×C₁₅₇	C₂	semidirect
23. Dic₃₁₄	D₄	semidirect
50. Dic₁₅₇	SD₁₆	semidirect
56. C₁₅₇	#460	semidirect
64. #102	D₁₅₇	semidirect

H ≅ G/N

type

1. C₂×#14578943

C₂

semidirect

2. C₂×#14578942

C₂

semidirect

4. #19097736

C₂

semidirect

5. #19097700

C₂

semidirect

6. #19097691

C₂

semidirect

7. #102×C₁₅₇

C₂

semidirect

23. Dic₃₁₄

D₄

semidirect

50. Dic₁₅₇

SD₁₆

semidirect

56. C₁₅₇

#460

semidirect

64. #102

D₁₅₇

semidirect

Subgroups — #23757472

Conjugacy classes of subgroups

Complements