Group #23757346 — subgroups

Subgroups — #23757346

Order 10048.

124 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 53 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×#14578937	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₈×D₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₆₂₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#11929434	C₂	1	1	yes	maximal, characteristic, complemented
5	#19097683	C₂	1	1	yes	maximal, characteristic, complemented
6	#19097683	C₂	1	1	yes	maximal, characteristic, complemented
7	#97×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₆₂₈	C₄	1	1	yes	characteristic
9	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₅₇	C₂²	1	1	yes	characteristic
11	#14578937	—	2	2	no	—
12	C₂×#11929434	C₂²	1	1	yes	characteristic
13	C₂×#11929434	C₂²	1	1	yes	characteristic
14	C₈×D₁₅₇	—	2	2	no	—
15	C₄⋊Dic₁₅₇	C₄	1	1	yes	characteristic
16	#14578931	C₄	1	1	yes	characteristic
17	#14578931	C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
21	C₂²×D₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₅₇	C₂×C₄	1	1	yes	characteristic
23	D₆₂₈	—	2	2	no	—
24	C₂×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
25	C₂×Dic₁₅₇	C₂×C₄	1	1	yes	characteristic
26	C₄×D₁₅₇	—	2	1	no	—
27	C₄×D₁₅₇	—	2	1	no	—
28	C₄×D₁₅₇	—	2	1	no	—
29	C₄×D₁₅₇	—	2	1	no	—
30	#11929434	—	2	1	no	—
31	#11929434	—	2	1	no	—
32	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₈×C₁₅₇	—	2	1	no	—
36	C₈×C₁₅₇	D₄	1	2	yes	complemented
37	C₂²×C₁₅₇	C₂²×C₄	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	—	2	1	no	—
42	C₂×D₁₅₇	—	2	1	no	—
43	C₂×D₁₅₇	M₁₆	1	2	yes	complemented
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	2	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	—	2	1	no	—
50	C₂×C₁₅₇	C₄×D₄	1	1	yes	characteristic
51	C₂×C₁₅₇	C₂×M₁₆	1	1	yes	characteristic
52	C₂×C₁₅₇	#131	1	1	yes	characteristic
53	D₁₅₇	—	2	2	no	—
54	D₁₅₇	—	4	1	no	—
55	C₁₅₇	#435	1	1	yes	characteristic, complemented, Sylow
56	#435	—	157	1	no	maximal, Sylow
57	C₂×M₁₆	—	157	1	no	—
58	C₂²×C₈	—	157	1	no	—
59	C₄×D₄	—	157	1	no	—
60	C₄⋊C₈	—	157	1	no	—
61	C₂²⋊C₈	—	157	1	no	—
62	C₂²⋊C₈	—	157	1	no	—
63	#97	D₁₅₇	1	1	yes	characteristic, complemented
64	C₂×D₄	—	157	1	no	—
65	C₂²×C₄	—	157	1	no	—
66	C₂²×C₄	—	157	1	no	—
67	M₁₆	—	314	2	no	—
68	C₂×C₈	—	314	2	no	—
69	C₂×C₈	—	157	1	no	—
70	C₂×C₈	—	157	1	no	—
71	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
72	C₂×C₈	C₂×D₁₅₇	1	1	yes	characteristic
73	C₄⋊C₄	—	157	1	no	—
74	C₂²⋊C₄	—	157	1	no	—
75	C₂²⋊C₄	—	157	1	no	—
76	C₄²	C₂×D₁₅₇	1	1	yes	characteristic
77	C₂³	—	157	1	no	—
78	C₂³	—	157	1	no	—
79	D₄	—	314	2	no	—
80	C₂×C₄	—	314	1	no	—
81	C₂×C₄	—	314	1	no	—
82	C₂×C₄	—	314	1	no	—
83	C₂×C₄	—	314	1	no	—
84	C₂×C₄	—	157	1	no	—
85	C₂×C₄	—	157	1	no	—
86	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
87	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	314	1	no	—
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	—	314	1	no	—
97	C₂²	—	314	1	no	—
98	C₂²	—	157	2	no	—
99	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	314	1	no	—
103	C₄	—	314	1	no	—
104	C₄	C₂×D₆₂₈	1	1	yes	characteristic
105	C₄	#14578954	1	1	yes	characteristic
106	C₂	—	314	2	no	—
107	C₂	—	628	1	no	—
108	C₂	C₄×D₆₂₈	1	1	yes	characteristic
109	C₂	M₁₆×D₁₅₇	1	1	yes	characteristic
110	C₂	#19097779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578937

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₅₇

C₂

yes

maximal, characteristic, complemented

C₄×D₆₂₈

C₂

yes

maximal, characteristic

C₄⋊#11929434

C₂

yes

maximal, characteristic, complemented

#19097683

C₂

yes

maximal, characteristic, complemented

#19097683

C₂

yes

maximal, characteristic, complemented

#97×C₁₅₇

C₂

yes

maximal, characteristic, complemented

C₂×D₆₂₈

C₄

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

C₂×C₄×D₁₅₇

C₂²

yes

characteristic

#14578937

—

C₂×#11929434

C₂²

yes

characteristic

C₂×#11929434

C₂²

yes

characteristic

C₈×D₁₅₇

—

C₄⋊Dic₁₅₇

C₄

yes

characteristic

#14578931

C₄

yes

characteristic

#14578931

C₄

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₈×C₁₅₇

C₂²

yes

characteristic

C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×D₁₅₇

C₂×C₄

yes

characteristic

C₂²×D₁₅₇

C₂×C₄

yes

characteristic

D₆₂₈

—

C₂×Dic₁₅₇

C₂×C₄

yes

characteristic

C₂×Dic₁₅₇

C₂×C₄

yes

characteristic

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

C₄×D₁₅₇

—

#11929434

—

#11929434

—

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

—

C₈×C₁₅₇

D₄

yes

complemented

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

—

C₂×D₁₅₇

M₁₆

yes

complemented

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₄×C₁₅₇

#41

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

C₄×D₄

yes

characteristic

C₂×C₁₅₇

C₂×M₁₆

yes

characteristic

C₂×C₁₅₇

#131

yes

characteristic

D₁₅₇

—

D₁₅₇

—

C₁₅₇

#435

yes

characteristic, complemented, Sylow

#435

—

157

maximal, Sylow

C₂×M₁₆

—

157

—

C₂²×C₈

—

157

—

C₄×D₄

—

157

—

C₄⋊C₈

—

157

—

C₂²⋊C₈

—

157

—

C₂²⋊C₈

—

157

—

#97

D₁₅₇

yes

characteristic, complemented

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

M₁₆

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₂×C₈

C₂×D₁₅₇

yes

characteristic

C₄⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₂²⋊C₄

—

157

—

C₄²

C₂×D₁₅₇

yes

characteristic

C₂³

—

157

—

C₂³

—

157

—

D₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

314

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₈

—

C₈

—

314

—

C₈

—

314

—

C₈

D₆₂₈

yes

complemented

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

314

—

C₂²

—

157

—

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

314

—

103

C₄

—

314

—

104

C₄

C₂×D₆₂₈

yes

characteristic

105

C₄

#14578954

yes

characteristic

106

C₂

—

314

—

107

C₂

—

628

—

108

C₂

C₄×D₆₂₈

yes

characteristic

109

C₂

M₁₆×D₁₅₇

yes

characteristic

110

C₂

#19097779

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₅₇
commutator, socle	C₂²×C₁₅₇	C₂²×C₄
Fitting	#97×C₁₅₇	C₂
prime radical	C₂×D₆₂₈	C₄
cosocle	C₂×C₄×C₁₅₇	C₂³
hypercenter	#97	D₁₅₇
hypocenter	C₁₅₇	#435
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₅₇

commutator, socle

C₂²×C₁₅₇

C₂²×C₄

Fitting

#97×C₁₅₇

C₂

prime radical

C₂×D₆₂₈

C₄

cosocle

C₂×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂²×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
1. C₂×#14578937	C₂	semidirect
2. C₂×C₈×D₁₅₇	C₂	semidirect
4. C₄⋊#11929434	C₂	semidirect
5. #19097683	C₂	semidirect
6. #19097683	C₂	semidirect
7. #97×C₁₅₇	C₂	semidirect
36. C₈×C₁₅₇	D₄	semidirect
43. C₂×D₁₅₇	M₁₆	semidirect
55. C₁₅₇	#435	semidirect
63. #97	D₁₅₇	semidirect
92. C₈	D₆₂₈	semidirect

H ≅ G/N

type

1. C₂×#14578937

C₂

semidirect

2. C₂×C₈×D₁₅₇

C₂

semidirect

4. C₄⋊#11929434

C₂

semidirect

5. #19097683

C₂

semidirect

6. #19097683

C₂

semidirect

7. #97×C₁₅₇

C₂

semidirect

36. C₈×C₁₅₇

D₄

semidirect

43. C₂×D₁₅₇

M₁₆

semidirect

55. C₁₅₇

#435

semidirect

63. #97

D₁₅₇

semidirect

92. C₈

D₆₂₈

semidirect

Subgroups — #23757346

Conjugacy classes of subgroups

Complements