Group #23758039 — subgroups

Subgroups — #23758039

Order 10048.

258 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 147 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	#19097779	C₂	1	4	yes	maximal, complemented
3	#19097777	C₂	1	4	yes	maximal, complemented
4	C₂×#14578938	C₂	1	1	yes	maximal, characteristic, complemented
5	M₁₆×D₁₅₇	C₂	1	4	yes	maximal, complemented
6	C₂×M₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#14578954	C₄	1	2	yes	—
8	#14578954	C₄	1	2	yes	—
9	#14578954	C₂²	1	2	yes	—
10	#14578954	C₂²	1	2	yes	—
11	C₂×Dic₃₁₄	C₄	1	1	yes	characteristic
12	C₂×D₆₂₈	C₄	1	1	yes	characteristic
13	C₂×C₂²⋊D₁₅₇	C₄	1	2	yes	—
14	C₂×C₄×D₁₅₇	C₂²	1	2	yes	—
15	#14578938	C₂²	1	2	yes	—
16	#14578938	C₂²	1	2	yes	complemented
17	#14578937	C₂²	1	8	yes	complemented
18	C₂×#11929434	C₂²	1	2	yes	complemented
19	C₈×D₁₅₇	C₂²	1	8	yes	complemented
20	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
21	M₁₆×C₁₅₇	C₂²	1	2	yes	—
22	M₁₆×C₁₅₇	C₂²	1	2	yes	complemented
23	C₂×C₈×C₁₅₇	C₂²	1	2	yes	complemented
24	C₂²×D₁₅₇	C₂×C₄	1	2	yes	—
25	Dic₃₁₄	C₂×C₄	1	2	yes	—
26	Dic₃₁₄	C₂×C₄	1	2	yes	—
27	D₆₂₈	C₂×C₄	1	2	yes	—
28	D₆₂₈	C₂×C₄	1	2	yes	—
29	C₂²⋊D₁₅₇	C₂×C₄	1	4	yes	—
30	C₂²⋊D₁₅₇	C₂×C₄	1	4	yes	—
31	C₂×Dic₁₅₇	C₂×C₄	1	2	yes	—
32	C₄×D₁₅₇	C₂×C₄	1	4	yes	—
33	C₄×D₁₅₇	C₂³	1	4	yes	—
34	#11929434	C₂³	1	4	yes	—
35	C₂³×C₁₅₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
42	C₈×C₁₅₇	C₂³	1	4	yes	—
43	C₂²×C₁₅₇	—	4	1	no	—
44	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×D₁₅₇	—	4	2	no	—
48	C₂×D₁₅₇	C₂²×C₄	1	4	yes	—
49	C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
50	C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
51	C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
52	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
53	Dic₁₅₇	C₂²×C₄	1	4	yes	—
54	C₂×C₁₅₇	—	2	1	no	—
55	C₂×C₁₅₇	—	2	1	no	—
56	C₂×C₁₅₇	—	2	1	no	—
57	C₂×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
58	D₁₅₇	—	2	4	no	—
59	C₁₅₇	#568	1	1	yes	characteristic, complemented, Sylow
60	#568	—	157	1	no	maximal, Sylow
61	C₂×#41	—	157	1	no	—
62	#131	—	157	4	no	—
63	#131	—	157	4	no	—
64	C₂×M₁₆	—	157	1	no	—
65	C₂×M₁₆	—	157	4	no	—
66	C₂×M₁₆	D₁₅₇	1	1	yes	characteristic, complemented
67	#41	—	157	2	no	—
68	#41	—	157	2	no	—
69	#41	—	157	2	no	—
70	#41	—	157	2	no	—
71	C₂×Q₈	—	157	1	no	—
72	C₂×D₄	—	157	2	no	—
73	C₂×D₄	—	157	1	no	—
74	C₂²×C₄	—	157	2	no	—
75	C₂²×C₄	C₂×D₁₅₇	1	1	yes	characteristic
76	M₁₆	—	157	2	no	—
77	M₁₆	—	157	2	no	—
78	M₁₆	—	157	8	no	—
79	M₁₆	C₂×D₁₅₇	1	2	yes	—
80	M₁₆	C₂×D₁₅₇	1	2	yes	complemented
81	C₂×C₈	—	157	2	no	—
82	C₂×C₈	—	157	8	no	—
83	C₂×C₈	C₂×D₁₅₇	1	2	yes	complemented
84	C₂³	—	157	2	no	—
85	C₂³	C₄×D₁₅₇	1	1	yes	characteristic
86	Q₈	—	157	2	no	—
87	Q₈	—	157	2	no	—
88	D₄	—	157	4	no	—
89	D₄	—	157	4	no	—
90	D₄	—	157	2	no	—
91	D₄	—	157	2	no	—
92	C₂×C₄	—	157	2	no	—
93	C₂×C₄	—	157	4	no	—
94	C₂×C₄	—	157	4	no	—
95	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
96	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
97	C₂×C₄	C₄×D₁₅₇	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
99	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
100	C₂×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
101	C₈	—	157	4	no	—
102	C₈	C₂²×D₁₅₇	1	4	yes	—
103	C₂²	—	4	1	no	—
104	C₂²	—	628	2	no	—
105	C₂²	—	157	4	no	—
106	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
107	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
108	C₂²	C₂×C₄×D₁₅₇	1	1	yes	characteristic
109	C₄	—	157	4	no	—
110	C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
111	C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
112	C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
113	C₄	C₂³×D₁₅₇	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	—	2	1	no	—
117	C₂	—	314	4	no	—
118	C₂	C₂²×C₄×D₁₅₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14578954

C₂

yes

maximal, characteristic

#19097779

C₂

yes

maximal, complemented

#19097777

C₂

yes

maximal, complemented

C₂×#14578938

C₂

yes

maximal, characteristic, complemented

M₁₆×D₁₅₇

C₂

yes

maximal, complemented

C₂×M₁₆×C₁₅₇

C₂

yes

maximal, characteristic, complemented

#14578954

C₄

yes

—

#14578954

C₄

yes

—

#14578954

C₂²

yes

—

#14578954

C₂²

yes

—

C₂×Dic₃₁₄

C₄

yes

characteristic

C₂×D₆₂₈

C₄

yes

characteristic

C₂×C₂²⋊D₁₅₇

C₄

yes

—

C₂×C₄×D₁₅₇

C₂²

yes

—

#14578938

C₂²

yes

—

#14578938

C₂²

yes

complemented

#14578937

C₂²

yes

complemented

C₂×#11929434

C₂²

yes

complemented

C₈×D₁₅₇

C₂²

yes

complemented

C₂²×C₄×C₁₅₇

C₂²

yes

characteristic

M₁₆×C₁₅₇

C₂²

yes

—

M₁₆×C₁₅₇

C₂²

yes

complemented

C₂×C₈×C₁₅₇

C₂²

yes

complemented

C₂²×D₁₅₇

C₂×C₄

yes

—

Dic₃₁₄

C₂×C₄

yes

—

Dic₃₁₄

C₂×C₄

yes

—

D₆₂₈

C₂×C₄

yes

—

D₆₂₈

C₂×C₄

yes

—

C₂²⋊D₁₅₇

C₂×C₄

yes

—

C₂²⋊D₁₅₇

C₂×C₄

yes

—

C₂×Dic₁₅₇

C₂×C₄

yes

—

C₄×D₁₅₇

C₂×C₄

yes

—

C₄×D₁₅₇

C₂³

yes

—

#11929434

C₂³

yes

—

C₂³×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₄×C₁₅₇

C₂³

yes

characteristic

C₈×C₁₅₇

C₂³

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂²×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×D₁₅₇

—

C₂×D₁₅₇

C₂²×C₄

yes

—

C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₄×C₁₅₇

C₂⁴

yes

characteristic

Dic₁₅₇

C₂²×C₄

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

C₂³×C₄

yes

characteristic

D₁₅₇

—

C₁₅₇

#568

yes

characteristic, complemented, Sylow

#568

—

157

maximal, Sylow

C₂×#41

—

157

—

#131

—

157

—

#131

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

—

157

—

C₂×M₁₆

D₁₅₇

yes

characteristic, complemented

#41

—

157

—

#41

—

157

—

#41

—

157

—

#41

—

157

—

C₂×Q₈

—

157

—

C₂×D₄

—

157

—

C₂×D₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂×D₁₅₇

yes

characteristic

M₁₆

—

157

—

M₁₆

—

157

—

M₁₆

—

157

—

M₁₆

C₂×D₁₅₇

yes

—

M₁₆

C₂×D₁₅₇

yes

complemented

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

C₂×D₁₅₇

yes

complemented

C₂³

—

157

—

C₂³

C₄×D₁₅₇

yes

characteristic

Q₈

—

157

—

Q₈

—

157

—

D₄

—

157

—

D₄

—

157

—

D₄

—

157

—

D₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

—

157

—

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₄×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

100

C₂×C₄

C₂²×D₁₅₇

yes

characteristic

101

C₈

—

157

—

102

C₈

C₂²×D₁₅₇

yes

—

103

C₂²

—

104

C₂²

—

628

—

105

C₂²

—

157

—

106

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

107

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

108

C₂²

C₂×C₄×D₁₅₇

yes

characteristic

109

C₄

—

157

—

110

C₄

C₂×C₄×D₁₅₇

yes

characteristic

111

C₄

C₂×C₄×D₁₅₇

yes

characteristic

112

C₄

C₂×C₄×D₁₅₇

yes

characteristic

113

C₄

C₂³×D₁₅₇

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

—

117

C₂

—

314

—

118

C₂

C₂²×C₄×D₁₅₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×D₁₅₇
commutator, socle	C₂×C₁₅₇	C₂³×C₄
Fitting	C₂×M₁₆×C₁₅₇	C₂
prime radical	C₂×#14578954	C₂
cosocle	C₄×C₁₅₇	C₂⁴
hypercenter	C₂×M₁₆	D₁₅₇
hypocenter	C₁₅₇	#568
LCS	G ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center, Frattini

C₄

C₂³×D₁₅₇

commutator, socle

C₂×C₁₅₇

C₂³×C₄

Fitting

C₂×M₁₆×C₁₅₇

C₂

prime radical

C₂×#14578954

C₂

cosocle

C₄×C₁₅₇

C₂⁴

hypercenter

C₂×M₁₆

D₁₅₇

C₁₅₇

G ▷ C₂×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
2. #19097779	C₂	semidirect
3. #19097777	C₂	semidirect
4. C₂×#14578938	C₂	semidirect
5. M₁₆×D₁₅₇	C₂	semidirect
6. C₂×M₁₆×C₁₅₇	C₂	semidirect
16. #14578938	C₂²	semidirect
17. #14578937	C₂²	semidirect
18. C₂×#11929434	C₂²	semidirect
19. C₈×D₁₅₇	C₂²	semidirect
22. M₁₆×C₁₅₇	C₂²	semidirect
23. C₂×C₈×C₁₅₇	C₂²	semidirect
59. C₁₅₇	#568	semidirect
66. C₂×M₁₆	D₁₅₇	semidirect
80. M₁₆	C₂×D₁₅₇	semidirect
83. C₂×C₈	C₂×D₁₅₇	semidirect

H ≅ G/N

type

2. #19097779

C₂

semidirect

3. #19097777

C₂

semidirect

4. C₂×#14578938

C₂

semidirect

5. M₁₆×D₁₅₇

C₂

semidirect

6. C₂×M₁₆×C₁₅₇

C₂

semidirect

16. #14578938

C₂²

semidirect

17. #14578937

C₂²

semidirect

18. C₂×#11929434

C₂²

semidirect

19. C₈×D₁₅₇

C₂²

semidirect

22. M₁₆×C₁₅₇

C₂²

semidirect

23. C₂×C₈×C₁₅₇

C₂²

semidirect

59. C₁₅₇

#568

semidirect

66. C₂×M₁₆

D₁₅₇

semidirect

80. M₁₆

C₂×D₁₅₇

semidirect

83. C₂×C₈

C₂×D₁₅₇

semidirect

Subgroups — #23758039

Conjugacy classes of subgroups

Complements