Group #21422186 — subgroups

Subgroups — #21422186

Order 7552.

136 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂²×C₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#16837357	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	C₄	1	1	yes	characteristic
5	#14273686	C₄	1	1	yes	characteristic
6	C₂×C₅₉⋊C₁₆	—	2	1	no	—
7	C₂×C₅₉⋊C₁₆	—	2	1	no	—
8	C₂×C₅₉⋊C₁₆	C₄	1	2	yes	complemented
9	#14273637	—	2	1	no	—
10	C₈×Dic₅₉	—	2	1	no	—
11	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
12	M₃₂×C₅₉	—	2	1	no	—
13	M₃₂×C₅₉	C₄	1	2	yes	complemented
14	C₂×C₁₆×C₅₉	—	2	1	no	—
15	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
19	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
20	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
22	C₂²⋊Dic₅₉	—	2	1	no	—
23	C₄⋊Dic₅₉	C₈	1	2	yes	—
24	C₄×Dic₅₉	—	2	1	no	—
25	#11745664	—	2	1	no	—
26	#11745664	C₈	1	2	yes	—
27	C₂×C₅₉⋊C₈	—	2	1	no	—
28	C₁₆×C₅₉	—	2	2	no	—
29	C₅₉⋊C₁₆	—	2	2	no	—
30	C₂³×C₅₉	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₉	M₁₆	1	1	yes	characteristic
37	C₂×Dic₅₉	—	2	2	no	—
38	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
39	C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
41	C₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
42	C₅₉⋊C₈	—	2	2	no	—
43	C₂²×C₅₉	—	2	2	no	—
44	C₂²×C₅₉	#97	1	1	yes	characteristic
45	C₂²×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
46	C₂²×C₅₉	C₄⋊C₈	1	1	yes	characteristic
47	C₄×C₅₉	#95	1	1	yes	characteristic
48	C₄×C₅₉	C₄×C₈	1	1	yes	characteristic
49	C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
50	C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
51	Dic₅₉	—	4	2	no	—
52	#1379	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	1	no	—
54	C₂×C₅₉	—	2	1	no	—
55	C₂×C₅₉	#336	1	1	yes	characteristic
56	C₂×C₅₉	#350	1	2	yes	—
57	C₂×M₃₂	D₅₉	1	1	yes	characteristic
58	C₂²×C₁₆	—	59	1	no	—
59	#405	—	59	1	no	—
60	C₅₉	#1379	1	1	yes	characteristic, complemented, Sylow
61	C₂×M₁₆	—	59	1	no	—
62	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
63	#117	—	59	1	no	—
64	M₃₂	—	2	1	no	—
65	M₃₂	Dic₅₉	1	2	yes	complemented
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	59	2	no	—
68	C₂×C₁₆	—	118	1	no	—
69	C₂×C₁₆	—	118	1	no	—
70	#97	—	118	1	no	—
71	C₄×C₈	—	118	1	no	—
72	C₂²×C₄	C₄×D₅₉	1	1	yes	characteristic
73	M₁₆	—	59	2	no	—
74	M₁₆	—	118	1	no	—
75	C₂×C₈	—	118	1	no	—
76	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
77	C₂×C₈	C₄×D₅₉	1	1	yes	characteristic
78	C₂×C₈	D₂₃₆	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₅₉	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
81	C₂×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
82	C₄⋊C₄	—	59	2	no	—
83	C₂²⋊C₄	—	118	1	no	—
84	C₄²	—	118	1	no	—
85	C₁₆	—	2	2	no	—
86	C₁₆	—	118	2	no	—
87	C₂³	#11745659	1	1	yes	characteristic
88	C₂×C₄	—	118	2	no	—
89	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
90	C₂×C₄	C₈×D₅₉	1	1	yes	characteristic
91	C₂×C₄	#11745659	1	1	yes	characteristic
92	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
93	C₂×C₄	Dic₅₉⋊C₄	1	1	yes	characteristic
94	C₂×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
95	C₈	—	118	2	no	—
96	C₈	Dic₅₉⋊C₄	1	1	yes	characteristic
97	C₈	C₄⋊Dic₅₉	1	1	yes	characteristic
98	C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
99	C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
100	C₂²	—	2	2	no	—
101	C₂²	#14273637	1	1	yes	characteristic
102	C₂²	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
103	C₂²	Dic₅₉⋊C₈	1	1	yes	characteristic
104	C₄	—	236	2	no	—
105	C₄	C₅₉⋊#95	1	1	yes	characteristic
106	C₄	C₈×Dic₅₉	1	1	yes	characteristic
107	C₄	(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
108	C₄	Dic₅₉⋊C₈	1	1	yes	characteristic
109	C₂	—	2	1	no	—
110	C₂	—	2	1	no	—
111	C₂	#16837156	1	1	yes	characteristic
112	C₂	#16837187	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅₉⋊C₁₆

C₂

yes

maximal, characteristic

#16837357

C₂

yes

maximal, characteristic

C₂×M₃₂×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₄

yes

characteristic

#14273686

C₄

yes

characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₄

yes

complemented

#14273637

—

C₈×Dic₅₉

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

M₃₂×C₅₉

—

M₃₂×C₅₉

C₄

yes

complemented

C₂×C₁₆×C₅₉

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

Q₈

yes

characteristic

C₂²⋊Dic₅₉

—

C₄⋊Dic₅₉

C₈

yes

—

C₄×Dic₅₉

—

#11745664

—

#11745664

C₈

yes

—

C₂×C₅₉⋊C₈

—

C₁₆×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

M₁₆

yes

characteristic

C₂×C₄×C₅₉

C₄²

yes

characteristic

C₂×C₄×C₅₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

M₁₆

yes

characteristic

C₂×Dic₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₂²⋊C₄

yes

characteristic

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₈×C₅₉

C₄⋊C₄

yes

characteristic

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

#97

yes

characteristic

C₂²×C₅₉

C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

C₄⋊C₈

yes

characteristic

C₄×C₅₉

#95

yes

characteristic

C₄×C₅₉

C₄×C₈

yes

characteristic

C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₄×C₅₉

C₄⋊C₈

yes

characteristic

Dic₅₉

—

#1379

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#336

yes

characteristic

C₂×C₅₉

#350

yes

—

C₂×M₃₂

D₅₉

yes

characteristic

C₂²×C₁₆

—

#405

—

C₅₉

#1379

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

#117

—

M₃₂

—

M₃₂

Dic₅₉

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

—

118

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

C₄×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

Dic₁₁₈

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

characteristic

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂×Dic₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

118

—

C₄²

—

118

—

C₁₆

—

C₁₆

—

118

—

C₂³

#11745659

yes

characteristic

C₂×C₄

—

118

—

C₂×C₄

C₈×D₅₉

yes

characteristic

C₂×C₄

C₈×D₅₉

yes

characteristic

C₂×C₄

#11745659

yes

characteristic

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

Dic₅₉⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₅₉)⋊C₄

yes

characteristic

C₈

—

118

—

C₈

Dic₅₉⋊C₄

yes

characteristic

C₈

C₄⋊Dic₅₉

yes

characteristic

C₈

(C₂×D₅₉)⋊C₄

yes

characteristic

C₈

C₂²⋊Dic₅₉

yes

characteristic

100

C₂²

—

101

C₂²

#14273637

yes

characteristic

102

C₂²

(C₂×D₅₉)⋊C₈

yes

characteristic

103

C₂²

Dic₅₉⋊C₈

yes

characteristic

104

C₄

—

236

—

105

C₄

C₅₉⋊#95

yes

characteristic

106

C₄

C₈×Dic₅₉

yes

characteristic

107

C₄

(C₂×D₅₉)⋊C₈

yes

characteristic

108

C₄

Dic₅₉⋊C₈

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

#16837156

yes

characteristic

112

C₂

#16837187

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₅₉
commutator	C₄×C₅₉	C₄×C₈
Frattini	C₂²×C₈	C₂×D₅₉
Fitting	C₂×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂²⋊C₈
hypercenter	C₂×M₃₂	D₅₉
hypocenter	C₅₉	#1379
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₈

C₂²⋊D₅₉

commutator

C₄×C₅₉

C₄×C₈

Frattini

C₂²×C₈

C₂×D₅₉

Fitting

C₂×M₃₂×C₅₉

C₂

socle

C₂²×C₅₉

C₂²⋊C₈

hypercenter

C₂×M₃₂

D₅₉

hypocenter

C₅₉

#1379

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
8. C₂×C₅₉⋊C₁₆	C₄	semidirect
13. M₃₂×C₅₉	C₄	semidirect
60. C₅₉	#1379	semidirect
65. M₃₂	Dic₅₉	semidirect

H ≅ G/N

type

8. C₂×C₅₉⋊C₁₆

C₄

semidirect

13. M₃₂×C₅₉

C₄

semidirect

60. C₅₉

#1379

semidirect

65. M₃₂

Dic₅₉

semidirect

Subgroups — #21422186

Conjugacy classes of subgroups

Complements