Group #21424003 — subgroups

Subgroups — #21424003

Order 7552.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#14273662	C₂	1	2	yes	maximal
2	C₄×#11745664	C₂	1	1	yes	maximal, characteristic
3	C₂×#14273638	C₂	1	1	yes	maximal, characteristic
4	#16837166	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
9	#14273662	C₄	1	8	yes	complemented
10	#14273638	—	2	1	no	—
11	#14273638	C₄	1	4	yes	—
12	C₄×C₅₉⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
15	#97×C₅₉	—	2	1	no	—
16	C₄×C₈×C₅₉	—	2	1	no	—
17	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
18	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
19	M₁₆×C₅₉	—	2	2	no	—
20	M₁₆×C₅₉	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₅₉	—	2	2	no	—
22	C₄²×C₅₉	C₂×C₄	1	2	yes	—
23	C₄²×C₅₉	D₄	1	1	yes	characteristic
24	C₄²×C₅₉	D₄	1	1	yes	characteristic
25	#11745664	—	2	2	no	—
26	#11745664	C₂×C₄	1	4	yes	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂×C₅₉⋊C₈	—	2	2	no	—
29	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
30	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
32	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₉	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₉	#41	1	2	yes	—
40	C₈×C₅₉	—	2	4	no	—
41	C₅₉⋊C₈	—	2	2	no	—
42	C₅₉⋊C₈	—	2	4	no	—
43	C₅₉⋊C₈	C₄²	1	4	yes	—
44	C₂²×C₅₉	—	2	2	no	—
45	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
47	C₄×C₅₉	—	2	2	no	—
48	C₄×C₅₉	—	2	2	no	—
49	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₅₉	#117	1	1	yes	characteristic
51	C₄×C₅₉	C₄×D₄	1	2	yes	—
52	#1778	—	59	1	no	maximal, Sylow
53	C₂×C₅₉	—	2	2	no	—
54	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₅₉	#430	1	2	yes	—
56	C₂×#108	—	59	2	no	—
57	C₄×M₁₆	—	59	1	no	—
58	C₄×M₁₆	D₅₉	1	1	yes	characteristic
59	C₂×#97	—	59	1	no	—
60	#341	—	59	2	no	—
61	C₅₉	#1778	1	1	yes	characteristic, complemented, Sylow
62	C₂×M₁₆	—	59	2	no	—
63	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
64	C₂²×C₈	—	59	1	no	—
65	C₂²×C₈	—	59	1	no	—
66	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
67	#108	—	59	8	no	—
68	#97	—	2	1	no	—
69	#97	—	59	4	no	—
70	#97	—	118	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	118	1	no	—
73	C₂²×C₄	Dic₁₁₈	1	2	yes	—
74	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	59	4	no	—
77	M₁₆	—	118	2	no	—
78	M₁₆	C₄×D₅₉	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	59	2	no	—
81	C₂×C₈	—	59	4	no	—
82	C₂×C₈	—	59	2	no	—
83	C₂×C₈	—	118	2	no	—
84	C₂×C₈	—	118	2	no	—
85	C₄²	C₄×D₅₉	1	2	yes	—
86	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
88	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₅₉⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
94	C₂×C₄	#11745685	1	2	yes	—
95	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
96	C₈	—	2	4	no	—
97	C₈	—	59	4	no	—
98	C₈	—	118	2	no	—
99	C₈	—	118	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₁₁₈	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₅₉	1	1	yes	characteristic
106	C₄	#14273684	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₅₉⋊C₄	1	1	yes	characteristic
110	C₂	#16837450	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273662

C₂

yes

maximal

C₄×#11745664

C₂

yes

maximal, characteristic

C₂×#14273638

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal

C₄×M₁₆×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

#14273662

C₄

yes

complemented

#14273638

—

#14273638

C₄

yes

—

C₄×C₅₉⋊C₈

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

#97×C₅₉

—

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

Q₈

yes

—

C₂²×C₄×C₅₉

C₂³

yes

characteristic

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₄²×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

#11745664

—

#11745664

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄²

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

—

#1778

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

#430

yes

—

C₂×#108

—

C₄×M₁₆

—

C₄×M₁₆

D₅₉

yes

characteristic

C₂×#97

—

#341

—

C₅₉

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₅₉

yes

—

—

—

—

118

—

C₄×C₈

—

C₄×C₈

—

118

—

C₂²×C₄

Dic₁₁₈

yes

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

C₄×D₅₉

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₄²

C₄×D₅₉

yes

—

C₄²

C₂²⋊D₅₉

yes

characteristic

C₄²

C₂²⋊D₅₉

yes

characteristic

C₂³

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₅₉⋊C₄

yes

—

C₂×C₄

Dic₅₉⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

#11745685

yes

—

C₂×C₄

C₂×C₂²⋊D₅₉

yes

characteristic

C₈

—

C₈

—

C₈

—

118

—

C₈

—

118

—

100

C₂²

—

101

C₂²

C₂×Dic₅₉⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₁₁₈

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₅₉

yes

characteristic

106

C₄

#14273684

yes

characteristic

107

C₄

C₄×C₂²⋊D₅₉

yes

—

108

C₂

—

109

C₂

C₄×Dic₅₉⋊C₄

yes

characteristic

110

C₂

#16837450

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₅₉
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₄×M₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	C₄×M₁₆	D₅₉
hypocenter	C₅₉	#1778
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₅₉

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₄×M₁₆×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

hypercenter

C₄×M₁₆

D₅₉

hypocenter

C₅₉

#1778

LCS

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #14273662	C₄	semidirect
61. C₅₉	#1778	semidirect

H ≅ G/N

type

9. #14273662

C₄

semidirect

61. C₅₉

#1778

semidirect

Subgroups — #21424003

Conjugacy classes of subgroups

Complements