Group #21424001 — subgroups

Subgroups — #21424001

Order 7552.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	M₁₆×Dic₅₉	C₂	1	2	yes	maximal
3	C₂×#14273637	C₂	1	1	yes	maximal, characteristic
4	#16837167	C₂	1	1	yes	maximal, characteristic
5	C₅₉×#341	C₂	1	1	yes	maximal, characteristic
6	C₂×#11745664	C₂²	1	2	yes	—
7	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
9	#14273662	C₄	1	8	yes	complemented
10	#14273637	—	2	2	no	—
11	#14273637	C₄	1	4	yes	—
12	C₈×Dic₅₉	—	2	2	no	—
13	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
16	C₂²×Dic₅₉	Q₈	1	2	yes	—
17	M₁₆×C₅₉	—	2	2	no	—
18	M₁₆×C₅₉	C₂×C₄	1	4	yes	—
19	C₂×C₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
22	C₄×Dic₅₉	C₂×C₄	1	2	yes	—
23	C₄×Dic₅₉	D₄	1	2	yes	—
24	#11745664	—	2	2	no	—
25	#11745664	C₂×C₄	1	4	yes	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	4	yes	—
29	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	#41	1	2	yes	—
34	C₂×Dic₅₉	—	4	2	no	—
35	C₂×Dic₅₉	C₄⋊C₄	1	4	yes	—
36	C₈×C₅₉	—	2	2	no	—
37	C₈×C₅₉	—	2	4	no	—
38	C₅₉⋊C₈	—	2	4	no	—
39	C₅₉⋊C₈	C₄²	1	4	yes	—
40	C₂²×C₅₉	—	2	2	no	—
41	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₅₉	C₄×Q₈	1	2	yes	—
43	C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₅₉	#117	1	1	yes	characteristic
45	C₄×C₅₉	C₄×D₄	1	2	yes	—
46	Dic₅₉	—	2	4	no	—
47	#1778	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	—	2	2	no	—
49	C₂×C₅₉	C₄×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₅₉	#430	1	2	yes	—
51	C₂×#108	—	59	2	no	—
52	C₄×M₁₆	—	59	2	no	—
53	C₂×#97	—	59	1	no	—
54	#341	—	59	1	no	—
55	#341	D₅₉	1	1	yes	characteristic
56	C₅₉	#1778	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₁₆	—	59	2	no	—
58	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
59	C₂²×C₈	—	59	1	no	—
60	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
61	C₂×C₄²	—	59	1	no	—
62	#108	—	59	8	no	—
63	#97	—	59	4	no	—
64	#97	—	118	2	no	—
65	C₄×C₈	—	118	2	no	—
66	C₂²×C₄	—	59	2	no	—
67	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	59	4	no	—
70	M₁₆	—	118	2	no	—
71	M₁₆	C₄×D₅₉	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	59	2	no	—
75	C₂×C₈	—	59	4	no	—
76	C₂×C₈	—	118	2	no	—
77	C₂×C₈	C₄×D₅₉	1	2	yes	—
78	C₄²	—	59	2	no	—
79	C₄²	—	59	2	no	—
80	C₂³	Q₈×D₅₉	1	1	yes	characteristic
81	C₂×C₄	—	59	4	no	—
82	C₂×C₄	—	236	2	no	—
83	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₅₉	1	2	yes	—
85	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
86	C₂×C₄	#11745687	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	59	4	no	—
90	C₈	—	118	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
93	C₂²	#14273692	1	2	yes	—
94	C₄	—	118	4	no	—
95	C₄	C₄²×D₅₉	1	1	yes	characteristic
96	C₄	#14273683	1	1	yes	characteristic
97	C₄	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#16837253	1	1	yes	characteristic
100	C₂	#16837449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14273662

C₂

yes

maximal

M₁₆×Dic₅₉

C₂

yes

maximal

C₂×#14273637

C₂

yes

maximal, characteristic

#16837167

C₂

yes

maximal, characteristic

C₅₉×#341

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

#14273662

C₄

yes

complemented

#14273637

—

#14273637

C₄

yes

—

C₈×Dic₅₉

—

C₂×M₁₆×C₅₉

C₂²

yes

—

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂²×C₄×C₅₉

C₂³

yes

characteristic

C₂²×Dic₅₉

Q₈

yes

—

M₁₆×C₅₉

—

M₁₆×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

D₄

yes

—

#11745664

—

#11745664

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

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

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

C₄²

yes

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

C₄×Q₈

yes

—

C₄×C₅₉

C₂×C₄²

yes

characteristic

C₄×C₅₉

#117

yes

characteristic

C₄×C₅₉

C₄×D₄

yes

—

Dic₅₉

—

#1778

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₉

#430

yes

—

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

#341

D₅₉

yes

characteristic

C₅₉

#1778

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

#97

—

118

—

C₄×C₈

—

118

—

C₂²×C₄

—

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

C₄×D₅₉

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

236

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

C₂×C₄×D₅₉

yes

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745687

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

118

—

C₂²

—

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

C₂²

#14273692

yes

—

C₄

—

118

—

C₄

C₄²×D₅₉

yes

characteristic

C₄

#14273683

yes

characteristic

C₄

(C₂²⋊D₅₉)⋊C₄

yes

—

C₂

—

C₂

#16837253

yes

characteristic

100

C₂

#16837449

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#341	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#341	D₅₉
hypocenter	C₅₉	#1778
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₄×C₅₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#341

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

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

H ≅ G/N

type

9. #14273662

C₄

semidirect

56. C₅₉

#1778

semidirect

Subgroups — #21424001

Conjugacy classes of subgroups

Complements