Group #21424232 — subgroups

Subgroups — #21424232

Order 7552.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#16837425	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837167	C₂	1	1	yes	maximal, characteristic
4	#16837166	C₂	1	2	yes	maximal
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	Dic₅₉⋊C₈	—	2	4	no	—
10	C₈×Dic₅₉	C₄	1	2	yes	—
11	C₈×Dic₅₉	C₄	1	2	yes	—
12	C₂×M₁₆×C₅₉	C₂²	1	2	yes	—
13	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
15	C₂²×Dic₅₉	Q₈	1	2	yes	—
16	M₁₆×C₅₉	—	2	2	no	—
17	M₁₆×C₅₉	—	2	2	no	—
18	C₂×C₈×C₅₉	—	2	2	no	—
19	C₂×C₈×C₅₉	—	2	2	no	—
20	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
21	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
23	C₄×Dic₅₉	D₄	1	2	yes	—
24	#11745664	—	2	2	no	—
25	#11745664	—	2	2	no	—
26	C₂×C₅₉⋊C₈	—	2	2	no	—
27	C₂×C₅₉⋊C₈	—	2	2	no	—
28	C₂×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
29	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	#41	1	2	yes	—
33	C₂×C₄×C₅₉	#41	1	2	yes	—
34	C₂×Dic₅₉	—	2	4	no	—
35	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₅₉	C₄⋊C₄	1	2	yes	—
37	C₈×C₅₉	—	2	2	no	—
38	C₈×C₅₉	—	4	2	no	—
39	C₅₉⋊C₈	—	2	2	no	—
40	C₅₉⋊C₈	—	4	2	no	—
41	C₂²×C₅₉	—	2	2	no	—
42	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₅₉	#125	1	2	yes	—
44	C₄×C₅₉	#117	1	1	yes	characteristic
45	C₄×C₅₉	#117	1	1	yes	characteristic
46	C₄×C₅₉	#124	1	2	yes	—
47	Dic₅₉	—	2	2	no	—
48	Dic₅₉	#108	1	4	yes	—
49	#1837	—	59	1	no	maximal, Sylow
50	C₂×C₅₉	—	2	2	no	—
51	C₂×C₅₉	#382	1	1	yes	characteristic
52	C₂×C₅₉	C₂×#108	1	2	yes	—
53	#423	—	59	2	no	—
54	C₂×C₄×C₈	—	59	1	no	—
55	#341	—	59	2	no	—
56	#341	—	59	1	no	—
57	#341	D₅₉	1	1	yes	characteristic
58	C₅₉	#1837	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₁₆	—	59	2	no	—
60	C₂×M₁₆	C₂×D₅₉	1	2	yes	—
61	C₂²×C₈	—	59	1	no	—
62	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
63	C₂×C₄²	—	59	1	no	—
64	C₄⋊C₈	—	118	4	no	—
65	C₄×C₈	—	59	2	no	—
66	C₄×C₈	—	59	2	no	—
67	C₂²×C₄	—	59	2	no	—
68	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	118	2	no	—
72	M₁₆	—	118	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	59	2	no	—
76	C₂×C₈	—	118	2	no	—
77	C₂×C₈	—	118	2	no	—
78	C₂×C₈	C₄×D₅₉	1	2	yes	—
79	C₄²	—	59	1	no	—
80	C₄²	—	59	1	no	—
81	C₄²	—	59	2	no	—
82	C₂³	Q₈×D₅₉	1	1	yes	characteristic
83	C₂×C₄	—	59	2	no	—
84	C₂×C₄	—	59	2	no	—
85	C₂×C₄	—	118	4	no	—
86	C₂×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
87	C₂×C₄	#11745685	1	2	yes	—
88	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
89	C₂×C₄	#11745687	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	118	2	no	—
93	C₈	—	236	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₅₉	1	1	yes	characteristic
96	C₂²	#14273717	1	2	yes	—
97	C₄	—	59	4	no	—
98	C₄	—	118	2	no	—
99	C₄	#14273683	1	1	yes	characteristic
100	C₄	#14273684	1	1	yes	characteristic
101	C₄	#14273713	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#16837266	1	1	yes	characteristic
104	C₂	#108×D₅₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16837425

C₂

yes

maximal

C₂×C₈×Dic₅₉

C₂

yes

maximal, characteristic

#16837167

C₂

yes

maximal, characteristic

#16837166

C₂

yes

maximal

C₅₉×#341

C₂

yes

maximal, characteristic

C₂×#11745664

C₂²

yes

—

C₂²×C₅₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₅₉

C₂²

yes

characteristic

Dic₅₉⋊C₈

—

C₈×Dic₅₉

C₄

yes

—

C₈×Dic₅₉

C₄

yes

—

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×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

C₂×C₄

yes

characteristic

C₄×Dic₅₉

D₄

yes

—

#11745664

—

#11745664

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

—

C₂×C₅₉⋊C₈

C₂×C₄

yes

—

C₂³×C₅₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

#41

yes

—

C₂×C₄×C₅₉

#41

yes

—

C₂×Dic₅₉

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₂×Dic₅₉

C₄⋊C₄

yes

—

C₈×C₅₉

—

C₈×C₅₉

—

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₉

#125

yes

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

yes

—

Dic₅₉

—

Dic₅₉

#108

yes

—

#1837

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

#382

yes

characteristic

C₂×C₅₉

C₂×#108

yes

—

#423

—

C₂×C₄×C₈

—

#341

—

#341

—

#341

D₅₉

yes

characteristic

C₅₉

#1837

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₉

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

118

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₄×D₅₉

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

118

—

C₂×C₄

C₂×C₄×D₅₉

yes

characteristic

C₂×C₄

#11745685

yes

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₂×C₄

#11745687

yes

—

C₈

—

C₈

—

C₈

—

118

—

C₈

—

236

—

C₂²

—

C₂²

C₄⋊C₄×D₅₉

yes

characteristic

C₂²

#14273717

yes

—

C₄

—

C₄

—

118

—

C₄

#14273683

yes

characteristic

100

C₄

#14273684

yes

characteristic

101

C₄

#14273713

yes

—

102

C₂

—

103

C₂

#16837266

yes

characteristic

104

C₂

#108×D₅₉

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₅₉×#341	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	#341	D₅₉
hypocenter	C₅₉	#1837
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₅₉

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₉

Fitting

C₅₉×#341

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

D₅₉

C₅₉

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
58. C₅₉	#1837	semidirect

H ≅ G/N

type

58. C₅₉

#1837

semidirect

Subgroups — #21424232

Conjugacy classes of subgroups

Complements