Group #21422174 — subgroups

Subgroups — #21422174

Order 7552.

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

normal subgroups only

This group has 160 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	C₂²×C₅₉⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅₉⋊C₁₆	—	2	2	no	—
4	C₂×C₅₉⋊C₁₆	—	2	2	no	—
5	C₂×C₅₉⋊C₁₆	C₄	1	4	yes	complemented
6	C₂²×C₈×C₅₉	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₅₉	C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₉	—	2	2	no	—
10	C₂²×C₄×C₅₉	C₈	1	2	yes	—
11	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₈×C₅₉	—	2	1	no	—
13	C₂×C₈×C₅₉	—	2	1	no	—
14	C₂×C₈×C₅₉	C₈	1	2	yes	—
15	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₅₉	D₄	1	1	yes	characteristic
17	C₂×C₈×C₅₉	D₄	1	2	yes	—
18	C₂×C₈×C₅₉	Q₈	1	1	yes	characteristic
19	C₄²×C₅₉	—	2	1	no	—
20	C₄²×C₅₉	—	2	1	no	—
21	C₅₉⋊C₁₆	—	2	4	no	—
22	C₂³×C₅₉	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₅₉	—	2	2	no	—
24	C₂×C₄×C₅₉	—	2	2	no	—
25	C₂×C₄×C₅₉	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₅₉	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₅₉	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₅₉	M₁₆	1	1	yes	characteristic
32	C₈×C₅₉	—	2	2	no	—
33	C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
34	C₈×C₅₉	C₄⋊C₄	1	2	yes	—
35	C₂²×C₅₉	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₅₉	C₄⋊C₈	1	1	yes	characteristic
38	C₂²×C₅₉	C₂×C₁₆	1	2	yes	—
39	C₂²×C₅₉	M₃₂	1	2	yes	—
40	C₄×C₅₉	—	2	4	no	—
41	C₄×C₅₉	#95	1	1	yes	characteristic
42	C₄×C₅₉	#97	1	1	yes	characteristic
43	C₄×C₅₉	C₂²⋊C₈	1	1	yes	characteristic
44	C₄×C₅₉	C₄⋊C₈	1	1	yes	characteristic
45	#1374	—	59	1	no	maximal, Sylow
46	C₂×C₅₉	#336	1	1	yes	characteristic
47	C₂×C₅₉	C₄×C₁₆	1	1	yes	characteristic
48	C₂×C₅₉	#346	1	1	yes	characteristic
49	C₂×C₅₉	C₂²⋊C₁₆	1	2	yes	—
50	C₂×C₅₉	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₁₆	—	59	2	no	—
52	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
53	C₅₉	#1374	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	Dic₅₉	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
56	C₂×C₄²	Dic₅₉	1	1	yes	characteristic
57	C₂×C₁₆	—	59	4	no	—
58	C₂×C₁₆	—	118	2	no	—
59	C₂×C₁₆	—	118	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₂²×C₄	C₅₉⋊C₈	1	2	yes	—
62	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₅₉⋊C₈	1	2	yes	—
66	C₂×C₈	Dic₁₁₈	1	1	yes	characteristic
67	C₂×C₈	C₄×D₅₉	1	2	yes	—
68	C₂×C₈	D₂₃₆	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊D₅₉	1	2	yes	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₁₆	—	118	4	no	—
73	C₂³	C₂×C₅₉⋊C₈	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₅₉⋊C₁₆	1	4	yes	complemented
77	C₂×C₄	C₂×C₅₉⋊C₈	1	1	yes	characteristic
78	C₂×C₄	#11745664	1	1	yes	characteristic
79	C₂×C₄	#11745664	1	1	yes	characteristic
80	C₂×C₄	C₄×Dic₅₉	1	1	yes	characteristic
81	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	Dic₅₉⋊C₄	1	2	yes	—
85	C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
86	C₂²	C₄×C₅₉⋊C₈	1	1	yes	characteristic
87	C₂²	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
88	C₂²	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
89	C₂²	C₂×C₅₉⋊C₁₆	1	2	yes	—
90	C₂²	#14273667	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	C₅₉⋊#95	1	1	yes	characteristic
93	C₄	#14273638	1	1	yes	characteristic
94	C₄	C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
95	C₄	C₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
96	C₂	#16837157	1	1	yes	characteristic
97	C₂	C₄×C₅₉⋊C₁₆	1	1	yes	characteristic
98	C₂	#16837178	1	1	yes	characteristic
99	C₂	C₂²⋊(C₅₉⋊C₁₆)	1	2	yes	—
100	C₂	C₄⋊(C₅₉⋊C₁₆)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₅₉⋊C₁₆

C₂

yes

maximal

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

—

C₂×C₅₉⋊C₁₆

C₄

yes

complemented

C₂²×C₈×C₅₉

C₄

yes

characteristic

C₂²×C₈×C₅₉

C₂²

yes

characteristic

C₂×C₄²×C₅₉

C₄

yes

characteristic

C₄×C₈×C₅₉

—

C₂²×C₄×C₅₉

C₈

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₈

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

D₄

yes

characteristic

C₂×C₈×C₅₉

D₄

yes

—

C₂×C₈×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

—

C₄²×C₅₉

—

C₅₉⋊C₁₆

—

C₂³×C₅₉

C₂×C₈

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₁₆

yes

complemented

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

M₁₆

yes

characteristic

C₂×C₄×C₅₉

M₁₆

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

C₂²⋊C₄

yes

—

C₈×C₅₉

C₄⋊C₄

yes

—

C₂²×C₅₉

C₄×C₈

yes

characteristic

C₂²×C₅₉

C₂²⋊C₈

yes

characteristic

C₂²×C₅₉

C₄⋊C₈

yes

characteristic

C₂²×C₅₉

C₂×C₁₆

yes

—

C₂²×C₅₉

M₃₂

yes

—

C₄×C₅₉

—

C₄×C₅₉

yes

C₄×C₅₉

yes

C₄×C₅₉

C₂²⋊C₈

yes

characteristic

C₄×C₅₉

C₄⋊C₈

yes

—

C₂×C₅₉

yes

C₂×C₅₉

C₄×C₁₆

yes

characteristic

C₂×C₅₉

#346

yes

characteristic

C₂×C₅₉

C₂²⋊C₁₆

yes

—

C₂×C₅₉

C₄⋊C₁₆

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

D₅₉

yes

characteristic

C₅₉

#1374

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₅₉

yes

characteristic

C₂²×C₈

C₂×D₅₉

yes

characteristic

C₂×C₄²

Dic₅₉

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

118

—

C₂×C₁₆

—

118

—

C₄×C₈

—

C₂²×C₄

C₅₉⋊C₈

yes

—

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₅₉⋊C₈

yes

—

C₂×C₈

Dic₁₁₈

yes

characteristic

C₂×C₈

C₄×D₅₉

yes

—

C₂×C₈

D₂₃₆

yes

characteristic

C₂×C₈

C₂²⋊D₅₉

yes

—

C₄²

—

C₄²

—

C₁₆

—

118

—

C₂³

C₂×C₅₉⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₅₉⋊C₁₆

yes

complemented

C₂×C₄

C₂×C₅₉⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄×Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂²⋊Dic₅₉

yes

characteristic

C₈

—

C₈

Dic₅₉⋊C₄

yes

—

C₈

(C₂×D₅₉)⋊C₄

yes

—

C₂²

C₄×C₅₉⋊C₈

yes

characteristic

C₂²

C₂²⋊(C₅₉⋊C₈)

yes

characteristic

C₂²

C₄⋊(C₅₉⋊C₈)

yes

characteristic

C₂²

C₂×C₅₉⋊C₁₆

yes

—

C₂²

#14273667

yes

—

C₄

—

C₄

C₅₉⋊#95

yes

characteristic

C₄

#14273638

yes

characteristic

C₄

C₂²⋊(C₅₉⋊C₈)

yes

characteristic

C₄

C₄⋊(C₅₉⋊C₈)

yes

characteristic

C₂

#16837157

yes

characteristic

C₂

C₄×C₅₉⋊C₁₆

yes

characteristic

C₂

#16837178

yes

characteristic

C₂

C₂²⋊(C₅₉⋊C₁₆)

yes

—

100

C₂

C₄⋊(C₅₉⋊C₁₆)

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂×D₅₉
commutator	C₂×C₅₉	C₄×C₁₆
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂³×C₅₉	C₂×C₈
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1374
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂×D₅₉

commutator

C₂×C₅₉

C₄×C₁₆

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂³×C₅₉

C₂×C₈

hypercenter

C₂×C₄×C₈

D₅₉

C₅₉

G ▷ C₂×C₅₉ ▷ C₅₉

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. C₂×C₅₉⋊C₁₆	C₄	semidirect
25. C₂×C₄×C₅₉	C₁₆	semidirect
53. C₅₉	#1374	semidirect
76. C₂×C₄	C₅₉⋊C₁₆	semidirect

H ≅ G/N

type

5. C₂×C₅₉⋊C₁₆

C₄

semidirect

25. C₂×C₄×C₅₉

C₁₆

semidirect

53. C₅₉

#1374

semidirect

76. C₂×C₄

C₅₉⋊C₁₆

semidirect

Subgroups — #21422174

Conjugacy classes of subgroups

Complements