Group #21422852 — subgroups

Subgroups — #21422852

Order 7552.

184 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 109 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	#16837426	C₂	1	2	yes	maximal
2	#16837153	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#11745664	—	2	2	no	—
5	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
6	C₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
7	C₂²×C₈×C₅₉	C₄	1	2	yes	—
8	C₂×C₄²×C₅₉	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
10	C₄×C₈×C₅₉	C₄	1	1	yes	characteristic
11	C₄×C₈×C₅₉	C₂²	1	2	yes	—
12	C₂²×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₅₉	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₉	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₅₉	—	2	4	no	—
16	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₅₉	C₂×C₄	1	2	yes	—
18	C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₅₉	D₄	1	1	yes	characteristic
20	C₄²×C₅₉	Q₈	1	1	yes	characteristic
21	C₄²×C₅₉	C₂³	1	1	yes	characteristic
22	#11745664	—	4	4	no	—
23	C₂×C₅₉⋊C₈	—	2	4	no	—
24	C₂³×C₅₉	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₅₉	—	2	2	no	—
26	C₂×C₄×C₅₉	—	2	2	no	—
27	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₉	SD₁₆	1	4	yes	—
31	C₂×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₅₉	—	2	2	no	—
35	C₈×C₅₉	M₁₆	1	4	yes	complemented
36	C₅₉⋊C₈	—	4	4	no	—
37	C₂²×C₅₉	—	2	1	no	—
38	C₂²×C₅₉	—	2	1	no	—
39	C₂²×C₅₉	#106	1	2	yes	—
40	C₂²×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₄×C₅₉	—	2	1	no	—
42	C₄×C₅₉	—	2	1	no	—
43	C₄×C₅₉	#106	1	2	yes	—
44	C₄×C₅₉	#108	1	2	yes	—
45	C₄×C₅₉	C₂×M₁₆	1	2	yes	—
46	C₄×C₅₉	C₂×SD₁₆	1	2	yes	—
47	#1566	—	59	1	no	maximal, Sylow
48	C₂×C₅₉	—	2	2	no	—
49	C₂×C₅₉	#423	1	1	yes	characteristic
50	C₂×C₅₉	C₂×#106	1	1	yes	characteristic
51	C₂×C₅₉	C₂×#108	1	1	yes	characteristic
52	#423	—	59	2	no	—
53	C₂×C₄×C₈	D₅₉	1	1	yes	characteristic
54	#334	—	59	4	no	—
55	C₅₉	#1566	1	1	yes	characteristic, complemented, Sylow
56	C₂×M₁₆	—	118	2	no	—
57	C₂²×C₈	Dic₅₉	1	2	yes	—
58	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
59	C₄⋊C₈	—	59	4	no	—
60	C₄⋊C₈	—	118	2	no	—
61	C₄×C₈	Dic₅₉	1	1	yes	characteristic
62	C₄×C₈	Dic₅₉	1	1	yes	characteristic
63	C₄×C₈	C₂×D₅₉	1	2	yes	—
64	C₂²×C₄	Dic₁₁₈	1	1	yes	characteristic
65	C₂²×C₄	D₂₃₆	1	1	yes	characteristic
66	C₂²×C₄	C₂×Dic₅₉	1	1	yes	characteristic
67	M₁₆	—	236	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₂×C₈	—	118	4	no	—
70	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
71	C₂×C₈	C₂×Dic₅₉	1	2	yes	—
72	C₄²	Dic₁₁₈	1	1	yes	characteristic
73	C₄²	D₂₃₆	1	1	yes	characteristic
74	C₄²	C₂×Dic₅₉	1	1	yes	characteristic
75	C₄²	C₂²×D₅₉	1	1	yes	characteristic
76	C₂³	C₄⋊Dic₅₉	1	1	yes	characteristic
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	#11745660	1	4	yes	—
80	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
81	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
82	C₂×C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
83	C₂×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
85	C₂×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	236	4	no	—
88	C₈	#11745664	1	4	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#14273659	1	2	yes	—
92	C₂²	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#14273659	1	2	yes	—
96	C₄	#14273663	1	2	yes	—
97	C₄	C₂×#11745664	1	2	yes	—
98	C₄	C₂×#11745660	1	2	yes	—
99	C₂	—	2	2	no	—
100	C₂	#16837426	1	1	yes	characteristic
101	C₂	C₂×#14273659	1	1	yes	characteristic
102	C₂	C₂×#14273663	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837426

C₂

yes

maximal

#16837153

C₂

yes

maximal, complemented

C₂×C₄×C₈×C₅₉

C₂

yes

maximal, characteristic

C₂×#11745664

—

C₄⋊(C₅₉⋊C₈)

—

C₄⋊(C₅₉⋊C₈)

C₂²

yes

—

C₂²×C₈×C₅₉

C₄

yes

—

C₂×C₄²×C₅₉

C₂²

yes

characteristic

C₄×C₈×C₅₉

C₄

yes

characteristic

C₄×C₈×C₅₉

C₄

yes

characteristic

C₄×C₈×C₅₉

C₂²

yes

—

C₂²×C₄×C₅₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₉

D₄

yes

characteristic

C₂²×C₄×C₅₉

Q₈

yes

characteristic

C₂×C₈×C₅₉

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₂×C₈×C₅₉

C₂×C₄

yes

—

C₄²×C₅₉

C₂×C₄

yes

characteristic

C₄²×C₅₉

D₄

yes

characteristic

C₄²×C₅₉

Q₈

yes

characteristic

C₄²×C₅₉

C₂³

yes

characteristic

#11745664

—

C₂×C₅₉⋊C₈

—

C₂³×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

—

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₉

SD₁₆

yes

—

C₂×C₄×C₅₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂×C₄×C₅₉

C₂×Q₈

yes

characteristic

C₈×C₅₉

—

C₈×C₅₉

M₁₆

yes

complemented

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#106

yes

—

C₂²×C₅₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₉

—

C₄×C₅₉

—

C₄×C₅₉

#106

yes

—

C₄×C₅₉

#108

yes

—

C₄×C₅₉

C₂×M₁₆

yes

—

C₄×C₅₉

C₂×SD₁₆

yes

—

#1566

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

yes

C₂×C₅₉

yes

C₂×C₅₉

yes

—

C₂×C₄×C₈

D₅₉

yes

characteristic

#334

—

C₅₉

#1566

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

118

—

C₂²×C₈

Dic₅₉

yes

—

C₂×C₄²

C₂×D₅₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

118

—

C₄×C₈

Dic₅₉

yes

characteristic

C₄×C₈

Dic₅₉

yes

characteristic

C₄×C₈

C₂×D₅₉

yes

—

C₂²×C₄

Dic₁₁₈

yes

characteristic

C₂²×C₄

D₂₃₆

yes

characteristic

C₂²×C₄

C₂×Dic₅₉

yes

characteristic

M₁₆

—

236

—

C₂×C₈

—

C₂×C₈

—

118

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₂×C₈

C₂×Dic₅₉

yes

—

C₄²

Dic₁₁₈

yes

characteristic

C₄²

D₂₃₆

yes

characteristic

C₄²

C₂×Dic₅₉

yes

characteristic

C₄²

C₂²×D₅₉

yes

characteristic

C₂³

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11745660

yes

—

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₄⋊Dic₅₉

yes

characteristic

C₂×C₄

C₂×Dic₁₁₈

yes

characteristic

C₂×C₄

C₂×D₂₃₆

yes

characteristic

C₂×C₄

C₂²×Dic₅₉

yes

characteristic

C₈

—

C₈

—

236

—

C₈

#11745664

yes

complemented

C₂²

—

C₂²

—

C₂²

#14273659

yes

—

C₂²

C₂×C₄⋊Dic₅₉

yes

characteristic

C₄

—

C₄

—

C₄

#14273659

yes

—

C₄

#14273663

yes

—

C₄

C₂×#11745664

yes

—

C₄

C₂×#11745660

yes

—

C₂

—

100

C₂

#16837426

yes

characteristic

101

C₂

C₂×#14273659

yes

characteristic

102

C₂

C₂×#14273663

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₄
Frattini	C₄²	C₂²×D₅₉
Fitting	C₂×C₄×C₈×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄⋊C₄
hypercenter	C₂×C₄×C₈	D₅₉
hypocenter	C₅₉	#1566
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₃₆

commutator

C₂×C₄×C₅₉

C₂²×C₄

Frattini

C₄²

C₂²×D₅₉

Fitting

C₂×C₄×C₈×C₅₉

C₂

socle

C₂²×C₅₉

C₂×C₄⋊C₄

hypercenter

C₂×C₄×C₈

D₅₉

hypocenter

C₅₉

#1566

LCS

G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16837153	C₂	semidirect
35. C₈×C₅₉	M₁₆	semidirect
55. C₅₉	#1566	semidirect
88. C₈	#11745664	semidirect

H ≅ G/N

type

2. #16837153

C₂

semidirect

35. C₈×C₅₉

M₁₆

semidirect

55. C₅₉

#1566

semidirect

88. C₈

#11745664

semidirect

Subgroups — #21422852

Conjugacy classes of subgroups

Complements