Group #21426213 — subgroups

Subgroups — #21426213

Order 7552.

148 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16837818	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837669	C₂	1	2	yes	maximal, complemented
3	#16837232	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837215	C₂	1	2	yes	maximal, complemented
5	C₂×D₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#14273753	—	2	2	no	—
7	#14273733	C₂²	1	1	yes	characteristic
8	#14273667	C₂²	1	1	yes	characteristic
9	#14273666	—	2	1	no	—
10	C₁₆×D₅₉	—	2	1	no	—
11	#14273663	C₂²	1	2	yes	—
12	#14273644	—	2	2	no	—
13	C₂×D₈×C₅₉	C₂²	1	2	yes	—
14	D₁₆×C₅₉	—	2	1	no	—
15	D₁₆×C₅₉	—	2	1	no	—
16	C₂×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₅₉	—	2	2	no	—
18	#11745685	D₄	1	1	yes	characteristic
19	D₈×C₅₉	—	2	2	no	—
20	D₈×C₅₉	—	2	2	no	—
21	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
22	#11745670	—	4	2	no	—
23	#11745669	—	4	2	no	—
24	#11745664	—	2	2	no	—
25	#11745664	D₄	1	1	yes	characteristic
26	#11745659	—	2	1	no	—
27	C₈×D₅₉	—	2	1	no	—
28	C₁₆×C₅₉	D₄	1	1	yes	characteristic
29	C₁₆×C₅₉	D₄	1	1	yes	characteristic
30	C₅₉⋊C₁₆	—	2	1	no	—
31	C₂³×C₅₉	—	4	2	no	—
32	D₄×C₅₉	—	4	2	no	—
33	D₄×C₅₉	—	4	2	no	—
34	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
35	C₂²⋊D₅₉	—	2	1	no	—
36	D₂₃₆	D₈	1	1	yes	characteristic
37	C₄×D₅₉	—	2	1	no	—
38	Dic₁₁₈	D₈	1	1	yes	characteristic
39	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
40	C₈×C₅₉	#41	1	1	yes	characteristic
41	C₅₉⋊C₈	—	2	1	no	—
42	C₅₉⋊C₈	—	4	2	no	—
43	C₂²×C₅₉	—	8	2	no	—
44	C₂²×C₅₉	—	16	2	no	—
45	C₂²×C₅₉	#135	1	1	yes	characteristic
46	C₂×D₅₉	—	2	1	no	—
47	C₄×C₅₉	#121	1	1	yes	characteristic
48	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
49	Dic₅₉	—	2	1	no	—
50	#2221	—	59	1	no	maximal, Sylow
51	C₂×C₅₉	—	2	1	no	—
52	C₂×C₅₉	—	16	2	no	—
53	C₂×C₅₉	#466	1	1	yes	characteristic
54	D₅₉	—	4	1	no	—
55	C₂×D₁₆	D₅₉	1	1	yes	characteristic, complemented
56	#504	—	59	1	no	—
57	#472	—	59	2	no	—
58	#368	—	59	1	no	—
59	#361	—	59	2	no	—
60	C₅₉	#2221	1	1	yes	characteristic, complemented, Sylow
61	AGL(1,ℤ/8)	—	118	2	no	—
62	C₂×D₈	C₂×D₅₉	1	2	yes	—
63	#131	—	59	1	no	—
64	D₁₆	—	2	1	no	—
65	D₁₆	—	2	1	no	—
66	M₃₂	—	59	1	no	—
67	M₃₂	—	118	1	no	—
68	C₂×C₁₆	—	118	1	no	—
69	C₂×C₁₆	C₂×D₅₉	1	1	yes	characteristic
70	#108	—	59	2	no	—
71	#100	—	118	2	no	—
72	#41	—	59	1	no	—
73	C₂×D₄	—	2	2	no	—
74	SD₁₆	—	236	2	no	—
75	D₈	—	2	2	no	—
76	D₈	—	2	2	no	—
77	D₈	—	236	2	no	—
78	M₁₆	—	59	1	no	—
79	M₁₆	—	118	1	no	—
80	M₁₆	—	118	2	no	—
81	C₂×C₈	—	118	1	no	—
82	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
83	C₁₆	—	118	1	no	—
84	C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
85	C₁₆	C₂²⋊D₅₉	1	1	yes	characteristic
86	C₂³	—	4	2	no	—
87	Q₈	—	59	1	no	—
88	D₄	—	4	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	59	1	no	—
91	D₄	—	118	1	no	—
92	C₂×C₄	—	118	1	no	—
93	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
94	C₈	—	118	1	no	—
95	C₈	—	236	2	no	—
96	C₈	#11745687	1	1	yes	characteristic
97	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
98	C₂²	—	8	2	no	—
99	C₂²	—	16	2	no	—
100	C₂²	—	118	1	no	—
101	C₂²	#14273746	1	1	yes	characteristic
102	C₄	—	118	1	no	—
103	C₄	#14273698	1	1	yes	characteristic
104	C₄	D₈×D₅₉	1	1	yes	characteristic
105	C₂	—	2	1	no	—
106	C₂	—	16	2	no	—
107	C₂	—	236	1	no	—
108	C₂	#16837635	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16837818

C₂

yes

maximal, characteristic, complemented

#16837669

C₂

yes

maximal, complemented

#16837232

C₂

yes

maximal, characteristic, complemented

#16837215

C₂

yes

maximal, complemented

C₂×D₁₆×C₅₉

C₂

yes

maximal, characteristic, complemented

#14273753

—

#14273733

C₂²

yes

characteristic

#14273667

C₂²

yes

characteristic

#14273666

—

C₁₆×D₅₉

—

#14273663

C₂²

yes

—

#14273644

—

C₂×D₈×C₅₉

C₂²

yes

—

D₁₆×C₅₉

—

D₁₆×C₅₉

—

C₂×C₁₆×C₅₉

C₂²

yes

characteristic

C₂×D₄×C₅₉

—

#11745685

D₄

yes

characteristic

D₈×C₅₉

—

D₈×C₅₉

—

C₂×C₈×C₅₉

C₂³

yes

—

—

—

D₄

yes

characteristic

#11745659

—

C₈×D₅₉

—

C₁₆×C₅₉

D₄

yes

characteristic

C₁₆×C₅₉

D₄

yes

characteristic

C₅₉⋊C₁₆

—

C₂³×C₅₉

—

D₄×C₅₉

—

D₄×C₅₉

—

C₂×C₄×C₅₉

C₂×D₄

yes

characteristic

C₂²⋊D₅₉

—

D₂₃₆

D₈

yes

characteristic

C₄×D₅₉

—

Dic₁₁₈

D₈

yes

characteristic

C₈×C₅₉

C₂×D₄

yes

characteristic

C₈×C₅₉

#41

yes

characteristic

C₅₉⋊C₈

—

C₅₉⋊C₈

—

C₂²×C₅₉

—

C₂²×C₅₉

—

C₂²×C₅₉

#135

yes

characteristic

C₂×D₅₉

—

C₄×C₅₉

#121

yes

characteristic

C₄×C₅₉

C₂×D₈

yes

characteristic

Dic₅₉

—

#2221

—

maximal, Sylow

C₂×C₅₉

—

C₂×C₅₉

—

C₂×C₅₉

#466

yes

characteristic

D₅₉

—

C₂×D₁₆

D₅₉

yes

characteristic, complemented

—

—

—

—

C₅₉

#2221

yes

characteristic, complemented, Sylow

AGL(1,ℤ/8)

—

118

—

C₂×D₈

C₂×D₅₉

yes

—

#131

—

D₁₆

—

D₁₆

—

M₃₂

—

M₃₂

—

118

—

C₂×C₁₆

—

118

—

C₂×C₁₆

C₂×D₅₉

yes

characteristic

#108

—

#100

—

118

—

#41

—

C₂×D₄

—

SD₁₆

—

236

—

D₈

—

D₈

—

D₈

—

236

—

M₁₆

—

M₁₆

—

118

—

M₁₆

—

118

—

C₂×C₈

—

118

—

C₂×C₈

C₂²×D₅₉

yes

characteristic

C₁₆

—

118

—

C₁₆

C₂²⋊D₅₉

yes

characteristic

C₁₆

C₂²⋊D₅₉

yes

characteristic

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

118

—

C₂×C₄

—

118

—

C₂×C₄

D₄×D₅₉

yes

characteristic

C₈

—

118

—

C₈

—

236

—

C₈

#11745687

yes

characteristic

C₈

C₂×C₂²⋊D₅₉

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

118

—

101

C₂²

#14273746

yes

characteristic

102

C₄

—

118

—

103

C₄

#14273698

yes

characteristic

104

C₄

D₈×D₅₉

yes

characteristic

105

C₂

—

106

C₂

—

107

C₂

—

236

—

108

C₂

#16837635

yes

characteristic

	subgroup	G/H
center	C₂	#16837635
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₂×D₁₆×C₅₉	C₂
socle	C₂×C₅₉	#466
hypercenter	C₂×D₁₆	D₅₉
hypocenter	C₅₉	#2221
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

subgroup

G/H

center

C₂

#16837635

commutator

C₂×C₈×C₅₉

C₂³

Frattini

C₂×C₈

C₂²×D₅₉

Fitting

C₂×D₁₆×C₅₉

C₂

socle

C₂×C₅₉

#466

hypercenter

C₂×D₁₆

D₅₉

hypocenter

C₅₉

#2221

LCS

G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #16837818	C₂	semidirect
2. #16837669	C₂	semidirect
3. #16837232	C₂	semidirect
4. #16837215	C₂	semidirect
5. C₂×D₁₆×C₅₉	C₂	semidirect
55. C₂×D₁₆	D₅₉	semidirect
60. C₅₉	#2221	semidirect

H ≅ G/N

type

1. #16837818

C₂

semidirect

2. #16837669

C₂

semidirect

3. #16837232

C₂

semidirect

4. #16837215

C₂

semidirect

5. C₂×D₁₆×C₅₉

C₂

semidirect

55. C₂×D₁₆

D₅₉

semidirect

60. C₅₉

#2221

semidirect

Subgroups — #21426213

Conjugacy classes of subgroups

Complements