Group #10601366 — subgroups

Subgroups — #10601366

Order 640.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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×C₅×#106	C₂	1	1	yes	maximal, characteristic
2	#65391	C₂	1	1	yes	maximal, characteristic
3	#65253	C₂	1	2	yes	maximal
4	#64772	C₂	1	1	yes	maximal, characteristic
5	#64750	C₂	1	2	yes	maximal
6	C₂²×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
8	#4072	C₂²	1	1	yes	characteristic
9	C₂×#681	C₂²	1	1	yes	characteristic
10	C₂×#681	C₂²	1	2	yes	—
11	C₅×#106	—	2	2	no	—
12	#3946	—	2	2	no	—
13	#3940	—	2	2	no	—
14	#3939	—	2	2	no	—
15	#2090	—	5	1	no	maximal, Sylow
16	C₂²×C₄×C₅	—	2	2	no	—
17	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
18	C₂×C₅×C₈	—	2	2	no	—
19	C₂×C₅×C₈	D₄	1	2	yes	—
20	C₅×C₄⋊C₄	—	2	2	no	—
21	C₅×C₄⋊C₄	—	2	2	no	—
22	C₂²⋊Dic₅	—	2	1	no	—
23	C₄⋊Dic₅	D₄	1	2	yes	—
24	C₄×Dic₅	—	2	1	no	—
25	#681	—	2	2	no	—
26	#681	—	2	2	no	—
27	#681	Q₈	1	4	yes	—
28	C₂×#168	—	2	2	no	—
29	C₂×#168	—	2	1	no	—
30	#428	—	5	2	no	—
31	C₂×#106	D₅	1	1	yes	characteristic
32	#424	—	5	1	no	—
33	#343	—	5	2	no	—
34	#341	—	5	1	no	—
35	C₂³×C₅	#41	1	1	yes	characteristic
36	C₂×C₄×C₅	—	4	2	no	—
37	C₂×C₄×C₅	—	8	2	no	—
38	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₅	#41	1	2	yes	—
42	C₂×Dic₅	—	2	2	no	—
43	C₅×C₈	—	2	2	no	—
44	#168	—	2	4	no	—
45	#168	—	4	1	no	—
46	C₂×M₁₆	—	5	2	no	—
47	C₂×M₁₆	—	5	1	no	—
48	C₂²×C₈	C₂×D₅	1	1	yes	characteristic
49	#117	—	5	1	no	—
50	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	—
51	#107	—	10	2	no	—
52	#106	—	2	2	no	—
53	#106	—	10	2	no	—
54	C₄⋊C₈	—	10	2	no	—
55	C₂²×C₅	—	2	2	no	—
56	C₂²×C₅	#121	1	1	yes	characteristic
57	C₂²×C₅	#125	1	2	yes	—
58	C₄×C₅	—	8	2	no	—
59	C₄×C₅	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₅	#123	1	1	yes	characteristic
61	C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
62	Dic₅	—	4	2	no	—
63	C₂²×C₄	—	2	2	no	—
64	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
65	M₁₆	—	5	4	no	—
66	M₁₆	—	10	2	no	—
67	M₁₆	—	10	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	10	2	no	—
70	C₂×C₈	—	10	1	no	—
71	C₂×C₈	C₂²⋊D₅	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	5	2	no	—
75	C₂²⋊C₄	—	10	1	no	—
76	C₄²	—	10	1	no	—
77	C₂×C₅	—	2	2	no	—
78	C₂×C₅	#398	1	1	yes	characteristic
79	C₂×C₅	#471	1	2	yes	—
80	C₂³	#711	1	1	yes	characteristic
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	10	2	no	—
84	C₂×C₄	#709	1	2	yes	—
85	C₂×C₄	D₄×D₅	1	1	yes	characteristic
86	C₂×C₄	Q₈×D₅	1	2	yes	—
87	C₂×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	10	4	no	—
90	C₈	—	20	1	no	—
91	C₅	#2090	1	1	yes	characteristic, complemented, Sylow
92	C₂²	—	2	2	no	—
93	C₂²	#4035	1	2	yes	—
94	C₂²	#4084	1	1	yes	characteristic
95	C₄	—	8	2	no	—
96	C₄	—	20	2	no	—
97	C₄	#4042	1	2	yes	—
98	C₄	#4075	1	1	yes	characteristic
99	C₄	#4090	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#65269	1	1	yes	characteristic
102	C₂	#65466	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅×#106

C₂

yes

maximal, characteristic

#65391

C₂

yes

maximal, characteristic

#65253

C₂

yes

maximal

#64772

C₂

yes

maximal, characteristic

#64750

C₂

yes

maximal

C₂²×C₅×C₈

C₂²

yes

characteristic

C₂×C₅×C₄⋊C₄

C₂²

yes

—

#4072

C₂²

yes

characteristic

C₂×#681

C₂²

yes

characteristic

C₂×#681

C₂²

yes

—

—

—

—

—

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂³

yes

characteristic

C₂×C₅×C₈

—

C₂×C₅×C₈

D₄

yes

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₂²⋊Dic₅

—

C₄⋊Dic₅

D₄

yes

—

C₄×Dic₅

—

#681

—

#681

—

#681

Q₈

yes

—

—

—

—

D₅

yes

—

—

—

C₂³×C₅

#41

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×D₄

yes

characteristic

C₂×C₄×C₅

C₂×Q₈

yes

—

C₂×C₄×C₅

#41

yes

—

C₂×Dic₅

—

C₅×C₈

—

#168

—

#168

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₅

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₅

yes

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂²×C₅

—

C₂²×C₅

#121

yes

characteristic

C₂²×C₅

#125

yes

—

C₄×C₅

—

C₄×C₅

C₂²⋊Q₈

yes

—

C₄×C₅

#123

yes

characteristic

C₄×C₅

C₄⋊Q₈

yes

characteristic

Dic₅

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₅

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂×C₅

—

C₂×C₅

#398

yes

characteristic

C₂×C₅

#471

yes

—

C₂³

#711

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#709

yes

—

C₂×C₄

D₄×D₅

yes

characteristic

C₂×C₄

Q₈×D₅

yes

—

C₂×C₄

C₂×C₂²⋊D₅

yes

characteristic

C₈

—

C₈

—

C₈

—

C₅

#2090

yes

characteristic, complemented, Sylow

C₂²

—

C₂²

#4035

yes

—

C₂²

#4084

yes

characteristic

C₄

—

C₄

—

C₄

#4042

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#65269

yes

characteristic

102

C₂

#65466

yes

—

	subgroup	G/H
center	C₂²	#4084
commutator, cosocle	C₂²×C₄×C₅	C₂³
Frattini	C₂²×C₄	C₂²×D₅
Fitting	C₂×C₅×#106	C₂
socle	C₂²×C₅	#121
prime radical	C₂³×C₅	#41
hypercenter	C₂×#106	D₅
hypocenter	C₅	#2090
LCS	G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#4084

commutator, cosocle

C₂²×C₄×C₅

C₂³

Frattini

C₂²×C₄

C₂²×D₅

Fitting

C₂×C₅×#106

C₂

C₂²×C₅

C₂³×C₅

D₅

C₅

G ▷ C₂²×C₄×C₅ ▷ C₂×C₅ ▷ C₅

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
91. C₅	#2090	semidirect

H ≅ G/N

type

91. C₅

#2090

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#106	2
#65391	2
#65253	2
#64772	2
#64750	2
#2090	5

subgroup

index

C₂×C₅×#106

Subgroups — #10601366

Conjugacy classes of subgroups

Complements

Maximal subgroups