Group #10592763 — subgroups

Subgroups — #10592763

Order 576.

270 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₃²×#125	C₂	1	1	yes	maximal, characteristic
2	#64078	C₂	1	1	yes	maximal, characteristic
3	#63635	C₂	1	2	yes	maximal
4	#63634	C₂	1	2	yes	maximal
5	#63630	C₂	1	1	yes	maximal, characteristic
6	#5064	—	3	4	no	maximal
7	C₂×#634	—	2	1	no	—
8	C₃²×C₄⋊C₄	—	2	1	no	—
9	C₃²×C₄⋊C₄	—	2	1	no	—
10	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₃²×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₃²×C₄²	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
14	C₄⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
15	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	2	yes	—
16	C₃×#125	D₃	1	4	yes	—
17	#869	—	3	4	no	—
18	#808	—	3	8	no	—
19	#807	—	3	8	no	—
20	#803	—	3	4	no	—
21	C₂×C₃²×C₄	—	2	1	no	—
22	C₂×C₃²×C₄	—	2	1	no	—
23	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
24	C₂×C₃²×C₄	D₄	1	1	yes	characteristic
25	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
26	C₂×C₃⋊Dic₃	—	2	2	no	—
27	#634	—	4	2	no	—
28	C₃⋊(C₃⋊C₈)	Q₈	1	4	yes	complemented
29	#501	—	9	1	no	Sylow
30	C₂×Dic₆	—	6	4	no	—
31	C₃×C₄⋊C₄	—	2	4	no	—
32	C₃×C₄⋊C₄	—	2	4	no	—
33	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
34	C₃×C₄⋊C₄	C₂×D₃	1	4	yes	—
35	C₃×C₄²	C₂×D₃	1	4	yes	—
36	C₄⋊Dic₃	—	3	8	no	—
37	C₄⋊Dic₃	—	6	4	no	—
38	C₂×C₃⋊C₈	—	3	8	no	—
39	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
40	C₃²×C₄	—	2	1	no	—
41	C₃²×C₄	—	2	1	no	—
42	C₃²×C₄	—	4	1	no	—
43	C₃²×C₄	—	4	1	no	—
44	C₃²×C₄	C₂×Q₈	1	2	yes	—
45	C₃⋊Dic₃	—	4	2	no	—
46	C₄⋊Q₈	—	9	1	no	—
47	#125	C₃⋊D₃	1	1	yes	characteristic
48	#107	—	9	2	no	—
49	#106	—	9	2	no	—
50	#97	—	9	1	no	—
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	#67	1	4	yes	—
54	C₂×C₃×C₄	#67	1	4	yes	—
55	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
56	C₂×Dic₃	—	6	8	no	—
57	Dic₆	—	12	8	no	—
58	C₃⋊C₈	—	3	16	no	—
59	C₂×C₃²	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic
61	C₂×C₃²	#137	1	1	yes	characteristic
62	C₂×Q₈	—	18	1	no	—
63	C₂×C₈	—	9	2	no	—
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	—	2	1	no	—
66	C₄⋊C₄	—	9	2	no	—
67	C₄⋊C₄	—	18	1	no	—
68	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
69	C₄⋊C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₂²×C₃	C₂×#67	1	4	yes	—
72	C₃×C₄	—	2	4	no	—
73	C₃×C₄	—	2	4	no	—
74	C₃×C₄	—	4	4	no	—
75	C₃×C₄	—	4	4	no	—
76	C₃×C₄	D₃×Q₈	1	8	yes	—
77	Dic₃	—	12	8	no	—
78	C₃²	#501	1	1	yes	characteristic, complemented, Sylow
79	Q₈	—	36	2	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	18	2	no	—
83	C₂×C₄	#638	1	1	yes	characteristic
84	C₂×C₄	#638	1	1	yes	characteristic
85	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
86	C₈	—	9	4	no	—
87	C₂×C₃	#944	1	4	yes	—
88	C₂×C₃	#949	1	4	yes	—
89	C₂×C₃	#951	1	4	yes	—
90	C₂²	C₂×#638	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	2	1	no	—
93	C₄	—	4	1	no	—
94	C₄	—	4	1	no	—
95	C₄	—	36	2	no	—
96	C₄	Q₈×C₃⋊D₃	1	2	yes	—
97	C₃	#5064	1	4	yes	complemented
98	C₂	#64153	1	1	yes	characteristic
99	C₂	#64158	1	1	yes	characteristic
100	C₂	#64160	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#125

C₂

yes

maximal, characteristic

#64078

C₂

yes

maximal, characteristic

#63635

C₂

yes

maximal

#63634

C₂

yes

maximal

#63630

C₂

yes

maximal, characteristic

#5064

—

maximal

C₂×#634

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂²

yes

characteristic

C₃²×C₄⋊C₄

C₂²

yes

characteristic

C₃²×C₄²

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

C₄⋊(C₃⋊Dic₃)

C₂²

yes

—

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

—

C₃×#125

D₃

yes

—

—

—

—

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

D₄

yes

characteristic

C₂×C₃²×C₄

D₄

yes

characteristic

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

—

#634

—

C₃⋊(C₃⋊C₈)

Q₈

yes

complemented

#501

—

Sylow

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×D₃

yes

—

C₃×C₄⋊C₄

C₂×D₃

yes

—

C₃×C₄²

C₂×D₃

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×Q₈

yes

—

C₃⋊Dic₃

—

C₄⋊Q₈

—

#125

C₃⋊D₃

yes

—

—

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

#67

yes

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₂×C₃²

C₄⋊Q₈

yes

C₂×C₃²

yes

C₂×C₃²

yes

C₂×Q₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₄

C₂×C₃⋊D₃

yes

characteristic

C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂²×C₃

C₂×#67

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

D₃×Q₈

yes

—

Dic₃

—

C₃²

#501

yes

characteristic, complemented, Sylow

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₈

—

C₂×C₃

#944

yes

—

C₂×C₃

#949

yes

—

C₂×C₃

#951

yes

—

C₂²

C₂×#638

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

Q₈×C₃⋊D₃

yes

—

C₃

yes

C₂

yes

C₂

yes

100

C₂

#64160

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#638
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₃²×#125	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	#125	C₃⋊D₃
hypocenter	C₃²	#501
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

subgroup

G/H

center

C₂²

C₂×#638

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₃²×#125

C₂

socle, prime radical

C₂²×C₃²

C₂×D₄

C₃⋊D₃

C₃²

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #125

N	H ≅ G/N	type
28. C₃⋊(C₃⋊C₈)	Q₈	semidirect
78. C₃²	#501	semidirect
97. C₃	#5064	semidirect

H ≅ G/N

type

28. C₃⋊(C₃⋊C₈)

Q₈

semidirect

78. C₃²

#501

semidirect

97. C₃

#5064

semidirect

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

subgroup	index
C₃²×#125	2
#64078	2
#63635	2
#63634	2
#63630	2
#5064	3

subgroup

index

C₃²×#125

Subgroups — #10592763

Conjugacy classes of subgroups

Complements

Maximal subgroups