Group #10592882 — subgroups

Subgroups — #10592882

Order 576.

366 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	#64153	C₂	1	2	yes	maximal
3	C₂×#3756	C₂	1	2	yes	maximal
4	C₂×#3746	C₂	1	1	yes	maximal, characteristic
5	C₈×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
6	#5183	—	3	4	no	maximal
7	C₂×C₃²×Q₈	C₂²	1	2	yes	—
8	C₂×#634	C₂²	1	2	yes	—
9	C₃²×Q₁₆	—	2	2	no	—
10	C₂×C₃²×C₈	C₂²	1	1	yes	characteristic
11	#3756	—	2	4	no	—
12	#3750	—	2	4	no	—
13	C₄×C₃⋊Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
15	#3746	—	2	2	no	—
16	C₂×C₃×Q₁₆	D₃	1	4	yes	—
17	#944	—	3	8	no	—
18	C₂×#216	—	3	8	no	—
19	C₂×Dic₁₂	—	3	4	no	—
20	C₈×Dic₃	—	3	4	no	—
21	C₃²×Q₈	—	2	2	no	—
22	C₃²×Q₈	—	2	2	no	—
23	C₂×C₃²×C₄	—	2	2	no	—
24	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃⋊Dic₃	—	2	2	no	—
26	C₂×C₃⋊Dic₃	D₄	1	2	yes	—
27	#634	—	2	2	no	—
28	#634	—	2	2	no	—
29	C₃²×C₈	D₄	1	2	yes	—
30	C₃⋊(C₃⋊C₈)	D₄	1	2	yes	—
31	#494	—	9	1	no	Sylow
32	C₂×C₃×Q₈	C₂×D₃	1	8	yes	—
33	C₂×Dic₆	—	3	8	no	—
34	C₃×Q₁₆	—	2	8	no	—
35	C₂×C₃×C₈	C₂×D₃	1	4	yes	—
36	#216	—	6	16	no	—
37	Dic₃⋊C₄	—	6	16	no	—
38	C₄×Dic₃	—	3	4	no	—
39	C₂×C₃⋊C₈	—	3	4	no	—
40	Dic₁₂	—	6	8	no	—
41	C₂²×C₃²	C₂×D₄	1	1	yes	characteristic
42	C₃²×C₄	—	4	2	no	—
43	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
44	C₃²×C₄	C₂×D₄	1	1	yes	characteristic
45	C₃⋊Dic₃	—	4	2	no	—
46	C₃⋊Dic₃	Q₁₆	1	4	yes	complemented
47	C₂×Q₁₆	—	9	2	no	—
48	C₂×Q₁₆	—	9	1	no	—
49	C₂×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
50	C₄⋊Q₈	—	9	2	no	—
51	C₄×C₈	—	9	1	no	—
52	C₃×Q₈	—	2	8	no	—
53	C₃×Q₈	—	2	8	no	—
54	C₂×C₃×C₄	—	2	8	no	—
55	C₂×C₃×C₄	C₂²×D₃	1	4	yes	—
56	C₂×Dic₃	—	3	8	no	—
57	C₂×Dic₃	—	6	8	no	—
58	Dic₆	—	6	8	no	—
59	Dic₆	—	6	8	no	—
60	C₃×C₈	#67	1	8	yes	—
61	C₃⋊C₈	—	3	8	no	—
62	C₂×C₃²	#127	1	1	yes	characteristic
63	C₂×C₃²	C₂×Q₁₆	1	2	yes	—
64	C₂×Q₈	—	9	2	no	—
65	C₂×Q₈	C₂×C₃⋊D₃	1	2	yes	—
66	Q₁₆	—	2	2	no	—
67	Q₁₆	—	18	4	no	—
68	Q₁₆	—	18	2	no	—
69	C₂×C₈	—	9	1	no	—
70	C₂×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
71	C₄⋊C₄	—	18	4	no	—
72	C₄²	—	9	1	no	—
73	C₂²×C₃	D₃×D₄	1	4	yes	—
74	C₃×C₄	—	4	8	no	—
75	C₃×C₄	D₃×D₄	1	4	yes	—
76	C₃×C₄	C₂×#67	1	4	yes	—
77	Dic₃	—	3	16	no	—
78	Dic₃	—	12	8	no	—
79	C₃²	#494	1	1	yes	characteristic, complemented, Sylow
80	Q₈	—	2	2	no	—
81	Q₈	—	2	2	no	—
82	Q₈	—	18	2	no	—
83	Q₈	—	18	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	9	2	no	—
86	C₂×C₄	—	18	2	no	—
87	C₂×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
88	C₈	—	9	2	no	—
89	C₈	#638	1	2	yes	—
90	C₂×C₃	D₃×Q₁₆	1	8	yes	—
91	C₂×C₃	#940	1	4	yes	—
92	C₂²	D₄×C₃⋊D₃	1	1	yes	characteristic
93	C₄	—	4	2	no	—
94	C₄	—	9	4	no	—
95	C₄	—	36	2	no	—
96	C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
97	C₄	C₂×#638	1	1	yes	characteristic
98	C₃	#5183	1	4	yes	complemented
99	C₂	Q₁₆×C₃⋊D₃	1	2	yes	—
100	C₂	#64149	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃²×Q₁₆

C₂

yes

maximal, characteristic

#64153

C₂

yes

maximal

C₂×#3756

C₂

yes

maximal

C₂×#3746

C₂

yes

maximal, characteristic

C₈×C₃⋊Dic₃

C₂

yes

maximal, characteristic

#5183

—

maximal

C₂×C₃²×Q₈

C₂²

yes

—

C₂×#634

C₂²

yes

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

C₂²

yes

characteristic

#3756

—

#3750

—

C₄×C₃⋊Dic₃

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₈)

C₂²

yes

characteristic

#3746

—

C₂×C₃×Q₁₆

D₃

yes

—

#944

—

C₂×#216

—

C₂×Dic₁₂

—

C₈×Dic₃

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂³

yes

characteristic

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

D₄

yes

—

#634

—

#634

—

C₃²×C₈

D₄

yes

—

C₃⋊(C₃⋊C₈)

D₄

yes

—

#494

—

Sylow

C₂×C₃×Q₈

C₂×D₃

yes

—

C₂×Dic₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₂×D₃

yes

—

#216

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂²×C₃²

C₂×D₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₂×D₄

yes

characteristic

C₃²×C₄

C₂×D₄

yes

characteristic

C₃⋊Dic₃

—

C₃⋊Dic₃

Q₁₆

yes

complemented

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₃⋊D₃

yes

characteristic

C₄⋊Q₈

—

C₄×C₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×D₃

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

#67

yes

—

C₃⋊C₈

—

C₂×C₃²

#127

yes

characteristic

C₂×C₃²

C₂×Q₁₆

yes

—

C₂×Q₈

—

C₂×Q₈

C₂×C₃⋊D₃

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

C₂×C₃⋊D₃

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂²×C₃

D₃×D₄

yes

—

C₃×C₄

—

C₃×C₄

D₃×D₄

yes

—

C₃×C₄

C₂×#67

yes

—

Dic₃

—

Dic₃

—

C₃²

#494

yes

characteristic, complemented, Sylow

Q₈

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₃⋊D₃

yes

characteristic

C₈

—

C₈

#638

yes

—

C₂×C₃

D₃×Q₁₆

yes

—

C₂×C₃

#940

yes

—

C₂²

D₄×C₃⋊D₃

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

D₄×C₃⋊D₃

yes

C₄

yes

C₃

yes

C₂

Q₁₆×C₃⋊D₃

yes

—

100

C₂

#64149

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×C₃⋊D₃
commutator, cosocle	C₂×C₃²×C₄	C₂³
Frattini	C₂×C₄	C₂²×C₃⋊D₃
Fitting	C₂×C₃²×Q₁₆	C₂
socle, prime radical	C₂²×C₃²	C₂×D₄
hypercenter	C₂×Q₁₆	C₃⋊D₃
hypocenter	C₃²	#494
LCS	G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

subgroup

G/H

center

C₂²

D₄×C₃⋊D₃

commutator, cosocle

C₂×C₃²×C₄

C₂³

Frattini

C₂×C₄

C₂²×C₃⋊D₃

Fitting

C₂×C₃²×Q₁₆

C₂

socle, prime radical

C₂²×C₃²

C₂×D₄

hypercenter

C₂×Q₁₆

C₃⋊D₃

hypocenter

C₃²

#494

LCS

G ▷ C₂×C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×Q₁₆

N	H ≅ G/N	type
46. C₃⋊Dic₃	Q₁₆	semidirect
79. C₃²	#494	semidirect
98. C₃	#5183	semidirect

H ≅ G/N

type

46. C₃⋊Dic₃

Q₁₆

semidirect

79. C₃²

#494

semidirect

98. C₃

#5183

semidirect

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

subgroup	index
C₂×C₃²×Q₁₆	2
#64153	2
C₂×#3756	2
C₂×#3746	2
C₈×C₃⋊Dic₃	2
#5183	3

subgroup

index

C₂×C₃²×Q₁₆

C₈×C₃⋊Dic₃

Subgroups — #10592882

Conjugacy classes of subgroups

Complements

Maximal subgroups