Group #12003809 — subgroups

Subgroups — #12003809

Order 1280.

350 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₅×#2845	C₂	1	1	yes	maximal, characteristic
2	#10600341	C₂	1	3	yes	maximal
3	#10598121	C₂	1	3	yes	maximal
4	C₅×#398	—	2	6	no	—
5	C₅×#382	C₂²	1	3	yes	—
6	#65224	C₂²	1	1	yes	characteristic
7	C₂×#3935	C₂²	1	3	yes	—
8	#64749	—	2	6	no	—
9	#10586	—	5	1	no	maximal, Sylow
10	C₂×C₅×C₄⋊C₄	—	2	6	no	—
11	C₂×C₅×C₄⋊C₄	D₄	1	6	yes	—
12	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
13	C₂×C₄⋊Dic₅	—	2	3	no	—
14	C₂²×#168	—	2	3	no	—
15	C₅×#95	—	2	2	no	—
16	C₅×#95	—	2	6	no	—
17	#3935	—	2	3	no	—
18	#3935	Q₈	1	6	yes	—
19	#2845	D₅	1	1	yes	characteristic
20	#1836	—	5	3	no	—
21	#1295	—	5	3	no	—
22	C₂²×C₄×C₅	—	2	6	no	—
23	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
24	C₂²×Dic₅	—	4	1	no	—
25	C₅×C₄⋊C₄	—	4	6	no	—
26	C₅×C₄⋊C₄	—	8	6	no	—
27	C₄²×C₅	C₂×Q₈	1	3	yes	—
28	C₄²×C₅	#41	1	1	yes	characteristic
29	C₄⋊Dic₅	—	4	6	no	—
30	C₂×#168	—	2	6	no	—
31	C₂×#168	—	4	3	no	—
32	C₂×#168	—	4	3	no	—
33	C₂×#97	—	5	3	no	—
34	#398	—	2	6	no	—
35	#384	—	5	1	no	—
36	#382	C₂×D₅	1	3	yes	—
37	#340	—	10	6	no	—
38	C₂³×C₅	#120	1	1	yes	characteristic
39	C₂×C₄×C₅	—	2	3	no	—
40	C₂×C₄×C₅	—	2	3	no	—
41	C₂×C₄×C₅	—	4	6	no	—
42	C₂×C₄×C₅	—	4	6	no	—
43	C₂×C₄×C₅	—	4	6	no	—
44	C₂×C₄×C₅	C₂²⋊Q₈	1	3	yes	—
45	C₂×C₄×C₅	C₄⋊Q₈	1	3	yes	—
46	C₂×Dic₅	—	8	3	no	—
47	#168	—	4	6	no	—
48	C₂²×C₈	—	10	3	no	—
49	C₂×C₄⋊C₄	—	2	6	no	—
50	C₂×C₄⋊C₄	—	10	3	no	—
51	C₂×C₄⋊C₄	C₂²⋊D₅	1	6	yes	—
52	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
53	#97	—	5	6	no	—
54	#97	—	10	3	no	—
55	#95	—	2	2	no	—
56	#95	—	2	6	no	—
57	C₂²×C₅	#395	1	1	yes	characteristic
58	C₂²×C₅	#454	1	3	yes	—
59	C₂²×C₅	#455	1	3	yes	—
60	C₄×C₅	—	2	3	no	—
61	C₄×C₅	—	2	3	no	—
62	C₄×C₅	—	8	6	no	—
63	Dic₅	—	16	1	no	—
64	C₂²×C₄	—	2	6	no	—
65	C₂²×C₄	—	20	1	no	—
66	C₂²×C₄	C₂×C₂²⋊D₅	1	3	yes	—
67	C₂×C₈	—	10	6	no	—
68	C₂×C₈	—	20	3	no	—
69	C₂×C₈	—	20	3	no	—
70	C₄⋊C₄	—	4	6	no	—
71	C₄⋊C₄	—	8	6	no	—
72	C₄⋊C₄	—	20	6	no	—
73	C₄²	Q₈×D₅	1	3	yes	—
74	C₄²	#713	1	1	yes	characteristic
75	C₂×C₅	#1658	1	1	yes	characteristic
76	C₂×C₅	#1662	1	3	yes	—
77	C₂×C₅	#2061	1	3	yes	—
78	C₂³	#4099	1	1	yes	characteristic
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	4	6	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	40	3	no	—
85	C₂×C₄	#4090	1	3	yes	—
86	C₂×C₄	#4092	1	3	yes	—
87	C₈	—	20	6	no	—
88	C₅	#10586	1	1	yes	characteristic, complemented, Sylow
89	C₂²	#65518	1	1	yes	characteristic
90	C₂²	#65531	1	3	yes	—
91	C₂²	#65534	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	80	1	no	—
96	C₂	#10602694	1	3	yes	—
97	C₂	#10602747	1	3	yes	—
98	C₂	#10602753	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅×#2845

C₂

yes

maximal, characteristic

#10600341

C₂

yes

maximal

#10598121

C₂

yes

maximal

C₅×#398

—

C₅×#382

C₂²

yes

—

#65224

C₂²

yes

characteristic

C₂×#3935

C₂²

yes

—

#64749

—

#10586

—

maximal, Sylow

C₂×C₅×C₄⋊C₄

—

C₂×C₅×C₄⋊C₄

D₄

yes

—

C₂×C₄²×C₅

C₂³

yes

characteristic

C₂×C₄⋊Dic₅

—

C₂²×#168

—

C₅×#95

—

C₅×#95

—

#3935

—

#3935

Q₈

yes

—

#2845

D₅

yes

characteristic

#1836

—

#1295

—

C₂²×C₄×C₅

—

C₂²×C₄×C₅

C₂×D₄

yes

—

C₂²×Dic₅

—

C₅×C₄⋊C₄

—

C₅×C₄⋊C₄

—

C₄²×C₅

C₂×Q₈

yes

—

C₄²×C₅

#41

yes

characteristic

C₄⋊Dic₅

—

C₂×#168

—

C₂×#168

—

C₂×#168

—

C₂×#97

—

#398

—

#384

—

#382

C₂×D₅

yes

—

#340

—

C₂³×C₅

#120

yes

characteristic

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

—

C₂×C₄×C₅

C₂²⋊Q₈

yes

—

C₂×C₄×C₅

C₄⋊Q₈

yes

—

C₂×Dic₅

—

#168

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₅

yes

—

C₂×C₄²

C₂²×D₅

yes

—

—

—

—

C₂²×C₅

#395

yes

characteristic

C₂²×C₅

#454

yes

—

C₂²×C₅

#455

yes

—

C₄×C₅

—

C₄×C₅

—

C₄×C₅

—

Dic₅

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊D₅

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Q₈×D₅

yes

—

C₄²

yes

C₂×C₅

yes

C₂×C₅

yes

—

C₂×C₅

#2061

yes

—

C₂³

#4099

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#4090

yes

—

C₂×C₄

#4092

yes

—

C₈

—

C₅

#10586

yes

characteristic, complemented, Sylow

C₂²

#65518

yes

characteristic

C₂²

#65531

yes

—

C₂²

#65534

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

#10602694

yes

—

C₂

#10602747

yes

—

C₂

#10602753

yes

characteristic

	subgroup	G/H
center	C₂³	#4099
commutator, cosocle	C₂×C₄²×C₅	C₂³
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#2845	C₂
socle, prime radical	C₂³×C₅	#120
hypercenter	#2845	D₅
hypocenter	C₅	#10586
LCS	G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

subgroup

G/H

center

C₂³

#4099

commutator, cosocle

C₂×C₄²×C₅

C₂³

Frattini

C₂×C₄²

C₂²×D₅

C₂

C₂³×C₅

D₅

C₅

G ▷ C₂×C₄²×C₅ ▷ C₂²×C₅ ▷ C₅

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2845

N	H ≅ G/N	type
88. C₅	#10586	semidirect

H ≅ G/N

type

88. C₅

#10586

semidirect

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

subgroup	index
C₅×#2845	2
#10600341	2
#10598121	2
#10586	5

subgroup

index

Subgroups — #12003809

Conjugacy classes of subgroups

Complements

Maximal subgroups