Subgroups — #11993034

Order 1280.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1455	C₂	1	1	yes	maximal, characteristic
2	#10598161	C₂	1	3	yes	maximal
3	#10598157	C₂	1	3	yes	maximal
4	C₄×C₅×M₁₆	C₂²	1	3	yes	—
5	#65226	C₂²	1	1	yes	characteristic
6	#65215	C₂²	1	3	yes	—
7	C₅×#322	C₄	1	3	yes	complemented
8	C₅×C₈²	C₄	1	1	yes	characteristic, complemented
9	#8804	—	5	1	no	maximal, Sylow
10	C₂×C₅×M₁₆	—	2	3	no	—
11	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
12	C₅×#97	—	2	3	no	—
13	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
14	C₄×C₅×C₈	C₂×C₄	1	3	yes	—
15	#3963	—	2	3	no	—
16	C₂²⋊#168	—	2	3	no	—
17	C₄⋊#168	—	2	3	no	—
18	#3935	—	2	3	no	—
19	#1455	D₅	1	1	yes	characteristic
20	#1291	—	5	3	no	—
21	#1288	—	5	3	no	—
22	C₂²×C₄×C₅	#41	1	3	yes	—
23	C₂²×Dic₅	—	4	1	no	—
24	C₅×M₁₆	—	4	6	no	—
25	C₂×C₅×C₈	—	2	3	no	—
26	C₂×C₅×C₈	—	2	3	no	—
27	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₅	#41	1	3	yes	—
29	C₂×#168	—	4	3	no	—
30	#432	—	5	3	no	—
31	C₄×M₁₆	C₂×D₅	1	3	yes	—
32	#383	—	5	1	no	—
33	#322	Dic₅	1	3	yes	complemented
34	C₈²	Dic₅	1	1	yes	characteristic, complemented
35	C₂³×C₅	#126	1	1	yes	characteristic
36	C₂×C₄×C₅	—	4	3	no	—
37	C₂×C₄×C₅	#117	1	3	yes	—
38	C₂×C₄×C₅	#126	1	3	yes	—
39	C₂×Dic₅	—	8	3	no	—
40	C₅×C₈	—	2	6	no	—
41	C₅×C₈	—	4	3	no	—
42	#168	—	8	3	no	—
43	C₂×M₁₆	—	2	3	no	—
44	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
45	C₄⋊C₈	—	10	3	no	—
46	C₂²⋊C₈	—	10	3	no	—
47	#97	—	2	3	no	—
48	#97	—	10	3	no	—
49	C₄×C₈	C₂×Dic₅	1	3	yes	—
50	C₄×C₈	C₂×Dic₅	1	3	yes	—
51	#95	—	10	3	no	—
52	C₂²×C₅	—	2	3	no	—
53	C₂²×C₅	#383	1	1	yes	characteristic
54	C₄×C₅	—	2	3	no	—
55	C₄×C₅	—	2	3	no	—
56	Dic₅	—	16	1	no	—
57	C₂²×C₄	—	20	1	no	—
58	C₂²×C₄	#709	1	3	yes	—
59	M₁₆	—	4	6	no	—
60	C₂×C₈	—	2	3	no	—
61	C₂×C₈	—	2	3	no	—
62	C₂×C₈	—	20	3	no	—
63	C₄²	#709	1	3	yes	—
64	C₄²	C₂²×Dic₅	1	1	yes	characteristic
65	C₂×C₅	—	4	1	no	—
66	C₂×C₅	#1841	1	3	yes	—
67	C₂³	#4022	1	1	yes	characteristic
68	C₂×C₄	—	4	3	no	—
69	C₂×C₄	—	40	3	no	—
70	C₂×C₄	#4022	1	3	yes	—
71	C₂×C₄	#4072	1	3	yes	—
72	C₈	—	2	6	no	—
73	C₈	—	4	3	no	—
74	C₈	—	40	3	no	—
75	C₅	#8804	1	1	yes	characteristic, complemented, Sylow
76	C₂²	—	2	3	no	—
77	C₂²	#65226	1	1	yes	characteristic
78	C₄	—	2	3	no	—
79	C₄	—	2	3	no	—
80	C₄	—	80	1	no	—
81	C₂	—	4	1	no	—
82	C₂	#10600664	1	3	yes	—

	subgroup	G/H
center	C₂²	#65226
commutator	C₄²×C₅	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅
Fitting	C₅×#1455	C₂
socle	C₂²×C₅	#383
prime radical	C₂³×C₅	#126
cosocle	C₂×C₄²×C₅	C₂³
hypercenter	#1455	D₅
hypocenter	C₅	#8804
LCS	G ▷ C₄²×C₅ ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1455

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. C₅×#322	C₄	semidirect
8. C₅×C₈²	C₄	semidirect
33. #322	Dic₅	semidirect
34. C₈²	Dic₅	semidirect
75. C₅	#8804	semidirect

Maximal subgroups

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

subgroup	index
C₅×#1455	2
#10598161	2
#10598157	2
#8804	5