Subgroups — #11987106

Order 1280.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×M₁₂₈	C₂	1	1	yes	maximal, characteristic
2	#10598080	C₂	1	1	yes	maximal, characteristic
3	#10598078	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×C₃₂	C₂²	1	1	yes	characteristic
5	C₁₆×Dic₅	C₄	1	1	yes	characteristic
6	C₂×#3926	C₄	1	1	yes	characteristic
7	C₅×C₆₄	C₄	1	2	yes	complemented
8	#64663	C₄	1	2	yes	complemented
9	#7511	—	5	1	no	maximal, Sylow
10	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
11	C₈×Dic₅	C₈	1	1	yes	characteristic
12	C₂×#672	C₈	1	1	yes	characteristic
13	C₅×C₃₂	C₂×C₄	1	1	yes	characteristic
14	C₅×C₃₂	C₂×C₄	1	1	yes	characteristic
15	#3926	C₈	1	2	yes	—
16	M₁₂₈	—	5	1	no	—
17	M₁₂₈	D₅	1	1	yes	characteristic
18	#1397	—	5	1	no	—
19	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
20	C₄×Dic₅	C₁₆	1	1	yes	characteristic
21	C₂×#168	C₁₆	1	1	yes	characteristic
22	C₅×C₁₆	C₄²	1	1	yes	characteristic
23	C₅×C₁₆	C₂×C₈	1	1	yes	characteristic
24	#672	C₁₆	1	2	yes	—
25	C₂×C₃₂	—	5	1	no	—
26	C₂×C₃₂	C₂×D₅	1	1	yes	characteristic
27	C₄×C₁₆	—	5	1	no	—
28	C₆₄	—	5	2	no	—
29	C₆₄	Dic₅	1	2	yes	complemented
30	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
31	C₂×Dic₅	—	2	1	no	—
32	C₅×C₈	C₄×C₈	1	1	yes	characteristic
33	C₅×C₈	C₂×C₁₆	1	1	yes	characteristic
34	#168	—	2	1	no	—
35	C₂×C₁₆	—	5	1	no	—
36	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
37	C₄×C₈	—	5	1	no	—
38	C₃₂	—	5	2	no	—
39	C₃₂	C₄×D₅	1	1	yes	characteristic
40	C₃₂	C₂×Dic₅	1	1	yes	characteristic
41	C₂²×C₅	M₆₄	1	1	yes	characteristic
42	C₄×C₅	C₄×C₁₆	1	1	yes	characteristic
43	C₄×C₅	M₆₄	1	1	yes	characteristic
44	Dic₅	—	4	1	no	—
45	C₂×C₈	—	5	1	no	—
46	C₂×C₈	C₈×D₅	1	1	yes	characteristic
47	C₄²	—	5	1	no	—
48	C₁₆	—	5	2	no	—
49	C₁₆	C₈×D₅	1	1	yes	characteristic
50	C₁₆	C₄×Dic₅	1	1	yes	characteristic
51	C₂×C₅	—	2	1	no	—
52	C₂×C₅	#1397	1	1	yes	characteristic
53	C₂×C₄	—	10	1	no	—
54	C₂×C₄	D₅×C₁₆	1	1	yes	characteristic
55	C₈	—	10	1	no	—
56	C₈	D₅×C₁₆	1	1	yes	characteristic
57	C₈	C₈×Dic₅	1	1	yes	characteristic
58	C₅	#7511	1	1	yes	characteristic, complemented, Sylow
59	C₂²	#64667	1	1	yes	characteristic
60	C₄	—	20	1	no	—
61	C₄	#64667	1	1	yes	characteristic
62	C₄	C₁₆×Dic₅	1	1	yes	characteristic
63	C₂	—	2	1	no	—
64	C₂	#10598080	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄×Dic₅
commutator	C₄×C₅	C₄×C₁₆
Frattini	C₂×C₃₂	C₂×D₅
Fitting	C₅×M₁₂₈	C₂
socle	C₂×C₅	#1397
prime radical	C₂²×C₅	M₆₄
cosocle	C₂×C₅×C₃₂	C₂²
hypercenter	M₁₂₈	D₅
hypocenter	C₅	#7511
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₁₆ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
7. C₅×C₆₄	C₄	semidirect
8. #64663	C₄	semidirect
29. C₆₄	Dic₅	semidirect
58. C₅	#7511	semidirect

Maximal subgroups

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

subgroup	index
C₅×M₁₂₈	2
#10598080	2
#10598078	2
#7511	5