Subgroups — #13014574

Order 1280.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10602091	C₂	1	1	yes	maximal, characteristic
2	#10598521	C₂	1	2	yes	maximal
3	C₅×#494	C₄	1	1	yes	characteristic
4	#65367	—	2	2	no	—
5	#64988	C₂²	1	1	yes	characteristic
6	#64855	—	2	2	no	—
7	C₈×#168	C₄	1	1	yes	characteristic
8	#7532	—	5	1	no	maximal, Sylow
9	C₂×C₅×Q₁₆	—	4	1	no	—
10	C₅×C₄⋊Q₈	—	2	1	no	—
11	C₂×Dic₂₀	—	4	1	no	—
12	#4015	D₄	1	2	yes	—
13	#4005	—	4	2	no	—
14	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	#3939	—	4	4	no	—
16	C₄×#168	—	2	1	no	—
17	#1713	—	5	1	no	—
18	#1411	—	5	2	no	—
19	C₂×C₅×Q₈	—	4	1	no	—
20	C₂×Dic₁₀	—	2	2	no	—
21	#704	—	8	2	no	—
22	C₅×Q₁₆	—	8	2	no	—
23	C₂×C₅×C₈	—	2	1	no	—
24	C₅×C₄⋊C₄	—	4	2	no	—
25	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
26	C₄⋊Dic₅	—	4	2	no	—
27	C₂×#168	—	2	2	no	—
28	Dic₂₀	—	8	2	no	—
29	#500	—	10	2	no	—
30	#494	—	5	1	no	—
31	#494	AGL(1,𝔽₅)	1	1	yes	characteristic
32	#356	—	10	2	no	—
33	C₈²	—	5	1	no	—
34	C₅×Q₈	—	8	2	no	—
35	C₂×C₄×C₅	—	2	1	no	—
36	C₂×C₄×C₅	—	8	1	no	—
37	C₂×C₄×C₅	#99	1	1	yes	characteristic
38	C₂×Dic₅	—	4	2	no	—
39	Dic₁₀	—	8	2	no	—
40	#170	—	16	2	no	—
41	C₅×C₈	—	2	2	no	—
42	#168	—	2	4	no	—
43	C₂×Q₁₆	—	4	1	no	—
44	C₂×Q₁₆	—	20	1	no	—
45	C₄⋊Q₈	—	2	1	no	—
46	C₄⋊Q₈	—	5	2	no	—
47	#107	—	20	4	no	—
48	#101	—	20	2	no	—
49	C₄×C₈	—	10	1	no	—
50	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
51	C₂²×C₅	#353	1	1	yes	characteristic
52	C₄×C₅	—	2	2	no	—
53	C₄×C₅	—	2	1	no	—
54	C₄×C₅	—	16	1	no	—
55	Dic₅	—	8	2	no	—
56	C₂×Q₈	—	4	1	no	—
57	C₂×Q₈	—	10	2	no	—
58	Q₁₆	—	8	2	no	—
59	Q₁₆	—	40	2	no	—
60	M₁₆	—	40	2	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	10	2	no	—
63	C₄⋊C₄	—	4	2	no	—
64	C₄⋊C₄	—	20	2	no	—
65	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂×C₅	—	2	1	no	—
67	C₂×C₅	#1408	1	1	yes	characteristic
68	Q₈	—	8	2	no	—
69	Q₈	—	40	2	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	—	20	2	no	—
73	C₂×C₄	#4011	1	1	yes	characteristic
74	C₈	—	2	2	no	—
75	C₈	—	10	4	no	—
76	C₈	—	80	2	no	—
77	C₅	#7532	1	1	yes	characteristic, complemented, Sylow
78	C₂²	#64922	1	1	yes	characteristic
79	C₄	—	2	2	no	—
80	C₄	—	2	1	no	—
81	C₄	—	16	1	no	—
82	C₄	—	40	2	no	—
83	C₂	—	2	1	no	—
84	C₂	#10598747	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10598747
commutator	C₄×C₅×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₅×#494	C₄
socle	C₂×C₅	#1408
prime radical	C₂²×C₅	#353
cosocle	#64988	C₂²
hypercenter	#494	AGL(1,𝔽₅)
hypocenter	C₅	#7532
LCS	G ▷ C₄×C₅×C₈ ▷ C₄²×C₅ ▷ C₂×C₄×C₅ ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
77. C₅	#7532	semidirect

Maximal subgroups

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

subgroup	index
#10602091	2
#10598521	2
#7532	5