Subgroups — #13013299

Order 1280.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10598513	C₂	1	1	yes	maximal, characteristic
2	#10598513	C₂	1	1	yes	maximal, characteristic
3	#10597989	C₂	1	1	yes	maximal, characteristic
4	C₄×#674	C₄	1	2	yes	—
5	C₄×#674	C₄	1	2	yes	—
6	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
7	C₄×#672	C₄	1	1	yes	characteristic
8	C₈×#168	C₂²	1	1	yes	characteristic
9	#7053	—	5	1	no	maximal, Sylow
10	C₂×#674	—	2	2	no	—
11	C₂×#674	—	2	2	no	—
12	C₂×C₅×C₁₆	C₈	1	2	yes	—
13	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×#672	C₈	1	2	yes	—
15	C₄×#168	C₂×C₄	1	1	yes	characteristic
16	C₄×#168	C₂×C₄	1	1	yes	characteristic
17	#1312	—	5	1	no	—
18	#1312	—	5	1	no	—
19	#1311	—	5	1	no	—
20	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
21	C₂×C₅×C₈	C₂×C₈	1	1	yes	characteristic
22	C₄²×C₅	C₄²	1	1	yes	characteristic
23	C₂×#168	M₁₆	1	2	yes	—
24	C₂×#168	M₁₆	1	2	yes	—
25	#674	—	4	2	no	—
26	#674	—	4	2	no	—
27	C₅×C₁₆	—	2	2	no	—
28	#672	C₁₆	1	4	yes	complemented
29	C₄×C₁₆	—	5	2	no	—
30	C₄×C₁₆	—	5	2	no	—
31	C₄×C₁₆	—	5	1	no	—
32	C₄×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
33	C₈²	—	5	1	no	—
34	C₂×C₄×C₅	C₄×C₈	1	1	yes	characteristic
35	C₂×C₄×C₅	#97	1	1	yes	characteristic
36	C₂×C₄×C₅	#97	1	1	yes	characteristic
37	C₅×C₈	C₂×C₁₆	1	2	yes	—
38	C₅×C₈	M₃₂	1	2	yes	—
39	#168	—	2	2	no	—
40	#168	—	2	2	no	—
41	C₂×C₁₆	—	5	2	no	—
42	C₂×C₁₆	—	10	2	no	—
43	C₂×C₁₆	—	10	2	no	—
44	C₂×C₁₆	#170	1	2	yes	—
45	C₄×C₈	—	5	1	no	—
46	C₄×C₈	—	5	1	no	—
47	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
48	C₂²×C₅	#322	1	1	yes	characteristic
49	C₄×C₅	C₄×C₁₆	1	1	yes	characteristic
50	C₄×C₅	#346	1	1	yes	characteristic
51	C₄×C₅	#347	1	2	yes	—
52	C₄×C₅	#347	1	2	yes	—
53	C₂×C₈	—	5	2	no	—
54	C₂×C₈	—	5	2	no	—
55	C₂×C₈	D₅⋊C₈	1	1	yes	characteristic
56	C₂×C₈	C₂×#170	1	1	yes	characteristic
57	C₄²	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₁₆	—	2	2	no	—
59	C₁₆	—	5	4	no	—
60	C₁₆	—	20	2	no	—
61	C₁₆	—	20	2	no	—
62	C₂×C₅	#1312	1	1	yes	characteristic
63	C₂×C₅	#1313	1	1	yes	characteristic
64	C₂×C₅	#1313	1	1	yes	characteristic
65	C₂×C₄	#3992	1	1	yes	characteristic
66	C₂×C₄	#3992	1	1	yes	characteristic
67	C₂×C₄	C₄×#170	1	1	yes	characteristic
68	C₈	—	10	2	no	—
69	C₈	—	10	2	no	—
70	C₈	D₅⋊C₁₆	1	2	yes	—
71	C₈	#3998	1	2	yes	—
72	C₅	#7053	1	1	yes	characteristic, complemented, Sylow
73	C₂²	#64879	1	1	yes	characteristic
74	C₄	#64845	1	2	yes	—
75	C₄	#64846	1	2	yes	—
76	C₄	#64859	1	1	yes	characteristic
77	C₄	#64885	1	1	yes	characteristic
78	C₂	#10598514	1	1	yes	characteristic
79	C₂	#10598638	1	1	yes	characteristic
80	C₂	#10598639	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₄×AGL(1,𝔽₅)
commutator	C₄×C₅	C₄×C₁₆
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₁₆	C₄
socle, prime radical	C₂²×C₅	#322
cosocle	C₈×#168	C₂²
hypercenter	C₄×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#7053
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
28. #672	C₁₆	semidirect
72. C₅	#7053	semidirect

Maximal subgroups

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

subgroup	index
#10598513	2
#10597989	2
#7053	5