Subgroups — #13014570

Order 1280.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10602090	C₂	1	1	yes	maximal, characteristic
2	#10598516	C₂	1	2	yes	maximal
3	C₅×#494	C₄	1	1	yes	characteristic, complemented
4	#65377	—	2	2	no	—
5	#64984	C₂²	1	1	yes	characteristic
6	#64853	—	2	2	no	—
7	C₈×#168	C₄	1	1	yes	characteristic, complemented
8	#7530	—	5	1	no	maximal, Sylow
9	C₂×C₅×Q₁₆	—	4	1	no	—
10	C₅×C₄⋊Q₈	—	2	1	no	—
11	C₂×D₄₀	—	4	1	no	—
12	#4020	D₄	1	2	yes	—
13	#3999	—	4	2	no	—
14	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
15	#3941	—	4	4	no	—
16	C₄×#168	—	2	1	no	—
17	#1709	—	5	1	no	—
18	#1408	—	5	2	no	—
19	C₂×C₅×Q₈	—	4	1	no	—
20	C₂×D₂₀	—	2	2	no	—
21	C₂×D₂₀	—	4	2	no	—
22	C₂²⋊AGL(1,𝔽₅)	—	8	2	no	—
23	C₅×Q₁₆	—	8	2	no	—
24	C₂×C₅×C₈	—	2	1	no	—
25	C₅×C₄⋊C₄	—	4	2	no	—
26	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×#168	—	2	2	no	—
28	D₄₀	—	8	2	no	—
29	#494	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
30	#493	—	5	1	no	—
31	#486	—	10	2	no	—
32	#353	—	10	2	no	—
33	C₈²	—	5	1	no	—
34	C₂²×D₅	—	4	2	no	—
35	C₂×AGL(1,𝔽₅)	—	16	2	no	—
36	C₅×Q₈	—	8	2	no	—
37	C₂×C₄×C₅	—	2	1	no	—
38	C₂×C₄×C₅	—	8	1	no	—
39	C₂×C₄×C₅	#99	1	1	yes	characteristic
40	D₂₀	—	8	2	no	—
41	D₂₀	—	8	4	no	—
42	C₅×C₈	—	2	2	no	—
43	#168	—	4	2	no	—
44	C₂×Q₁₆	—	4	1	no	—
45	C₂×D₈	—	20	1	no	—
46	C₄⋊Q₈	—	2	1	no	—
47	#127	—	5	2	no	—
48	#102	—	20	4	no	—
49	#99	—	20	2	no	—
50	C₄×C₈	—	10	1	no	—
51	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂²×C₅	#353	1	1	yes	characteristic
53	C₂×D₅	—	8	2	no	—
54	C₂×D₅	—	16	2	no	—
55	AGL(1,𝔽₅)	—	32	2	no	—
56	C₄×C₅	—	2	2	no	—
57	C₄×C₅	—	2	1	no	—
58	C₄×C₅	—	16	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×D₄	—	10	2	no	—
61	C₂×D₄	—	20	2	no	—
62	Q₁₆	—	8	2	no	—
63	D₈	—	40	2	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	10	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₂²⋊C₄	—	40	2	no	—
68	C₄²	C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
69	C₂×C₅	—	2	1	no	—
70	C₂×C₅	#1408	1	1	yes	characteristic
71	D₅	—	16	2	no	—
72	C₂³	—	20	2	no	—
73	Q₈	—	8	2	no	—
74	D₄	—	40	2	no	—
75	D₄	—	40	4	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	8	1	no	—
78	C₂×C₄	—	80	2	no	—
79	C₂×C₄	#4011	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₈	—	20	2	no	—
82	C₅	#7530	1	1	yes	characteristic, complemented, Sylow
83	C₂²	—	40	2	no	—
84	C₂²	—	80	2	no	—
85	C₂²	#64922	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	—	16	1	no	—
89	C₄	—	160	2	no	—
90	C₂	—	2	1	no	—
91	C₂	—	80	2	no	—
92	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, cosocle	#64984	C₂²
hypercenter	#494	AGL(1,𝔽₅)
hypocenter	C₅	#7530
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
3. C₅×#494	C₄	semidirect
7. C₈×#168	C₄	semidirect
29. #494	AGL(1,𝔽₅)	semidirect
82. C₅	#7530	semidirect

Maximal subgroups

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

subgroup	index
#10602090	2
#10598516	2
#7530	5