Subgroups — #13014352

Order 1280.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#3928	C₂	1	2	yes	maximal
2	C₁₆×#168	C₂	1	1	yes	maximal, characteristic
3	C₂×#3928	—	2	2	no	—
4	C₄×C₅×C₁₆	C₄	1	1	yes	characteristic
5	C₄×#672	C₂²	1	1	yes	characteristic
6	C₈×#168	C₄	1	1	yes	characteristic
7	#7458	—	5	1	no	maximal, Sylow
8	C₂×C₅×C₁₆	—	2	1	no	—
9	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
10	C₂×#672	D₄	1	1	yes	characteristic
11	C₂×#672	Q₈	1	1	yes	characteristic
12	C₄×#168	C₈	1	1	yes	characteristic
13	C₄×#168	C₈	1	1	yes	characteristic
14	#3928	—	4	2	no	—
15	C₄⋊C₃₂	—	5	2	no	—
16	C₈×C₁₆	—	5	1	no	—
17	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₅×C₈	M₁₆	1	1	yes	characteristic
19	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
20	C₂×#168	—	2	1	no	—
21	C₂×#168	C₁₆	1	1	yes	characteristic
22	C₂×#168	C₁₆	1	1	yes	characteristic
23	C₅×C₁₆	—	2	2	no	—
24	#672	—	2	1	no	—
25	#672	D₈	1	1	yes	characteristic
26	#672	Q₁₆	1	1	yes	characteristic
27	C₂×C₃₂	—	10	2	no	—
28	C₄×C₁₆	—	5	1	no	—
29	C₄×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
30	C₈²	—	5	1	no	—
31	C₂×C₄×C₅	C₄⋊C₈	1	1	yes	characteristic
32	C₂×C₄×C₅	C₂×C₁₆	1	1	yes	characteristic
33	C₂×C₄×C₅	M₃₂	1	1	yes	characteristic
34	C₅×C₈	—	2	1	no	—
35	C₅×C₈	#107	1	1	yes	characteristic
36	C₅×C₈	#108	1	1	yes	characteristic
37	#168	—	2	2	no	—
38	#168	—	2	1	no	—
39	#168	C₃₂	1	2	yes	complemented
40	C₂×C₁₆	—	2	1	no	—
41	C₂×C₁₆	—	5	1	no	—
42	C₂×C₁₆	—	5	1	no	—
43	C₄×C₈	—	5	1	no	—
44	C₄×C₈	—	5	1	no	—
45	C₄×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
46	C₃₂	—	20	2	no	—
47	C₂²×C₅	C₄⋊C₁₆	1	1	yes	characteristic
48	C₄×C₅	—	2	1	no	—
49	C₄×C₅	#335	1	1	yes	characteristic
50	C₄×C₅	#364	1	1	yes	characteristic
51	C₄×C₅	C₂×C₃₂	1	1	yes	characteristic
52	C₄×C₅	M₆₄	1	1	yes	characteristic
53	C₂×C₈	—	5	1	no	—
54	C₂×C₈	—	5	1	no	—
55	C₂×C₈	—	10	1	no	—
56	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₂×C₈	#704	1	1	yes	characteristic
58	C₄²	D₅⋊C₈	1	1	yes	characteristic
59	C₁₆	—	2	2	no	—
60	C₁₆	—	5	1	no	—
61	C₁₆	—	5	1	no	—
62	C₁₆	—	10	1	no	—
63	C₂×C₅	#1370	1	1	yes	characteristic
64	C₂×C₅	C₄⋊C₃₂	1	1	yes	characteristic
65	C₂×C₅	#1422	1	1	yes	characteristic
66	C₂×C₄	D₅⋊C₁₆	1	1	yes	characteristic
67	C₂×C₄	#3990	1	1	yes	characteristic
68	C₂×C₄	#4004	1	1	yes	characteristic
69	C₈	—	2	1	no	—
70	C₈	—	5	2	no	—
71	C₈	—	10	2	no	—
72	C₈	—	10	1	no	—
73	C₈	#3994	1	1	yes	characteristic
74	C₈	#3995	1	1	yes	characteristic
75	C₅	#7458	1	1	yes	characteristic, complemented, Sylow
76	C₂²	#64888	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	D₅⋊C₃₂	1	1	yes	characteristic
79	C₄	#64842	1	1	yes	characteristic
80	C₄	#64861	1	1	yes	characteristic
81	C₄	#64884	1	1	yes	characteristic
82	C₂	#10598525	1	1	yes	characteristic
83	C₂	#10598655	1	1	yes	characteristic
84	C₂	#10598658	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄⋊AGL(1,𝔽₅)
commutator	C₄×C₅	C₂×C₃₂
Frattini	C₄×C₈	C₂×AGL(1,𝔽₅)
Fitting	C₄×C₅×C₁₆	C₄
socle, prime radical	C₂²×C₅	C₄⋊C₁₆
cosocle	C₄×#672	C₂²
hypercenter	C₄×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	#7458
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
39. #168	C₃₂	semidirect
75. C₅	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#3928	2
C₁₆×#168	2
#7458	5