Subgroups — #65788

Order 320.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₅	C₂	1	1	yes	maximal, characteristic
2	#4004	C₂	1	2	yes	maximal
3	#4002	C₂	1	2	yes	maximal
4	C₄⋊#170	C₂	1	1	yes	maximal, characteristic
5	C₄×#170	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×Q₈	C₄	1	1	yes	characteristic
7	C₂×#170	C₂²	1	1	yes	characteristic
8	C₂×#170	C₂²	1	2	yes	—
9	C₂×#170	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₅	C₄	1	2	yes	—
11	C₄⋊Dic₅	C₄	1	1	yes	characteristic
12	C₄×Dic₅	C₂²	1	1	yes	characteristic
13	C₄×Dic₅	C₂²	1	2	yes	—
14	#446	—	5	1	no	maximal, Sylow
15	C₅×Q₈	—	2	2	no	—
16	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
19	C₂×Dic₅	C₂×C₄	1	2	yes	—
20	C₂×Dic₅	C₂³	1	1	yes	characteristic
21	#170	—	2	1	no	—
22	#170	—	2	2	no	—
23	#170	Q₈	1	2	yes	complemented
24	C₄×Q₈	—	5	1	no	—
25	C₄⋊C₈	—	5	1	no	—
26	C₄⋊C₈	—	5	2	no	—
27	#97	—	5	2	no	—
28	C₄×C₈	—	5	1	no	—
29	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₅	—	2	2	no	—
31	C₄×C₅	M₁₆	1	2	yes	—
32	Dic₅	—	2	1	no	—
33	Dic₅	—	2	2	no	—
34	Dic₅	C₂×Q₈	1	1	yes	characteristic
35	Dic₅	#41	1	1	yes	characteristic
36	C₂×Q₈	AGL(1,𝔽₅)	1	1	yes	characteristic
37	C₂×C₈	—	5	1	no	—
38	C₂×C₈	—	5	1	no	—
39	C₂×C₈	—	5	2	no	—
40	C₄⋊C₄	—	5	2	no	—
41	C₄⋊C₄	—	5	1	no	—
42	C₄²	—	5	1	no	—
43	C₄²	—	5	2	no	—
44	C₂×C₅	C₄×Q₈	1	1	yes	characteristic
45	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
46	C₂×C₅	#131	1	1	yes	characteristic
47	Q₈	—	2	2	no	—
48	C₂×C₄	—	5	1	no	—
49	C₂×C₄	—	5	1	no	—
50	C₂×C₄	—	5	2	no	—
51	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
52	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
53	C₈	—	5	2	no	—
54	C₈	—	10	1	no	—
55	C₈	—	10	2	no	—
56	C₅	#446	1	1	yes	characteristic, complemented, Sylow
57	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	5	1	no	—
60	C₄	—	5	1	no	—
61	C₄	—	10	1	no	—
62	C₄	—	10	2	no	—
63	C₄	#704	1	2	yes	—
64	C₂	#4133	1	1	yes	characteristic
65	C₂	Q₈×AGL(1,𝔽₅)	1	1	yes	characteristic
66	C₂	C₂×#704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator, socle, prime radical	C₂²×C₅	C₂²×C₄
Fitting	C₂×C₅×Q₈	C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂×Q₈	AGL(1,𝔽₅)
hypocenter	C₅	#446
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
23. #170	Q₈	semidirect
56. C₅	#446	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₅	2
#4004	2
#4002	2
C₄⋊#170	2
C₄×#170	2
#446	5