Subgroups — #65691

Order 320.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#704	C₂	1	1	yes	maximal, characteristic
2	C₂²×#170	C₂	1	1	yes	maximal, characteristic
3	#4023	C₂	1	1	yes	maximal, characteristic
4	#4002	C₂	1	2	yes	maximal, complemented
5	C₄×#170	C₂	1	2	yes	maximal, complemented
6	C₂²×Dic₅	C₂²	1	1	yes	characteristic
7	#704	C₄	1	4	yes	complemented
8	C₂×#170	C₄	1	4	yes	complemented
9	C₂×#170	C₂²	1	1	yes	characteristic
10	C₂×#170	C₂²	1	1	yes	characteristic
11	C₂×#170	C₂²	1	2	yes	—
12	C₅×C₂²⋊C₄	C₄	1	1	yes	characteristic
13	C₂²⋊Dic₅	C₄	1	1	yes	characteristic
14	#683	C₄	1	2	yes	—
15	C₄×Dic₅	C₂²	1	2	yes	—
16	#405	—	5	1	no	maximal, Sylow
17	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
19	C₂×Dic₅	C₂×C₄	1	2	yes	—
20	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
21	C₂×Dic₅	C₂×C₄	1	2	yes	—
22	C₂×Dic₅	C₂×C₄	1	2	yes	—
23	C₂×Dic₅	C₂³	1	1	yes	characteristic
24	#170	C₂×C₄	1	2	yes	—
25	#170	C₂×C₄	1	2	yes	—
26	#170	C₂×C₄	1	4	yes	—
27	C₂×M₁₆	—	5	1	no	—
28	C₂²×C₈	—	5	1	no	—
29	#117	—	5	1	no	—
30	#97	—	5	2	no	—
31	C₄×C₈	—	5	2	no	—
32	C₂²×C₅	—	2	2	no	—
33	C₂²×C₅	C₄²	1	2	yes	—
34	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
35	C₄×C₅	—	2	2	no	—
36	Dic₅	—	2	2	no	—
37	Dic₅	C₄²	1	2	yes	—
38	Dic₅	C₂²×C₄	1	1	yes	characteristic
39	Dic₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄	—	5	1	no	—
41	M₁₆	—	5	4	no	—
42	C₂×C₈	—	5	1	no	—
43	C₂×C₈	—	5	2	no	—
44	C₂×C₈	—	5	1	no	—
45	C₂×C₈	—	5	4	no	—
46	C₄⋊C₄	—	5	2	no	—
47	C₂²⋊C₄	—	5	1	no	—
48	C₂²⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
49	C₄²	—	5	2	no	—
50	C₂×C₅	—	2	2	no	—
51	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₅	#131	1	2	yes	—
53	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
54	C₂×C₄	—	5	2	no	—
55	C₂×C₄	—	5	2	no	—
56	C₂×C₄	—	5	1	no	—
57	C₂×C₄	—	5	1	no	—
58	C₂×C₄	—	5	2	no	—
59	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
60	C₈	—	5	2	no	—
61	C₈	—	5	2	no	—
62	C₈	—	5	4	no	—
63	C₅	#405	1	1	yes	characteristic, complemented, Sylow
64	C₂²	—	2	2	no	—
65	C₂²	C₄×AGL(1,𝔽₅)	1	2	yes	—
66	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	5	1	no	—
69	C₄	—	5	1	no	—
70	C₄	—	5	2	no	—
71	C₄	—	10	2	no	—
72	C₂	—	2	2	no	—
73	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
74	C₂	#4131	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Fitting	C₅×C₂²⋊C₄	C₄
socle	C₂²×C₅	C₂²×C₄
prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²⋊C₄	AGL(1,𝔽₅)
hypocenter	C₅	#405
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. #4002	C₂	semidirect
5. C₄×#170	C₂	semidirect
7. #704	C₄	semidirect
8. C₂×#170	C₄	semidirect
63. C₅	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#704	2
C₂²×#170	2
#4023	2
#4002	2
C₄×#170	2
#405	5