Subgroups — C₂×#4002 #65749

Order 320.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#170	C₂	1	2	yes	maximal
2	C₂×C₄×Dic₅	C₂	1	1	yes	maximal, characteristic
3	#4002	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂²×Dic₅	C₄	1	1	yes	characteristic
6	C₂²×Dic₅	C₂²	1	1	yes	characteristic
7	C₂×#170	C₄	1	8	yes	complemented
8	C₂×#170	C₂²	1	4	yes	—
9	C₄×Dic₅	C₄	1	2	yes	—
10	C₄×Dic₅	C₂²	1	2	yes	—
11	C₂×#97	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅	—	2	2	no	—
14	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
15	C₂×Dic₅	—	2	2	no	—
16	C₂×Dic₅	C₂×C₄	1	2	yes	—
17	C₂×Dic₅	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₅	C₂×C₄	1	4	yes	—
19	C₂×Dic₅	C₂³	1	1	yes	characteristic
20	#170	C₂×C₄	1	8	yes	complemented
21	C₂²×C₈	—	5	2	no	—
22	C₂×C₄²	—	5	1	no	—
23	#97	—	5	4	no	—
24	C₂²×C₅	C₄²	1	2	yes	—
25	C₂²×C₅	M₁₆	1	2	yes	—
26	C₂²×C₅	M₁₆	1	2	yes	—
27	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₅	—	2	2	no	—
29	Dic₅	—	2	2	no	—
30	Dic₅	C₄²	1	2	yes	—
31	Dic₅	C₂²×C₄	1	2	yes	—
32	C₂²×C₄	—	5	1	no	—
33	C₂²×C₄	—	5	1	no	—
34	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
35	C₂×C₈	—	5	4	no	—
36	C₂×C₈	—	5	8	no	—
37	C₄²	—	5	2	no	—
38	C₄²	—	5	2	no	—
39	C₂×C₅	#97	1	4	yes	complemented
40	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
43	C₂³	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	5	4	no	—
46	C₂×C₄	—	5	1	no	—
47	C₂×C₄	—	5	1	no	—
48	C₂×C₄	—	5	2	no	—
49	C₂×C₄	—	10	2	no	—
50	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
51	C₈	—	5	8	no	—
52	C₅	C₂×#97	1	1	yes	characteristic, complemented, Sylow
53	C₂²	#700	1	2	yes	—
54	C₂²	C₄×AGL(1,𝔽₅)	1	2	yes	—
55	C₂²	#704	1	2	yes	—
56	C₂²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	5	2	no	—
59	C₄	—	5	2	no	—
60	C₄	—	10	2	no	—
61	C₂	#4002	1	4	yes	direct product factor
62	C₂	C₂×#700	1	1	yes	characteristic
63	C₂	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
64	C₂	C₂×#704	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂×C₄²
Frattini	C₂²	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×Dic₅	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#97
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. #4002	C₂	direct (N × H)
7. C₂×#170	C₄	semidirect
20. #170	C₂×C₄	semidirect
39. C₂×C₅	#97	semidirect
52. C₅	C₂×#97	semidirect
61. C₂	#4002	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#170	2
C₂×C₄×Dic₅	2
#4002	2
C₂×#97	5

Subgroups — C2×#4002 #65749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4002 #65749