Subgroups — C₂×#3998 #65743

Order 320.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#168	C₂	1	1	yes	maximal, characteristic
2	#3998	C₂	1	4	yes	maximal, direct product factor
3	C₂×#674	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂×#168	C₄	1	2	yes	—
6	C₂×#168	C₄	1	1	yes	characteristic
7	C₂×#168	C₂²	1	1	yes	characteristic
8	C₂×#168	C₂²	1	2	yes	—
9	#674	C₂²	1	4	yes	complemented
10	C₂×M₃₂	—	5	1	no	maximal, Sylow
11	C₂³×C₅	C₈	1	1	yes	characteristic
12	C₂×C₄×C₅	C₈	1	2	yes	—
13	C₂×C₄×C₅	C₈	1	1	yes	characteristic
14	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
15	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
16	#168	C₂×C₄	1	1	yes	characteristic
17	#168	C₂×C₄	1	2	yes	—
18	#168	C₂³	1	1	yes	characteristic
19	C₂²×C₈	—	5	1	no	—
20	M₃₂	—	5	4	no	—
21	C₂×C₁₆	—	5	2	no	—
22	C₂²×C₅	—	2	2	no	—
23	C₂²×C₅	C₂×C₈	1	2	yes	—
24	C₂²×C₅	C₂×C₈	1	1	yes	characteristic
25	C₄×C₅	C₂×C₈	1	2	yes	—
26	C₄×C₅	C₂×C₈	1	1	yes	characteristic
27	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
29	C₂×C₈	—	5	2	no	—
30	C₂×C₈	—	5	2	no	—
31	C₂×C₈	—	5	1	no	—
32	C₂×C₈	—	5	1	no	—
33	C₁₆	—	5	4	no	—
34	C₂×C₅	—	2	2	no	—
35	C₂×C₅	M₃₂	1	2	yes	complemented
36	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
37	C₂³	#170	1	1	yes	characteristic
38	C₂×C₄	#170	1	1	yes	characteristic
39	C₂×C₄	#170	1	2	yes	—
40	C₂×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
41	C₂×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
42	C₈	—	5	1	no	—
43	C₈	—	5	2	no	—
44	C₈	—	5	1	no	—
45	C₅	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
46	C₂²	—	2	2	no	—
47	C₂²	C₂×#170	1	2	yes	—
48	C₂²	C₂×#170	1	1	yes	characteristic
49	C₄	C₂×#170	1	2	yes	—
50	C₄	C₂×#170	1	1	yes	characteristic
51	C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	#3998	1	2	yes	direct product factor
54	C₂	C₂²×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂²×C₈
Frattini	C₄	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₄×C₅	C₄
socle	C₂²×C₅	C₂×C₈
prime radical	C₂³×C₅	C₈
cosocle	#168	C₂³
hypercenter	C₂²×C₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×M₃₂
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #3998	C₂	direct (N × H)
3. C₂×#674	C₂	semidirect
9. #674	C₂²	semidirect
35. C₂×C₅	M₃₂	semidirect
45. C₅	C₂×M₃₂	semidirect
53. C₂	#3998	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×#168	2
#3998	2
C₂×#674	2
C₂×M₃₂	5

Subgroups — C2×#3998 #65743

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3998 #65743