Subgroups — C₂²×#681 #66115

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#681	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#168	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅	C₄	1	6	yes	—
6	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
7	C₂²×C₄×C₅	C₂²	1	6	yes	—
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	#681	C₂²	1	16	yes	direct product factor
10	C₂×#168	C₂²	1	12	yes	complemented
11	C₂²×M₁₆	—	5	1	no	maximal, Sylow
12	C₂³×C₅	—	2	4	no	—
13	C₂³×C₅	C₂×C₄	1	6	yes	—
14	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₅	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
18	C₂×C₄×C₅	C₂³	1	3	yes	—
19	C₂×C₄×C₅	C₂³	1	4	yes	—
20	#168	C₂³	1	8	yes	complemented
21	C₂³×C₄	D₅	1	1	yes	characteristic
22	C₂×M₁₆	—	5	12	no	—
23	C₂²×C₈	—	5	2	no	—
24	C₂²×C₅	—	2	12	no	—
25	C₂²×C₅	M₁₆	1	4	yes	complemented
26	C₂²×C₅	C₂²×C₄	1	4	yes	—
27	C₂²×C₅	C₂²×C₄	1	3	yes	—
28	C₄×C₅	C₂²×C₄	1	3	yes	—
29	C₄×C₅	C₂²×C₄	1	4	yes	—
30	C₄×C₅	C₂⁴	1	1	yes	characteristic
31	C₂⁴	Dic₅	1	1	yes	characteristic
32	C₂²×C₄	Dic₅	1	1	yes	characteristic
33	C₂²×C₄	Dic₅	1	6	yes	—
34	C₂²×C₄	C₂×D₅	1	6	yes	—
35	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
36	M₁₆	—	5	16	no	—
37	C₂×C₈	—	5	12	no	—
38	C₂×C₅	—	2	4	no	—
39	C₂×C₅	C₂×M₁₆	1	6	yes	complemented
40	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
41	C₂³	—	2	4	no	—
42	C₂³	C₂×Dic₅	1	6	yes	—
43	C₂³	C₂×Dic₅	1	1	yes	characteristic
44	C₂×C₄	C₂×Dic₅	1	12	yes	—
45	C₂×C₄	C₂×Dic₅	1	6	yes	—
46	C₂×C₄	C₂×Dic₅	1	3	yes	—
47	C₂×C₄	C₂²×D₅	1	3	yes	—
48	C₂×C₄	C₂²×D₅	1	4	yes	—
49	C₈	—	5	8	no	—
50	C₅	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
51	C₂²	—	2	12	no	—
52	C₂²	#681	1	4	yes	direct product factor
53	C₂²	C₂²×Dic₅	1	4	yes	—
54	C₂²	C₂²×Dic₅	1	3	yes	—
55	C₄	C₂²×Dic₅	1	4	yes	—
56	C₄	C₂²×Dic₅	1	3	yes	—
57	C₄	C₂³×D₅	1	1	yes	characteristic
58	C₂	—	2	4	no	—
59	C₂	C₂×#681	1	6	yes	direct product factor
60	C₂	C₂³×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅
commutator	C₂×C₅	C₂³×C₄
Frattini	C₄	C₂³×D₅
Fitting	C₂³×C₄×C₅	C₂
socle	C₂³×C₅	C₂×C₄
prime radical	C₂⁴×C₅	C₄
cosocle	C₄×C₅	C₂⁴
hypercenter	C₂³×C₄	D₅
hypocenter	C₅	C₂²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×#681	C₂	direct (N × H)
3. C₂²×#168	C₂	semidirect
9. #681	C₂²	direct (N × H)
10. C₂×#168	C₂²	semidirect
20. #168	C₂³	semidirect
25. C₂²×C₅	M₁₆	semidirect
39. C₂×C₅	C₂×M₁₆	semidirect
50. C₅	C₂²×M₁₆	semidirect
52. C₂²	#681	direct (N × H)
59. C₂	C₂×#681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₅	2
C₂×#681	2
C₂²×#168	2
C₂²×M₁₆	5

Subgroups — C22×#681 #66115

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#681 #66115