Subgroups — C₂²×#688 #66141

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₂₀	C₂	1	1	yes	maximal, characteristic
3	C₂×#688	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₅	—	2	1	no	—
6	C₂×C₅×Q₈	—	2	3	no	—
7	C₂×C₅×Q₈	C₂²	1	6	yes	complemented
8	C₂²×C₄×C₅	—	2	1	no	—
9	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂×D₂₀	—	2	3	no	—
11	C₂×D₂₀	C₂²	1	6	yes	—
12	#688	C₂²	1	16	yes	direct product factor
13	C₂×#168	C₂²	1	6	yes	complemented
14	C₂²×SD₁₆	—	5	1	no	maximal, Sylow
15	C₂³×C₅	D₄	1	1	yes	characteristic
16	C₂²×D₅	—	2	6	no	—
17	C₂²×D₅	—	4	4	no	—
18	C₅×Q₈	—	2	6	no	—
19	C₅×Q₈	C₂³	1	4	yes	complemented
20	C₂×C₄×C₅	—	2	6	no	—
21	C₂×C₄×C₅	D₄	1	3	yes	—
22	C₂×C₄×C₅	C₂³	1	3	yes	—
23	D₂₀	—	2	6	no	—
24	D₂₀	C₂³	1	4	yes	—
25	#168	C₂³	1	4	yes	complemented
26	C₂²×Q₈	D₅	1	1	yes	characteristic, complemented
27	C₂²×D₄	—	5	1	no	—
28	C₂×SD₁₆	—	5	12	no	—
29	C₂²×C₈	—	5	1	no	—
30	C₂²×C₅	SD₁₆	1	4	yes	complemented
31	C₂²×C₅	C₂×D₄	1	3	yes	—
32	C₂×D₅	—	2	4	no	—
33	C₂×D₅	—	4	12	no	—
34	C₄×C₅	—	2	4	no	—
35	C₄×C₅	C₂×D₄	1	3	yes	—
36	C₄×C₅	C₂⁴	1	1	yes	characteristic
37	C₂⁴	—	10	1	no	—
38	C₂×Q₈	—	2	3	no	—
39	C₂×Q₈	C₂×D₅	1	6	yes	complemented
40	C₂×D₄	—	5	6	no	—
41	C₂×D₄	—	10	3	no	—
42	C₂²×C₄	—	2	1	no	—
43	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
44	SD₁₆	—	5	16	no	—
45	C₂×C₈	—	5	6	no	—
46	C₂×C₅	C₂×SD₁₆	1	6	yes	complemented
47	C₂×C₅	C₂²×D₄	1	1	yes	characteristic
48	D₅	—	4	4	no	—
49	C₂³	—	10	6	no	—
50	C₂³	—	20	4	no	—
51	C₂³	C₂²⋊D₅	1	1	yes	characteristic
52	Q₈	—	2	6	no	—
53	Q₈	C₂²×D₅	1	4	yes	complemented
54	D₄	—	5	4	no	—
55	D₄	—	10	6	no	—
56	C₂×C₄	—	2	6	no	—
57	C₂×C₄	C₂²⋊D₅	1	3	yes	—
58	C₂×C₄	C₂²×D₅	1	3	yes	—
59	C₈	—	5	4	no	—
60	C₅	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
61	C₂²	—	10	4	no	—
62	C₂²	—	20	12	no	—
63	C₂²	#688	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₅	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×C₂²⋊D₅	1	3	yes	—
67	C₄	C₂³×D₅	1	1	yes	characteristic
68	C₂	—	20	4	no	—
69	C₂	C₂×#688	1	6	yes	direct product factor
70	C₂	C₂²×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₅
commutator, cosocle	C₄×C₅	C₂⁴
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×Q₈	C₂
socle	C₂³×C₅	D₄
prime radical	C₂²×D₂₀	C₂
hypercenter	C₂²×Q₈	D₅
hypocenter	C₅	C₂²×SD₁₆
LCS	G ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Q₈	C₂	semidirect
3. C₂×#688	C₂	direct (N × H)
4. C₂²×#168	C₂	semidirect
7. C₂×C₅×Q₈	C₂²	semidirect
12. #688	C₂²	direct (N × H)
13. C₂×#168	C₂²	semidirect
19. C₅×Q₈	C₂³	semidirect
25. #168	C₂³	semidirect
26. C₂²×Q₈	D₅	semidirect
30. C₂²×C₅	SD₁₆	semidirect
39. C₂×Q₈	C₂×D₅	semidirect
46. C₂×C₅	C₂×SD₁₆	semidirect
53. Q₈	C₂²×D₅	semidirect
60. C₅	C₂²×SD₁₆	semidirect
63. C₂²	#688	direct (N × H)
69. C₂	C₂×#688	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×C₅×Q₈	2
C₂²×D₂₀	2
C₂×#688	2
C₂²×#168	2
C₂²×SD₁₆	5

Subgroups — C22×#688 #66141

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#688 #66141