Subgroups — C₄×C₅×C₄⋊C₄ #65541

Order 320.

194 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
2	C₂×C₄²×C₅	C₂	1	2	yes	maximal
3	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
4	C₅×#95	C₂	1	2	yes	maximal
5	C₂²×C₄×C₅	C₂²	1	4	yes	—
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
9	C₅×C₄⋊C₄	C₄	1	8	yes	direct product factor
10	C₄²×C₅	—	2	4	no	—
11	C₄²×C₅	C₄	1	4	yes	complemented
12	C₄×C₄⋊C₄	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
13	C₂³×C₅	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₅	—	2	4	no	—
15	C₂×C₄×C₅	—	2	2	no	—
16	C₂×C₄×C₅	—	2	4	no	—
17	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅	C₂×C₄	1	8	yes	—
19	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₅	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₅	D₄	1	2	yes	—
23	C₂×C₄×C₅	Q₈	1	2	yes	—
24	C₂×C₄⋊C₄	C₂×C₅	1	2	yes	—
25	C₂×C₄²	C₂×C₅	1	2	yes	—
26	C₂×C₄²	C₂×C₅	1	1	yes	characteristic
27	#95	C₂×C₅	1	2	yes	—
28	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₅	C₂²×C₄	1	2	yes	—
30	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₅	C₂×Q₈	1	1	yes	characteristic
32	C₂²×C₅	#41	1	2	yes	—
33	C₄×C₅	—	2	2	no	—
34	C₄×C₅	—	2	8	no	—
35	C₄×C₅	C₄²	1	4	yes	complemented
36	C₄×C₅	C₄⋊C₄	1	4	yes	direct product factor
37	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
38	C₂²×C₄	C₂²×C₅	1	4	yes	—
39	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
40	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
41	C₄⋊C₄	C₄×C₅	1	8	yes	direct product factor
42	C₄²	—	2	4	no	—
43	C₄²	C₄×C₅	1	4	yes	complemented
44	C₂×C₅	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₅	#117	1	1	yes	characteristic
47	C₂×C₅	C₄×D₄	1	2	yes	—
48	C₂×C₅	C₄×Q₈	1	2	yes	—
49	C₂³	C₂³×C₅	1	1	yes	characteristic
50	C₂×C₄	—	2	4	no	—
51	C₂×C₄	—	2	2	no	—
52	C₂×C₄	—	2	4	no	—
53	C₂×C₄	C₂×C₄×C₅	1	2	yes	—
54	C₂×C₄	C₂×C₄×C₅	1	8	yes	—
55	C₂×C₄	C₂×C₄×C₅	1	2	yes	—
56	C₂×C₄	C₂×C₄×C₅	1	4	yes	—
57	C₂×C₄	C₂×C₄×C₅	1	2	yes	—
58	C₂×C₄	C₅×D₄	1	2	yes	—
59	C₂×C₄	C₅×Q₈	1	2	yes	—
60	C₅	C₄×C₄⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	C₂²×C₄×C₅	1	1	yes	characteristic
62	C₂²	C₂²×C₄×C₅	1	2	yes	—
63	C₂²	C₂×C₅×D₄	1	1	yes	characteristic
64	C₂²	C₂×C₅×Q₈	1	1	yes	characteristic
65	C₂²	C₅×#41	1	2	yes	—
66	C₄	—	2	2	no	—
67	C₄	—	2	8	no	—
68	C₄	C₄²×C₅	1	4	yes	complemented
69	C₄	C₅×C₄⋊C₄	1	4	yes	direct product factor
70	C₂	C₂×C₄²×C₅	1	1	yes	characteristic
71	C₂	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
72	C₂	C₅×#117	1	1	yes	characteristic
73	C₂	C₄×C₅×D₄	1	2	yes	—
74	C₂	C₄×C₅×Q₈	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²
commutator	C₂	C₂×C₄²×C₅
Frattini, cosocle	C₂³	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₅×C₄⋊C₄	C₄	direct (N × H)
11. C₄²×C₅	C₄	semidirect
12. C₄×C₄⋊C₄	C₅	direct (N × H)
35. C₄×C₅	C₄²	semidirect
36. C₄×C₅	C₄⋊C₄	direct (N × H)
41. C₄⋊C₄	C₄×C₅	direct (N × H)
43. C₄²	C₄×C₅	semidirect
60. C₅	C₄×C₄⋊C₄	direct (N × H)
68. C₄	C₄²×C₅	semidirect
69. C₄	C₅×C₄⋊C₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₄⋊C₄	2
C₂×C₄²×C₅	2
C₅×#95	2
C₄×C₄⋊C₄	5

Subgroups — C4×C5×C4⋊C4 #65541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₅×C₄⋊C₄ #65541