Subgroups — C₂×C₅×#103 #65578

Order 320.

162 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
2	C₂²×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#103	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₅×Q₈	—	2	2	no	—
6	C₂×C₅×Q₈	—	2	1	no	—
7	C₂×C₅×Q₈	C₄	1	4	yes	complemented
8	C₂×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₄×C₅	—	2	1	no	—
10	C₂²×C₄×C₅	—	2	1	no	—
11	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₈	—	2	2	no	—
13	C₂×C₅×C₈	C₂²	1	2	yes	—
14	C₅×C₄⋊C₄	—	2	1	no	—
15	C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×#103	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₅	D₄	1	1	yes	characteristic
18	C₅×Q₈	—	2	4	no	—
19	C₅×Q₈	—	2	2	no	—
20	C₅×Q₈	C₂×C₄	1	4	yes	complemented
21	C₂×C₄×C₅	—	2	4	no	—
22	C₂×C₄×C₅	—	2	2	no	—
23	C₂×C₄×C₅	—	2	2	no	—
24	C₂×C₄×C₅	—	4	2	no	—
25	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄×C₅	D₄	1	2	yes	—
28	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
29	C₅×C₈	—	2	2	no	—
30	C₂²×Q₈	C₂×C₅	1	1	yes	characteristic
31	C₂²×C₈	C₂×C₅	1	1	yes	characteristic
32	C₂×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
33	#103	C₂×C₅	1	4	yes	direct product factor
34	C₂²×C₅	C₂²⋊C₄	1	2	yes	—
35	C₂²×C₅	SD₁₆	1	2	yes	—
36	C₂²×C₅	Q₁₆	1	2	yes	—
37	C₂²×C₅	C₂×D₄	1	1	yes	characteristic
38	C₄×C₅	—	2	4	no	—
39	C₄×C₅	—	4	2	no	—
40	C₄×C₅	C₂²⋊C₄	1	2	yes	—
41	C₄×C₅	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂×Q₈	—	2	2	no	—
44	C₂×Q₈	—	2	1	no	—
45	C₂×Q₈	C₄×C₅	1	4	yes	complemented
46	C₂×Q₈	C₂²×C₅	1	2	yes	—
47	C₂²×C₄	—	2	1	no	—
48	C₂²×C₄	—	2	1	no	—
49	C₂²×C₄	C₂²×C₅	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	C₂²×C₅	1	2	yes	—
52	C₄⋊C₄	—	2	1	no	—
53	C₄⋊C₄	C₂²×C₅	1	2	yes	—
54	C₂×C₅	#103	1	4	yes	direct product factor
55	C₂×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₅	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₅	C₂×Q₁₆	1	1	yes	characteristic
58	C₂³	C₅×D₄	1	1	yes	characteristic
59	Q₈	—	2	4	no	—
60	Q₈	—	2	2	no	—
61	Q₈	C₂×C₄×C₅	1	4	yes	complemented
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	C₂×C₄×C₅	1	2	yes	—
67	C₂×C₄	C₅×D₄	1	1	yes	characteristic
68	C₂×C₄	C₅×D₄	1	2	yes	—
69	C₂×C₄	C₂³×C₅	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₅	C₂×#103	1	1	yes	characteristic, direct product factor, Sylow
72	C₂²	C₅×C₂²⋊C₄	1	2	yes	—
73	C₂²	C₅×SD₁₆	1	2	yes	—
74	C₂²	C₅×Q₁₆	1	2	yes	—
75	C₂²	C₂×C₅×D₄	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	4	2	no	—
78	C₄	C₅×C₂²⋊C₄	1	2	yes	—
79	C₄	C₂²×C₄×C₅	1	1	yes	characteristic
80	C₄	C₂×C₅×D₄	1	1	yes	characteristic
81	C₂	C₅×#103	1	4	yes	direct product factor
82	C₂	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
83	C₂	C₂×C₅×SD₁₆	1	1	yes	characteristic
84	C₂	C₂×C₅×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	D₄
commutator	C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄	C₂³×C₅
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#103	C₂	direct (N × H)
7. C₂×C₅×Q₈	C₄	semidirect
16. C₂×#103	C₅	direct (N × H)
20. C₅×Q₈	C₂×C₄	semidirect
33. #103	C₂×C₅	direct (N × H)
45. C₂×Q₈	C₄×C₅	semidirect
54. C₂×C₅	#103	direct (N × H)
61. Q₈	C₂×C₄×C₅	semidirect
71. C₅	C₂×#103	direct (N × H)
81. C₂	C₅×#103	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×C₅×C₈	2
C₂×C₅×C₄⋊C₄	2
C₅×#103	2
C₂×#103	5

Subgroups — C2×C5×#103 #65578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#103 #65578