Subgroups — C₂×C₄⋊#168 #65212

Order 320.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#168	C₂	1	2	yes	maximal
3	C₄⋊#168	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₅	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₄²×C₅	C₄	1	2	yes	—
8	C₄²×C₅	C₂²	1	2	yes	—
9	C₂×#168	—	2	4	no	—
10	C₂×#168	C₂²	1	4	yes	—
11	C₂×C₄⋊C₈	—	5	1	no	maximal, Sylow
12	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄×C₅	—	2	2	no	—
14	C₂×C₄×C₅	C₈	1	4	yes	complemented
15	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
16	C₂×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂×C₄×C₅	D₄	1	2	yes	—
19	C₂×C₄×C₅	Q₈	1	2	yes	—
20	C₂×C₄×C₅	C₂³	1	1	yes	characteristic
21	#168	—	2	4	no	—
22	C₂²×C₈	—	5	2	no	—
23	C₂×C₄²	D₅	1	1	yes	characteristic
24	C₄⋊C₈	—	5	4	no	—
25	C₂²×C₅	C₄⋊C₄	1	2	yes	—
26	C₂²×C₅	C₂×C₈	1	2	yes	—
27	C₂²×C₅	M₁₆	1	2	yes	—
28	C₂²×C₅	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₅	—	2	2	no	—
30	C₄×C₅	C₄⋊C₄	1	2	yes	—
31	C₄×C₅	C₂×C₈	1	4	yes	complemented
32	C₄×C₅	C₂×D₄	1	1	yes	characteristic
33	C₄×C₅	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄	Dic₅	1	1	yes	characteristic
35	C₂²×C₄	Dic₅	1	1	yes	characteristic
36	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
37	C₂×C₈	—	5	4	no	—
38	C₂×C₈	—	10	4	no	—
39	C₄²	Dic₅	1	2	yes	—
40	C₄²	C₂×D₅	1	2	yes	—
41	C₂×C₅	C₄⋊C₈	1	4	yes	complemented
42	C₂×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₅	C₂²×C₈	1	1	yes	characteristic
44	C₂×C₅	C₂×M₁₆	1	1	yes	characteristic
45	C₂³	C₂×Dic₅	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	#168	1	4	yes	complemented
48	C₂×C₄	Dic₁₀	1	2	yes	—
49	C₂×C₄	D₂₀	1	2	yes	—
50	C₂×C₄	C₂×Dic₅	1	2	yes	—
51	C₂×C₄	C₂×Dic₅	1	2	yes	—
52	C₂×C₄	C₂×Dic₅	1	1	yes	characteristic
53	C₂×C₄	C₂²×D₅	1	1	yes	characteristic
54	C₈	—	10	4	no	—
55	C₅	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
56	C₂²	C₂×#168	1	2	yes	—
57	C₂²	#681	1	2	yes	—
58	C₂²	C₄⋊Dic₅	1	2	yes	—
59	C₂²	C₂²×Dic₅	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₂×#168	1	4	yes	complemented
62	C₄	C₄⋊Dic₅	1	2	yes	—
63	C₄	C₂×Dic₁₀	1	1	yes	characteristic
64	C₄	C₂×D₂₀	1	1	yes	characteristic
65	C₂	C₄⋊#168	1	4	yes	direct product factor
66	C₂	C₂²×#168	1	1	yes	characteristic
67	C₂	C₂×#681	1	1	yes	characteristic
68	C₂	C₂×C₄⋊Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅
commutator	C₂×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×D₅
Fitting	C₂×C₄²×C₅	C₂
socle, prime radical	C₂³×C₅	C₂×C₄
cosocle	C₂×C₄×C₅	C₂³
hypercenter	C₂×C₄²	D₅
hypocenter	C₅	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₄⋊#168	C₂	direct (N × H)
14. C₂×C₄×C₅	C₈	semidirect
31. C₄×C₅	C₂×C₈	semidirect
41. C₂×C₅	C₄⋊C₈	semidirect
47. C₂×C₄	#168	semidirect
55. C₅	C₂×C₄⋊C₈	semidirect
61. C₄	C₂×#168	semidirect
65. C₂	C₄⋊#168	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×C₄²×C₅	2
C₂²×#168	2
C₄⋊#168	2
C₂×C₄⋊C₈	5

Subgroups — C2×C4⋊#168 #65212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊#168 #65212