Subgroups — C₂²×#676 #66071

Order 320.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#676	C₂	1	12	yes	maximal, direct product factor
5	C₂³×D₅	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₅	C₂²	1	1	yes	characteristic
7	C₂²×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₄×D₅	C₄	1	6	yes	—
9	C₂×C₄×D₅	C₂²	1	6	yes	—
10	C₂×C₅×C₈	C₂²	1	6	yes	complemented
11	C₂×#168	C₂²	1	6	yes	complemented
12	#676	C₂²	1	16	yes	direct product factor
13	C₂²×M₁₆	—	5	1	no	maximal, Sylow
14	C₂³×C₅	C₂×C₄	1	1	yes	characteristic
15	C₂²×D₅	—	2	4	no	—
16	C₂²×D₅	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₅	C₂×C₄	1	3	yes	—
18	C₂×C₄×C₅	C₂³	1	3	yes	—
19	C₂×Dic₅	C₂×C₄	1	6	yes	—
20	C₄×D₅	C₂×C₄	1	12	yes	—
21	C₄×D₅	C₂³	1	4	yes	—
22	C₅×C₈	C₂³	1	4	yes	complemented
23	#168	C₂³	1	4	yes	complemented
24	C₂³×C₄	—	5	1	no	—
25	C₂×M₁₆	—	5	12	no	—
26	C₂²×C₈	—	5	1	no	—
27	C₂²×C₈	D₅	1	1	yes	characteristic, complemented
28	C₂²×C₅	M₁₆	1	4	yes	complemented
29	C₂²×C₅	C₂²×C₄	1	3	yes	—
30	C₂×D₅	—	2	12	no	—
31	C₂×D₅	C₂²×C₄	1	4	yes	—
32	C₄×C₅	C₂²×C₄	1	3	yes	—
33	C₄×C₅	C₂⁴	1	1	yes	characteristic
34	Dic₅	C₂²×C₄	1	4	yes	—
35	C₂⁴	—	5	1	no	—
36	C₂²×C₄	—	5	1	no	—
37	C₂²×C₄	—	5	6	no	—
38	C₂²×C₄	—	5	6	no	—
39	C₂²×C₄	C₂×D₅	1	1	yes	characteristic
40	M₁₆	—	5	16	no	—
41	C₂×C₈	—	5	6	no	—
42	C₂×C₈	C₂×D₅	1	6	yes	complemented
43	C₂×C₅	C₂×M₁₆	1	6	yes	complemented
44	C₂×C₅	C₂³×C₄	1	1	yes	characteristic
45	D₅	—	2	4	no	—
46	C₂³	—	5	6	no	—
47	C₂³	—	10	4	no	—
48	C₂³	C₄×D₅	1	1	yes	characteristic
49	C₂×C₄	—	5	6	no	—
50	C₂×C₄	—	5	12	no	—
51	C₂×C₄	—	5	4	no	—
52	C₂×C₄	C₄×D₅	1	3	yes	—
53	C₂×C₄	C₂²×D₅	1	3	yes	—
54	C₈	—	5	4	no	—
55	C₈	C₂²×D₅	1	4	yes	complemented
56	C₅	C₂²×M₁₆	1	1	yes	characteristic, complemented, Sylow
57	C₂²	—	5	4	no	—
58	C₂²	—	10	12	no	—
59	C₂²	#676	1	4	yes	direct product factor
60	C₂²	C₂×C₄×D₅	1	3	yes	—
61	C₄	—	5	4	no	—
62	C₄	C₂×C₄×D₅	1	3	yes	—
63	C₄	C₂³×D₅	1	1	yes	characteristic
64	C₂	—	10	4	no	—
65	C₂	C₂×#676	1	6	yes	direct product factor
66	C₂	C₂²×C₄×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₅
commutator	C₂×C₅	C₂³×C₄
Frattini	C₄	C₂³×D₅
Fitting	C₂²×C₅×C₈	C₂
socle	C₂³×C₅	C₂×C₄
prime radical	C₂³×D₅	C₄
cosocle	C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₅
hypocenter	C₅	C₂²×M₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×C₈	C₂	semidirect
3. C₂²×#168	C₂	semidirect
4. C₂×#676	C₂	direct (N × H)
10. C₂×C₅×C₈	C₂²	semidirect
11. C₂×#168	C₂²	semidirect
12. #676	C₂²	direct (N × H)
22. C₅×C₈	C₂³	semidirect
23. #168	C₂³	semidirect
27. C₂²×C₈	D₅	semidirect
28. C₂²×C₅	M₁₆	semidirect
42. C₂×C₈	C₂×D₅	semidirect
43. C₂×C₅	C₂×M₁₆	semidirect
55. C₈	C₂²×D₅	semidirect
56. C₅	C₂²×M₁₆	semidirect
59. C₂²	#676	direct (N × H)
65. C₂	C₂×#676	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×C₄×D₅	2
C₂²×C₅×C₈	2
C₂²×#168	2
C₂×#676	2
C₂²×M₁₆	5

Subgroups — C22×#676 #66071

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#676 #66071