Subgroups — C₄×#585 #62651

Order 256.

5673 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585	C₂	1	1	yes	maximal, characteristic
2	C₄×2₋¹⁺⁴	C₂	1	12	yes	maximal, complemented
3	C₄×H(5,𝔽₂)	C₂	1	20	yes	maximal, complemented
4	#3428	C₂	1	15	yes	maximal, complemented
5	C₂×C₄×#41	C₂	1	15	yes	maximal, complemented
6	#585	C₄	1	32	yes	direct product factor
7	C₂×2₋¹⁺⁴	C₂²	1	6	yes	—
8	C₂×H(5,𝔽₂)	C₂²	1	10	yes	—
9	C₂²×#41	C₂²	1	15	yes	—
10	#520	C₂²	1	120	yes	complemented
11	#519	C₂²	1	30	yes	complemented
12	#518	C₂²	1	90	yes	complemented
13	C₄×#41	C₂²	1	20	yes	complemented
14	C₄×#41	C₂²	1	60	yes	complemented
15	C₄×#41	C₂²	1	120	yes	complemented
16	C₂×C₄×Q₈	C₂²	1	30	yes	complemented
17	C₂×C₄×D₄	C₂²	1	90	yes	complemented
18	C₂×#117	C₂²	1	45	yes	complemented
19	C₂²×C₄²	C₂²	1	15	yes	complemented
20	2₋¹⁺⁴	C₂×C₄	1	96	yes	complemented
21	H(5,𝔽₂)	C₂×C₄	1	160	yes	complemented
22	C₂×#41	C₂×C₄	1	240	yes	complemented
23	C₂×#41	C₂³	1	20	yes	—
24	C₂×#41	C₂³	1	60	yes	—
25	C₂²×Q₈	C₂³	1	15	yes	—
26	C₂²×D₄	C₂³	1	45	yes	—
27	C₂³×C₄	C₂³	1	30	yes	—
28	C₂³×C₄	C₂³	1	15	yes	—
29	C₄×Q₈	C₂³	1	40	yes	—
30	C₄×Q₈	C₂³	1	120	yes	—
31	C₄×D₄	C₂³	1	120	yes	—
32	C₄×D₄	C₂³	1	360	yes	—
33	#117	C₂³	1	60	yes	—
34	#117	C₂³	1	180	yes	—
35	C₂×C₄⋊C₄	C₂³	1	90	yes	—
36	C₂×C₂²⋊C₄	C₂³	1	90	yes	—
37	C₂×C₄²	C₂³	1	45	yes	—
38	C₂×C₄²	C₂³	1	90	yes	—
39	C₂×C₄²	C₂³	1	15	yes	—
40	C₂⁴	C₂⁴	1	15	yes	—
41	#41	C₂²×C₄	1	160	yes	complemented
42	#41	C₂²×C₄	1	480	yes	complemented
43	C₂×Q₈	C₂²×C₄	1	120	yes	complemented
44	C₂×Q₈	C₂⁴	1	20	yes	—
45	C₂×D₄	C₂²×C₄	1	360	yes	complemented
46	C₂×D₄	C₂⁴	1	60	yes	—
47	C₂²×C₄	—	4	60	no	—
48	C₂²×C₄	C₂²×C₄	1	120	yes	complemented
49	C₂²×C₄	C₂⁴	1	30	yes	—
50	C₂²×C₄	C₂⁴	1	90	yes	—
51	C₂²×C₄	C₂⁴	1	15	yes	—
52	C₂²×C₄	C₂⁴	1	45	yes	—
53	C₄⋊C₄	C₂⁴	1	120	yes	—
54	C₂²⋊C₄	C₂⁴	1	120	yes	—
55	C₄²	C₂⁴	1	15	yes	—
56	C₄²	C₂⁴	1	30	yes	—
57	C₄²	C₂⁴	1	90	yes	—
58	C₄²	C₂⁴	1	1	yes	characteristic
59	C₂³	—	4	30	no	—
60	C₂³	C₂³×C₄	1	60	yes	—
61	C₂³	C₂⁵	1	15	yes	—
62	Q₈	C₂³×C₄	1	80	yes	—
63	D₄	C₂³×C₄	1	240	yes	—
64	C₂×C₄	—	2	60	no	—
65	C₂×C₄	—	4	90	no	—
66	C₂×C₄	C₂³×C₄	1	60	yes	—
67	C₂×C₄	C₂³×C₄	1	180	yes	—
68	C₂×C₄	C₂⁵	1	30	yes	—
69	C₂×C₄	C₂⁵	1	1	yes	characteristic
70	C₂×C₄	C₂⁵	1	15	yes	—
71	C₂×C₄	C₂⁵	1	2	yes	—
72	C₂²	—	2	30	no	—
73	C₂²	—	4	60	no	—
74	C₂²	C₂⁴×C₄	1	30	yes	—
75	C₂²	C₂⁶	1	1	yes	characteristic
76	C₄	—	2	30	no	—
77	C₄	C₂⁴×C₄	1	2	yes	—
78	C₄	C₂⁴×C₄	1	30	yes	—
79	C₄	#585	1	4	yes	direct product factor
80	C₂	—	2	30	no	—
81	C₂	C₂⁵×C₄	1	1	yes	characteristic
82	C₂	C₂×#585	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₄×2₋¹⁺⁴	C₂	semidirect
3. C₄×H(5,𝔽₂)	C₂	semidirect
4. #3428	C₂	semidirect
5. C₂×C₄×#41	C₂	semidirect
6. #585	C₄	direct (N × H)
10. #520	C₂²	semidirect
11. #519	C₂²	semidirect
12. #518	C₂²	semidirect
13. C₄×#41	C₂²	semidirect
14. C₄×#41	C₂²	semidirect
15. C₄×#41	C₂²	semidirect
16. C₂×C₄×Q₈	C₂²	semidirect
17. C₂×C₄×D₄	C₂²	semidirect
18. C₂×#117	C₂²	semidirect
19. C₂²×C₄²	C₂²	semidirect
20. 2₋¹⁺⁴	C₂×C₄	semidirect
21. H(5,𝔽₂)	C₂×C₄	semidirect
22. C₂×#41	C₂×C₄	semidirect
41. #41	C₂²×C₄	semidirect
42. #41	C₂²×C₄	semidirect
43. C₂×Q₈	C₂²×C₄	semidirect
45. C₂×D₄	C₂²×C₄	semidirect
48. C₂²×C₄	C₂²×C₄	semidirect
79. C₄	#585	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×#585	2
C₄×2₋¹⁺⁴	2
C₄×H(5,𝔽₂)	2
#3428	2
C₂×C₄×#41	2

Subgroups — C4×#585 #62651

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#585 #62651