Subgroups — C₂×#3573 #63075

Order 256.

5429 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	#3573	C₂	1	32	yes	maximal, direct product factor
3	C₂×#568	C₂	1	15	yes	maximal, complemented
4	C₂²×#131	C₂	1	15	yes	maximal, complemented
5	#585	C₄	1	16	yes	—
6	#585	C₂²	1	16	yes	—
7	C₂×2₋¹⁺⁴	C₄	1	6	yes	—
8	C₂×H(5,𝔽₂)	C₄	1	10	yes	—
9	C₂²×#41	C₂²	1	15	yes	—
10	#568	C₂²	1	240	yes	complemented
11	C₂×#131	C₂²	1	20	yes	complemented
12	C₂×#131	C₂²	1	60	yes	complemented
13	C₂×#131	C₂²	1	240	yes	complemented
14	C₂²×M₁₆	C₂²	1	45	yes	complemented
15	C₂³×C₈	C₂²	1	15	yes	complemented
16	2₋¹⁺⁴	C₂×C₄	1	96	yes	—
17	H(5,𝔽₂)	C₂×C₄	1	160	yes	—
18	C₂×#41	C₂×C₄	1	60	yes	—
19	C₂×#41	C₂×C₄	1	120	yes	—
20	C₂×#41	C₂³	1	20	yes	—
21	C₂×#41	C₂³	1	120	yes	—
22	C₂²×Q₈	C₂×C₄	1	15	yes	—
23	C₂²×D₄	C₂×C₄	1	45	yes	—
24	C₂³×C₄	C₂³	1	15	yes	—
25	#131	C₂³	1	160	yes	complemented
26	#131	C₂³	1	480	yes	complemented
27	C₂×M₁₆	C₂³	1	60	yes	—
28	C₂×M₁₆	C₂³	1	360	yes	complemented
29	C₂²×C₈	C₂³	1	45	yes	—
30	C₂²×C₈	C₂³	1	15	yes	—
31	C₂²×C₈	C₂³	1	120	yes	complemented
32	C₂⁴	C₂²×C₄	1	15	yes	—
33	#41	C₂²×C₄	1	80	yes	—
34	#41	C₂²×C₄	1	480	yes	—
35	#41	C₂⁴	1	80	yes	—
36	C₂×Q₈	C₂²×C₄	1	20	yes	—
37	C₂×Q₈	C₂²×C₄	1	120	yes	—
38	C₂×D₄	C₂²×C₄	1	60	yes	—
39	C₂×D₄	C₂²×C₄	1	360	yes	—
40	C₂²×C₄	C₂²×C₄	1	45	yes	—
41	C₂²×C₄	C₂²×C₄	1	60	yes	—
42	C₂²×C₄	C₂⁴	1	15	yes	—
43	C₂²×C₄	C₂⁴	1	60	yes	—
44	M₁₆	C₂⁴	1	240	yes	—
45	C₂×C₈	C₂⁴	1	15	yes	—
46	C₂×C₈	C₂⁴	1	60	yes	—
47	C₂×C₈	C₂⁴	1	180	yes	—
48	C₂×C₈	C₂⁴	1	1	yes	characteristic
49	C₂³	—	4	30	no	—
50	C₂³	C₂³×C₄	1	15	yes	—
51	C₂³	C₂³×C₄	1	60	yes	—
52	Q₈	C₂³×C₄	1	80	yes	—
53	D₄	C₂³×C₄	1	240	yes	—
54	C₂×C₄	C₂³×C₄	1	15	yes	—
55	C₂×C₄	C₂³×C₄	1	30	yes	—
56	C₂×C₄	C₂³×C₄	1	180	yes	—
57	C₂×C₄	C₂⁵	1	1	yes	characteristic
58	C₂×C₄	C₂⁵	1	30	yes	—
59	C₈	C₂⁵	1	2	yes	—
60	C₈	C₂⁵	1	30	yes	—
61	C₂²	—	2	30	no	—
62	C₂²	—	4	60	no	—
63	C₂²	C₂⁴×C₄	1	30	yes	—
64	C₂²	C₂⁴×C₄	1	1	yes	characteristic
65	C₄	C₂⁴×C₄	1	1	yes	characteristic
66	C₄	C₂⁴×C₄	1	30	yes	—
67	C₄	C₂⁶	1	1	yes	characteristic
68	C₂	—	2	30	no	—
69	C₂	#3573	1	2	yes	direct product factor
70	C₂	C₂⁵×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₄	C₂⁶
socle	C₂²	C₂⁴×C₄
prime radical	C₂×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #3573	C₂	direct (N × H)
3. C₂×#568	C₂	semidirect
4. C₂²×#131	C₂	semidirect
10. #568	C₂²	semidirect
11. C₂×#131	C₂²	semidirect
12. C₂×#131	C₂²	semidirect
13. C₂×#131	C₂²	semidirect
14. C₂²×M₁₆	C₂²	semidirect
15. C₂³×C₈	C₂²	semidirect
25. #131	C₂³	semidirect
26. #131	C₂³	semidirect
28. C₂×M₁₆	C₂³	semidirect
31. C₂²×C₈	C₂³	semidirect
69. C₂	#3573	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
#3573	2
C₂×#568	2
C₂²×#131	2

Subgroups — C2×#3573 #63075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#3573 #63075