Subgroups — C₂³×#131 #63073

Order 256.

5711 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41	C₂	1	1	yes	maximal, characteristic
2	C₂²×#131	C₂	1	56	yes	maximal, direct product factor
3	C₂³×M₁₆	C₂	1	3	yes	maximal, complemented
4	C₂⁴×C₈	C₂	1	3	yes	maximal, complemented
5	C₂²×#41	C₄	1	28	yes	—
6	C₂²×#41	C₂²	1	28	yes	—
7	C₂³×Q₈	C₄	1	1	yes	characteristic
8	C₂³×D₄	C₄	1	3	yes	—
9	C₂⁴×C₄	C₂²	1	3	yes	—
10	C₂×#131	C₂²	1	448	yes	direct product factor
11	C₂²×M₁₆	C₂²	1	84	yes	complemented
12	C₂³×C₈	C₂²	1	3	yes	—
13	C₂³×C₈	C₂²	1	1	yes	characteristic
14	C₂³×C₈	C₂²	1	84	yes	complemented
15	C₂⁵	C₂×C₄	1	3	yes	—
16	C₂×#41	C₂×C₄	1	336	yes	—
17	C₂×#41	C₂³	1	112	yes	—
18	C₂²×Q₈	C₂×C₄	1	28	yes	—
19	C₂²×D₄	C₂×C₄	1	84	yes	—
20	C₂³×C₄	C₂×C₄	1	3	yes	—
21	C₂³×C₄	C₂×C₄	1	42	yes	—
22	C₂³×C₄	C₂³	1	1	yes	characteristic
23	C₂³×C₄	C₂³	1	42	yes	—
24	#131	C₂³	1	512	yes	direct product factor
25	C₂×M₁₆	C₂³	1	336	yes	complemented
26	C₂²×C₈	C₂³	1	42	yes	—
27	C₂²×C₈	C₂³	1	14	yes	—
28	C₂²×C₈	C₂³	1	336	yes	complemented
29	C₂⁴	—	2	24	no	—
30	C₂⁴	C₂²×C₄	1	42	yes	—
31	C₂⁴	C₂²×C₄	1	1	yes	characteristic
32	#41	C₂²×C₄	1	448	yes	—
33	#41	C₂⁴	1	64	yes	—
34	C₂×Q₈	C₂²×C₄	1	112	yes	—
35	C₂×D₄	C₂²×C₄	1	336	yes	—
36	C₂²×C₄	C₂²×C₄	1	7	yes	—
37	C₂²×C₄	C₂²×C₄	1	42	yes	—
38	C₂²×C₄	C₂²×C₄	1	252	yes	—
39	C₂²×C₄	C₂⁴	1	7	yes	—
40	C₂²×C₄	C₂⁴	1	84	yes	—
41	M₁₆	C₂⁴	1	192	yes	complemented
42	C₂×C₈	C₂⁴	1	28	yes	—
43	C₂×C₈	C₂⁴	1	84	yes	—
44	C₂×C₈	C₂⁴	1	192	yes	complemented
45	C₂³	—	2	168	no	—
46	C₂³	#131	1	8	yes	direct product factor
47	C₂³	C₂³×C₄	1	84	yes	—
48	C₂³	C₂³×C₄	1	7	yes	—
49	Q₈	C₂³×C₄	1	64	yes	—
50	D₄	C₂³×C₄	1	192	yes	—
51	C₂×C₄	C₂³×C₄	1	21	yes	—
52	C₂×C₄	C₂³×C₄	1	84	yes	—
53	C₂×C₄	C₂³×C₄	1	168	yes	—
54	C₂×C₄	C₂⁵	1	7	yes	—
55	C₂×C₄	C₂⁵	1	24	yes	—
56	C₈	C₂⁵	1	8	yes	—
57	C₈	C₂⁵	1	24	yes	—
58	C₂²	—	2	168	no	—
59	C₂²	C₂×#131	1	28	yes	direct product factor
60	C₂²	C₂⁴×C₄	1	24	yes	—
61	C₂²	C₂⁴×C₄	1	7	yes	—
62	C₄	C₂⁴×C₄	1	7	yes	—
63	C₄	C₂⁴×C₄	1	24	yes	—
64	C₄	C₂⁶	1	1	yes	characteristic
65	C₂	—	2	24	no	—
66	C₂	C₂²×#131	1	14	yes	direct product factor
67	C₂	C₂⁵×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₈	C₂²
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₄	C₂⁶
socle	C₂⁴	C₂²×C₄
prime radical	C₂³×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×#131	C₂	direct (N × H)
3. C₂³×M₁₆	C₂	semidirect
4. C₂⁴×C₈	C₂	semidirect
10. C₂×#131	C₂²	direct (N × H)
11. C₂²×M₁₆	C₂²	semidirect
14. C₂³×C₈	C₂²	semidirect
24. #131	C₂³	direct (N × H)
25. C₂×M₁₆	C₂³	semidirect
28. C₂²×C₈	C₂³	semidirect
41. M₁₆	C₂⁴	semidirect
44. C₂×C₈	C₂⁴	semidirect
46. C₂³	#131	direct (N × H)
59. C₂²	C₂×#131	direct (N × H)
66. C₂	C₂²×#131	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#41	2
C₂²×#131	2
C₂³×M₁₆	2
C₂⁴×C₈	2

Subgroups — C23×#131 #63073

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#131 #63073