Subgroups — C₂³×#16860811 #44565164

Order 32128.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860811	C₂	1	56	yes	maximal, direct product factor
2	C₂³×D₁₀₀₄	C₂	1	3	yes	maximal, complemented
3	C₂³×C₄×D₂₅₁	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₂₅₁	C₂²	1	3	yes	—
6	C₂×#16860811	C₂²	1	448	yes	direct product factor
7	C₂²×D₁₀₀₄	C₂²	1	84	yes	complemented
8	C₂³×Dic₂₅₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₂₅₁	C₂²	1	84	yes	complemented
10	C₂²×Q₈×C₂₅₁	C₂²	1	28	yes	complemented
11	C₂³×C₄×C₂₅₁	C₂²	1	3	yes	—
12	C₂³×D₂₅₁	—	2	24	no	—
13	C₂³×D₂₅₁	C₂³	1	42	yes	—
14	#16860811	C₂³	1	512	yes	direct product factor
15	C₂×D₁₀₀₄	C₂³	1	336	yes	complemented
16	C₂²×Dic₂₅₁	C₂³	1	14	yes	—
17	C₂×C₄×D₂₅₁	C₂³	1	336	yes	complemented
18	C₂⁴×C₂₅₁	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₂₅₁	C₂³	1	112	yes	complemented
20	C₂²×C₄×C₂₅₁	C₂³	1	42	yes	—
21	C₂²×D₂₅₁	—	2	168	no	—
22	C₂²×D₂₅₁	C₂⁴	1	84	yes	—
23	D₁₀₀₄	C₂⁴	1	192	yes	complemented
24	C₂×Dic₂₅₁	C₂⁴	1	28	yes	—
25	C₄×D₂₅₁	C₂⁴	1	192	yes	complemented
26	C₂³×C₂₅₁	#41	1	8	yes	complemented
27	C₂³×C₂₅₁	C₂⁴	1	7	yes	—
28	Q₈×C₂₅₁	C₂⁴	1	64	yes	complemented
29	C₂×C₄×C₂₅₁	C₂⁴	1	84	yes	—
30	C₂²×C₂₅₁	C₂×#41	1	28	yes	complemented
31	C₂²×C₂₅₁	C₂⁵	1	7	yes	—
32	C₂×D₂₅₁	—	2	168	no	—
33	C₂×D₂₅₁	C₂⁵	1	24	yes	—
34	C₄×C₂₅₁	C₂⁵	1	24	yes	—
35	Dic₂₅₁	C₂⁵	1	8	yes	—
36	C₂×C₂₅₁	C₂²×#41	1	14	yes	complemented
37	C₂×C₂₅₁	C₂⁶	1	1	yes	characteristic
38	D₂₅₁	—	2	24	no	—
39	C₂₅₁	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
40	C₂³×#41	—	251	1	no	maximal, Sylow
41	C₂²×#41	—	251	56	no	—
42	C₂³×Q₈	D₂₅₁	1	1	yes	characteristic, complemented
43	C₂³×D₄	—	251	3	no	—
44	C₂⁴×C₄	—	251	3	no	—
45	C₂⁵	—	251	3	no	—
46	C₂×#41	—	251	448	no	—
47	C₂²×Q₈	C₂×D₂₅₁	1	28	yes	complemented
48	C₂²×D₄	—	251	84	no	—
49	C₂³×C₄	—	251	1	no	—
50	C₂³×C₄	—	251	84	no	—
51	C₂³×C₄	C₂×D₂₅₁	1	3	yes	—
52	C₂⁴	—	502	24	no	—
53	C₂⁴	—	251	42	no	—
54	C₂⁴	C₂²×D₂₅₁	1	1	yes	characteristic
55	#41	—	251	512	no	—
56	C₂×Q₈	C₂²×D₂₅₁	1	112	yes	complemented
57	C₂×D₄	—	251	336	no	—
58	C₂²×C₄	—	251	14	no	—
59	C₂²×C₄	—	251	336	no	—
60	C₂²×C₄	C₂²×D₂₅₁	1	42	yes	—
61	C₂³	—	502	168	no	—
62	C₂³	—	251	84	no	—
63	C₂³	#16860811	1	8	yes	direct product factor
64	C₂³	C₂³×D₂₅₁	1	7	yes	—
65	Q₈	C₂³×D₂₅₁	1	64	yes	complemented
66	D₄	—	251	192	no	—
67	C₂×C₄	—	251	28	no	—
68	C₂×C₄	—	251	192	no	—
69	C₂×C₄	C₂³×D₂₅₁	1	84	yes	—
70	C₂²	—	502	168	no	—
71	C₂²	—	251	24	no	—
72	C₂²	C₂×#16860811	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₂₅₁	1	7	yes	—
74	C₄	—	251	8	no	—
75	C₄	C₂⁴×D₂₅₁	1	24	yes	—
76	C₂	—	502	24	no	—
77	C₂	C₂²×#16860811	1	14	yes	direct product factor
78	C₂	C₂⁵×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₂₅₁
commutator, cosocle	C₂×C₂₅₁	C₂⁶
Frattini	C₂	C₂⁵×D₂₅₁
Fitting	C₂³×Q₈×C₂₅₁	C₂
socle	C₂⁴×C₂₅₁	C₂³
hypercenter	C₂³×Q₈	D₂₅₁
hypocenter	C₂₅₁	C₂³×#41
LCS	G ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#16860811	C₂	direct (N × H)
2. C₂³×D₁₀₀₄	C₂	semidirect
3. C₂³×C₄×D₂₅₁	C₂	semidirect
4. C₂³×Q₈×C₂₅₁	C₂	semidirect
6. C₂×#16860811	C₂²	direct (N × H)
7. C₂²×D₁₀₀₄	C₂²	semidirect
9. C₂²×C₄×D₂₅₁	C₂²	semidirect
10. C₂²×Q₈×C₂₅₁	C₂²	semidirect
14. #16860811	C₂³	direct (N × H)
15. C₂×D₁₀₀₄	C₂³	semidirect
17. C₂×C₄×D₂₅₁	C₂³	semidirect
19. C₂×Q₈×C₂₅₁	C₂³	semidirect
23. D₁₀₀₄	C₂⁴	semidirect
25. C₄×D₂₅₁	C₂⁴	semidirect
26. C₂³×C₂₅₁	#41	semidirect
28. Q₈×C₂₅₁	C₂⁴	semidirect
30. C₂²×C₂₅₁	C₂×#41	semidirect
36. C₂×C₂₅₁	C₂²×#41	semidirect
39. C₂₅₁	C₂³×#41	semidirect
42. C₂³×Q₈	D₂₅₁	semidirect
47. C₂²×Q₈	C₂×D₂₅₁	semidirect
56. C₂×Q₈	C₂²×D₂₅₁	semidirect
63. C₂³	#16860811	direct (N × H)
65. Q₈	C₂³×D₂₅₁	semidirect
72. C₂²	C₂×#16860811	direct (N × H)
77. C₂	C₂²×#16860811	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×#16860811	2
C₂³×D₁₀₀₄	2
C₂³×C₄×D₂₅₁	2
C₂³×Q₈×C₂₅₁	2
C₂³×#41	251

Subgroups — C23×#16860811 #44565164

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#16860811 #44565164