Subgroups — C₂³×#14523514 #30680760

Order 16768.

6520 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523514	C₂	1	56	yes	maximal, direct product factor
2	C₂³×D₅₂₄	C₂	1	3	yes	maximal, complemented
3	C₂³×C₄×D₁₃₁	C₂	1	3	yes	maximal, complemented
4	C₂³×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁴×D₁₃₁	C₂²	1	3	yes	—
6	C₂×#14523514	C₂²	1	448	yes	direct product factor
7	C₂²×D₅₂₄	C₂²	1	84	yes	complemented
8	C₂³×Dic₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄×D₁₃₁	C₂²	1	84	yes	complemented
10	C₂²×Q₈×C₁₃₁	C₂²	1	28	yes	complemented
11	C₂³×C₄×C₁₃₁	C₂²	1	3	yes	—
12	C₂³×D₁₃₁	—	2	24	no	—
13	C₂³×D₁₃₁	C₂³	1	42	yes	—
14	#14523514	C₂³	1	512	yes	direct product factor
15	C₂×D₅₂₄	C₂³	1	336	yes	complemented
16	C₂²×Dic₁₃₁	C₂³	1	14	yes	—
17	C₂×C₄×D₁₃₁	C₂³	1	336	yes	complemented
18	C₂⁴×C₁₃₁	C₂³	1	1	yes	characteristic
19	C₂×Q₈×C₁₃₁	C₂³	1	112	yes	complemented
20	C₂²×C₄×C₁₃₁	C₂³	1	42	yes	—
21	C₂²×D₁₃₁	—	2	168	no	—
22	C₂²×D₁₃₁	C₂⁴	1	84	yes	—
23	D₅₂₄	C₂⁴	1	192	yes	complemented
24	C₂×Dic₁₃₁	C₂⁴	1	28	yes	—
25	C₄×D₁₃₁	C₂⁴	1	192	yes	complemented
26	C₂³×C₁₃₁	#41	1	8	yes	complemented
27	C₂³×C₁₃₁	C₂⁴	1	7	yes	—
28	Q₈×C₁₃₁	C₂⁴	1	64	yes	complemented
29	C₂×C₄×C₁₃₁	C₂⁴	1	84	yes	—
30	C₂²×C₁₃₁	C₂×#41	1	28	yes	complemented
31	C₂²×C₁₃₁	C₂⁵	1	7	yes	—
32	C₂×D₁₃₁	—	2	168	no	—
33	C₂×D₁₃₁	C₂⁵	1	24	yes	—
34	C₄×C₁₃₁	C₂⁵	1	24	yes	—
35	Dic₁₃₁	C₂⁵	1	8	yes	—
36	C₂×C₁₃₁	C₂²×#41	1	14	yes	complemented
37	C₂×C₁₃₁	C₂⁶	1	1	yes	characteristic
38	D₁₃₁	—	2	24	no	—
39	C₁₃₁	C₂³×#41	1	1	yes	characteristic, complemented, Sylow
40	C₂³×#41	—	131	1	no	maximal, Sylow
41	C₂²×#41	—	131	56	no	—
42	C₂³×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
43	C₂³×D₄	—	131	3	no	—
44	C₂⁴×C₄	—	131	3	no	—
45	C₂⁵	—	131	3	no	—
46	C₂×#41	—	131	448	no	—
47	C₂²×Q₈	C₂×D₁₃₁	1	28	yes	complemented
48	C₂²×D₄	—	131	84	no	—
49	C₂³×C₄	—	131	1	no	—
50	C₂³×C₄	—	131	84	no	—
51	C₂³×C₄	C₂×D₁₃₁	1	3	yes	—
52	C₂⁴	—	262	24	no	—
53	C₂⁴	—	131	42	no	—
54	C₂⁴	C₂²×D₁₃₁	1	1	yes	characteristic
55	#41	—	131	512	no	—
56	C₂×Q₈	C₂²×D₁₃₁	1	112	yes	complemented
57	C₂×D₄	—	131	336	no	—
58	C₂²×C₄	—	131	14	no	—
59	C₂²×C₄	—	131	336	no	—
60	C₂²×C₄	C₂²×D₁₃₁	1	42	yes	—
61	C₂³	—	262	168	no	—
62	C₂³	—	131	84	no	—
63	C₂³	#14523514	1	8	yes	direct product factor
64	C₂³	C₂³×D₁₃₁	1	7	yes	—
65	Q₈	C₂³×D₁₃₁	1	64	yes	complemented
66	D₄	—	131	192	no	—
67	C₂×C₄	—	131	28	no	—
68	C₂×C₄	—	131	192	no	—
69	C₂×C₄	C₂³×D₁₃₁	1	84	yes	—
70	C₂²	—	262	168	no	—
71	C₂²	—	131	24	no	—
72	C₂²	C₂×#14523514	1	28	yes	direct product factor
73	C₂²	C₂⁴×D₁₃₁	1	7	yes	—
74	C₄	—	131	8	no	—
75	C₄	C₂⁴×D₁₃₁	1	24	yes	—
76	C₂	—	262	24	no	—
77	C₂	C₂²×#14523514	1	14	yes	direct product factor
78	C₂	C₂⁵×D₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂²×D₁₃₁
commutator, cosocle	C₂×C₁₃₁	C₂⁶
Frattini	C₂	C₂⁵×D₁₃₁
Fitting	C₂³×Q₈×C₁₃₁	C₂
socle	C₂⁴×C₁₃₁	C₂³
hypercenter	C₂³×Q₈	D₁₃₁
hypocenter	C₁₃₁	C₂³×#41
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523514	C₂	direct (N × H)
2. C₂³×D₅₂₄	C₂	semidirect
3. C₂³×C₄×D₁₃₁	C₂	semidirect
4. C₂³×Q₈×C₁₃₁	C₂	semidirect
6. C₂×#14523514	C₂²	direct (N × H)
7. C₂²×D₅₂₄	C₂²	semidirect
9. C₂²×C₄×D₁₃₁	C₂²	semidirect
10. C₂²×Q₈×C₁₃₁	C₂²	semidirect
14. #14523514	C₂³	direct (N × H)
15. C₂×D₅₂₄	C₂³	semidirect
17. C₂×C₄×D₁₃₁	C₂³	semidirect
19. C₂×Q₈×C₁₃₁	C₂³	semidirect
23. D₅₂₄	C₂⁴	semidirect
25. C₄×D₁₃₁	C₂⁴	semidirect
26. C₂³×C₁₃₁	#41	semidirect
28. Q₈×C₁₃₁	C₂⁴	semidirect
30. C₂²×C₁₃₁	C₂×#41	semidirect
36. C₂×C₁₃₁	C₂²×#41	semidirect
39. C₁₃₁	C₂³×#41	semidirect
42. C₂³×Q₈	D₁₃₁	semidirect
47. C₂²×Q₈	C₂×D₁₃₁	semidirect
56. C₂×Q₈	C₂²×D₁₃₁	semidirect
63. C₂³	#14523514	direct (N × H)
65. Q₈	C₂³×D₁₃₁	semidirect
72. C₂²	C₂×#14523514	direct (N × H)
77. C₂	C₂²×#14523514	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×#14523514	2
C₂³×D₅₂₄	2
C₂³×C₄×D₁₃₁	2
C₂³×Q₈×C₁₃₁	2
C₂³×#41	131

Subgroups — C23×#14523514 #30680760

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523514 #30680760