Subgroups — C₂³×Dic₅₂₄ #30680684

Order 16768.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₆₂	C₂	1	2	yes	maximal
2	C₂²×Dic₅₂₄	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₆₂	—	2	14	no	—
5	C₂²×Dic₂₆₂	C₂²	1	28	yes	—
6	C₂³×Dic₁₃₁	—	2	2	no	—
7	C₂×Dic₅₂₄	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₃₁	C₂²	1	14	yes	—
10	C₂×Dic₂₆₂	—	2	84	no	—
11	C₂×Dic₂₆₂	C₂³	1	56	yes	—
12	C₂²×Dic₁₃₁	—	2	28	no	—
13	Dic₅₂₄	C₂³	1	64	yes	direct product factor
14	C₂⁴×C₁₃₁	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₃₁	D₄	1	7	yes	—
16	C₂²×C₄×C₁₃₁	C₂³	1	7	yes	—
17	C₂×C₈×C₁₃₁	C₂³	1	28	yes	—
18	Dic₂₆₂	—	2	56	no	—
19	Dic₂₆₂	C₂⁴	1	16	yes	—
20	C₂×Dic₁₃₁	—	2	56	no	—
21	C₂³×C₁₃₁	Q₁₆	1	8	yes	complemented
22	C₂³×C₁₃₁	C₂×D₄	1	7	yes	—
23	C₂×C₄×C₁₃₁	C₂×D₄	1	21	yes	—
24	C₂×C₄×C₁₃₁	C₂⁴	1	7	yes	—
25	C₈×C₁₃₁	C₂⁴	1	8	yes	—
26	C₂²×C₁₃₁	C₂×Q₁₆	1	28	yes	complemented
27	C₂²×C₁₃₁	C₂²×D₄	1	7	yes	—
28	C₄×C₁₃₁	C₂²×D₄	1	7	yes	—
29	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
30	Dic₁₃₁	—	2	16	no	—
31	C₂×C₁₃₁	C₂²×Q₁₆	1	14	yes	complemented
32	C₂×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
33	C₁₃₁	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂³×Q₁₆	—	131	1	no	maximal, Sylow
35	C₂³×Q₈	—	131	2	no	—
36	C₂²×Q₁₆	—	131	28	no	—
37	C₂³×C₈	D₁₃₁	1	1	yes	characteristic
38	C₂²×Q₈	—	262	14	no	—
39	C₂²×Q₈	—	131	28	no	—
40	C₂³×C₄	—	262	2	no	—
41	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
42	C₂×Q₁₆	—	131	112	no	—
43	C₂²×C₈	C₂×D₁₃₁	1	14	yes	—
44	C₂⁴	D₅₂₄	1	1	yes	characteristic
45	C₂×Q₈	—	262	84	no	—
46	C₂×Q₈	—	131	56	no	—
47	C₂²×C₄	—	262	28	no	—
48	C₂²×C₄	D₅₂₄	1	7	yes	—
49	C₂²×C₄	C₂²×D₁₃₁	1	7	yes	—
50	Q₁₆	—	131	64	no	—
51	C₂×C₈	C₂²×D₁₃₁	1	28	yes	—
52	C₂³	Dic₅₂₄	1	8	yes	direct product factor
53	C₂³	C₂×D₅₂₄	1	7	yes	—
54	Q₈	—	262	56	no	—
55	Q₈	—	131	16	no	—
56	C₂×C₄	—	262	56	no	—
57	C₂×C₄	C₂×D₅₂₄	1	21	yes	—
58	C₂×C₄	C₂³×D₁₃₁	1	7	yes	—
59	C₈	C₂³×D₁₃₁	1	8	yes	—
60	C₂²	C₂×Dic₅₂₄	1	28	yes	direct product factor
61	C₂²	C₂²×D₅₂₄	1	7	yes	—
62	C₄	—	262	16	no	—
63	C₄	C₂²×D₅₂₄	1	7	yes	—
64	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
65	C₂	C₂²×Dic₅₂₄	1	14	yes	direct product factor
66	C₂	C₂³×D₅₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₂₄
commutator, cosocle	C₄×C₁₃₁	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂³×C₈×C₁₃₁	C₂
socle, prime radical	C₂⁴×C₁₃₁	D₄
hypercenter	C₂³×C₈	D₁₃₁
hypocenter	C₁₃₁	C₂³×Q₁₆
LCS	G ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×Dic₅₂₄	C₂	direct (N × H)
7. C₂×Dic₅₂₄	C₂²	direct (N × H)
13. Dic₅₂₄	C₂³	direct (N × H)
21. C₂³×C₁₃₁	Q₁₆	semidirect
26. C₂²×C₁₃₁	C₂×Q₁₆	semidirect
31. C₂×C₁₃₁	C₂²×Q₁₆	semidirect
33. C₁₃₁	C₂³×Q₁₆	semidirect
52. C₂³	Dic₅₂₄	direct (N × H)
60. C₂²	C₂×Dic₅₂₄	direct (N × H)
65. C₂	C₂²×Dic₅₂₄	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×Dic₂₆₂	2
C₂²×Dic₅₂₄	2
C₂³×C₈×C₁₃₁	2
C₂³×Q₁₆	131

Subgroups — C23×Dic524 #30680684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Dic₅₂₄ #30680684