Subgroups — C₂³×Dic₅₅₆ #31821127

Order 17792.

1464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₂₇₈	C₂	1	2	yes	maximal
2	C₂²×Dic₅₅₆	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₈×C₁₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₇₈	—	2	14	no	—
5	C₂²×Dic₂₇₈	C₂²	1	28	yes	—
6	C₂³×Dic₁₃₉	—	2	2	no	—
7	C₂×Dic₅₅₆	C₂²	1	112	yes	direct product factor
8	C₂³×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₃₉	C₂²	1	14	yes	—
10	C₂×Dic₂₇₈	—	2	84	no	—
11	C₂×Dic₂₇₈	C₂³	1	56	yes	—
12	C₂²×Dic₁₃₉	—	2	28	no	—
13	Dic₅₅₆	C₂³	1	64	yes	direct product factor
14	C₂⁴×C₁₃₉	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₃₉	D₄	1	7	yes	—
16	C₂²×C₄×C₁₃₉	C₂³	1	7	yes	—
17	C₂×C₈×C₁₃₉	C₂³	1	28	yes	—
18	Dic₂₇₈	—	2	56	no	—
19	Dic₂₇₈	C₂⁴	1	16	yes	—
20	C₂×Dic₁₃₉	—	2	56	no	—
21	C₂³×C₁₃₉	Q₁₆	1	8	yes	complemented
22	C₂³×C₁₃₉	C₂×D₄	1	7	yes	—
23	C₂×C₄×C₁₃₉	C₂×D₄	1	21	yes	—
24	C₂×C₄×C₁₃₉	C₂⁴	1	7	yes	—
25	C₈×C₁₃₉	C₂⁴	1	8	yes	—
26	C₂²×C₁₃₉	C₂×Q₁₆	1	28	yes	complemented
27	C₂²×C₁₃₉	C₂²×D₄	1	7	yes	—
28	C₄×C₁₃₉	C₂²×D₄	1	7	yes	—
29	C₄×C₁₃₉	C₂⁵	1	1	yes	characteristic
30	Dic₁₃₉	—	2	16	no	—
31	C₂×C₁₃₉	C₂²×Q₁₆	1	14	yes	complemented
32	C₂×C₁₃₉	C₂³×D₄	1	1	yes	characteristic
33	C₁₃₉	C₂³×Q₁₆	1	1	yes	characteristic, complemented, Sylow
34	C₂³×Q₁₆	—	139	1	no	maximal, Sylow
35	C₂³×Q₈	—	139	2	no	—
36	C₂²×Q₁₆	—	139	28	no	—
37	C₂³×C₈	D₁₃₉	1	1	yes	characteristic
38	C₂²×Q₈	—	278	14	no	—
39	C₂²×Q₈	—	139	28	no	—
40	C₂³×C₄	—	278	2	no	—
41	C₂³×C₄	C₂×D₁₃₉	1	1	yes	characteristic
42	C₂×Q₁₆	—	139	112	no	—
43	C₂²×C₈	C₂×D₁₃₉	1	14	yes	—
44	C₂⁴	D₅₅₆	1	1	yes	characteristic
45	C₂×Q₈	—	278	84	no	—
46	C₂×Q₈	—	139	56	no	—
47	C₂²×C₄	—	278	28	no	—
48	C₂²×C₄	D₅₅₆	1	7	yes	—
49	C₂²×C₄	C₂²×D₁₃₉	1	7	yes	—
50	Q₁₆	—	139	64	no	—
51	C₂×C₈	C₂²×D₁₃₉	1	28	yes	—
52	C₂³	Dic₅₅₆	1	8	yes	direct product factor
53	C₂³	C₂×D₅₅₆	1	7	yes	—
54	Q₈	—	278	56	no	—
55	Q₈	—	139	16	no	—
56	C₂×C₄	—	278	56	no	—
57	C₂×C₄	C₂×D₅₅₆	1	21	yes	—
58	C₂×C₄	C₂³×D₁₃₉	1	7	yes	—
59	C₈	C₂³×D₁₃₉	1	8	yes	—
60	C₂²	C₂×Dic₅₅₆	1	28	yes	direct product factor
61	C₂²	C₂²×D₅₅₆	1	7	yes	—
62	C₄	—	278	16	no	—
63	C₄	C₂²×D₅₅₆	1	7	yes	—
64	C₄	C₂⁴×D₁₃₉	1	1	yes	characteristic
65	C₂	C₂²×Dic₅₅₆	1	14	yes	direct product factor
66	C₂	C₂³×D₅₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₅₅₆
commutator, cosocle	C₄×C₁₃₉	C₂⁵
Frattini	C₄	C₂⁴×D₁₃₉
Fitting	C₂³×C₈×C₁₃₉	C₂
socle, prime radical	C₂⁴×C₁₃₉	D₄
hypercenter	C₂³×C₈	D₁₃₉
hypocenter	C₁₃₉	C₂³×Q₁₆
LCS	G ▷ C₄×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
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×Dic₅₅₆	C₂	direct (N × H)
7. C₂×Dic₅₅₆	C₂²	direct (N × H)
13. Dic₅₅₆	C₂³	direct (N × H)
21. C₂³×C₁₃₉	Q₁₆	semidirect
26. C₂²×C₁₃₉	C₂×Q₁₆	semidirect
31. C₂×C₁₃₉	C₂²×Q₁₆	semidirect
33. C₁₃₉	C₂³×Q₁₆	semidirect
52. C₂³	Dic₅₅₆	direct (N × H)
60. C₂²	C₂×Dic₅₅₆	direct (N × H)
65. C₂	C₂²×Dic₅₅₆	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×Dic₂₇₈	2
C₂²×Dic₅₅₆	2
C₂³×C₈×C₁₃₉	2
C₂³×Q₁₆	139

Subgroups — C23×Dic556 #31821127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Dic₅₅₆ #31821127