Subgroups — C₂²×Dic₁₂₅₆ #34113249

Order 20096.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₆₂₈	C₂	1	2	yes	maximal
2	C₂×Dic₁₂₅₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₁₄	—	2	2	no	—
5	C₂×Dic₆₂₈	—	2	6	no	—
6	C₂×Dic₆₂₈	C₂²	1	12	yes	—
7	Dic₁₂₅₆	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₅₇	C₂²	1	6	yes	—
10	C₂×Dic₃₁₄	—	2	12	no	—
11	C₂×Dic₃₁₄	—	4	6	no	—
12	C₂²×Dic₁₅₇	—	4	2	no	—
13	Dic₆₂₈	—	2	12	no	—
14	Dic₆₂₈	C₂³	1	8	yes	—
15	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₅₇	D₄	1	3	yes	—
17	C₂×C₈×C₁₅₇	C₂³	1	3	yes	—
18	C₁₆×C₁₅₇	C₂³	1	4	yes	—
19	Dic₃₁₄	—	2	8	no	—
20	Dic₃₁₄	—	4	12	no	—
21	C₂×Dic₁₅₇	—	4	12	no	—
22	C₂³×C₁₅₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₁₅₇	D₈	1	3	yes	—
24	C₂×C₄×C₁₅₇	C₂×D₄	1	3	yes	—
25	C₈×C₁₅₇	C₂×D₄	1	3	yes	—
26	C₈×C₁₅₇	C₂⁴	1	1	yes	characteristic
27	C₂²×C₁₅₇	Q₃₂	1	4	yes	complemented
28	C₂²×C₁₅₇	C₂×D₈	1	3	yes	—
29	C₄×C₁₅₇	C₂×D₈	1	3	yes	—
30	C₄×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
31	Dic₁₅₇	—	4	8	no	—
32	C₂×C₁₅₇	C₂×Q₃₂	1	6	yes	complemented
33	C₂×C₁₅₇	C₂²×D₈	1	1	yes	characteristic
34	C₁₅₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
35	C₂²×Q₃₂	—	157	1	no	maximal, Sylow
36	C₂²×Q₁₆	—	157	2	no	—
37	C₂×Q₃₂	—	157	12	no	—
38	C₂²×C₁₆	D₁₅₇	1	1	yes	characteristic
39	C₂²×Q₈	—	314	2	no	—
40	C₂×Q₁₆	—	314	6	no	—
41	C₂×Q₁₆	—	157	12	no	—
42	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
43	Q₃₂	—	157	16	no	—
44	C₂×C₁₆	C₂×D₁₅₇	1	6	yes	—
45	C₂×Q₈	—	314	12	no	—
46	C₂×Q₈	—	628	6	no	—
47	C₂²×C₄	—	628	2	no	—
48	C₂²×C₄	D₆₂₈	1	1	yes	characteristic
49	Q₁₆	—	314	12	no	—
50	Q₁₆	—	157	8	no	—
51	C₂×C₈	D₆₂₈	1	3	yes	—
52	C₂×C₈	C₂²×D₁₅₇	1	3	yes	—
53	C₁₆	C₂²×D₁₅₇	1	4	yes	—
54	C₂³	D₁₂₅₆	1	1	yes	characteristic
55	Q₈	—	314	8	no	—
56	Q₈	—	628	12	no	—
57	C₂×C₄	—	628	12	no	—
58	C₂×C₄	D₁₂₅₆	1	3	yes	—
59	C₂×C₄	C₂×D₆₂₈	1	3	yes	—
60	C₈	C₂×D₆₂₈	1	3	yes	—
61	C₈	C₂³×D₁₅₇	1	1	yes	characteristic
62	C₂²	Dic₁₂₅₆	1	4	yes	direct product factor
63	C₂²	C₂×D₁₂₅₆	1	3	yes	—
64	C₄	—	628	8	no	—
65	C₄	C₂×D₁₂₅₆	1	3	yes	—
66	C₄	C₂²×D₆₂₈	1	1	yes	characteristic
67	C₂	C₂×Dic₁₂₅₆	1	6	yes	direct product factor
68	C₂	C₂²×D₁₂₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₁₂₅₆
commutator, cosocle	C₈×C₁₅₇	C₂⁴
Frattini	C₈	C₂³×D₁₅₇
Fitting	C₂²×C₁₆×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	D₈
hypercenter	C₂²×C₁₆	D₁₅₇
hypocenter	C₁₅₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₂₅₆	C₂	direct (N × H)
7. Dic₁₂₅₆	C₂²	direct (N × H)
27. C₂²×C₁₅₇	Q₃₂	semidirect
32. C₂×C₁₅₇	C₂×Q₃₂	semidirect
34. C₁₅₇	C₂²×Q₃₂	semidirect
62. C₂²	Dic₁₂₅₆	direct (N × H)
67. C₂	C₂×Dic₁₂₅₆	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×Dic₆₂₈	2
C₂×Dic₁₂₅₆	2
C₂²×C₁₆×C₁₅₇	2
C₂²×Q₃₂	157

Subgroups — C22×Dic1256 #34113249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₁₂₅₆ #34113249