Subgroups — C₂×Dic₁₅₆ #11928967

Order 1248.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₇₈	C₂	1	2	yes	maximal
2	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
3	Dic₁₅₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×Dic₅₂	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×Dic₃₉	—	2	2	no	—
7	Dic₇₈	—	2	2	no	—
8	Dic₇₈	C₂²	1	4	yes	—
9	C₃×C₈×C₁₃	C₂²	1	2	yes	—
10	C₂×Dic₂₆	—	3	2	no	—
11	C₂×C₈×C₁₃	D₃	1	1	yes	characteristic
12	Dic₅₂	—	3	4	no	—
13	C₂²×C₃×C₁₃	D₄	1	1	yes	characteristic
14	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
15	C₃×C₄×C₁₃	C₂³	1	1	yes	characteristic
16	Dic₃₉	—	2	4	no	—
17	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
18	C₂×Dic₁₃	—	6	2	no	—
19	Dic₂₆	—	3	4	no	—
20	Dic₂₆	—	6	2	no	—
21	C₈×C₁₃	C₂×D₃	1	2	yes	—
22	C₂×Dic₁₂	—	13	1	no	maximal, Hall
23	C₂×C₃×C₁₃	Q₁₆	1	2	yes	complemented
24	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₁₃	D₁₂	1	1	yes	characteristic
26	C₄×C₁₃	D₁₂	1	1	yes	characteristic
27	C₄×C₁₃	C₂²×D₃	1	1	yes	characteristic
28	Dic₁₃	—	6	4	no	—
29	C₂×Dic₆	—	13	2	no	—
30	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic
31	Dic₁₂	—	13	4	no	—
32	C₃×C₁₃	C₂×Q₁₆	1	1	yes	characteristic, complemented, Hall
33	C₂×Q₁₆	—	39	1	no	Sylow
34	C₂×C₁₃	Dic₁₂	1	2	yes	complemented
35	C₂×C₁₃	C₂×D₁₂	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
37	C₂×Dic₃	—	26	2	no	—
38	Dic₆	—	13	4	no	—
39	Dic₆	—	26	2	no	—
40	C₃×C₈	C₂×D₁₃	1	2	yes	—
41	C₂×Q₈	—	39	2	no	—
42	Q₁₆	—	39	4	no	—
43	C₂×C₈	D₃₉	1	1	yes	characteristic
44	C₁₃	C₂×Dic₁₂	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₃	D₅₂	1	1	yes	characteristic
46	C₃×C₄	D₅₂	1	1	yes	characteristic
47	C₃×C₄	C₂²×D₁₃	1	1	yes	characteristic
48	Dic₃	—	26	4	no	—
49	Q₈	—	39	4	no	—
50	Q₈	—	78	2	no	—
51	C₂×C₄	—	78	2	no	—
52	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
53	C₈	C₂×D₃₉	1	2	yes	—
54	C₂×C₃	Dic₅₂	1	2	yes	complemented
55	C₂×C₃	C₂×D₅₂	1	1	yes	characteristic
56	C₂²	D₁₅₆	1	1	yes	characteristic
57	C₄	—	78	4	no	—
58	C₄	D₁₅₆	1	1	yes	characteristic
59	C₄	C₂²×D₃₉	1	1	yes	characteristic
60	C₃	C₂×Dic₅₂	1	1	yes	characteristic, complemented, Sylow
61	C₂	Dic₁₅₆	1	2	yes	direct product factor
62	C₂	C₂×D₁₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₅₆
commutator, cosocle	C₃×C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₃₉
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle, prime radical	C₂²×C₃×C₁₃	D₄
hypercenter	C₂×C₈	D₃₉
hypocenter	C₃×C₁₃	C₂×Q₁₆
LCS	G ▷ C₃×C₄×C₁₃ ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
3. Dic₁₅₆	C₂	direct (N × H)
23. C₂×C₃×C₁₃	Q₁₆	semidirect
32. C₃×C₁₃	C₂×Q₁₆	semidirect
34. C₂×C₁₃	Dic₁₂	semidirect
44. C₁₃	C₂×Dic₁₂	semidirect
54. C₂×C₃	Dic₅₂	semidirect
60. C₃	C₂×Dic₅₂	semidirect
61. C₂	Dic₁₅₆	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₇₈	2
C₂×C₃×C₈×C₁₃	2
Dic₁₅₆	2
C₂×Dic₅₂	3
C₂×Dic₁₂	13

Subgroups — C2×Dic156 #11928967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₁₅₆ #11928967