Subgroups — C₂×Dic₁₆₉₆ #27305142

Order 13568.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₄₈	C₂	1	2	yes	maximal
2	Dic₁₆₉₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₅₃×C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₂₄	—	2	2	no	—
5	Dic₈₄₈	—	2	2	no	—
6	Dic₈₄₈	C₂²	1	4	yes	—
7	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
8	C₅₃×C₆₄	C₂²	1	2	yes	—
9	C₂×Dic₂₁₂	—	4	2	no	—
10	Dic₄₂₄	—	2	4	no	—
11	Dic₄₂₄	—	4	2	no	—
12	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
13	C₃₂×C₅₃	D₄	1	1	yes	characteristic
14	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
15	C₂×Dic₁₀₆	—	8	2	no	—
16	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
17	Dic₂₁₂	—	4	4	no	—
18	Dic₂₁₂	—	8	2	no	—
19	C₁₆×C₅₃	D₈	1	1	yes	characteristic
20	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
21	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
22	C₂×Dic₅₃	—	16	2	no	—
23	Dic₁₀₆	—	8	4	no	—
24	Dic₁₀₆	—	16	2	no	—
25	C₈×C₅₃	D₁₆	1	1	yes	characteristic
26	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
27	C₂×Q₁₂₈	—	53	1	no	maximal, Sylow
28	C₂²×C₅₃	D₃₂	1	1	yes	characteristic
29	C₄×C₅₃	D₃₂	1	1	yes	characteristic
30	C₄×C₅₃	C₂×D₁₆	1	1	yes	characteristic
31	Dic₅₃	—	16	4	no	—
32	C₂×Q₆₄	—	53	2	no	—
33	Q₁₂₈	—	53	4	no	—
34	C₂×C₆₄	D₅₃	1	1	yes	characteristic
35	C₂×C₅₃	Q₁₂₈	1	2	yes	complemented
36	C₂×C₅₃	C₂×D₃₂	1	1	yes	characteristic
37	C₂×Q₃₂	—	106	2	no	—
38	Q₆₄	—	53	4	no	—
39	Q₆₄	—	106	2	no	—
40	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
41	C₆₄	C₂×D₅₃	1	2	yes	—
42	C₅₃	C₂×Q₁₂₈	1	1	yes	characteristic, complemented, Sylow
43	C₂×Q₁₆	—	212	2	no	—
44	Q₃₂	—	106	4	no	—
45	Q₃₂	—	212	2	no	—
46	C₂×C₁₆	D₂₁₂	1	1	yes	characteristic
47	C₃₂	D₂₁₂	1	1	yes	characteristic
48	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
49	C₂×Q₈	—	424	2	no	—
50	Q₁₆	—	424	2	no	—
51	Q₁₆	—	212	4	no	—
52	C₂×C₈	D₄₂₄	1	1	yes	characteristic
53	C₁₆	D₄₂₄	1	1	yes	characteristic
54	C₁₆	C₂×D₂₁₂	1	1	yes	characteristic
55	Q₈	—	848	2	no	—
56	Q₈	—	424	4	no	—
57	C₂×C₄	—	848	2	no	—
58	C₂×C₄	D₈₄₈	1	1	yes	characteristic
59	C₈	D₈₄₈	1	1	yes	characteristic
60	C₈	C₂×D₄₂₄	1	1	yes	characteristic
61	C₂²	D₁₆₉₆	1	1	yes	characteristic
62	C₄	—	848	4	no	—
63	C₄	D₁₆₉₆	1	1	yes	characteristic
64	C₄	C₂×D₈₄₈	1	1	yes	characteristic
65	C₂	Dic₁₆₉₆	1	2	yes	direct product factor
66	C₂	C₂×D₁₆₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₁₆₉₆
commutator, cosocle	C₃₂×C₅₃	C₂³
Frattini	C₃₂	C₂²×D₅₃
Fitting	C₂×C₅₃×C₆₄	C₂
socle, prime radical	C₂²×C₅₃	D₃₂
hypercenter	C₂×C₆₄	D₅₃
hypocenter	C₅₃	C₂×Q₁₂₈
LCS	G ▷ C₃₂×C₅₃ ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. Dic₁₆₉₆	C₂	direct (N × H)
35. C₂×C₅₃	Q₁₂₈	semidirect
42. C₅₃	C₂×Q₁₂₈	semidirect
65. C₂	Dic₁₆₉₆	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×Dic₈₄₈	2
Dic₁₆₉₆	2
C₂×C₅₃×C₆₄	2
C₂×Q₁₂₈	53

Subgroups — C2×Dic1696 #27305142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₁₆₉₆ #27305142