Subgroups — C₂²×Dic₈₅₆ #28349574

Order 13696.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₈	C₂	1	2	yes	maximal
2	C₂×Dic₈₅₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₂₁₄	—	2	2	no	—
5	C₂×Dic₄₂₈	—	2	6	no	—
6	C₂×Dic₄₂₈	C₂²	1	12	yes	—
7	Dic₈₅₆	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₀₇	C₂²	1	6	yes	—
10	C₂×Dic₂₁₄	—	2	12	no	—
11	C₂×Dic₂₁₄	—	4	6	no	—
12	C₂²×Dic₁₀₇	—	4	2	no	—
13	Dic₄₂₈	—	2	12	no	—
14	Dic₄₂₈	C₂³	1	8	yes	—
15	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
16	C₂×C₈×C₁₀₇	D₄	1	3	yes	—
17	C₂×C₈×C₁₀₇	C₂³	1	3	yes	—
18	C₁₆×C₁₀₇	C₂³	1	4	yes	—
19	C₂³×C₁₀₇	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₀₇	D₈	1	3	yes	—
21	C₂×C₄×C₁₀₇	C₂×D₄	1	3	yes	—
22	C₂×Dic₁₀₇	—	4	12	no	—
23	Dic₂₁₄	—	2	8	no	—
24	Dic₂₁₄	—	4	12	no	—
25	C₈×C₁₀₇	C₂×D₄	1	3	yes	—
26	C₈×C₁₀₇	C₂⁴	1	1	yes	characteristic
27	C₂²×C₁₀₇	Q₃₂	1	4	yes	complemented
28	C₂²×C₁₀₇	C₂×D₈	1	3	yes	—
29	C₄×C₁₀₇	C₂×D₈	1	3	yes	—
30	C₄×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
31	Dic₁₀₇	—	4	8	no	—
32	C₂×C₁₀₇	C₂×Q₃₂	1	6	yes	complemented
33	C₂×C₁₀₇	C₂²×D₈	1	1	yes	characteristic
34	C₂²×Q₃₂	—	107	1	no	maximal, Sylow
35	C₁₀₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
36	C₂²×Q₁₆	—	107	2	no	—
37	C₂×Q₃₂	—	107	12	no	—
38	C₂²×C₁₆	D₁₀₇	1	1	yes	characteristic
39	C₂²×Q₈	—	214	2	no	—
40	C₂×Q₁₆	—	107	12	no	—
41	C₂×Q₁₆	—	214	6	no	—
42	C₂²×C₈	C₂×D₁₀₇	1	1	yes	characteristic
43	Q₃₂	—	107	16	no	—
44	C₂×C₁₆	C₂×D₁₀₇	1	6	yes	—
45	C₂×Q₈	—	428	6	no	—
46	C₂×Q₈	—	214	12	no	—
47	C₂²×C₄	—	428	2	no	—
48	C₂²×C₄	D₄₂₈	1	1	yes	characteristic
49	Q₁₆	—	107	8	no	—
50	Q₁₆	—	214	12	no	—
51	C₂×C₈	D₄₂₈	1	3	yes	—
52	C₂×C₈	C₂²×D₁₀₇	1	3	yes	—
53	C₁₆	C₂²×D₁₀₇	1	4	yes	—
54	C₂³	D₈₅₆	1	1	yes	characteristic
55	Q₈	—	428	12	no	—
56	Q₈	—	214	8	no	—
57	C₂×C₄	—	428	12	no	—
58	C₂×C₄	D₈₅₆	1	3	yes	—
59	C₂×C₄	C₂×D₄₂₈	1	3	yes	—
60	C₈	C₂×D₄₂₈	1	3	yes	—
61	C₈	C₂³×D₁₀₇	1	1	yes	characteristic
62	C₂²	Dic₈₅₆	1	4	yes	direct product factor
63	C₂²	C₂×D₈₅₆	1	3	yes	—
64	C₄	—	428	8	no	—
65	C₄	C₂×D₈₅₆	1	3	yes	—
66	C₄	C₂²×D₄₂₈	1	1	yes	characteristic
67	C₂	C₂×Dic₈₅₆	1	6	yes	direct product factor
68	C₂	C₂²×D₈₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈₅₆
commutator, cosocle	C₈×C₁₀₇	C₂⁴
Frattini	C₈	C₂³×D₁₀₇
Fitting	C₂²×C₁₆×C₁₀₇	C₂
socle, prime radical	C₂³×C₁₀₇	D₈
hypercenter	C₂²×C₁₆	D₁₀₇
hypocenter	C₁₀₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₁₀₇ ▷ C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
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×Dic₈₅₆	C₂	direct (N × H)
7. Dic₈₅₆	C₂²	direct (N × H)
27. C₂²×C₁₀₇	Q₃₂	semidirect
32. C₂×C₁₀₇	C₂×Q₃₂	semidirect
35. C₁₀₇	C₂²×Q₃₂	semidirect
62. C₂²	Dic₈₅₆	direct (N × H)
67. C₂	C₂×Dic₈₅₆	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×Dic₄₂₈	2
C₂×Dic₈₅₆	2
C₂²×C₁₆×C₁₀₇	2
C₂²×Q₃₂	107

Subgroups — C22×Dic856 #28349574

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₈₅₆ #28349574