Subgroups — C₂²×Dic₂₂₆₄ #49134374

Order 36224.

360 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₁₁₃₂	C₂	1	2	yes	maximal
2	C₂×Dic₂₂₆₄	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₁₆×C₂₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₅₆₆	—	2	2	no	—
5	C₂×Dic₁₁₃₂	—	2	6	no	—
6	C₂×Dic₁₁₃₂	C₂²	1	12	yes	—
7	Dic₂₂₆₄	C₂²	1	16	yes	direct product factor
8	C₂²×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₂₈₃	C₂²	1	6	yes	—
10	C₂×Dic₅₆₆	—	2	12	no	—
11	C₂×Dic₅₆₆	—	4	6	no	—
12	C₂²×Dic₂₈₃	—	4	2	no	—
13	Dic₁₁₃₂	—	2	12	no	—
14	Dic₁₁₃₂	C₂³	1	8	yes	—
15	C₂²×C₄×C₂₈₃	D₄	1	1	yes	characteristic
16	C₂×C₈×C₂₈₃	D₄	1	3	yes	—
17	C₂×C₈×C₂₈₃	C₂³	1	3	yes	—
18	C₁₆×C₂₈₃	C₂³	1	4	yes	—
19	Dic₅₆₆	—	2	8	no	—
20	Dic₅₆₆	—	4	12	no	—
21	C₂×Dic₂₈₃	—	4	12	no	—
22	C₂³×C₂₈₃	D₈	1	1	yes	characteristic
23	C₂×C₄×C₂₈₃	D₈	1	3	yes	—
24	C₂×C₄×C₂₈₃	C₂×D₄	1	3	yes	—
25	C₈×C₂₈₃	C₂×D₄	1	3	yes	—
26	C₈×C₂₈₃	C₂⁴	1	1	yes	characteristic
27	C₂²×C₂₈₃	Q₃₂	1	4	yes	complemented
28	C₂²×C₂₈₃	C₂×D₈	1	3	yes	—
29	C₄×C₂₈₃	C₂×D₈	1	3	yes	—
30	C₄×C₂₈₃	C₂²×D₄	1	1	yes	characteristic
31	Dic₂₈₃	—	4	8	no	—
32	C₂×C₂₈₃	C₂×Q₃₂	1	6	yes	complemented
33	C₂×C₂₈₃	C₂²×D₈	1	1	yes	characteristic
34	C₂₈₃	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
35	C₂²×Q₃₂	—	283	1	no	maximal, Sylow
36	C₂²×Q₁₆	—	283	2	no	—
37	C₂×Q₃₂	—	283	12	no	—
38	C₂²×C₁₆	D₂₈₃	1	1	yes	characteristic
39	C₂²×Q₈	—	566	2	no	—
40	C₂×Q₁₆	—	283	12	no	—
41	C₂×Q₁₆	—	566	6	no	—
42	C₂²×C₈	C₂×D₂₈₃	1	1	yes	characteristic
43	Q₃₂	—	283	16	no	—
44	C₂×C₁₆	C₂×D₂₈₃	1	6	yes	—
45	C₂×Q₈	—	566	12	no	—
46	C₂×Q₈	—	1132	6	no	—
47	C₂²×C₄	—	1132	2	no	—
48	C₂²×C₄	D₁₁₃₂	1	1	yes	characteristic
49	Q₁₆	—	283	8	no	—
50	Q₁₆	—	566	12	no	—
51	C₂×C₈	D₁₁₃₂	1	3	yes	—
52	C₂×C₈	C₂²×D₂₈₃	1	3	yes	—
53	C₁₆	C₂²×D₂₈₃	1	4	yes	—
54	C₂³	D₂₂₆₄	1	1	yes	characteristic
55	Q₈	—	566	8	no	—
56	Q₈	—	1132	12	no	—
57	C₂×C₄	—	1132	12	no	—
58	C₂×C₄	D₂₂₆₄	1	3	yes	—
59	C₂×C₄	C₂×D₁₁₃₂	1	3	yes	—
60	C₈	C₂×D₁₁₃₂	1	3	yes	—
61	C₈	C₂³×D₂₈₃	1	1	yes	characteristic
62	C₂²	Dic₂₂₆₄	1	4	yes	direct product factor
63	C₂²	C₂×D₂₂₆₄	1	3	yes	—
64	C₄	—	1132	8	no	—
65	C₄	C₂×D₂₂₆₄	1	3	yes	—
66	C₄	C₂²×D₁₁₃₂	1	1	yes	characteristic
67	C₂	C₂×Dic₂₂₆₄	1	6	yes	direct product factor
68	C₂	C₂²×D₂₂₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₂₂₆₄
commutator, cosocle	C₈×C₂₈₃	C₂⁴
Frattini	C₈	C₂³×D₂₈₃
Fitting	C₂²×C₁₆×C₂₈₃	C₂
socle, prime radical	C₂³×C₂₈₃	D₈
hypercenter	C₂²×C₁₆	D₂₈₃
hypocenter	C₂₈₃	C₂²×Q₃₂
LCS	G ▷ C₈×C₂₈₃ ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
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×Dic₂₂₆₄	C₂	direct (N × H)
7. Dic₂₂₆₄	C₂²	direct (N × H)
27. C₂²×C₂₈₃	Q₃₂	semidirect
32. C₂×C₂₈₃	C₂×Q₃₂	semidirect
34. C₂₈₃	C₂²×Q₃₂	semidirect
62. C₂²	Dic₂₂₆₄	direct (N × H)
67. C₂	C₂×Dic₂₂₆₄	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×Dic₁₁₃₂	2
C₂×Dic₂₂₆₄	2
C₂²×C₁₆×C₂₈₃	2
C₂²×Q₃₂	283

Subgroups — C22×Dic2264 #49134374

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Dic₂₂₆₄ #49134374