Subgroups — C₄×D₂₂₆₄ #31826549

Order 18112.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₆₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₁₃₂	C₂	1	2	yes	maximal, complemented
3	#22583544	C₂	1	1	yes	maximal, characteristic, complemented
4	#22583532	C₂	1	2	yes	maximal, complemented
5	C₄×C₈×C₂₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₁₃₂	C₂²	1	2	yes	—
7	C₂×C₄×D₂₈₃	—	2	2	no	—
8	D₂₂₆₄	C₄	1	4	yes	direct product factor
9	C₄⋊Dic₂₈₃	C₂²	1	2	yes	—
10	(C₂×D₂₈₃)⋊C₄	—	2	2	no	—
11	C₂×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₂₈₃	C₂²	1	1	yes	characteristic
13	C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
14	C₂²×D₂₈₃	—	2	2	no	—
15	D₁₁₃₂	—	2	2	no	—
16	D₁₁₃₂	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₂₈₃	—	2	2	no	—
18	C₄×D₂₈₃	—	4	4	no	—
19	C₂×C₄×C₂₈₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₂₈₃	D₄	1	1	yes	characteristic
21	C₂×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
22	C₈×C₂₈₃	—	2	1	no	—
23	C₈×C₂₈₃	C₂×C₄	1	2	yes	complemented
24	C₂²×C₂₈₃	C₂×D₄	1	1	yes	characteristic
25	C₂×D₂₈₃	—	2	4	no	—
26	C₂×D₂₈₃	—	4	2	no	—
27	C₂×D₂₈₃	—	4	2	no	—
28	C₄×C₂₈₃	—	2	1	no	—
29	C₄×C₂₈₃	D₈	1	2	yes	complemented
30	C₄×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₂₈₃	#41	1	1	yes	characteristic
32	Dic₂₈₃	—	4	2	no	—
33	C₂×C₂₈₃	C₄×D₄	1	1	yes	characteristic
34	C₂×C₂₈₃	C₂×D₈	1	1	yes	characteristic
35	C₂×C₂₈₃	#135	1	1	yes	characteristic
36	D₂₈₃	—	4	4	no	—
37	C₂₈₃	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
38	C₄×D₈	—	283	1	no	maximal, Sylow
39	C₂×D₈	—	283	1	no	—
40	C₄×D₄	—	283	2	no	—
41	#107	—	283	1	no	—
42	#102	—	283	2	no	—
43	C₄×C₈	D₂₈₃	1	1	yes	characteristic, complemented
44	C₂×D₄	—	283	2	no	—
45	C₂²×C₄	—	566	2	no	—
46	D₈	—	283	4	no	—
47	C₂×C₈	C₂×D₂₈₃	1	1	yes	characteristic
48	C₂×C₈	C₂×D₂₈₃	1	1	yes	characteristic
49	C₄⋊C₄	—	283	2	no	—
50	C₂²⋊C₄	—	566	2	no	—
51	C₄²	C₂×D₂₈₃	1	1	yes	characteristic
52	C₂³	—	566	2	no	—
53	D₄	—	283	4	no	—
54	D₄	—	566	2	no	—
55	C₂×C₄	—	566	2	no	—
56	C₂×C₄	—	1132	4	no	—
57	C₂×C₄	D₁₁₃₂	1	1	yes	characteristic
58	C₂×C₄	D₁₁₃₂	1	1	yes	characteristic
59	C₂×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₄×D₂₈₃	1	2	yes	complemented
62	C₂²	—	566	4	no	—
63	C₂²	—	1132	2	no	—
64	C₂²	—	1132	2	no	—
65	C₂²	C₂×D₁₁₃₂	1	1	yes	characteristic
66	C₄	—	2	1	no	—
67	C₄	—	1132	2	no	—
68	C₄	D₂₂₆₄	1	2	yes	direct product factor
69	C₄	C₂×C₄×D₂₈₃	1	1	yes	characteristic
70	C₄	#17985392	1	1	yes	characteristic
71	C₂	—	1132	4	no	—
72	C₂	C₄×D₁₁₃₂	1	1	yes	characteristic
73	C₂	C₂×D₂₂₆₄	1	1	yes	characteristic
74	C₂	#22583632	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₁₃₂
commutator	C₄×C₂₈₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₂₈₃
Fitting	C₄×C₈×C₂₈₃	C₂
socle	C₂²×C₂₈₃	C₂×D₄
prime radical	C₂×D₂₂₆₄	C₂
cosocle	C₂×C₄×C₂₈₃	C₂³
hypercenter	C₄×C₈	D₂₈₃
hypocenter	C₂₈₃	C₄×D₈
LCS	G ▷ C₄×C₂₈₃ ▷ C₂×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₁₃₂	C₂	semidirect
3. #22583544	C₂	semidirect
4. #22583532	C₂	semidirect
5. C₄×C₈×C₂₈₃	C₂	semidirect
8. D₂₂₆₄	C₄	direct (N × H)
16. D₁₁₃₂	C₂×C₄	semidirect
23. C₈×C₂₈₃	C₂×C₄	semidirect
29. C₄×C₂₈₃	D₈	semidirect
37. C₂₈₃	C₄×D₈	semidirect
43. C₄×C₈	D₂₈₃	semidirect
61. C₈	C₄×D₂₈₃	semidirect
68. C₄	D₂₂₆₄	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×D₂₂₆₄	2
C₄×D₁₁₃₂	2
#22583544	2
#22583532	2
C₄×C₈×C₂₈₃	2
C₄×D₈	283

Subgroups — C4×D2264 #31826549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×D₂₂₆₄ #31826549