Subgroups — #41×D₂₂₃ #20297345

Order 7136.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16813171	C₂	1	1	yes	maximal, characteristic, complemented
2	#16813170	C₂	1	3	yes	maximal, complemented
3	#16813169	C₂	1	3	yes	maximal, complemented
4	Q₈×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×D₂₂₃	C₂	1	3	yes	maximal, complemented
6	C₂×C₄×D₂₂₃	C₂	1	3	yes	maximal, complemented
7	#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₂₂₃	C₂²	1	3	yes	—
9	Dic₄₄₆	C₂²	1	3	yes	complemented
10	D₈₉₂	C₂²	1	3	yes	complemented
11	C₂²⋊D₂₂₃	C₂²	1	6	yes	complemented
12	C₂×Dic₂₂₃	C₂²	1	3	yes	complemented
13	C₄×D₂₂₃	C₂²	1	1	yes	characteristic
14	C₄×D₂₂₃	C₂²	1	3	yes	—
15	C₄×D₂₂₃	C₂²	1	3	yes	complemented
16	C₄×D₂₂₃	C₂²	1	3	yes	complemented
17	Q₈×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
18	D₄×C₂₂₃	C₂²	1	3	yes	complemented
19	C₂×C₄×C₂₂₃	C₂²	1	3	yes	complemented
20	C₂²×C₂₂₃	C₂³	1	3	yes	—
21	C₂×D₂₂₃	—	2	6	no	—
22	C₂×D₂₂₃	C₂³	1	3	yes	—
23	C₂×D₂₂₃	C₂³	1	1	yes	characteristic
24	C₄×C₂₂₃	C₂³	1	3	yes	—
25	C₄×C₂₂₃	C₂³	1	1	yes	characteristic
26	Dic₂₂₃	C₂³	1	1	yes	characteristic
27	Dic₂₂₃	C₂³	1	3	yes	—
28	C₂×C₂₂₃	—	2	3	no	—
29	C₂×C₂₂₃	C₂⁴	1	1	yes	characteristic
30	D₂₂₃	—	2	3	no	—
31	D₂₂₃	#41	1	2	yes	direct product factor
32	C₂₂₃	C₂×#41	1	1	yes	characteristic, complemented, Sylow
33	C₂×#41	—	223	1	no	maximal, Sylow
34	#41	—	223	1	no	—
35	#41	—	223	3	no	—
36	#41	—	223	3	no	—
37	#41	D₂₂₃	1	1	yes	characteristic, direct product factor
38	C₂×Q₈	—	223	1	no	—
39	C₂×D₄	—	223	3	no	—
40	C₂²×C₄	—	223	3	no	—
41	C₂³	—	223	3	no	—
42	Q₈	—	223	3	no	—
43	Q₈	C₂×D₂₂₃	1	1	yes	characteristic, complemented
44	D₄	—	223	6	no	—
45	D₄	—	223	3	no	—
46	D₄	C₂×D₂₂₃	1	3	yes	complemented
47	C₂×C₄	—	223	3	no	—
48	C₂×C₄	—	223	3	no	—
49	C₂×C₄	—	223	3	no	—
50	C₂×C₄	—	223	1	no	—
51	C₂×C₄	—	223	3	no	—
52	C₂×C₄	C₂×D₂₂₃	1	3	yes	complemented
53	C₂²	—	446	6	no	—
54	C₂²	—	223	3	no	—
55	C₂²	—	223	1	no	—
56	C₂²	C₂²×D₂₂₃	1	3	yes	—
57	C₄	—	223	1	no	—
58	C₄	—	223	3	no	—
59	C₄	C₂²×D₂₂₃	1	3	yes	—
60	C₄	C₂²×D₂₂₃	1	1	yes	characteristic
61	C₂	—	2	3	no	—
62	C₂	—	446	3	no	—
63	C₂	—	223	2	no	—
64	C₂	C₂³×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×D₂₂₃
commutator, socle	C₂×C₂₂₃	C₂⁴
Frattini	C₂	C₂³×D₂₂₃
Fitting	#41×C₂₂₃	C₂
hypercenter	#41	D₂₂₃
hypocenter	C₂₂₃	C₂×#41
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #16813171	C₂	semidirect
2. #16813170	C₂	semidirect
3. #16813169	C₂	semidirect
4. Q₈×D₂₂₃	C₂	semidirect
5. D₄×D₂₂₃	C₂	semidirect
6. C₂×C₄×D₂₂₃	C₂	semidirect
7. #41×C₂₂₃	C₂	semidirect
9. Dic₄₄₆	C₂²	semidirect
10. D₈₉₂	C₂²	semidirect
11. C₂²⋊D₂₂₃	C₂²	semidirect
12. C₂×Dic₂₂₃	C₂²	semidirect
15. C₄×D₂₂₃	C₂²	semidirect
16. C₄×D₂₂₃	C₂²	semidirect
17. Q₈×C₂₂₃	C₂²	semidirect
18. D₄×C₂₂₃	C₂²	semidirect
19. C₂×C₄×C₂₂₃	C₂²	semidirect
31. D₂₂₃	#41	direct (N × H)
32. C₂₂₃	C₂×#41	semidirect
37. #41	D₂₂₃	direct (N × H)
43. Q₈	C₂×D₂₂₃	semidirect
46. D₄	C₂×D₂₂₃	semidirect
52. C₂×C₄	C₂×D₂₂₃	semidirect

Subgroups — #41×D223 #20297345

Conjugacy classes of subgroups

Complements

Subgroups — #41×D₂₂₃ #20297345