Subgroups — #28384417

Order 14272.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297352	C₂	1	5	yes	maximal, complemented
2	#20297350	C₂	1	10	yes	maximal, complemented
3	C₂×#16813171	C₂	1	5	yes	maximal, complemented
4	#41×D₂₂₃	C₂	1	10	yes	maximal, complemented
5	2₋¹⁺⁴×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	C₂²	1	10	yes	complemented
7	#16813171	C₂²	1	20	yes	complemented
8	#16813170	C₂²	1	30	yes	complemented
9	#16813169	C₂²	1	10	yes	complemented
10	Q₈×D₂₂₃	C₂²	1	10	yes	complemented
11	C₂×D₈₉₂	C₂²	1	15	yes	complemented
12	D₄×D₂₂₃	C₂²	1	30	yes	complemented
13	C₂×C₄×D₂₂₃	C₂²	1	15	yes	complemented
14	#41×C₂₂₃	C₂²	1	10	yes	complemented
15	C₂×Q₈×C₂₂₃	C₂²	1	5	yes	complemented
16	C₂²×D₂₂₃	C₂³	1	15	yes	—
17	Dic₄₄₆	C₂³	1	10	yes	—
18	D₈₉₂	C₂³	1	30	yes	—
19	C₂²⋊D₂₂₃	C₂³	1	20	yes	—
20	C₂×Dic₂₂₃	C₂³	1	5	yes	—
21	C₄×D₂₂₃	C₂³	1	10	yes	—
22	C₄×D₂₂₃	C₂³	1	30	yes	—
23	Q₈×C₂₂₃	C₂³	1	10	yes	—
24	D₄×C₂₂₃	C₂³	1	10	yes	—
25	C₂×C₄×C₂₂₃	C₂³	1	15	yes	—
26	C₂²×C₂₂₃	C₂⁴	1	5	yes	—
27	C₂×D₂₂₃	—	4	15	no	—
28	C₂×D₂₂₃	C₂⁴	1	10	yes	—
29	C₄×C₂₂₃	C₂⁴	1	10	yes	—
30	Dic₂₂₃	C₂⁴	1	1	yes	characteristic
31	Dic₂₂₃	C₂⁴	1	5	yes	—
32	C₂×C₂₂₃	—	2	5	no	—
33	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
34	D₂₂₃	—	2	10	no	—
35	C₂₂₃	#585	1	1	yes	characteristic, complemented, Sylow
36	#585	—	223	1	no	maximal, Sylow
37	2₋¹⁺⁴	—	223	5	no	—
38	2₋¹⁺⁴	D₂₂₃	1	1	yes	characteristic, complemented
39	H(5,𝔽₂)	—	223	10	no	—
40	C₂×#41	—	223	10	no	—
41	C₂×#41	—	223	5	no	—
42	#41	—	223	10	no	—
43	#41	—	223	20	no	—
44	#41	—	223	10	no	—
45	#41	—	223	30	no	—
46	#41	C₂×D₂₂₃	1	10	yes	complemented
47	C₂×Q₈	—	223	10	no	—
48	C₂×Q₈	C₂×D₂₂₃	1	5	yes	complemented
49	C₂×D₄	—	223	30	no	—
50	C₂×D₄	—	223	15	no	—
51	C₂²×C₄	—	223	15	no	—
52	C₂³	—	223	15	no	—
53	Q₈	—	223	10	no	—
54	Q₈	C₂²×D₂₂₃	1	10	yes	—
55	D₄	—	223	20	no	—
56	D₄	—	223	30	no	—
57	D₄	C₂²×D₂₂₃	1	10	yes	—
58	C₂×C₄	—	223	5	no	—
59	C₂×C₄	—	223	10	no	—
60	C₂×C₄	—	223	30	no	—
61	C₂×C₄	C₂²×D₂₂₃	1	15	yes	—
62	C₂²	—	892	15	no	—
63	C₂²	—	223	10	no	—
64	C₂²	C₂³×D₂₂₃	1	5	yes	—
65	C₄	—	223	1	no	—
66	C₄	—	223	5	no	—
67	C₄	C₂³×D₂₂₃	1	10	yes	—
68	C₂	—	2	5	no	—
69	C₂	—	446	10	no	—
70	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₂₂₃
commutator, socle, cosocle	C₂×C₂₂₃	C₂⁵
Fitting	2₋¹⁺⁴×C₂₂₃	C₂
hypercenter	2₋¹⁺⁴	D₂₂₃
hypocenter	C₂₂₃	#585
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ 2₋¹⁺⁴

Complements

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

N	H ≅ G/N	type
1. #20297352	C₂	semidirect
2. #20297350	C₂	semidirect
3. C₂×#16813171	C₂	semidirect
4. #41×D₂₂₃	C₂	semidirect
5. 2₋¹⁺⁴×C₂₂₃	C₂	semidirect
6. #16813171	C₂²	semidirect
7. #16813171	C₂²	semidirect
8. #16813170	C₂²	semidirect
9. #16813169	C₂²	semidirect
10. Q₈×D₂₂₃	C₂²	semidirect
11. C₂×D₈₉₂	C₂²	semidirect
12. D₄×D₂₂₃	C₂²	semidirect
13. C₂×C₄×D₂₂₃	C₂²	semidirect
14. #41×C₂₂₃	C₂²	semidirect
15. C₂×Q₈×C₂₂₃	C₂²	semidirect
35. C₂₂₃	#585	semidirect
38. 2₋¹⁺⁴	D₂₂₃	semidirect
46. #41	C₂×D₂₂₃	semidirect
48. C₂×Q₈	C₂×D₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20297352	2
#20297350	2
C₂×#16813171	2
#41×D₂₂₃	2
2₋¹⁺⁴×C₂₂₃	2
#585	223