Subgroups — #28384415

Order 14272.

810 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297351	C₂	1	6	yes	maximal, complemented
2	#20297349	C₂	1	9	yes	maximal, complemented
3	C₂×#16813169	C₂	1	9	yes	maximal, complemented
4	#41×D₂₂₃	C₂	1	6	yes	maximal, complemented
5	H(5,𝔽₂)×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#16813171	C₂²	1	2	yes	complemented
7	#16813170	C₂²	1	18	yes	complemented
8	#16813169	C₂²	1	18	yes	complemented
9	#16813169	C₂²	1	36	yes	complemented
10	C₂×Dic₄₄₆	C₂²	1	9	yes	complemented
11	Q₈×D₂₂₃	C₂²	1	6	yes	complemented
12	C₂×C₂²⋊D₂₂₃	C₂²	1	18	yes	complemented
13	D₄×D₂₂₃	C₂²	1	18	yes	complemented
14	C₂²×Dic₂₂₃	C₂²	1	6	yes	complemented
15	C₂×C₄×D₂₂₃	C₂²	1	9	yes	complemented
16	#41×C₂₂₃	C₂²	1	6	yes	complemented
17	C₂×D₄×C₂₂₃	C₂²	1	9	yes	complemented
18	C₂²×D₂₂₃	C₂³	1	9	yes	—
19	Dic₄₄₆	C₂³	1	18	yes	—
20	D₈₉₂	C₂³	1	6	yes	—
21	C₂²⋊D₂₂₃	C₂³	1	36	yes	—
22	C₂×Dic₂₂₃	C₂³	1	9	yes	—
23	C₂×Dic₂₂₃	C₂³	1	18	yes	—
24	C₄×D₂₂₃	C₂³	1	6	yes	—
25	C₄×D₂₂₃	C₂³	1	18	yes	—
26	C₂³×C₂₂₃	C₂³	1	6	yes	—
27	Q₈×C₂₂₃	C₂³	1	2	yes	—
28	D₄×C₂₂₃	C₂³	1	18	yes	—
29	C₂×C₄×C₂₂₃	C₂³	1	9	yes	—
30	C₂²×C₂₂₃	—	4	6	no	—
31	C₂²×C₂₂₃	C₂⁴	1	9	yes	—
32	C₂×D₂₂₃	—	4	9	no	—
33	C₂×D₂₂₃	C₂⁴	1	6	yes	—
34	C₄×C₂₂₃	C₂⁴	1	6	yes	—
35	Dic₂₂₃	C₂⁴	1	1	yes	characteristic
36	Dic₂₂₃	C₂⁴	1	9	yes	—
37	C₂×C₂₂₃	—	2	9	no	—
38	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
39	D₂₂₃	—	2	6	no	—
40	C₂₂₃	#585	1	1	yes	characteristic, complemented, Sylow
41	#585	—	223	1	no	maximal, Sylow
42	2₋¹⁺⁴	—	223	6	no	—
43	H(5,𝔽₂)	—	223	9	no	—
44	H(5,𝔽₂)	D₂₂₃	1	1	yes	characteristic, complemented
45	C₂×#41	—	223	6	no	—
46	C₂×#41	—	223	9	no	—
47	#41	—	223	2	no	—
48	#41	—	223	18	no	—
49	#41	—	223	36	no	—
50	#41	—	223	18	no	—
51	#41	C₂×D₂₂₃	1	6	yes	complemented
52	C₂×Q₈	—	223	6	no	—
53	C₂×Q₈	—	223	9	no	—
54	C₂×D₄	—	223	18	no	—
55	C₂×D₄	—	223	18	no	—
56	C₂×D₄	C₂×D₂₂₃	1	9	yes	complemented
57	C₂²×C₄	—	223	6	no	—
58	C₂²×C₄	—	223	9	no	—
59	C₂³	—	223	9	no	—
60	C₂³	C₂²×D₂₂₃	1	6	yes	—
61	Q₈	—	223	18	no	—
62	Q₈	C₂²×D₂₂₃	1	2	yes	—
63	D₄	—	223	36	no	—
64	D₄	—	223	6	no	—
65	D₄	C₂²×D₂₂₃	1	18	yes	—
66	C₂×C₄	—	223	9	no	—
67	C₂×C₄	—	223	18	no	—
68	C₂×C₄	—	223	6	no	—
69	C₂×C₄	—	223	18	no	—
70	C₂×C₄	C₂²×D₂₂₃	1	9	yes	—
71	C₂²	—	4	6	no	—
72	C₂²	—	892	9	no	—
73	C₂²	—	223	6	no	—
74	C₂²	C₂³×D₂₂₃	1	9	yes	—
75	C₄	—	223	1	no	—
76	C₄	—	223	9	no	—
77	C₄	C₂³×D₂₂₃	1	6	yes	—
78	C₂	—	2	9	no	—
79	C₂	—	446	6	no	—
80	C₂	C₂⁴×D₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₂₂₃
commutator, socle, cosocle	C₂×C₂₂₃	C₂⁵
Fitting	H(5,𝔽₂)×C₂₂₃	C₂
hypercenter	H(5,𝔽₂)	D₂₂₃
hypocenter	C₂₂₃	#585
LCS	G ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #20297351	C₂	semidirect
2. #20297349	C₂	semidirect
3. C₂×#16813169	C₂	semidirect
4. #41×D₂₂₃	C₂	semidirect
5. H(5,𝔽₂)×C₂₂₃	C₂	semidirect
6. #16813171	C₂²	semidirect
7. #16813170	C₂²	semidirect
8. #16813169	C₂²	semidirect
9. #16813169	C₂²	semidirect
10. C₂×Dic₄₄₆	C₂²	semidirect
11. Q₈×D₂₂₃	C₂²	semidirect
12. C₂×C₂²⋊D₂₂₃	C₂²	semidirect
13. D₄×D₂₂₃	C₂²	semidirect
14. C₂²×Dic₂₂₃	C₂²	semidirect
15. C₂×C₄×D₂₂₃	C₂²	semidirect
16. #41×C₂₂₃	C₂²	semidirect
17. C₂×D₄×C₂₂₃	C₂²	semidirect
40. C₂₂₃	#585	semidirect
44. H(5,𝔽₂)	D₂₂₃	semidirect
51. #41	C₂×D₂₂₃	semidirect
56. C₂×D₄	C₂×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297351	2
#20297349	2
C₂×#16813169	2
#41×D₂₂₃	2
H(5,𝔽₂)×C₂₂₃	2
#585	223