Subgroups — H(5,𝔽₂)×D₂₂₃ #28384409

Order 14272.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297350	C₂	1	6	yes	maximal, complemented
2	#20297349	C₂	1	9	yes	maximal, complemented
3	#41×D₂₂₃	C₂	1	6	yes	maximal, complemented
4	C₂×D₄×D₂₂₃	C₂	1	9	yes	maximal, complemented
5	H(5,𝔽₂)×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₂₂₃	C₂²	1	6	yes	complemented
7	#16813171	C₂²	1	6	yes	complemented
8	#16813170	C₂²	1	18	yes	complemented
9	#16813169	C₂²	1	18	yes	complemented
10	Q₈×D₂₂₃	C₂²	1	2	yes	complemented
11	C₂×D₈₉₂	C₂²	1	9	yes	complemented
12	C₂×C₂²⋊D₂₂₃	C₂²	1	18	yes	complemented
13	D₄×D₂₂₃	C₂²	1	18	yes	complemented
14	D₄×D₂₂₃	C₂²	1	36	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	—	4	12	no	—
19	C₂²×D₂₂₃	C₂³	1	9	yes	—
20	C₂²×D₂₂₃	C₂³	1	18	yes	complemented
21	Dic₄₄₆	C₂³	1	6	yes	complemented
22	D₈₉₂	C₂³	1	18	yes	complemented
23	C₂²⋊D₂₂₃	C₂³	1	36	yes	complemented
24	C₂×Dic₂₂₃	C₂³	1	9	yes	complemented
25	C₄×D₂₂₃	C₂³	1	6	yes	—
26	C₄×D₂₂₃	C₂³	1	18	yes	complemented
27	C₂³×C₂₂₃	C₂³	1	6	yes	complemented
28	Q₈×C₂₂₃	C₂³	1	2	yes	complemented
29	D₄×C₂₂₃	C₂³	1	18	yes	complemented
30	C₂×C₄×C₂₂₃	C₂³	1	9	yes	complemented
31	C₂²×C₂₂₃	—	4	6	no	—
32	C₂²×C₂₂₃	C₂⁴	1	9	yes	—
33	C₂×D₂₂₃	—	2	18	no	—
34	C₂×D₂₂₃	—	4	18	no	—
35	C₂×D₂₂₃	C₂⁴	1	9	yes	—
36	C₂×D₂₂₃	C₂⁴	1	1	yes	characteristic
37	C₄×C₂₂₃	C₂⁴	1	6	yes	—
38	Dic₂₂₃	C₂⁴	1	6	yes	—
39	C₂×C₂₂₃	—	2	9	no	—
40	C₂×C₂₂₃	C₂⁵	1	1	yes	characteristic
41	D₂₂₃	—	2	9	no	—
42	D₂₂₃	H(5,𝔽₂)	1	2	yes	direct product factor
43	C₂₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
44	C₂×H(5,𝔽₂)	—	223	1	no	maximal, Sylow
45	H(5,𝔽₂)	—	223	6	no	—
46	H(5,𝔽₂)	—	223	9	no	—
47	H(5,𝔽₂)	D₂₂₃	1	1	yes	characteristic, direct product factor
48	C₂×#41	—	223	6	no	—
49	C₂²×D₄	—	223	9	no	—
50	C₂⁴	—	223	6	no	—
51	#41	—	223	6	no	—
52	#41	—	223	18	no	—
53	#41	—	223	18	no	—
54	#41	C₂×D₂₂₃	1	6	yes	complemented
55	C₂×Q₈	—	223	2	no	—
56	C₂×D₄	—	223	18	no	—
57	C₂×D₄	—	223	18	no	—
58	C₂×D₄	—	223	36	no	—
59	C₂×D₄	—	223	9	no	—
60	C₂×D₄	C₂×D₂₂₃	1	9	yes	complemented
61	C₂²×C₄	—	223	9	no	—
62	C₂³	—	892	12	no	—
63	C₂³	—	223	18	no	—
64	C₂³	—	223	9	no	—
65	C₂³	C₂²×D₂₂₃	1	6	yes	complemented
66	Q₈	—	223	6	no	—
67	Q₈	C₂²×D₂₂₃	1	2	yes	complemented
68	D₄	—	223	36	no	—
69	D₄	—	223	18	no	—
70	D₄	C₂²×D₂₂₃	1	18	yes	complemented
71	C₂×C₄	—	223	9	no	—
72	C₂×C₄	—	223	6	no	—
73	C₂×C₄	—	223	18	no	—
74	C₂×C₄	C₂²×D₂₂₃	1	9	yes	complemented
75	C₂²	—	4	6	no	—
76	C₂²	—	892	18	no	—
77	C₂²	—	446	18	no	—
78	C₂²	—	223	9	no	—
79	C₂²	—	223	1	no	—
80	C₂²	C₂³×D₂₂₃	1	9	yes	—
81	C₄	—	223	6	no	—
82	C₄	C₂³×D₂₂₃	1	6	yes	—
83	C₂	—	2	9	no	—
84	C₂	—	446	9	no	—
85	C₂	—	223	2	no	—
86	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	C₂×H(5,𝔽₂)
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. #20297350	C₂	semidirect
2. #20297349	C₂	semidirect
3. #41×D₂₂₃	C₂	semidirect
4. C₂×D₄×D₂₂₃	C₂	semidirect
5. H(5,𝔽₂)×C₂₂₃	C₂	semidirect
6. C₂³×D₂₂₃	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. C₂×C₂²⋊D₂₂₃	C₂²	semidirect
13. D₄×D₂₂₃	C₂²	semidirect
14. D₄×D₂₂₃	C₂²	semidirect
15. C₂×C₄×D₂₂₃	C₂²	semidirect
16. #41×C₂₂₃	C₂²	semidirect
17. C₂×D₄×C₂₂₃	C₂²	semidirect
20. C₂²×D₂₂₃	C₂³	semidirect
21. Dic₄₄₆	C₂³	semidirect
22. D₈₉₂	C₂³	semidirect
23. C₂²⋊D₂₂₃	C₂³	semidirect
24. C₂×Dic₂₂₃	C₂³	semidirect
26. C₄×D₂₂₃	C₂³	semidirect
27. C₂³×C₂₂₃	C₂³	semidirect
28. Q₈×C₂₂₃	C₂³	semidirect
29. D₄×C₂₂₃	C₂³	semidirect
30. C₂×C₄×C₂₂₃	C₂³	semidirect
42. D₂₂₃	H(5,𝔽₂)	direct (N × H)
43. C₂₂₃	C₂×H(5,𝔽₂)	semidirect
47. H(5,𝔽₂)	D₂₂₃	direct (N × H)
54. #41	C₂×D₂₂₃	semidirect
60. C₂×D₄	C₂×D₂₂₃	semidirect
65. C₂³	C₂²×D₂₂₃	semidirect
67. Q₈	C₂²×D₂₂₃	semidirect
70. D₄	C₂²×D₂₂₃	semidirect
74. C₂×C₄	C₂²×D₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
#20297350	2
#20297349	2
#41×D₂₂₃	2
C₂×D₄×D₂₂₃	2
H(5,𝔽₂)×C₂₂₃	2
C₂×H(5,𝔽₂)	223

Subgroups — H(5,𝔽2)×D223 #28384409

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×D₂₂₃ #28384409