Subgroups — H(5,𝔽₄) #72228232

Order 1024.

430453 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10586807	C₂	1	180	yes	maximal, complemented
2	#10586796	C₂	1	75	yes	maximal, complemented
3	#63069	C₂²	1	2160	yes	complemented
4	#63048	C₂²	1	600	yes	complemented
5	#63046	C₂²	1	3600	yes	complemented
6	#63045	C₂²	1	1800	yes	complemented
7	#62841	C₂²	1	900	yes	complemented
8	#62824	C₂²	1	1350	yes	complemented
9	#62817	C₂²	1	300	yes	complemented
10	#62694	C₂²	1	60	yes	complemented
11	C₂²×H(3,𝔽₄)	C₂²	1	25	yes	complemented
12	#3568	C₂³	1	7200	yes	complemented
13	#3561	C₂³	1	10800	yes	complemented
14	#3558	C₂³	1	10800	yes	complemented
15	#3550	C₂³	1	2700	yes	complemented
16	#3549	C₂³	1	900	yes	complemented
17	#3545	C₂³	1	10800	yes	complemented
18	#3540	C₂³	1	10800	yes	complemented
19	#3532	C₂³	1	7200	yes	complemented
20	#3531	C₂³	1	1080	yes	complemented
21	#3529	C₂³	1	1800	yes	complemented
22	C₂×H(3,𝔽₄)	C₂³	1	1200	yes	complemented
23	#3518	C₂³	1	3600	yes	complemented
24	#3514	C₂³	1	10800	yes	complemented
25	#3510	C₂³	1	5400	yes	complemented
26	#3500	C₂³	1	1800	yes	complemented
27	#3490	C₂³	1	5400	yes	complemented
28	#3486	C₂³	1	900	yes	complemented
29	#3484	C₂³	1	2700	yes	complemented
30	#3444	C₂³	1	900	yes	complemented
31	C₂²×#124	C₂³	1	225	yes	complemented
32	C₂²×#120	C₂³	1	150	yes	complemented
33	C₂⁶	C₂⁴	1	10	yes	complemented
34	C₂×2₋¹⁺⁴	C₂⁴	1	180	yes	—
35	C₂×H(5,𝔽₂)	C₂⁴	1	300	yes	complemented
36	C₂³×Q₈	C₂⁴	1	75	yes	complemented
37	C₂³×D₄	C₂⁴	1	225	yes	—
38	#564	C₂⁴	1	72	yes	complemented
39	#563	C₂⁴	1	5400	yes	complemented
40	#562	C₂⁴	1	5400	yes	complemented
41	H(3,𝔽₄)	C₂⁴	1	200	yes	complemented
42	H(3,𝔽₄)	C₂⁴	1	2400	yes	complemented
43	H(3,𝔽₄)	C₂⁴	1	3600	yes	complemented
44	#560	C₂⁴	1	3600	yes	complemented
45	#559	C₂⁴	1	5400	yes	complemented
46	Q₈²	C₂⁴	1	600	yes	complemented
47	#556	C₂⁴	1	10800	yes	complemented
48	#555	C₂⁴	1	5400	yes	complemented
49	#551	C₂⁴	1	21600	yes	complemented
50	D₄×Q₈	C₂⁴	1	3600	yes	complemented
51	#548	C₂⁴	1	7200	yes	complemented
52	#546	C₂⁴	1	21600	yes	complemented
53	D₄²	C₂⁴	1	5400	yes	complemented
54	#543	C₂⁴	1	1800	yes	complemented
55	#542	C₂⁴	1	21600	yes	complemented
56	#538	C₂⁴	1	21600	yes	complemented
57	#536	C₂⁴	1	5400	yes	complemented
58	#535	C₂⁴	1	5400	yes	—
59	#534	C₂⁴	1	5400	yes	complemented
60	#533	C₂⁴	1	2700	yes	complemented
61	#532	C₂⁴	1	2700	yes	—
62	#529	C₂⁴	1	10800	yes	complemented
63	C₂×#124	C₂⁴	1	5400	yes	complemented
64	#525	C₂⁴	1	5400	yes	complemented
65	C₂×#120	C₂⁴	1	3600	yes	complemented
66	#519	C₂⁴	1	900	yes	complemented
67	#518	C₂⁴	1	2700	yes	complemented
68	C₄×#41	C₂⁴	1	1800	yes	complemented
69	C₂²×C₂²⋊C₄	C₂⁴	1	450	yes	complemented
70	C₂²×C₄²	C₂⁴	1	75	yes	complemented
71	C₂⁵	—	16	30	no	—
72	C₂⁵	C₂⁵	1	150	yes	—
73	2₋¹⁺⁴	—	16	180	no	—
74	H(5,𝔽₂)	—	16	300	no	—
75	C₂×#41	C₂⁵	1	1800	yes	—
76	C₂²×Q₈	—	16	75	no	—
77	C₂²×Q₈	C₂⁵	1	900	yes	—
78	C₂²×D₄	—	16	225	no	—
79	C₂²×D₄	C₂⁵	1	2700	yes	—
80	C₂³×C₄	C₂⁵	1	225	yes	—
81	C₄⋊Q₈	C₂⁵	1	2700	yes	—
82	#127	C₂⁵	1	900	yes	—
83	#126	C₂⁵	1	7200	yes	—
84	#125	C₂⁵	1	1080	yes	—
85	#124	C₂⁵	1	1800	yes	—
86	#124	C₂⁵	1	10800	yes	—
87	#123	C₂⁵	1	10800	yes	—
88	C₂²⋊Q₈	C₂⁵	1	10800	yes	—
89	#121	C₂⁵	1	10800	yes	—
90	#120	C₂⁵	1	1200	yes	—
91	#120	C₂⁵	1	7200	yes	—
92	C₄×Q₈	C₂⁵	1	3600	yes	—
93	C₄×D₄	C₂⁵	1	10800	yes	—
94	#117	C₂⁵	1	5400	yes	—
95	C₂×C₂²⋊C₄	C₂⁵	1	5400	yes	—
96	C₂×C₄²	C₂⁵	1	900	yes	—
97	C₂⁴	—	8	450	no	—
98	C₂⁴	—	16	160	no	—
99	C₂⁴	C₂⁶	1	300	yes	—
100	C₂⁴	C₂⁶	1	25	yes	—
101	#41	—	8	1800	no	—
102	C₂×Q₈	—	8	900	no	—
103	C₂×Q₈	C₂⁶	1	600	yes	—
104	C₂×D₄	—	8	2700	no	—
105	C₂×D₄	C₂⁶	1	1800	yes	—
106	C₂²×C₄	—	8	225	no	—
107	C₂²×C₄	C₂⁶	1	1350	yes	—
108	C₄⋊C₄	C₂⁶	1	2160	yes	—
109	C₂²⋊C₄	C₂⁶	1	3600	yes	—
110	C₄²	C₂⁶	1	60	yes	—
111	C₄²	C₂⁶	1	900	yes	—
112	C₂³	—	4	75	no	—
113	C₂³	—	4	900	no	—
114	C₂³	—	16	600	no	—
115	C₂³	C₂⁷	1	75	yes	—
116	Q₈	—	4	600	no	—
117	D₄	—	4	1800	no	—
118	C₂×C₄	—	4	1350	no	—
119	C₂×C₄	C₂⁷	1	180	yes	—
120	C₂²	—	2	225	no	—
121	C₂²	—	4	100	no	—
122	C₂²	—	16	300	no	—
123	C₂²	C₂⁸	1	1	yes	characteristic
124	C₄	—	2	180	no	—
125	C₂	—	4	75	no	—
126	C₂	H(9,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10586807	C₂	semidirect
2. #10586796	C₂	semidirect
3. #63069	C₂²	semidirect
4. #63048	C₂²	semidirect
5. #63046	C₂²	semidirect
6. #63045	C₂²	semidirect
7. #62841	C₂²	semidirect
8. #62824	C₂²	semidirect
9. #62817	C₂²	semidirect
10. #62694	C₂²	semidirect
11. C₂²×H(3,𝔽₄)	C₂²	semidirect
12. #3568	C₂³	semidirect
13. #3561	C₂³	semidirect
14. #3558	C₂³	semidirect
15. #3550	C₂³	semidirect
16. #3549	C₂³	semidirect
17. #3545	C₂³	semidirect
18. #3540	C₂³	semidirect
19. #3532	C₂³	semidirect
20. #3531	C₂³	semidirect
21. #3529	C₂³	semidirect
22. C₂×H(3,𝔽₄)	C₂³	semidirect
23. #3518	C₂³	semidirect
24. #3514	C₂³	semidirect
25. #3510	C₂³	semidirect
26. #3500	C₂³	semidirect
27. #3490	C₂³	semidirect
28. #3486	C₂³	semidirect
29. #3484	C₂³	semidirect
30. #3444	C₂³	semidirect
31. C₂²×#124	C₂³	semidirect
32. C₂²×#120	C₂³	semidirect
33. C₂⁶	C₂⁴	semidirect
35. C₂×H(5,𝔽₂)	C₂⁴	semidirect
36. C₂³×Q₈	C₂⁴	semidirect
38. #564	C₂⁴	semidirect
39. #563	C₂⁴	semidirect
40. #562	C₂⁴	semidirect
41. H(3,𝔽₄)	C₂⁴	semidirect
42. H(3,𝔽₄)	C₂⁴	semidirect
43. H(3,𝔽₄)	C₂⁴	semidirect
44. #560	C₂⁴	semidirect
45. #559	C₂⁴	semidirect
46. Q₈²	C₂⁴	semidirect
47. #556	C₂⁴	semidirect
48. #555	C₂⁴	semidirect
49. #551	C₂⁴	semidirect
50. D₄×Q₈	C₂⁴	semidirect
51. #548	C₂⁴	semidirect
52. #546	C₂⁴	semidirect
53. D₄²	C₂⁴	semidirect
54. #543	C₂⁴	semidirect
55. #542	C₂⁴	semidirect
56. #538	C₂⁴	semidirect
57. #536	C₂⁴	semidirect
59. #534	C₂⁴	semidirect
60. #533	C₂⁴	semidirect
62. #529	C₂⁴	semidirect
63. C₂×#124	C₂⁴	semidirect
64. #525	C₂⁴	semidirect
65. C₂×#120	C₂⁴	semidirect
66. #519	C₂⁴	semidirect
67. #518	C₂⁴	semidirect
68. C₄×#41	C₂⁴	semidirect
69. C₂²×C₂²⋊C₄	C₂⁴	semidirect
70. C₂²×C₄²	C₂⁴	semidirect

Maximal subgroups

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

subgroup	index
#10586807	2
#10586796	2

Subgroups — H(5,𝔽4) #72228232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₄) #72228232