Subgroups — C₂²×C₃×H(5,𝔽₂) #87581

Order 384.

6560 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×H(5,𝔽₂)	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃×#41	C₂	1	6	yes	maximal, complemented
3	C₂³×C₃×D₄	C₂	1	9	yes	maximal, complemented
4	C₂²×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁵×C₃	C₂²	1	6	yes	complemented
6	C₃×H(5,𝔽₂)	C₂²	1	256	yes	direct product factor
7	C₂×C₃×#41	C₂²	1	144	yes	complemented
8	C₂²×C₃×Q₈	C₂²	1	2	yes	complemented
9	C₂²×C₃×D₄	C₂²	1	18	yes	complemented
10	C₂²×C₃×D₄	C₂²	1	216	yes	complemented
11	C₂³×C₃×C₄	C₂²	1	9	yes	complemented
12	C₂×H(5,𝔽₂)	C₂×C₃	1	48	yes	direct product factor
13	C₂²×#41	C₂×C₃	1	6	yes	complemented
14	C₂³×D₄	C₂×C₃	1	9	yes	complemented
15	C₂⁴×C₃	—	4	24	no	—
16	C₂⁴×C₃	C₂³	1	9	yes	—
17	C₂⁴×C₃	C₂³	1	72	yes	complemented
18	C₃×#41	C₂³	1	384	yes	complemented
19	C₂×C₃×Q₈	C₂³	1	24	yes	complemented
20	C₂×C₃×D₄	C₂³	1	216	yes	complemented
21	C₂×C₃×D₄	C₂³	1	576	yes	complemented
22	C₂²×C₃×C₄	C₂³	1	6	yes	—
23	C₂²×C₃×C₄	C₂³	1	108	yes	complemented
24	C₂⁵	C₂²×C₃	1	6	yes	complemented
25	H(5,𝔽₂)	C₂²×C₃	1	256	yes	direct product factor
26	C₂×#41	C₂²×C₃	1	144	yes	complemented
27	C₂²×Q₈	C₂²×C₃	1	2	yes	complemented
28	C₂²×D₄	C₂²×C₃	1	18	yes	complemented
29	C₂²×D₄	C₂²×C₃	1	216	yes	complemented
30	C₂³×C₄	C₂²×C₃	1	9	yes	complemented
31	C₂³×C₃	—	2	36	no	—
32	C₂³×C₃	—	4	144	no	—
33	C₂³×C₃	C₂⁴	1	54	yes	—
34	C₂³×C₃	C₂⁴	1	1	yes	characteristic
35	C₂³×C₃	C₂⁴	1	96	yes	complemented
36	C₃×Q₈	C₂⁴	1	32	yes	complemented
37	C₃×D₄	C₂⁴	1	288	yes	complemented
38	C₂×C₃×C₄	C₂⁴	1	36	yes	—
39	C₂×C₃×C₄	C₂⁴	1	144	yes	complemented
40	C₂⁴	—	4	24	no	—
41	C₂⁴	C₂³×C₃	1	9	yes	—
42	C₂⁴	C₂³×C₃	1	72	yes	complemented
43	#41	C₂³×C₃	1	384	yes	complemented
44	C₂×Q₈	C₂³×C₃	1	24	yes	complemented
45	C₂×D₄	C₂³×C₃	1	216	yes	complemented
46	C₂×D₄	C₂³×C₃	1	576	yes	complemented
47	C₂²×C₄	C₂³×C₃	1	6	yes	—
48	C₂²×C₄	C₂³×C₃	1	108	yes	complemented
49	C₂²×C₃	—	2	108	no	—
50	C₂²×C₃	—	4	96	no	—
51	C₂²×C₃	H(5,𝔽₂)	1	4	yes	direct product factor
52	C₂²×C₃	C₂⁵	1	36	yes	—
53	C₂²×C₃	C₂⁵	1	3	yes	—
54	C₃×C₄	C₂⁵	1	24	yes	—
55	C₂³	—	2	36	no	—
56	C₂³	—	4	144	no	—
57	C₂³	C₂⁴×C₃	1	54	yes	—
58	C₂³	C₂⁴×C₃	1	1	yes	characteristic
59	C₂³	C₂⁴×C₃	1	96	yes	complemented
60	Q₈	C₂⁴×C₃	1	32	yes	complemented
61	D₄	C₂⁴×C₃	1	288	yes	complemented
62	C₂×C₄	C₂⁴×C₃	1	36	yes	—
63	C₂×C₄	C₂⁴×C₃	1	144	yes	complemented
64	C₂×C₃	—	2	36	no	—
65	C₂×C₃	C₂×H(5,𝔽₂)	1	6	yes	direct product factor
66	C₂×C₃	C₂⁶	1	1	yes	characteristic
67	C₂²	—	2	108	no	—
68	C₂²	—	4	96	no	—
69	C₂²	C₃×H(5,𝔽₂)	1	4	yes	direct product factor
70	C₂²	C₂⁵×C₃	1	36	yes	—
71	C₂²	C₂⁵×C₃	1	3	yes	—
72	C₄	C₂⁵×C₃	1	24	yes	—
73	C₃	C₂²×H(5,𝔽₂)	1	1	yes	characteristic, direct product factor, Sylow
74	C₂	—	2	36	no	—
75	C₂	C₂×C₃×H(5,𝔽₂)	1	6	yes	direct product factor
76	C₂	C₂⁶×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁶×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×H(5,𝔽₂)	C₂	direct (N × H)
2. C₂²×C₃×#41	C₂	semidirect
3. C₂³×C₃×D₄	C₂	semidirect
4. C₂²×H(5,𝔽₂)	C₃	direct (N × H)
5. C₂⁵×C₃	C₂²	semidirect
6. C₃×H(5,𝔽₂)	C₂²	direct (N × H)
7. C₂×C₃×#41	C₂²	semidirect
8. C₂²×C₃×Q₈	C₂²	semidirect
9. C₂²×C₃×D₄	C₂²	semidirect
10. C₂²×C₃×D₄	C₂²	semidirect
11. C₂³×C₃×C₄	C₂²	semidirect
12. C₂×H(5,𝔽₂)	C₂×C₃	direct (N × H)
13. C₂²×#41	C₂×C₃	semidirect
14. C₂³×D₄	C₂×C₃	semidirect
17. C₂⁴×C₃	C₂³	semidirect
18. C₃×#41	C₂³	semidirect
19. C₂×C₃×Q₈	C₂³	semidirect
20. C₂×C₃×D₄	C₂³	semidirect
21. C₂×C₃×D₄	C₂³	semidirect
23. C₂²×C₃×C₄	C₂³	semidirect
24. C₂⁵	C₂²×C₃	semidirect
25. H(5,𝔽₂)	C₂²×C₃	direct (N × H)
26. C₂×#41	C₂²×C₃	semidirect
27. C₂²×Q₈	C₂²×C₃	semidirect
28. C₂²×D₄	C₂²×C₃	semidirect
29. C₂²×D₄	C₂²×C₃	semidirect
30. C₂³×C₄	C₂²×C₃	semidirect
35. C₂³×C₃	C₂⁴	semidirect
36. C₃×Q₈	C₂⁴	semidirect
37. C₃×D₄	C₂⁴	semidirect
39. C₂×C₃×C₄	C₂⁴	semidirect
42. C₂⁴	C₂³×C₃	semidirect
43. #41	C₂³×C₃	semidirect
44. C₂×Q₈	C₂³×C₃	semidirect
45. C₂×D₄	C₂³×C₃	semidirect
46. C₂×D₄	C₂³×C₃	semidirect
48. C₂²×C₄	C₂³×C₃	semidirect
51. C₂²×C₃	H(5,𝔽₂)	direct (N × H)
59. C₂³	C₂⁴×C₃	semidirect
60. Q₈	C₂⁴×C₃	semidirect
61. D₄	C₂⁴×C₃	semidirect
63. C₂×C₄	C₂⁴×C₃	semidirect
65. C₂×C₃	C₂×H(5,𝔽₂)	direct (N × H)
69. C₂²	C₃×H(5,𝔽₂)	direct (N × H)
73. C₃	C₂²×H(5,𝔽₂)	direct (N × H)
75. C₂	C₂×C₃×H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×H(5,𝔽₂)	2
C₂²×C₃×#41	2
C₂³×C₃×D₄	2
C₂²×H(5,𝔽₂)	3

Subgroups — C22×C3×H(5,𝔽2) #87581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×H(5,𝔽₂) #87581