Subgroups — C₂³×ΓL(1,𝔽₂₅) #87146

Order 384.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×ΓL(1,𝔽₂₅)	C₂	1	28	yes	maximal, direct product factor
5	C₂³×M₁₆	—	3	1	no	maximal, Sylow
6	C₂⁴×D₃	C₄	1	1	yes	characteristic
7	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂³×Dic₃	C₄	1	1	yes	characteristic
9	C₂²×C₄×D₃	C₄	1	14	yes	—
10	C₂²×C₄×D₃	C₂²	1	14	yes	—
11	C₂²×C₃×C₈	C₂²	1	14	yes	complemented
12	C₂²×C₃⋊C₈	C₂²	1	14	yes	complemented
13	C₂×ΓL(1,𝔽₂₅)	C₂²	1	112	yes	direct product factor
14	C₂⁴×C₄	—	3	1	no	—
15	C₂²×M₁₆	—	3	28	no	—
16	C₂³×C₈	—	3	1	no	—
17	C₂³×C₈	D₃	1	1	yes	characteristic, complemented
18	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
19	C₂³×D₃	—	2	8	no	—
20	C₂³×D₃	C₂×C₄	1	14	yes	—
21	C₂²×C₃×C₄	C₂×C₄	1	7	yes	—
22	C₂²×C₃×C₄	C₂³	1	7	yes	—
23	C₂²×Dic₃	C₂×C₄	1	14	yes	—
24	C₂×C₄×D₃	C₂×C₄	1	84	yes	—
25	C₂×C₄×D₃	C₂³	1	28	yes	—
26	C₂×C₃×C₈	C₂³	1	28	yes	complemented
27	C₂×C₃⋊C₈	C₂³	1	28	yes	complemented
28	ΓL(1,𝔽₂₅)	C₂³	1	64	yes	direct product factor
29	C₂⁵	—	3	1	no	—
30	C₂³×C₄	—	3	1	no	—
31	C₂³×C₄	—	3	14	no	—
32	C₂³×C₄	—	3	14	no	—
33	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
34	C₂×M₁₆	—	3	112	no	—
35	C₂²×C₈	—	3	14	no	—
36	C₂²×C₈	C₂×D₃	1	14	yes	complemented
37	C₂³×C₃	M₁₆	1	8	yes	complemented
38	C₂³×C₃	C₂²×C₄	1	7	yes	—
39	C₂²×D₃	—	2	56	no	—
40	C₂²×D₃	C₂²×C₄	1	28	yes	—
41	C₂×C₃×C₄	C₂²×C₄	1	21	yes	—
42	C₂×C₃×C₄	C₂⁴	1	7	yes	—
43	C₂×Dic₃	C₂²×C₄	1	28	yes	—
44	C₄×D₃	C₂²×C₄	1	56	yes	—
45	C₄×D₃	C₂⁴	1	8	yes	—
46	C₃×C₈	C₂⁴	1	8	yes	complemented
47	C₃⋊C₈	C₂⁴	1	8	yes	complemented
48	C₂⁴	—	3	14	no	—
49	C₂⁴	—	6	8	no	—
50	C₂⁴	C₄×D₃	1	1	yes	characteristic
51	C₂²×C₄	—	3	14	no	—
52	C₂²×C₄	—	3	84	no	—
53	C₂²×C₄	—	3	28	no	—
54	C₂²×C₄	C₄×D₃	1	7	yes	—
55	C₂²×C₄	C₂²×D₃	1	7	yes	—
56	M₁₆	—	3	64	no	—
57	C₂×C₈	—	3	28	no	—
58	C₂×C₈	C₂²×D₃	1	28	yes	complemented
59	C₂²×C₃	C₂×M₁₆	1	28	yes	complemented
60	C₂²×C₃	C₂³×C₄	1	7	yes	—
61	C₂×D₃	—	2	56	no	—
62	C₂×D₃	C₂³×C₄	1	8	yes	—
63	C₃×C₄	C₂³×C₄	1	7	yes	—
64	C₃×C₄	C₂⁵	1	1	yes	characteristic
65	Dic₃	C₂³×C₄	1	8	yes	—
66	C₂³	—	3	28	no	—
67	C₂³	—	6	56	no	—
68	C₂³	ΓL(1,𝔽₂₅)	1	8	yes	direct product factor
69	C₂³	C₂×C₄×D₃	1	7	yes	—
70	C₂×C₄	—	3	28	no	—
71	C₂×C₄	—	3	56	no	—
72	C₂×C₄	—	3	8	no	—
73	C₂×C₄	C₂×C₄×D₃	1	21	yes	—
74	C₂×C₄	C₂³×D₃	1	7	yes	—
75	C₈	—	3	8	no	—
76	C₈	C₂³×D₃	1	8	yes	complemented
77	C₂×C₃	C₂²×M₁₆	1	14	yes	complemented
78	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
79	D₃	—	2	8	no	—
80	C₂²	—	3	8	no	—
81	C₂²	—	6	56	no	—
82	C₂²	C₂×ΓL(1,𝔽₂₅)	1	28	yes	direct product factor
83	C₂²	C₂²×C₄×D₃	1	7	yes	—
84	C₄	—	3	8	no	—
85	C₄	C₂²×C₄×D₃	1	7	yes	—
86	C₄	C₂⁴×D₃	1	1	yes	characteristic
87	C₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	6	8	no	—
89	C₂	C₂²×ΓL(1,𝔽₂₅)	1	14	yes	direct product factor
90	C₂	C₂³×C₄×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₃
Fitting	C₂³×C₃×C₈	C₂
socle	C₂⁴×C₃	C₂×C₄
prime radical	C₂⁴×D₃	C₄
cosocle	C₃×C₄	C₂⁵
hypercenter	C₂³×C₈	D₃
hypocenter	C₃	C₂³×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃×C₈	C₂	semidirect
3. C₂³×C₃⋊C₈	C₂	semidirect
4. C₂²×ΓL(1,𝔽₂₅)	C₂	direct (N × H)
11. C₂²×C₃×C₈	C₂²	semidirect
12. C₂²×C₃⋊C₈	C₂²	semidirect
13. C₂×ΓL(1,𝔽₂₅)	C₂²	direct (N × H)
17. C₂³×C₈	D₃	semidirect
26. C₂×C₃×C₈	C₂³	semidirect
27. C₂×C₃⋊C₈	C₂³	semidirect
28. ΓL(1,𝔽₂₅)	C₂³	direct (N × H)
36. C₂²×C₈	C₂×D₃	semidirect
37. C₂³×C₃	M₁₆	semidirect
46. C₃×C₈	C₂⁴	semidirect
47. C₃⋊C₈	C₂⁴	semidirect
58. C₂×C₈	C₂²×D₃	semidirect
59. C₂²×C₃	C₂×M₁₆	semidirect
68. C₂³	ΓL(1,𝔽₂₅)	direct (N × H)
76. C₈	C₂³×D₃	semidirect
77. C₂×C₃	C₂²×M₁₆	semidirect
82. C₂²	C₂×ΓL(1,𝔽₂₅)	direct (N × H)
87. C₃	C₂³×M₁₆	semidirect
89. C₂	C₂²×ΓL(1,𝔽₂₅)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×D₃	2
C₂³×C₃×C₈	2
C₂³×C₃⋊C₈	2
C₂²×ΓL(1,𝔽₂₅)	2
C₂³×M₁₆	3

Subgroups — C23×ΓL(1,𝔽25) #87146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×ΓL(1,𝔽₂₅) #87146