Subgroups — AΣL(1,𝔽₁₆)×C₂₃₉ #29504255

Order 15296.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#120×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#100×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#99×C₂₃₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×C₂₃₉	C₄	1	1	yes	characteristic, complemented
5	C₂×D₄×C₂₃₉	—	2	1	no	—
6	C₂×D₄×C₂₃₉	C₂²	1	1	yes	characteristic
7	M₁₆×C₂₃₉	—	2	1	no	—
8	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
9	C₂²⋊C₄×C₂₃₉	—	2	1	no	—
10	C₂²⋊C₄×C₂₃₉	C₄	1	1	yes	characteristic, complemented
11	C₂³×C₂₃₉	—	2	1	no	—
12	C₂³×C₂₃₉	—	4	2	no	—
13	C₂³×C₂₃₉	—	4	1	no	—
14	C₂³×C₂₃₉	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₂₃₉	D₄	1	1	yes	characteristic
16	D₄×C₂₃₉	—	4	1	no	—
17	D₄×C₂₃₉	—	4	2	no	—
18	C₂×C₄×C₂₃₉	—	2	1	no	—
19	C₂×C₄×C₂₃₉	—	4	1	no	—
20	C₂×C₄×C₂₃₉	D₄	1	1	yes	characteristic
21	C₈×C₂₃₉	—	4	1	no	—
22	C₂²×C₂₃₉	—	2	2	no	—
23	C₂²×C₂₃₉	—	2	1	no	—
24	C₂²×C₂₃₉	—	2	1	no	—
25	C₂²×C₂₃₉	—	4	1	no	—
26	C₂²×C₂₃₉	—	4	1	no	—
27	C₂²×C₂₃₉	—	4	1	no	—
28	C₂²×C₂₃₉	—	4	2	no	—
29	C₂²×C₂₃₉	—	4	2	no	—
30	C₂²×C₂₃₉	—	4	1	no	—
31	C₂²×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₂₃₉	—	2	1	no	—
33	C₄×C₂₃₉	—	4	1	no	—
34	C₄×C₂₃₉	—	8	1	no	—
35	C₂×C₂₃₉	—	2	1	no	—
36	C₂×C₂₃₉	—	4	1	no	—
37	C₂×C₂₃₉	—	4	2	no	—
38	C₂×C₂₃₉	—	4	1	no	—
39	C₂×C₂₃₉	#99	1	1	yes	characteristic
40	C₂₃₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, direct product factor, Sylow
41	AΣL(1,𝔽₁₆)	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	#120	C₂×C₂₃₉	1	1	yes	characteristic
43	#100	C₂×C₂₃₉	1	1	yes	characteristic, complemented
44	#99	C₂×C₂₃₉	1	1	yes	characteristic, complemented
45	C₂⁴	C₄×C₂₃₉	1	1	yes	characteristic, complemented
46	C₂×D₄	—	2	1	no	—
47	C₂×D₄	C₂²×C₂₃₉	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₂²⋊C₄	—	2	1	no	—
51	C₂²⋊C₄	C₄×C₂₃₉	1	1	yes	characteristic, complemented
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	C₂×C₄×C₂₃₉	1	1	yes	characteristic
56	C₂³	D₄×C₂₃₉	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	4	2	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	D₄×C₂₃₉	1	1	yes	characteristic
62	C₈	—	4	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	1	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	4	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	—	4	1	no	—
72	C₂²	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	4	1	no	—
75	C₄	—	8	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	2	no	—
79	C₂	—	4	1	no	—
80	C₂	#99×C₂₃₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₃₉	#99
commutator	C₂³	C₂×C₄×C₂₃₉
Frattini, cosocle	C₂×D₄	C₂²×C₂₃₉
prime radical	#120×C₂₃₉	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₃₉ ◁ C₂²×C₂₃₉ ◁ C₂×D₄×C₂₃₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #100×C₂₃₉	C₂	semidirect
3. #99×C₂₃₉	C₂	semidirect
4. C₂⁴×C₂₃₉	C₄	semidirect
10. C₂²⋊C₄×C₂₃₉	C₄	semidirect
40. C₂₃₉	AΣL(1,𝔽₁₆)	direct (N × H)
41. AΣL(1,𝔽₁₆)	C₂₃₉	direct (N × H)
43. #100	C₂×C₂₃₉	semidirect
44. #99	C₂×C₂₃₉	semidirect
45. C₂⁴	C₄×C₂₃₉	semidirect
51. C₂²⋊C₄	C₄×C₂₃₉	semidirect

Maximal subgroups

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

subgroup	index
#120×C₂₃₉	2
#100×C₂₃₉	2
#99×C₂₃₉	2
AΣL(1,𝔽₁₆)	239

Subgroups — AΣL(1,𝔽16)×C239 #29504255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — AΣL(1,𝔽₁₆)×C₂₃₉ #29504255