Subgroups — M₁₆×#3862 #72245213

Order 2352.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#3862	C₂	1	1	yes	maximal, characteristic
2	C₈×#3862	C₂	1	2	yes	maximal, complemented
3	C₇²×M₁₆	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	C₂²×#3862	C₄	1	1	yes	characteristic
5	C₄×#3862	C₄	1	1	yes	characteristic
6	C₄×#3862	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₇²	C₂×C₃	1	1	yes	characteristic
8	C₇²×C₈	C₂×C₃	1	2	yes	complemented
9	M₁₆×ΓL(1,𝔽₈)	—	7	2	no	maximal
10	C₂×#3862	—	2	1	no	—
11	C₂×#3862	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₇²	C₃×C₄	1	1	yes	characteristic
13	C₄×C₇²	C₃×C₄	1	1	yes	characteristic
14	C₄×C₇²	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₄×ΓL(1,𝔽₈)	—	7	2	no	—
16	C₈×ΓL(1,𝔽₈)	—	7	4	no	—
17	#3862	M₁₆	1	1	yes	characteristic, direct product factor, Hall
18	C₇×M₁₆	—	3	2	no	—
19	C₇×M₁₆	ΓL(1,𝔽₈)	1	2	yes	complemented
20	C₂×C₇²	—	2	1	no	—
21	C₂×C₇²	C₂×C₃×C₄	1	1	yes	characteristic
22	C₂²×ΓL(1,𝔽₈)	—	7	2	no	—
23	C₄×ΓL(1,𝔽₈)	—	7	2	no	—
24	C₄×ΓL(1,𝔽₈)	—	7	2	no	—
25	C₂×C₄×C₇	—	3	2	no	—
26	C₂×C₄×C₇	C₂×ΓL(1,𝔽₈)	1	2	yes	—
27	C₇×C₈	—	3	4	no	—
28	C₇×C₈	C₂×ΓL(1,𝔽₈)	1	4	yes	complemented
29	C₇²	C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
30	C₃×M₁₆	—	49	1	no	Hall
31	C₂×ΓL(1,𝔽₈)	—	7	2	no	—
32	C₂×ΓL(1,𝔽₈)	—	14	2	no	—
33	C₂²×C₇	—	3	2	no	—
34	C₂²×C₇	C₄×ΓL(1,𝔽₈)	1	2	yes	—
35	C₄×C₇	—	3	2	no	—
36	C₄×C₇	—	3	2	no	—
37	C₄×C₇	C₄×ΓL(1,𝔽₈)	1	2	yes	—
38	C₄×C₇	C₂²×ΓL(1,𝔽₈)	1	2	yes	—
39	C₂×C₃×C₄	—	49	1	no	—
40	C₃×C₈	—	49	2	no	—
41	ΓL(1,𝔽₈)	—	7	2	no	—
42	M₁₆	#3862	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₇	—	2	2	no	—
44	C₂×C₇	—	3	2	no	—
45	C₂×C₇	—	6	2	no	—
46	C₂×C₇	C₂×C₄×ΓL(1,𝔽₈)	1	2	yes	—
47	C₂²×C₃	—	49	1	no	—
48	C₃×C₄	—	49	1	no	—
49	C₃×C₄	—	49	1	no	—
50	C₂×C₄	C₂×#3862	1	1	yes	characteristic
51	C₈	C₂×#3862	1	2	yes	complemented
52	C₇	—	3	2	no	—
53	C₇	M₁₆×ΓL(1,𝔽₈)	1	2	yes	complemented
54	C₂×C₃	—	49	1	no	—
55	C₂×C₃	—	98	1	no	—
56	C₂²	C₄×#3862	1	1	yes	characteristic
57	C₄	C₄×#3862	1	1	yes	characteristic
58	C₄	C₂²×#3862	1	1	yes	characteristic
59	C₃	—	49	1	no	Sylow
60	C₂	—	2	1	no	—
61	C₂	C₂×C₄×#3862	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×#3862
commutator, socle	C₂×C₇²	C₂×C₃×C₄
Fitting	C₇²×M₁₆	C₃
prime radical	C₂²×#3862	C₄
cosocle	C₄×C₇²	C₂²×C₃
hypercenter	M₁₆	#3862
hypocenter	C₇²	C₃×M₁₆
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₈×#3862	C₂	semidirect
3. C₇²×M₁₆	C₃	semidirect
8. C₇²×C₈	C₂×C₃	semidirect
17. #3862	M₁₆	direct (N × H)
19. C₇×M₁₆	ΓL(1,𝔽₈)	semidirect
28. C₇×C₈	C₂×ΓL(1,𝔽₈)	semidirect
29. C₇²	C₃×M₁₆	semidirect
42. M₁₆	#3862	direct (N × H)
51. C₈	C₂×#3862	semidirect
53. C₇	M₁₆×ΓL(1,𝔽₈)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#3862	2
C₈×#3862	2
C₇²×M₁₆	3
M₁₆×ΓL(1,𝔽₈)	7

Subgroups — M16×#3862 #72245213

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#3862 #72245213