Subgroups — C₂×M₁₆×C₇₂₇ #37551439

Order 23264.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₇₂₇	C₂	1	1	yes	maximal, characteristic
2	M₁₆×C₇₂₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×C₇₂₇	C₂	1	2	yes	maximal, complemented
4	C₂³×C₇₂₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₇₂₇	C₄	1	1	yes	characteristic
6	C₂×C₄×C₇₂₇	C₄	1	2	yes	—
7	C₂×C₄×C₇₂₇	C₂²	1	2	yes	—
8	C₂×C₄×C₇₂₇	C₂²	1	1	yes	characteristic
9	C₈×C₇₂₇	C₂²	1	4	yes	complemented
10	C₂²×C₇₂₇	—	2	2	no	—
11	C₂²×C₇₂₇	C₂×C₄	1	2	yes	—
12	C₂²×C₇₂₇	C₂×C₄	1	1	yes	characteristic
13	C₄×C₇₂₇	C₂×C₄	1	2	yes	—
14	C₄×C₇₂₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₇₂₇	C₂³	1	1	yes	characteristic
16	C₂×C₇₂₇	—	2	2	no	—
17	C₂×C₇₂₇	M₁₆	1	2	yes	direct product factor
18	C₂×C₇₂₇	C₂²×C₄	1	1	yes	characteristic
19	C₇₂₇	C₂×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×M₁₆	C₇₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₄	C₂×C₇₂₇	1	1	yes	characteristic
22	M₁₆	C₂×C₇₂₇	1	4	yes	direct product factor
23	C₂×C₈	C₂×C₇₂₇	1	2	yes	complemented
24	C₂³	C₄×C₇₂₇	1	1	yes	characteristic
25	C₂×C₄	C₄×C₇₂₇	1	2	yes	—
26	C₂×C₄	C₄×C₇₂₇	1	1	yes	characteristic
27	C₂×C₄	C₂²×C₇₂₇	1	1	yes	characteristic
28	C₂×C₄	C₂²×C₇₂₇	1	2	yes	—
29	C₈	C₂²×C₇₂₇	1	4	yes	complemented
30	C₂²	—	2	2	no	—
31	C₂²	C₂×C₄×C₇₂₇	1	2	yes	—
32	C₂²	C₂×C₄×C₇₂₇	1	1	yes	characteristic
33	C₄	C₂×C₄×C₇₂₇	1	1	yes	characteristic
34	C₄	C₂×C₄×C₇₂₇	1	2	yes	—
35	C₄	C₂³×C₇₂₇	1	1	yes	characteristic
36	C₂	—	2	2	no	—
37	C₂	M₁₆×C₇₂₇	1	2	yes	direct product factor
38	C₂	C₂²×C₄×C₇₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇₂₇	C₂²
commutator	C₂	C₂²×C₄×C₇₂₇
Frattini, cosocle	C₄	C₂³×C₇₂₇
socle	C₂²×C₇₂₇	C₂×C₄
prime radical	C₂³×C₇₂₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₁₆×C₇₂₇	C₂	direct (N × H)
3. C₂×C₈×C₇₂₇	C₂	semidirect
9. C₈×C₇₂₇	C₂²	semidirect
17. C₂×C₇₂₇	M₁₆	direct (N × H)
19. C₇₂₇	C₂×M₁₆	direct (N × H)
20. C₂×M₁₆	C₇₂₇	direct (N × H)
22. M₁₆	C₂×C₇₂₇	direct (N × H)
23. C₂×C₈	C₂×C₇₂₇	semidirect
29. C₈	C₂²×C₇₂₇	semidirect
37. C₂	M₁₆×C₇₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₇₂₇	2
M₁₆×C₇₂₇	2
C₂×C₈×C₇₂₇	2
C₂×M₁₆	727

Subgroups — C2×M16×C727 #37551439

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×C₇₂₇ #37551439