Subgroups — M₃₂₇₆₈ #72210205

Order 32768.

43 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈₁₉₂	C₂	1	1	yes	maximal, characteristic
2	C₁₆₃₈₄	C₂	1	2	yes	maximal, complemented
3	C₂×C₄₀₉₆	C₄	1	1	yes	characteristic
4	C₈₁₉₂	C₄	1	1	yes	characteristic
5	C₈₁₉₂	C₂²	1	1	yes	characteristic
6	C₂×C₂₀₄₈	C₈	1	1	yes	characteristic
7	C₄₀₉₆	C₈	1	1	yes	characteristic
8	C₄₀₉₆	C₂×C₄	1	1	yes	characteristic
9	C₂×C₁₀₂₄	C₁₆	1	1	yes	characteristic
10	C₂₀₄₈	C₁₆	1	1	yes	characteristic
11	C₂₀₄₈	C₂×C₈	1	1	yes	characteristic
12	C₂×C₅₁₂	C₃₂	1	1	yes	characteristic
13	C₁₀₂₄	C₃₂	1	1	yes	characteristic
14	C₁₀₂₄	C₂×C₁₆	1	1	yes	characteristic
15	C₂×C₂₅₆	C₆₄	1	1	yes	characteristic
16	C₅₁₂	C₆₄	1	1	yes	characteristic
17	C₅₁₂	C₂×C₃₂	1	1	yes	characteristic
18	C₂×C₁₂₈	C₁₂₈	1	1	yes	characteristic
19	C₂₅₆	C₁₂₈	1	1	yes	characteristic
20	C₂₅₆	C₂×C₆₄	1	1	yes	characteristic
21	C₂×C₆₄	C₂₅₆	1	1	yes	characteristic
22	C₁₂₈	C₂₅₆	1	1	yes	characteristic
23	C₁₂₈	C₂×C₁₂₈	1	1	yes	characteristic
24	C₂×C₃₂	C₅₁₂	1	1	yes	characteristic
25	C₆₄	C₅₁₂	1	1	yes	characteristic
26	C₆₄	C₂×C₂₅₆	1	1	yes	characteristic
27	C₂×C₁₆	C₁₀₂₄	1	1	yes	characteristic
28	C₃₂	C₁₀₂₄	1	1	yes	characteristic
29	C₃₂	C₂×C₅₁₂	1	1	yes	characteristic
30	C₂×C₈	C₂₀₄₈	1	1	yes	characteristic
31	C₁₆	C₂₀₄₈	1	1	yes	characteristic
32	C₁₆	C₂×C₁₀₂₄	1	1	yes	characteristic
33	C₂×C₄	C₄₀₉₆	1	1	yes	characteristic
34	C₈	C₄₀₉₆	1	1	yes	characteristic
35	C₈	C₂×C₂₀₄₈	1	1	yes	characteristic
36	C₂²	C₈₁₉₂	1	1	yes	characteristic
37	C₄	C₈₁₉₂	1	1	yes	characteristic
38	C₄	C₂×C₄₀₉₆	1	1	yes	characteristic
39	C₂	—	2	1	no	—
40	C₂	C₂×C₈₁₉₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₈₁₉₂	C₂²
commutator, socle	C₂	C₂×C₈₁₉₂
prime radical	C₂²	C₈₁₉₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈₁₉₂ ◁ G
LECS	G ▷ C₈₁₉₂ ▷ C₄₀₉₆ ▷ C₂₀₄₈ ▷ C₁₀₂₄ ▷ C₅₁₂ ▷ C₂₅₆ ▷ C₁₂₈ ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₁₆₃₈₄	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈₁₉₂	2
C₁₆₃₈₄	2

Subgroups — M32768 #72210205

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₃₂₇₆₈ #72210205