Subgroups — C₂×M₃₂×C₂₅₇ #30656372

Order 16448.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×C₂₅₇	C₂	1	1	yes	maximal, characteristic
2	M₃₂×C₂₅₇	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₆×C₂₅₇	C₂	1	2	yes	maximal, complemented
4	C₂²×C₄×C₂₅₇	C₄	1	1	yes	characteristic
5	C₂×C₈×C₂₅₇	C₄	1	1	yes	characteristic
6	C₂×C₈×C₂₅₇	C₄	1	2	yes	—
7	C₂×C₈×C₂₅₇	C₂²	1	2	yes	—
8	C₂×C₈×C₂₅₇	C₂²	1	1	yes	characteristic
9	C₁₆×C₂₅₇	C₂²	1	4	yes	complemented
10	C₂³×C₂₅₇	C₈	1	1	yes	characteristic
11	C₂×C₄×C₂₅₇	C₈	1	2	yes	—
12	C₂×C₄×C₂₅₇	C₈	1	1	yes	characteristic
13	C₂×C₄×C₂₅₇	C₂×C₄	1	2	yes	—
14	C₂×C₄×C₂₅₇	C₂×C₄	1	1	yes	characteristic
15	C₈×C₂₅₇	C₂×C₄	1	2	yes	—
16	C₈×C₂₅₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₂₅₇	C₂³	1	1	yes	characteristic
18	C₂²×C₂₅₇	—	2	2	no	—
19	C₂²×C₂₅₇	C₂×C₈	1	2	yes	—
20	C₂²×C₂₅₇	C₂×C₈	1	1	yes	characteristic
21	C₄×C₂₅₇	C₂×C₈	1	2	yes	—
22	C₄×C₂₅₇	C₂×C₈	1	1	yes	characteristic
23	C₄×C₂₅₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×C₂₅₇	—	2	2	no	—
25	C₂×C₂₅₇	M₃₂	1	2	yes	direct product factor
26	C₂×C₂₅₇	C₂²×C₈	1	1	yes	characteristic
27	C₂₅₇	C₂×M₃₂	1	1	yes	characteristic, direct product factor, Sylow
28	C₂×M₃₂	C₂₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₈	C₂×C₂₅₇	1	1	yes	characteristic
30	M₃₂	C₂×C₂₅₇	1	4	yes	direct product factor
31	C₂×C₁₆	C₂×C₂₅₇	1	2	yes	complemented
32	C₂²×C₄	C₄×C₂₅₇	1	1	yes	characteristic
33	C₂×C₈	C₄×C₂₅₇	1	2	yes	—
34	C₂×C₈	C₄×C₂₅₇	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₂₅₇	1	1	yes	characteristic
36	C₂×C₈	C₂²×C₂₅₇	1	2	yes	—
37	C₁₆	C₂²×C₂₅₇	1	4	yes	complemented
38	C₂³	C₈×C₂₅₇	1	1	yes	characteristic
39	C₂×C₄	C₈×C₂₅₇	1	2	yes	—
40	C₂×C₄	C₈×C₂₅₇	1	1	yes	characteristic
41	C₂×C₄	C₂×C₄×C₂₅₇	1	2	yes	—
42	C₂×C₄	C₂×C₄×C₂₅₇	1	1	yes	characteristic
43	C₈	C₂×C₄×C₂₅₇	1	1	yes	characteristic
44	C₈	C₂×C₄×C₂₅₇	1	2	yes	—
45	C₈	C₂³×C₂₅₇	1	1	yes	characteristic
46	C₂²	—	2	2	no	—
47	C₂²	C₂×C₈×C₂₅₇	1	2	yes	—
48	C₂²	C₂×C₈×C₂₅₇	1	1	yes	characteristic
49	C₄	C₂×C₈×C₂₅₇	1	1	yes	characteristic
50	C₄	C₂×C₈×C₂₅₇	1	2	yes	—
51	C₄	C₂²×C₄×C₂₅₇	1	1	yes	characteristic
52	C₂	—	2	2	no	—
53	C₂	M₃₂×C₂₅₇	1	2	yes	direct product factor
54	C₂	C₂²×C₈×C₂₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₅₇	C₂²
commutator	C₂	C₂²×C₈×C₂₅₇
Frattini, cosocle	C₈	C₂³×C₂₅₇
socle	C₂²×C₂₅₇	C₂×C₈
prime radical	C₂³×C₂₅₇	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₃₂×C₂₅₇	C₂	direct (N × H)
3. C₂×C₁₆×C₂₅₇	C₂	semidirect
9. C₁₆×C₂₅₇	C₂²	semidirect
25. C₂×C₂₅₇	M₃₂	direct (N × H)
27. C₂₅₇	C₂×M₃₂	direct (N × H)
28. C₂×M₃₂	C₂₅₇	direct (N × H)
30. M₃₂	C₂×C₂₅₇	direct (N × H)
31. C₂×C₁₆	C₂×C₂₅₇	semidirect
37. C₁₆	C₂²×C₂₅₇	semidirect
53. C₂	M₃₂×C₂₅₇	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×C₈×C₂₅₇	2
M₃₂×C₂₅₇	2
C₂×C₁₆×C₂₅₇	2
C₂×M₃₂	257

Subgroups — C2×M32×C257 #30656372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₃₂×C₂₅₇ #30656372