Subgroups — C₂²×M₁₆×C₁₅₇ #23756971

Order 10048.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₁₅₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₁₅₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₈×C₁₅₇	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₁₅₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₁₅₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₁₅₇	C₄	1	6	yes	—
7	C₂²×C₄×C₁₅₇	C₂²	1	6	yes	—
8	C₂²×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
9	M₁₆×C₁₅₇	C₂²	1	16	yes	direct product factor
10	C₂×C₈×C₁₅₇	C₂²	1	12	yes	complemented
11	C₂³×C₁₅₇	—	2	4	no	—
12	C₂³×C₁₅₇	C₂×C₄	1	6	yes	—
13	C₂³×C₁₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₅₇	C₂×C₄	1	12	yes	—
15	C₂×C₄×C₁₅₇	C₂×C₄	1	6	yes	—
16	C₂×C₄×C₁₅₇	C₂×C₄	1	3	yes	—
17	C₂×C₄×C₁₅₇	C₂³	1	3	yes	—
18	C₂×C₄×C₁₅₇	C₂³	1	4	yes	—
19	C₈×C₁₅₇	C₂³	1	8	yes	complemented
20	C₂²×C₁₅₇	—	2	12	no	—
21	C₂²×C₁₅₇	M₁₆	1	4	yes	direct product factor
22	C₂²×C₁₅₇	C₂²×C₄	1	4	yes	—
23	C₂²×C₁₅₇	C₂²×C₄	1	3	yes	—
24	C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
25	C₄×C₁₅₇	C₂²×C₄	1	3	yes	—
26	C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
27	C₂×C₁₅₇	—	2	4	no	—
28	C₂×C₁₅₇	C₂×M₁₆	1	6	yes	direct product factor
29	C₂×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
30	C₁₅₇	C₂²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²×M₁₆	C₁₅₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄	C₂×C₁₅₇	1	1	yes	characteristic
33	C₂×M₁₆	C₂×C₁₅₇	1	12	yes	direct product factor
34	C₂²×C₈	C₂×C₁₅₇	1	2	yes	complemented
35	C₂⁴	C₄×C₁₅₇	1	1	yes	characteristic
36	C₂²×C₄	C₄×C₁₅₇	1	6	yes	—
37	C₂²×C₄	C₄×C₁₅₇	1	1	yes	characteristic
38	C₂²×C₄	C₂²×C₁₅₇	1	1	yes	characteristic
39	C₂²×C₄	C₂²×C₁₅₇	1	6	yes	—
40	M₁₆	C₂²×C₁₅₇	1	16	yes	direct product factor
41	C₂×C₈	C₂²×C₁₅₇	1	12	yes	complemented
42	C₂³	—	2	4	no	—
43	C₂³	C₂×C₄×C₁₅₇	1	6	yes	—
44	C₂³	C₂×C₄×C₁₅₇	1	1	yes	characteristic
45	C₂×C₄	C₂×C₄×C₁₅₇	1	12	yes	—
46	C₂×C₄	C₂×C₄×C₁₅₇	1	3	yes	—
47	C₂×C₄	C₂×C₄×C₁₅₇	1	6	yes	—
48	C₂×C₄	C₂³×C₁₅₇	1	3	yes	—
49	C₂×C₄	C₂³×C₁₅₇	1	4	yes	—
50	C₈	C₂³×C₁₅₇	1	8	yes	complemented
51	C₂²	—	2	12	no	—
52	C₂²	M₁₆×C₁₅₇	1	4	yes	direct product factor
53	C₂²	C₂²×C₄×C₁₅₇	1	4	yes	—
54	C₂²	C₂²×C₄×C₁₅₇	1	3	yes	—
55	C₄	C₂²×C₄×C₁₅₇	1	3	yes	—
56	C₄	C₂²×C₄×C₁₅₇	1	4	yes	—
57	C₄	C₂⁴×C₁₅₇	1	1	yes	characteristic
58	C₂	—	2	4	no	—
59	C₂	C₂×M₁₆×C₁₅₇	1	6	yes	direct product factor
60	C₂	C₂³×C₄×C₁₅₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₅₇	C₂²
commutator	C₂	C₂³×C₄×C₁₅₇
Frattini, cosocle	C₄	C₂⁴×C₁₅₇
socle	C₂³×C₁₅₇	C₂×C₄
prime radical	C₂⁴×C₁₅₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₅₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×M₁₆×C₁₅₇	C₂	direct (N × H)
3. C₂²×C₈×C₁₅₇	C₂	semidirect
9. M₁₆×C₁₅₇	C₂²	direct (N × H)
10. C₂×C₈×C₁₅₇	C₂²	semidirect
19. C₈×C₁₅₇	C₂³	semidirect
21. C₂²×C₁₅₇	M₁₆	direct (N × H)
28. C₂×C₁₅₇	C₂×M₁₆	direct (N × H)
30. C₁₅₇	C₂²×M₁₆	direct (N × H)
31. C₂²×M₁₆	C₁₅₇	direct (N × H)
33. C₂×M₁₆	C₂×C₁₅₇	direct (N × H)
34. C₂²×C₈	C₂×C₁₅₇	semidirect
40. M₁₆	C₂²×C₁₅₇	direct (N × H)
41. C₂×C₈	C₂²×C₁₅₇	semidirect
50. C₈	C₂³×C₁₅₇	semidirect
52. C₂²	M₁₆×C₁₅₇	direct (N × H)
59. C₂	C₂×M₁₆×C₁₅₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₁₅₇	2
C₂×M₁₆×C₁₅₇	2
C₂²×C₈×C₁₅₇	2
C₂²×M₁₆	157

Subgroups — C22×M16×C157 #23756971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₁₆×C₁₅₇ #23756971