Subgroups — C₂²×C₇×M₁₆ #90587

Order 448.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×M₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₇×C₈	C₂	1	2	yes	maximal, complemented
4	C₂⁴×C₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₇	C₄	1	6	yes	—
7	C₂²×C₄×C₇	C₂²	1	6	yes	—
8	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
9	C₇×M₁₆	C₂²	1	16	yes	direct product factor
10	C₂×C₇×C₈	C₂²	1	12	yes	complemented
11	C₂²×M₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
12	C₂³×C₇	—	2	4	no	—
13	C₂³×C₇	C₂×C₄	1	6	yes	—
14	C₂³×C₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×C₇	C₂×C₄	1	12	yes	—
16	C₂×C₄×C₇	C₂×C₄	1	6	yes	—
17	C₂×C₄×C₇	C₂×C₄	1	3	yes	—
18	C₂×C₄×C₇	C₂³	1	3	yes	—
19	C₂×C₄×C₇	C₂³	1	4	yes	—
20	C₇×C₈	C₂³	1	8	yes	complemented
21	C₂³×C₄	C₂×C₇	1	1	yes	characteristic
22	C₂×M₁₆	C₂×C₇	1	12	yes	direct product factor
23	C₂²×C₈	C₂×C₇	1	2	yes	complemented
24	C₂²×C₇	—	2	12	no	—
25	C₂²×C₇	M₁₆	1	4	yes	direct product factor
26	C₂²×C₇	C₂²×C₄	1	4	yes	—
27	C₂²×C₇	C₂²×C₄	1	3	yes	—
28	C₄×C₇	C₂²×C₄	1	4	yes	—
29	C₄×C₇	C₂²×C₄	1	3	yes	—
30	C₄×C₇	C₂⁴	1	1	yes	characteristic
31	C₂⁴	C₄×C₇	1	1	yes	characteristic
32	C₂²×C₄	C₄×C₇	1	6	yes	—
33	C₂²×C₄	C₄×C₇	1	1	yes	characteristic
34	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
35	C₂²×C₄	C₂²×C₇	1	6	yes	—
36	M₁₆	C₂²×C₇	1	16	yes	direct product factor
37	C₂×C₈	C₂²×C₇	1	12	yes	complemented
38	C₂×C₇	—	2	4	no	—
39	C₂×C₇	C₂×M₁₆	1	6	yes	direct product factor
40	C₂×C₇	C₂³×C₄	1	1	yes	characteristic
41	C₂³	—	2	4	no	—
42	C₂³	C₂×C₄×C₇	1	6	yes	—
43	C₂³	C₂×C₄×C₇	1	1	yes	characteristic
44	C₂×C₄	C₂×C₄×C₇	1	12	yes	—
45	C₂×C₄	C₂×C₄×C₇	1	6	yes	—
46	C₂×C₄	C₂×C₄×C₇	1	3	yes	—
47	C₂×C₄	C₂³×C₇	1	3	yes	—
48	C₂×C₄	C₂³×C₇	1	4	yes	—
49	C₈	C₂³×C₇	1	8	yes	complemented
50	C₇	C₂²×M₁₆	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²	—	2	12	no	—
52	C₂²	C₇×M₁₆	1	4	yes	direct product factor
53	C₂²	C₂²×C₄×C₇	1	4	yes	—
54	C₂²	C₂²×C₄×C₇	1	3	yes	—
55	C₄	C₂²×C₄×C₇	1	3	yes	—
56	C₄	C₂²×C₄×C₇	1	4	yes	—
57	C₄	C₂⁴×C₇	1	1	yes	characteristic
58	C₂	—	2	4	no	—
59	C₂	C₂×C₇×M₁₆	1	6	yes	direct product factor
60	C₂	C₂³×C₄×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂²
commutator	C₂	C₂³×C₄×C₇
Frattini, cosocle	C₄	C₂⁴×C₇
socle	C₂³×C₇	C₂×C₄
prime radical	C₂⁴×C₇	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×M₁₆	C₂	direct (N × H)
3. C₂²×C₇×C₈	C₂	semidirect
9. C₇×M₁₆	C₂²	direct (N × H)
10. C₂×C₇×C₈	C₂²	semidirect
11. C₂²×M₁₆	C₇	direct (N × H)
20. C₇×C₈	C₂³	semidirect
22. C₂×M₁₆	C₂×C₇	direct (N × H)
23. C₂²×C₈	C₂×C₇	semidirect
25. C₂²×C₇	M₁₆	direct (N × H)
36. M₁₆	C₂²×C₇	direct (N × H)
37. C₂×C₈	C₂²×C₇	semidirect
39. C₂×C₇	C₂×M₁₆	direct (N × H)
49. C₈	C₂³×C₇	semidirect
50. C₇	C₂²×M₁₆	direct (N × H)
52. C₂²	C₇×M₁₆	direct (N × H)
59. C₂	C₂×C₇×M₁₆	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×C₇	2
C₂×C₇×M₁₆	2
C₂²×C₇×C₈	2
C₂²×M₁₆	7

Subgroups — C22×C7×M16 #90587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×M₁₆ #90587