Subgroups — C₂²×C₇×C₁₆ #90148

Order 448.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₄×C₇	C₄	1	1	yes	characteristic
4	C₂×C₇×C₈	C₄	1	3	yes	—
5	C₂×C₇×C₈	C₂²	1	3	yes	—
6	C₇×C₁₆	C₂²	1	4	yes	direct product factor
7	C₂²×C₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₇	C₈	1	1	yes	characteristic
9	C₂×C₄×C₇	C₈	1	3	yes	—
10	C₂×C₄×C₇	C₂×C₄	1	3	yes	—
11	C₇×C₈	C₂×C₄	1	3	yes	—
12	C₇×C₈	C₂³	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
14	C₂×C₁₆	C₂×C₇	1	6	yes	direct product factor
15	C₂²×C₇	C₁₆	1	4	yes	direct product factor
16	C₂²×C₇	C₂×C₈	1	3	yes	—
17	C₄×C₇	C₂×C₈	1	3	yes	—
18	C₄×C₇	C₂²×C₄	1	1	yes	characteristic
19	C₂²×C₄	C₄×C₇	1	1	yes	characteristic
20	C₂×C₈	C₄×C₇	1	3	yes	—
21	C₂×C₈	C₂²×C₇	1	3	yes	—
22	C₁₆	C₂²×C₇	1	4	yes	direct product factor
23	C₂×C₇	C₂×C₁₆	1	6	yes	direct product factor
24	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
25	C₂³	C₇×C₈	1	1	yes	characteristic
26	C₂×C₄	C₇×C₈	1	3	yes	—
27	C₂×C₄	C₂×C₄×C₇	1	3	yes	—
28	C₈	C₂×C₄×C₇	1	3	yes	—
29	C₈	C₂³×C₇	1	1	yes	characteristic
30	C₇	C₂²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
31	C₂²	C₇×C₁₆	1	4	yes	direct product factor
32	C₂²	C₂×C₇×C₈	1	3	yes	—
33	C₄	C₂×C₇×C₈	1	3	yes	—
34	C₄	C₂²×C₄×C₇	1	1	yes	characteristic
35	C₂	C₂×C₇×C₁₆	1	6	yes	direct product factor
36	C₂	C₂²×C₇×C₈	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂³×C₇
socle, prime radical	C₂³×C₇	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×C₁₆	C₂	direct (N × H)
6. C₇×C₁₆	C₂²	direct (N × H)
7. C₂²×C₁₆	C₇	direct (N × H)
14. C₂×C₁₆	C₂×C₇	direct (N × H)
15. C₂²×C₇	C₁₆	direct (N × H)
22. C₁₆	C₂²×C₇	direct (N × H)
23. C₂×C₇	C₂×C₁₆	direct (N × H)
30. C₇	C₂²×C₁₆	direct (N × H)
31. C₂²	C₇×C₁₆	direct (N × H)
35. C₂	C₂×C₇×C₁₆	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×C₇×C₈	2
C₂×C₇×C₁₆	2
C₂²×C₁₆	7

Subgroups — C22×C7×C16 #90148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×C₁₆ #90148