Subgroups — C₂²×C₇×Q₁₆ #90592

Order 448.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×Q₈	C₂	1	2	yes	maximal
2	C₂×C₇×Q₁₆	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₇×C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×Q₈	—	2	6	no	—
5	C₂×C₇×Q₈	C₂²	1	12	yes	—
6	C₂²×C₄×C₇	—	2	2	no	—
7	C₂²×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₇×Q₁₆	C₂²	1	16	yes	direct product factor
9	C₂×C₇×C₈	C₂²	1	6	yes	—
10	C₂²×Q₁₆	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂³×C₇	D₄	1	1	yes	characteristic
12	C₇×Q₈	—	2	12	no	—
13	C₇×Q₈	C₂³	1	8	yes	—
14	C₂×C₄×C₇	—	2	12	no	—
15	C₂×C₄×C₇	D₄	1	3	yes	—
16	C₂×C₄×C₇	C₂³	1	3	yes	—
17	C₇×C₈	C₂³	1	4	yes	—
18	C₂²×Q₈	C₂×C₇	1	2	yes	—
19	C₂×Q₁₆	C₂×C₇	1	12	yes	direct product factor
20	C₂²×C₈	C₂×C₇	1	1	yes	characteristic
21	C₂²×C₇	Q₁₆	1	4	yes	direct product factor
22	C₂²×C₇	C₂×D₄	1	3	yes	—
23	C₄×C₇	—	2	8	no	—
24	C₄×C₇	C₂×D₄	1	3	yes	—
25	C₄×C₇	C₂⁴	1	1	yes	characteristic
26	C₂×Q₈	—	2	6	no	—
27	C₂×Q₈	C₂²×C₇	1	12	yes	—
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	C₂²×C₇	1	1	yes	characteristic
30	Q₁₆	C₂²×C₇	1	16	yes	direct product factor
31	C₂×C₈	C₂²×C₇	1	6	yes	—
32	C₂×C₇	C₂×Q₁₆	1	6	yes	direct product factor
33	C₂×C₇	C₂²×D₄	1	1	yes	characteristic
34	C₂³	C₇×D₄	1	1	yes	characteristic
35	Q₈	—	2	12	no	—
36	Q₈	C₂³×C₇	1	8	yes	—
37	C₂×C₄	—	2	12	no	—
38	C₂×C₄	C₇×D₄	1	3	yes	—
39	C₂×C₄	C₂³×C₇	1	3	yes	—
40	C₈	C₂³×C₇	1	4	yes	—
41	C₇	C₂²×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	C₇×Q₁₆	1	4	yes	direct product factor
43	C₂²	C₂×C₇×D₄	1	3	yes	—
44	C₄	—	2	8	no	—
45	C₄	C₂×C₇×D₄	1	3	yes	—
46	C₄	C₂⁴×C₇	1	1	yes	characteristic
47	C₂	C₂×C₇×Q₁₆	1	6	yes	direct product factor
48	C₂	C₂²×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×Q₁₆	C₂	direct (N × H)
8. C₇×Q₁₆	C₂²	direct (N × H)
10. C₂²×Q₁₆	C₇	direct (N × H)
19. C₂×Q₁₆	C₂×C₇	direct (N × H)
21. C₂²×C₇	Q₁₆	direct (N × H)
30. Q₁₆	C₂²×C₇	direct (N × H)
32. C₂×C₇	C₂×Q₁₆	direct (N × H)
41. C₇	C₂²×Q₁₆	direct (N × H)
42. C₂²	C₇×Q₁₆	direct (N × H)
47. C₂	C₂×C₇×Q₁₆	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×Q₈	2
C₂×C₇×Q₁₆	2
C₂²×C₇×C₈	2
C₂²×Q₁₆	7

Subgroups — C22×C7×Q16 #90592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₇×Q₁₆ #90592