Subgroups — C₇×C₈×Q₈ #90091

Order 448.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₇×C₄⋊C₈	C₂	1	3	yes	maximal
3	C₄×C₇×C₈	C₂	1	3	yes	maximal
4	C₂×C₇×Q₈	C₄	1	1	yes	characteristic
5	C₂×C₇×C₈	C₂²	1	3	yes	—
6	C₂×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₇×C₄⋊C₄	C₄	1	3	yes	—
8	C₄²×C₇	C₂²	1	3	yes	—
9	C₈×Q₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₇×Q₈	C₈	1	4	yes	direct product factor
11	C₂×C₄×C₇	C₂×C₄	1	3	yes	—
12	C₂×C₄×C₇	C₂×C₄	1	3	yes	—
13	C₂×C₄×C₇	C₂³	1	1	yes	characteristic
14	C₇×C₈	—	2	3	no	—
15	C₇×C₈	Q₈	1	2	yes	direct product factor
16	C₄×Q₈	C₂×C₇	1	1	yes	characteristic
17	C₄⋊C₈	C₂×C₇	1	3	yes	—
18	C₄×C₈	C₂×C₇	1	3	yes	—
19	C₂²×C₇	C₂²×C₄	1	1	yes	characteristic
20	C₄×C₇	—	2	3	no	—
21	C₄×C₇	C₂×C₈	1	6	yes	—
22	C₄×C₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₇	#41	1	1	yes	characteristic
24	C₂×Q₈	C₄×C₇	1	1	yes	characteristic
25	C₂×C₈	C₂²×C₇	1	3	yes	—
26	C₂×C₈	C₂²×C₇	1	1	yes	characteristic
27	C₄⋊C₄	C₄×C₇	1	3	yes	—
28	C₄²	C₂²×C₇	1	3	yes	—
29	C₂×C₇	C₄×Q₈	1	1	yes	characteristic
30	C₂×C₇	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₇	#131	1	1	yes	characteristic
32	Q₈	C₇×C₈	1	4	yes	direct product factor
33	C₂×C₄	C₂×C₄×C₇	1	3	yes	—
34	C₂×C₄	C₂×C₄×C₇	1	3	yes	—
35	C₂×C₄	C₂³×C₇	1	1	yes	characteristic
36	C₈	—	2	3	no	—
37	C₈	C₇×Q₈	1	2	yes	direct product factor
38	C₇	C₈×Q₈	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²	C₂²×C₄×C₇	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×C₇×C₈	1	6	yes	—
42	C₄	C₂×C₇×Q₈	1	1	yes	characteristic
43	C₄	C₇×#41	1	1	yes	characteristic
44	C₂	C₄×C₇×Q₈	1	1	yes	characteristic
45	C₂	C₂²×C₇×C₈	1	1	yes	characteristic
46	C₂	C₇×#131	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₇×C₈	C₂²
commutator	C₂	C₂²×C₇×C₈
Frattini, cosocle	C₂×C₄	C₂³×C₇
socle, prime radical	C₂²×C₇	C₂²×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₈×Q₈	C₇	direct (N × H)
10. C₇×Q₈	C₈	direct (N × H)
15. C₇×C₈	Q₈	direct (N × H)
32. Q₈	C₇×C₈	direct (N × H)
37. C₈	C₇×Q₈	direct (N × H)
38. C₇	C₈×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×Q₈	2
C₇×C₄⋊C₈	2
C₄×C₇×C₈	2
C₈×Q₈	7

Subgroups — C7×C8×Q8 #90091

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×C₈×Q₈ #90091