Subgroups — C₂×C₇²×Q₁₆ #13096614

Order 1568.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇²×Q₈	C₂	1	2	yes	maximal
2	C₇²×Q₁₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₇²×C₈	C₂	1	1	yes	maximal, characteristic
4	C₇²×Q₈	—	2	2	no	—
5	C₇²×Q₈	C₂²	1	4	yes	—
6	C₂×C₄×C₇²	—	2	2	no	—
7	C₂×C₄×C₇²	C₂²	1	1	yes	characteristic
8	C₇²×C₈	C₂²	1	2	yes	—
9	C₂×C₇×Q₁₆	C₇	1	8	yes	maximal, direct product factor
10	C₂²×C₇²	D₄	1	1	yes	characteristic
11	C₄×C₇²	—	2	4	no	—
12	C₄×C₇²	D₄	1	1	yes	characteristic
13	C₄×C₇²	C₂³	1	1	yes	characteristic
14	C₂×C₇×Q₈	C₂×C₇	1	16	yes	—
15	C₇×Q₁₆	C₂×C₇	1	32	yes	direct product factor
16	C₂×C₇×C₈	C₂×C₇	1	8	yes	—
17	C₂×C₇²	Q₁₆	1	2	yes	direct product factor
18	C₂×C₇²	C₂×D₄	1	1	yes	characteristic
19	C₇×Q₈	—	2	16	no	—
20	C₇×Q₈	C₂²×C₇	1	32	yes	—
21	C₂×C₄×C₇	—	2	16	no	—
22	C₂×C₄×C₇	C₂²×C₇	1	8	yes	—
23	C₇×C₈	C₂²×C₇	1	16	yes	—
24	C₇²	C₂×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
25	C₂×Q₁₆	C₇²	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₇	C₇×D₄	1	8	yes	—
27	C₄×C₇	—	2	32	no	—
28	C₄×C₇	C₇×D₄	1	8	yes	—
29	C₄×C₇	C₂³×C₇	1	8	yes	—
30	C₂×Q₈	C₂×C₇²	1	2	yes	—
31	Q₁₆	C₂×C₇²	1	4	yes	direct product factor
32	C₂×C₈	C₂×C₇²	1	1	yes	characteristic
33	C₂×C₇	C₇×Q₁₆	1	16	yes	direct product factor
34	C₂×C₇	C₂×C₇×D₄	1	8	yes	—
35	Q₈	—	2	2	no	—
36	Q₈	C₂²×C₇²	1	4	yes	—
37	C₂×C₄	—	2	2	no	—
38	C₂×C₄	C₂²×C₇²	1	1	yes	characteristic
39	C₈	C₂²×C₇²	1	2	yes	—
40	C₇	C₂×C₇×Q₁₆	1	8	yes	direct product factor
41	C₂²	C₇²×D₄	1	1	yes	characteristic
42	C₄	—	2	4	no	—
43	C₄	C₇²×D₄	1	1	yes	characteristic
44	C₄	C₂³×C₇²	1	1	yes	characteristic
45	C₂	C₇²×Q₁₆	1	2	yes	direct product factor
46	C₂	C₂×C₇²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₇²	D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇² ◁ C₂×C₄×C₇² ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇²×Q₁₆	C₂	direct (N × H)
9. C₂×C₇×Q₁₆	C₇	direct (N × H)
15. C₇×Q₁₆	C₂×C₇	direct (N × H)
17. C₂×C₇²	Q₁₆	direct (N × H)
24. C₇²	C₂×Q₁₆	direct (N × H)
25. C₂×Q₁₆	C₇²	direct (N × H)
31. Q₁₆	C₂×C₇²	direct (N × H)
33. C₂×C₇	C₇×Q₁₆	direct (N × H)
40. C₇	C₂×C₇×Q₁₆	direct (N × H)
45. C₂	C₇²×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
C₂×C₇²×Q₈	2
C₇²×Q₁₆	2
C₂×C₇²×C₈	2
C₂×C₇×Q₁₆	7

Subgroups — C2×C72×Q16 #13096614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇²×Q₁₆ #13096614