Subgroups — C₃×C₇×Q₈² #13061978

Order 1344.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₃×C₄×C₇×Q₈	C₂	1	6	yes	maximal
3	C₇×Q₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₇×Q₈	C₂²	1	6	yes	—
5	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
6	C₃×C₇×C₄⋊C₄	C₂²	1	18	yes	—
7	C₃×C₄²×C₇	C₂²	1	9	yes	—
8	C₇×C₄⋊Q₈	C₂×C₃	1	9	yes	—
9	C₄×C₇×Q₈	C₂×C₃	1	6	yes	—
10	C₃×Q₈²	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₃×C₇×Q₈	—	4	6	no	—
12	C₃×C₇×Q₈	Q₈	1	8	yes	direct product factor
13	C₂×C₃×C₄×C₇	C₂³	1	9	yes	—
14	C₂×C₃×C₄×C₇	C₂³	1	6	yes	—
15	C₂×C₇×Q₈	C₂²×C₃	1	6	yes	—
16	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
17	C₇×C₄⋊C₄	C₂²×C₃	1	18	yes	—
18	C₄²×C₇	C₂²×C₃	1	9	yes	—
19	C₃×C₄⋊Q₈	C₂×C₇	1	9	yes	—
20	C₃×C₄×Q₈	C₂×C₇	1	6	yes	—
21	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
22	C₃×C₄×C₇	—	2	9	no	—
23	C₃×C₄×C₇	C₂×Q₈	1	12	yes	—
24	Q₈²	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
25	C₇×Q₈	—	4	6	no	—
26	C₇×Q₈	C₃×Q₈	1	8	yes	direct product factor
27	C₂×C₄×C₇	C₂³×C₃	1	9	yes	—
28	C₂×C₄×C₇	C₂³×C₃	1	6	yes	—
29	C₂×C₃×Q₈	C₂²×C₇	1	6	yes	—
30	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
31	C₃×C₄⋊C₄	C₂²×C₇	1	18	yes	—
32	C₃×C₄²	C₂²×C₇	1	9	yes	—
33	C₂×C₃×C₇	C₂²×Q₈	1	2	yes	—
34	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
35	C₄⋊Q₈	C₂×C₃×C₇	1	9	yes	—
36	C₄×Q₈	C₂×C₃×C₇	1	6	yes	—
37	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
38	C₄×C₇	—	2	9	no	—
39	C₄×C₇	C₂×C₃×Q₈	1	12	yes	—
40	C₃×Q₈	—	4	6	no	—
41	C₃×Q₈	C₇×Q₈	1	8	yes	direct product factor
42	C₂×C₃×C₄	C₂³×C₇	1	9	yes	—
43	C₂×C₃×C₄	C₂³×C₇	1	6	yes	—
44	C₃×C₇	Q₈²	1	1	yes	characteristic, direct product factor, Hall
45	C₂×Q₈	C₂²×C₃×C₇	1	6	yes	—
46	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
47	C₄⋊C₄	C₂²×C₃×C₇	1	18	yes	—
48	C₄²	C₂²×C₃×C₇	1	9	yes	—
49	C₂×C₇	C₂²×C₃×Q₈	1	2	yes	—
50	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
52	C₃×C₄	—	2	9	no	—
53	C₃×C₄	C₂×C₇×Q₈	1	12	yes	—
54	Q₈	—	4	6	no	—
55	Q₈	C₃×C₇×Q₈	1	8	yes	direct product factor
56	C₂×C₄	C₂³×C₃×C₇	1	9	yes	—
57	C₂×C₄	C₂³×C₃×C₇	1	6	yes	—
58	C₇	C₃×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×C₃	C₂²×C₇×Q₈	1	2	yes	—
60	C₂×C₃	C₇×H(5,𝔽₂)	1	1	yes	characteristic
61	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
62	C₄	—	2	9	no	—
63	C₄	C₂×C₃×C₇×Q₈	1	12	yes	—
64	C₃	C₇×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
65	C₂	C₂²×C₃×C₇×Q₈	1	2	yes	—
66	C₂	C₃×C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×Q₈²	C₃	direct (N × H)
10. C₃×Q₈²	C₇	direct (N × H)
12. C₃×C₇×Q₈	Q₈	direct (N × H)
24. Q₈²	C₃×C₇	direct (N × H)
26. C₇×Q₈	C₃×Q₈	direct (N × H)
41. C₃×Q₈	C₇×Q₈	direct (N × H)
44. C₃×C₇	Q₈²	direct (N × H)
55. Q₈	C₃×C₇×Q₈	direct (N × H)
58. C₇	C₃×Q₈²	direct (N × H)
64. C₃	C₇×Q₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×C₄⋊Q₈	2
C₃×C₄×C₇×Q₈	2
C₇×Q₈²	3
C₃×Q₈²	7

Subgroups — C3×C7×Q82 #13061978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×Q₈² #13061978