Subgroups — C₂×C₇×Q₈² #11744429

Order 896.

1112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈²	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₇×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂×C₄×C₇×Q₈	C₂	1	6	yes	maximal
4	C₂²×C₇×Q₈	C₂²	1	6	yes	—
5	C₂²×C₇×Q₈	C₂²	1	2	yes	—
6	C₇×C₄⋊Q₈	C₂²	1	72	yes	—
7	C₄×C₇×Q₈	C₂²	1	48	yes	—
8	C₂×C₇×C₄⋊C₄	C₂²	1	18	yes	—
9	C₂×C₄²×C₇	C₂²	1	9	yes	—
10	C₂×Q₈²	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₇×Q₈	—	4	12	no	—
12	C₂×C₇×Q₈	Q₈	1	16	yes	direct product factor
13	C₂×C₇×Q₈	C₂³	1	24	yes	—
14	C₂×C₇×Q₈	C₂³	1	8	yes	—
15	C₂²×C₄×C₇	C₂³	1	9	yes	—
16	C₂²×C₄×C₇	C₂³	1	6	yes	—
17	C₇×C₄⋊C₄	C₂³	1	72	yes	—
18	C₄²×C₇	C₂³	1	36	yes	—
19	Q₈²	C₂×C₇	1	16	yes	direct product factor
20	C₂×C₄⋊Q₈	C₂×C₇	1	9	yes	—
21	C₂×C₄×Q₈	C₂×C₇	1	6	yes	—
22	C₂³×C₇	C₂⁴	1	1	yes	characteristic
23	C₇×Q₈	—	4	24	no	—
24	C₇×Q₈	C₂×Q₈	1	32	yes	direct product factor
25	C₂×C₄×C₇	—	2	18	no	—
26	C₂×C₄×C₇	C₂×Q₈	1	24	yes	—
27	C₂×C₄×C₇	C₂⁴	1	18	yes	—
28	C₂×C₄×C₇	C₂⁴	1	12	yes	—
29	C₂²×Q₈	C₂²×C₇	1	6	yes	—
30	C₂²×Q₈	C₂²×C₇	1	2	yes	—
31	C₄⋊Q₈	C₂²×C₇	1	72	yes	—
32	C₄×Q₈	C₂²×C₇	1	48	yes	—
33	C₂×C₄⋊C₄	C₂²×C₇	1	18	yes	—
34	C₂×C₄²	C₂²×C₇	1	9	yes	—
35	C₂²×C₇	C₂²×Q₈	1	4	yes	—
36	C₂²×C₇	H(5,𝔽₂)	1	2	yes	—
37	C₂²×C₇	C₂⁵	1	1	yes	characteristic
38	C₄×C₇	—	2	18	no	—
39	C₄×C₇	C₂²×Q₈	1	24	yes	—
40	C₂×Q₈	—	4	12	no	—
41	C₂×Q₈	C₇×Q₈	1	16	yes	direct product factor
42	C₂×Q₈	C₂³×C₇	1	24	yes	—
43	C₂×Q₈	C₂³×C₇	1	8	yes	—
44	C₂²×C₄	C₂³×C₇	1	9	yes	—
45	C₂²×C₄	C₂³×C₇	1	6	yes	—
46	C₄⋊C₄	C₂³×C₇	1	72	yes	—
47	C₄²	C₂³×C₇	1	36	yes	—
48	C₂×C₇	Q₈²	1	4	yes	direct product factor
49	C₂×C₇	C₂³×Q₈	1	2	yes	—
50	C₂×C₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
51	C₂³	C₂⁴×C₇	1	1	yes	characteristic
52	Q₈	—	4	24	no	—
53	Q₈	C₂×C₇×Q₈	1	32	yes	direct product factor
54	C₂×C₄	—	2	18	no	—
55	C₂×C₄	C₂×C₇×Q₈	1	24	yes	—
56	C₂×C₄	C₂⁴×C₇	1	18	yes	—
57	C₂×C₄	C₂⁴×C₇	1	12	yes	—
58	C₇	C₂×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
59	C₂²	C₂²×C₇×Q₈	1	4	yes	—
60	C₂²	C₇×H(5,𝔽₂)	1	2	yes	—
61	C₂²	C₂⁵×C₇	1	1	yes	characteristic
62	C₄	—	2	18	no	—
63	C₄	C₂²×C₇×Q₈	1	24	yes	—
64	C₂	C₇×Q₈²	1	4	yes	direct product factor
65	C₂	C₂³×C₇×Q₈	1	2	yes	—
66	C₂	C₂×C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×Q₈²	C₂	direct (N × H)
10. C₂×Q₈²	C₇	direct (N × H)
12. C₂×C₇×Q₈	Q₈	direct (N × H)
19. Q₈²	C₂×C₇	direct (N × H)
24. C₇×Q₈	C₂×Q₈	direct (N × H)
41. C₂×Q₈	C₇×Q₈	direct (N × H)
48. C₂×C₇	Q₈²	direct (N × H)
53. Q₈	C₂×C₇×Q₈	direct (N × H)
58. C₇	C₂×Q₈²	direct (N × H)
64. C₂	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
C₇×Q₈²	2
C₂×C₇×C₄⋊Q₈	2
C₂×C₄×C₇×Q₈	2
C₂×Q₈²	7

Subgroups — C2×C7×Q82 #11744429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×Q₈² #11744429