Subgroups — C₂×Q₈²×C₂₁₁ #41011097

Order 27008.

1112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈²×C₂₁₁	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₄⋊Q₈×C₂₁₁	C₂	1	9	yes	maximal
3	C₂×C₄×Q₈×C₂₁₁	C₂	1	6	yes	maximal
4	C₂²×Q₈×C₂₁₁	C₂²	1	6	yes	—
5	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
6	C₄⋊Q₈×C₂₁₁	C₂²	1	72	yes	—
7	C₄×Q₈×C₂₁₁	C₂²	1	48	yes	—
8	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	18	yes	—
9	C₂×C₄²×C₂₁₁	C₂²	1	9	yes	—
10	C₂×Q₈×C₂₁₁	—	4	12	no	—
11	C₂×Q₈×C₂₁₁	Q₈	1	16	yes	direct product factor
12	C₂×Q₈×C₂₁₁	C₂³	1	24	yes	—
13	C₂×Q₈×C₂₁₁	C₂³	1	8	yes	—
14	C₂²×C₄×C₂₁₁	C₂³	1	9	yes	—
15	C₂²×C₄×C₂₁₁	C₂³	1	6	yes	—
16	C₄⋊C₄×C₂₁₁	C₂³	1	72	yes	—
17	C₄²×C₂₁₁	C₂³	1	36	yes	—
18	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
19	Q₈×C₂₁₁	—	4	24	no	—
20	Q₈×C₂₁₁	C₂×Q₈	1	32	yes	direct product factor
21	C₂×C₄×C₂₁₁	—	2	18	no	—
22	C₂×C₄×C₂₁₁	C₂×Q₈	1	24	yes	—
23	C₂×C₄×C₂₁₁	C₂⁴	1	18	yes	—
24	C₂×C₄×C₂₁₁	C₂⁴	1	12	yes	—
25	C₂²×C₂₁₁	C₂²×Q₈	1	4	yes	—
26	C₂²×C₂₁₁	H(5,𝔽₂)	1	2	yes	—
27	C₂²×C₂₁₁	C₂⁵	1	1	yes	characteristic
28	C₄×C₂₁₁	—	2	18	no	—
29	C₄×C₂₁₁	C₂²×Q₈	1	24	yes	—
30	C₂×C₂₁₁	Q₈²	1	4	yes	direct product factor
31	C₂×C₂₁₁	C₂³×Q₈	1	2	yes	—
32	C₂×C₂₁₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
33	C₂₁₁	C₂×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
34	C₂×Q₈²	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	Q₈²	C₂×C₂₁₁	1	16	yes	direct product factor
36	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	9	yes	—
37	C₂×C₄×Q₈	C₂×C₂₁₁	1	6	yes	—
38	C₂²×Q₈	C₂²×C₂₁₁	1	6	yes	—
39	C₂²×Q₈	C₂²×C₂₁₁	1	2	yes	—
40	C₄⋊Q₈	C₂²×C₂₁₁	1	72	yes	—
41	C₄×Q₈	C₂²×C₂₁₁	1	48	yes	—
42	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	18	yes	—
43	C₂×C₄²	C₂²×C₂₁₁	1	9	yes	—
44	C₂×Q₈	—	4	12	no	—
45	C₂×Q₈	Q₈×C₂₁₁	1	16	yes	direct product factor
46	C₂×Q₈	C₂³×C₂₁₁	1	24	yes	—
47	C₂×Q₈	C₂³×C₂₁₁	1	8	yes	—
48	C₂²×C₄	C₂³×C₂₁₁	1	9	yes	—
49	C₂²×C₄	C₂³×C₂₁₁	1	6	yes	—
50	C₄⋊C₄	C₂³×C₂₁₁	1	72	yes	—
51	C₄²	C₂³×C₂₁₁	1	36	yes	—
52	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
53	Q₈	—	4	24	no	—
54	Q₈	C₂×Q₈×C₂₁₁	1	32	yes	direct product factor
55	C₂×C₄	—	2	18	no	—
56	C₂×C₄	C₂×Q₈×C₂₁₁	1	24	yes	—
57	C₂×C₄	C₂⁴×C₂₁₁	1	18	yes	—
58	C₂×C₄	C₂⁴×C₂₁₁	1	12	yes	—
59	C₂²	C₂²×Q₈×C₂₁₁	1	4	yes	—
60	C₂²	H(5,𝔽₂)×C₂₁₁	1	2	yes	—
61	C₂²	C₂⁵×C₂₁₁	1	1	yes	characteristic
62	C₄	—	2	18	no	—
63	C₄	C₂²×Q₈×C₂₁₁	1	24	yes	—
64	C₂	Q₈²×C₂₁₁	1	4	yes	direct product factor
65	C₂	C₂³×Q₈×C₂₁₁	1	2	yes	—
66	C₂	C₂×H(5,𝔽₂)×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₁₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈²×C₂₁₁	C₂	direct (N × H)
11. C₂×Q₈×C₂₁₁	Q₈	direct (N × H)
20. Q₈×C₂₁₁	C₂×Q₈	direct (N × H)
30. C₂×C₂₁₁	Q₈²	direct (N × H)
33. C₂₁₁	C₂×Q₈²	direct (N × H)
34. C₂×Q₈²	C₂₁₁	direct (N × H)
35. Q₈²	C₂×C₂₁₁	direct (N × H)
45. C₂×Q₈	Q₈×C₂₁₁	direct (N × H)
54. Q₈	C₂×Q₈×C₂₁₁	direct (N × H)
64. C₂	Q₈²×C₂₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈²×C₂₁₁	2
C₂×C₄⋊Q₈×C₂₁₁	2
C₂×C₄×Q₈×C₂₁₁	2
C₂×Q₈²	211

Subgroups — C2×Q82×C211 #41011097

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Q₈²×C₂₁₁ #41011097