Subgroups — C₄²×Q₈×C₂₁₁ #41009892

Order 27008.

684 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Q₈×C₂₁₁	C₂	1	3	yes	maximal
2	C₄×C₄⋊C₄×C₂₁₁	C₂	1	9	yes	maximal
3	C₄³×C₂₁₁	C₂	1	3	yes	maximal
4	C₂²×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
5	C₄×Q₈×C₂₁₁	C₄	1	24	yes	direct product factor
6	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	9	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	9	yes	—
8	C₂×C₄²×C₂₁₁	C₂²	1	9	yes	—
9	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
10	#95×C₂₁₁	C₂²	1	6	yes	—
11	C₂×Q₈×C₂₁₁	C₂×C₄	1	12	yes	—
12	C₂²×C₄×C₂₁₁	C₂³	1	9	yes	—
13	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
14	C₂²×C₄×C₂₁₁	C₂³	1	3	yes	—
15	C₄⋊C₄×C₂₁₁	C₂×C₄	1	36	yes	—
16	C₄²×C₂₁₁	—	2	18	no	—
17	C₄²×C₂₁₁	C₂×C₄	1	36	yes	—
18	C₄²×C₂₁₁	Q₈	1	4	yes	direct product factor
19	C₂³×C₂₁₁	C₂⁴	1	1	yes	characteristic
20	Q₈×C₂₁₁	C₄²	1	16	yes	direct product factor
21	C₂×C₄×C₂₁₁	—	2	18	no	—
22	C₂×C₄×C₂₁₁	C₂²×C₄	1	18	yes	—
23	C₂×C₄×C₂₁₁	C₂²×C₄	1	18	yes	—
24	C₂×C₄×C₂₁₁	C₂²×C₄	1	6	yes	—
25	C₂×C₄×C₂₁₁	C₂×Q₈	1	6	yes	—
26	C₂×C₄×C₂₁₁	#41	1	6	yes	—
27	C₂²×C₂₁₁	C₂³×C₄	1	3	yes	—
28	C₂²×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
29	C₂²×C₂₁₁	C₂×#41	1	3	yes	—
30	C₄×C₂₁₁	—	2	18	no	—
31	C₄×C₂₁₁	C₂×C₄²	1	12	yes	—
32	C₄×C₂₁₁	C₄×Q₈	1	12	yes	direct product factor
33	C₂×C₂₁₁	C₂²×C₄²	1	1	yes	characteristic
34	C₂×C₂₁₁	C₂×C₄×Q₈	1	3	yes	—
35	C₂×C₂₁₁	C₄×#41	1	3	yes	—
36	C₂₁₁	C₄²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
37	C₄²×Q₈	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₂×C₄×Q₈	C₂×C₂₁₁	1	3	yes	—
39	C₄×C₄⋊C₄	C₂×C₂₁₁	1	9	yes	—
40	C₄³	C₂×C₂₁₁	1	3	yes	—
41	C₂²×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
42	C₄×Q₈	C₄×C₂₁₁	1	24	yes	direct product factor
43	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	9	yes	—
44	C₂×C₄²	C₂²×C₂₁₁	1	9	yes	—
45	C₂×C₄²	C₂²×C₂₁₁	1	9	yes	—
46	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
47	#95	C₂²×C₂₁₁	1	6	yes	—
48	C₂×Q₈	C₂×C₄×C₂₁₁	1	12	yes	—
49	C₂²×C₄	C₂³×C₂₁₁	1	9	yes	—
50	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
51	C₂²×C₄	C₂³×C₂₁₁	1	3	yes	—
52	C₄⋊C₄	C₂×C₄×C₂₁₁	1	36	yes	—
53	C₄²	—	2	18	no	—
54	C₄²	C₂×C₄×C₂₁₁	1	36	yes	—
55	C₄²	Q₈×C₂₁₁	1	4	yes	direct product factor
56	C₂³	C₂⁴×C₂₁₁	1	1	yes	characteristic
57	Q₈	C₄²×C₂₁₁	1	16	yes	direct product factor
58	C₂×C₄	—	2	18	no	—
59	C₂×C₄	C₂²×C₄×C₂₁₁	1	18	yes	—
60	C₂×C₄	C₂²×C₄×C₂₁₁	1	18	yes	—
61	C₂×C₄	C₂²×C₄×C₂₁₁	1	6	yes	—
62	C₂×C₄	C₂×Q₈×C₂₁₁	1	6	yes	—
63	C₂×C₄	#41×C₂₁₁	1	6	yes	—
64	C₂²	C₂³×C₄×C₂₁₁	1	3	yes	—
65	C₂²	C₂²×Q₈×C₂₁₁	1	1	yes	characteristic
66	C₂²	C₂×#41×C₂₁₁	1	3	yes	—
67	C₄	—	2	18	no	—
68	C₄	C₂×C₄²×C₂₁₁	1	12	yes	—
69	C₄	C₄×Q₈×C₂₁₁	1	12	yes	direct product factor
70	C₂	C₂²×C₄²×C₂₁₁	1	1	yes	characteristic
71	C₂	C₂×C₄×Q₈×C₂₁₁	1	3	yes	—
72	C₂	C₄×#41×C₂₁₁	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²×C₂₁₁	C₂²
commutator	C₂	C₂²×C₄²×C₂₁₁
Frattini, cosocle	C₂³	C₂⁴×C₂₁₁
socle, prime radical	C₂³×C₂₁₁	C₂⁴
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₄×Q₈×C₂₁₁	C₄	direct (N × H)
18. C₄²×C₂₁₁	Q₈	direct (N × H)
20. Q₈×C₂₁₁	C₄²	direct (N × H)
32. C₄×C₂₁₁	C₄×Q₈	direct (N × H)
36. C₂₁₁	C₄²×Q₈	direct (N × H)
37. C₄²×Q₈	C₂₁₁	direct (N × H)
42. C₄×Q₈	C₄×C₂₁₁	direct (N × H)
55. C₄²	Q₈×C₂₁₁	direct (N × H)
57. Q₈	C₄²×C₂₁₁	direct (N × H)
69. C₄	C₄×Q₈×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
C₂×C₄×Q₈×C₂₁₁	2
C₄×C₄⋊C₄×C₂₁₁	2
C₄³×C₂₁₁	2
C₄²×Q₈	211

Subgroups — C42×Q8×C211 #41009892

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×Q₈×C₂₁₁ #41009892