Subgroups — #1416×C₂₁₁ #41009036

Order 27008.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#366×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆×C₂₁₁	—	2	1	no	—
5	Q₃₂×C₂₁₁	—	2	1	no	—
6	Q₃₂×C₂₁₁	C₄	1	2	yes	complemented
7	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
8	#107×C₂₁₁	—	2	1	no	—
9	C₃₂×C₂₁₁	—	2	1	no	—
10	C₂×Q₈×C₂₁₁	—	4	1	no	—
11	Q₁₆×C₂₁₁	—	2	2	no	—
12	Q₁₆×C₂₁₁	—	4	1	no	—
13	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
14	C₄⋊C₄×C₂₁₁	—	4	1	no	—
15	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
17	Q₈×C₂₁₁	—	4	2	no	—
18	Q₈×C₂₁₁	—	8	1	no	—
19	C₂×C₄×C₂₁₁	—	8	1	no	—
20	C₂×C₄×C₂₁₁	—	8	1	no	—
21	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
22	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₁₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
25	C₄×C₂₁₁	—	8	2	no	—
26	C₄×C₂₁₁	—	16	1	no	—
27	C₄×C₂₁₁	#102	1	1	yes	characteristic
28	C₄×C₂₁₁	SD₃₂	1	1	yes	characteristic
29	C₂×C₂₁₁	#357	1	1	yes	characteristic
30	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
31	C₂×C₂₁₁	Q₆₄	1	1	yes	characteristic
32	C₂₁₁	#1416	1	1	yes	characteristic, direct product factor, Sylow
33	#1416	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×Q₃₂	C₂×C₂₁₁	1	1	yes	characteristic
35	C₂×C₃₂	C₂×C₂₁₁	1	1	yes	characteristic
36	#366	C₂×C₂₁₁	1	1	yes	characteristic
37	C₂×Q₁₆	—	2	1	no	—
38	Q₃₂	—	2	1	no	—
39	Q₃₂	C₄×C₂₁₁	1	2	yes	complemented
40	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
41	#107	—	2	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×Q₈	—	4	1	no	—
44	Q₁₆	—	2	2	no	—
45	Q₁₆	—	4	1	no	—
46	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
47	C₄⋊C₄	—	4	1	no	—
48	C₁₆	C₂×C₄×C₂₁₁	1	1	yes	characteristic
49	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
50	Q₈	—	4	2	no	—
51	Q₈	—	8	1	no	—
52	C₂×C₄	—	8	1	no	—
53	C₂×C₄	—	8	1	no	—
54	C₂×C₄	D₈×C₂₁₁	1	1	yes	characteristic
55	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
56	C₈	SD₁₆×C₂₁₁	1	1	yes	characteristic
57	C₂²	D₁₆×C₂₁₁	1	1	yes	characteristic
58	C₄	—	8	2	no	—
59	C₄	—	16	1	no	—
60	C₄	#102×C₂₁₁	1	1	yes	characteristic
61	C₄	SD₃₂×C₂₁₁	1	1	yes	characteristic
62	C₂	#357×C₂₁₁	1	1	yes	characteristic
63	C₂	SD₆₄×C₂₁₁	1	1	yes	characteristic
64	C₂	Q₆₄×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₁₁	D₁₆
commutator	C₁₆	C₂×C₄×C₂₁₁
Frattini, cosocle	C₂×C₁₆	C₂²×C₂₁₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ C₂×C₈×C₂₁₁ ◁ C₂×C₁₆×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₃₂×C₂₁₁	C₄	semidirect
32. C₂₁₁	#1416	direct (N × H)
33. #1416	C₂₁₁	direct (N × H)
39. Q₃₂	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×C₂₁₁	2
C₂×C₃₂×C₂₁₁	2
#366×C₂₁₁	2
#1416	211

Subgroups — #1416×C211 #41009036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1416×C₂₁₁ #41009036