Subgroups — #1415×C₂₁₁ #41009035

Order 27008.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#366×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈×C₂₁₁	—	2	1	no	—
5	D₁₆×C₂₁₁	—	2	1	no	—
6	D₁₆×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×D₄×C₂₁₁	—	4	1	no	—
11	D₈×C₂₁₁	—	2	2	no	—
12	D₈×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	C₂³×C₂₁₁	—	8	1	no	—
18	D₄×C₂₁₁	—	4	2	no	—
19	D₄×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	—	8	2	no	—
25	C₂²×C₂₁₁	—	16	1	no	—
26	C₂²×C₂₁₁	—	16	1	no	—
27	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
28	C₄×C₂₁₁	—	16	1	no	—
29	C₄×C₂₁₁	#102	1	1	yes	characteristic
30	C₄×C₂₁₁	SD₃₂	1	1	yes	characteristic
31	C₂×C₂₁₁	—	16	2	no	—
32	C₂×C₂₁₁	#357	1	1	yes	characteristic
33	C₂×C₂₁₁	D₃₂	1	1	yes	characteristic
34	C₂×C₂₁₁	SD₆₄	1	1	yes	characteristic
35	C₂₁₁	#1415	1	1	yes	characteristic, direct product factor, Sylow
36	#1415	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×D₁₆	C₂×C₂₁₁	1	1	yes	characteristic
38	C₂×C₃₂	C₂×C₂₁₁	1	1	yes	characteristic, complemented
39	#366	C₂×C₂₁₁	1	1	yes	characteristic, complemented
40	C₂×D₈	—	2	1	no	—
41	D₁₆	—	2	1	no	—
42	D₁₆	C₄×C₂₁₁	1	2	yes	complemented
43	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
44	#107	—	2	1	no	—
45	C₃₂	—	2	1	no	—
46	C₂×D₄	—	4	1	no	—
47	D₈	—	2	2	no	—
48	D₈	—	4	1	no	—
49	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
50	C₄⋊C₄	—	4	1	no	—
51	C₁₆	C₂×C₄×C₂₁₁	1	1	yes	characteristic
52	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
53	C₂³	—	8	1	no	—
54	D₄	—	4	2	no	—
55	D₄	—	8	1	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	D₈×C₂₁₁	1	1	yes	characteristic
58	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
59	C₈	SD₁₆×C₂₁₁	1	1	yes	characteristic
60	C₂²	—	8	2	no	—
61	C₂²	—	16	1	no	—
62	C₂²	—	16	1	no	—
63	C₂²	D₁₆×C₂₁₁	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	#102×C₂₁₁	1	1	yes	characteristic
66	C₄	SD₃₂×C₂₁₁	1	1	yes	characteristic
67	C₂	—	16	2	no	—
68	C₂	#357×C₂₁₁	1	1	yes	characteristic
69	C₂	D₃₂×C₂₁₁	1	1	yes	characteristic
70	C₂	SD₆₄×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₁₁	D₁₆
commutator	C₁₆	C₂×C₄×C₂₁₁
Frattini, cosocle	C₂×C₁₆	C₂²×C₂₁₁
prime radical	C₂×D₁₆×C₂₁₁	C₂
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
2. C₂×C₃₂×C₂₁₁	C₂	semidirect
3. #366×C₂₁₁	C₂	semidirect
6. D₁₆×C₂₁₁	C₄	semidirect
35. C₂₁₁	#1415	direct (N × H)
36. #1415	C₂₁₁	direct (N × H)
38. C₂×C₃₂	C₂×C₂₁₁	semidirect
39. #366	C₂×C₂₁₁	semidirect
42. D₁₆	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×D₁₆×C₂₁₁	2
C₂×C₃₂×C₂₁₁	2
#366×C₂₁₁	2
#1415	211

Subgroups — #1415×C211 #41009035

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1415×C₂₁₁ #41009035