Subgroups — #1401×C₂₁₁ #41009021

Order 27008.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	M₆₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#131×C₂₁₁	C₄	1	1	yes	characteristic
5	M₃₂×C₂₁₁	—	2	1	no	—
6	M₃₂×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₁₁	—	2	1	no	—
8	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₁₁	—	2	1	no	—
10	C₃₂×C₂₁₁	—	2	1	no	—
11	#41×C₂₁₁	C₈	1	1	yes	characteristic
12	M₁₆×C₂₁₁	—	2	1	no	—
13	M₁₆×C₂₁₁	C₈	1	1	yes	characteristic
14	C₂×C₈×C₂₁₁	—	2	1	no	—
15	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₁₁	—	2	1	no	—
17	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
18	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
19	Q₈×C₂₁₁	C₁₆	1	1	yes	characteristic
20	D₄×C₂₁₁	—	2	1	no	—
21	D₄×C₂₁₁	C₁₆	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	—	2	1	no	—
23	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
24	C₈×C₂₁₁	—	2	1	no	—
25	C₈×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₁₁	M₁₆	1	1	yes	characteristic
27	C₂²×C₂₁₁	—	2	1	no	—
28	C₂²×C₂₁₁	M₃₂	1	1	yes	characteristic
29	C₄×C₂₁₁	—	2	1	no	—
30	C₄×C₂₁₁	C₂²⋊C₈	1	1	yes	characteristic
31	C₄×C₂₁₁	C₂×C₁₆	1	1	yes	characteristic
32	C₂×C₂₁₁	—	2	1	no	—
33	C₂×C₂₁₁	—	4	1	no	—
34	C₂×C₂₁₁	C₂²⋊C₁₆	1	1	yes	characteristic
35	C₂₁₁	#1401	1	1	yes	characteristic, direct product factor, Sylow
36	#1401	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	#504	C₂×C₂₁₁	1	1	yes	characteristic
38	M₆₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
39	C₂×C₃₂	C₂×C₂₁₁	1	1	yes	characteristic, complemented
40	#131	C₄×C₂₁₁	1	1	yes	characteristic
41	M₃₂	—	2	1	no	—
42	M₃₂	C₄×C₂₁₁	1	1	yes	characteristic
43	C₂×C₁₆	—	2	1	no	—
44	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	C₈×C₂₁₁	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	M₁₆	C₈×C₂₁₁	1	1	yes	characteristic
50	C₂×C₈	—	2	1	no	—
51	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
52	C₁₆	—	2	1	no	—
53	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
54	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
55	Q₈	C₁₆×C₂₁₁	1	1	yes	characteristic
56	D₄	—	2	1	no	—
57	D₄	C₁₆×C₂₁₁	1	1	yes	characteristic
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	C₂×C₈×C₂₁₁	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
62	C₈	M₁₆×C₂₁₁	1	1	yes	characteristic
63	C₂²	—	2	1	no	—
64	C₂²	M₃₂×C₂₁₁	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	C₂²⋊C₈×C₂₁₁	1	1	yes	characteristic
67	C₄	C₂×C₁₆×C₂₁₁	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	1	no	—
70	C₂	C₂²⋊C₁₆×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₂₁₁	D₄
commutator	C₄	C₂×C₁₆×C₂₁₁
Frattini, cosocle	C₂×C₁₆	C₂²×C₂₁₁
socle	C₂×C₂₁₁	C₂²⋊C₁₆
prime radical	#41×C₂₁₁	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₁₁ ◁ C₂×C₁₆×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. M₆₄×C₂₁₁	C₂	semidirect
3. C₂×C₃₂×C₂₁₁	C₂	semidirect
35. C₂₁₁	#1401	direct (N × H)
36. #1401	C₂₁₁	direct (N × H)
38. M₆₄	C₂×C₂₁₁	semidirect
39. C₂×C₃₂	C₂×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#504×C₂₁₁	2
M₆₄×C₂₁₁	2
C₂×C₃₂×C₂₁₁	2
#1401	211

Subgroups — #1401×C211 #41009021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1401×C₂₁₁ #41009021