Subgroups — #1419×C₂₁₁ #41009039

Order 27008.

78 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	M₆₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×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	—
7	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
8	#108×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	M₁₆×C₂₁₁	—	4	1	no	—
14	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
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	D₈	1	1	yes	characteristic
21	C₈×C₂₁₁	—	8	1	no	—
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	2	no	—
26	C₂²×C₂₁₁	SD₃₂	1	1	yes	characteristic
27	C₄×C₂₁₁	#102	1	1	yes	characteristic
28	C₄×C₂₁₁	D₁₆	1	1	yes	characteristic
29	C₂×C₂₁₁	—	2	1	no	—
30	C₂×C₂₁₁	—	16	2	no	—
31	C₂×C₂₁₁	#357	1	1	yes	characteristic
32	C₂₁₁	#1419	1	1	yes	characteristic, direct product factor, Sylow
33	#1419	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×D₁₆	C₂×C₂₁₁	1	1	yes	characteristic
35	M₆₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
36	#368	C₂×C₂₁₁	1	1	yes	characteristic, complemented
37	C₂×D₈	—	2	1	no	—
38	D₁₆	—	2	1	no	—
39	D₁₆	C₄×C₂₁₁	1	2	yes	—
40	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
41	#108	—	2	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	4	1	no	—
44	D₈	—	2	2	no	—
45	D₈	—	4	1	no	—
46	M₁₆	—	4	1	no	—
47	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
48	C₁₆	C₂×C₄×C₂₁₁	1	1	yes	characteristic
49	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
50	C₂³	—	8	1	no	—
51	D₄	—	4	2	no	—
52	D₄	—	8	1	no	—
53	C₂×C₄	D₈×C₂₁₁	1	1	yes	characteristic
54	C₈	—	8	1	no	—
55	C₈	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
56	C₈	SD₁₆×C₂₁₁	1	1	yes	characteristic
57	C₂²	—	8	2	no	—
58	C₂²	—	16	2	no	—
59	C₂²	SD₃₂×C₂₁₁	1	1	yes	characteristic
60	C₄	#102×C₂₁₁	1	1	yes	characteristic
61	C₄	D₁₆×C₂₁₁	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	16	2	no	—
64	C₂	#357×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	#357
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 ◁ C₂×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. M₆₄×C₂₁₁	C₂	semidirect
3. #368×C₂₁₁	C₂	semidirect
32. C₂₁₁	#1419	direct (N × H)
33. #1419	C₂₁₁	direct (N × H)
35. M₆₄	C₂×C₂₁₁	semidirect
36. #368	C₂×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
M₆₄×C₂₁₁	2
#368×C₂₁₁	2
#1419	211

Subgroups — #1419×C211 #41009039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1419×C₂₁₁ #41009039