Subgroups — #1842×C₂₁₁ #41009462

Order 27008.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×C₂₁₁	C₂	1	6	yes	maximal
2	C₄³×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₁₁	C₂²	1	4	yes	—
4	C₂×C₄²×C₂₁₁	C₄	1	3	yes	—
5	C₂×C₄²×C₂₁₁	C₄	1	1	yes	characteristic
6	C₂×C₄²×C₂₁₁	C₂²	1	3	yes	—
7	C₄⋊C₈×C₂₁₁	—	2	12	no	—
8	C₂²×C₄×C₂₁₁	C₂×C₄	1	3	yes	—
9	C₂²×C₄×C₂₁₁	C₂×C₄	1	3	yes	—
10	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
11	C₂×C₈×C₂₁₁	—	2	12	no	—
12	C₄²×C₂₁₁	—	2	6	no	—
13	C₄²×C₂₁₁	C₈	1	4	yes	complemented
14	C₄²×C₂₁₁	D₄	1	6	yes	—
15	C₄²×C₂₁₁	Q₈	1	6	yes	—
16	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₄×C₂₁₁	—	2	6	no	—
18	C₂×C₄×C₂₁₁	C₄⋊C₄	1	6	yes	—
19	C₂×C₄×C₂₁₁	C₄⋊C₄	1	6	yes	—
20	C₂×C₄×C₂₁₁	C₂×C₈	1	6	yes	—
21	C₂×C₄×C₂₁₁	M₁₆	1	6	yes	—
22	C₂×C₄×C₂₁₁	C₂×D₄	1	3	yes	—
23	C₂×C₄×C₂₁₁	C₂×Q₈	1	3	yes	—
24	C₈×C₂₁₁	—	4	4	no	—
25	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	3	yes	—
26	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
27	C₂²×C₂₁₁	C₂×M₁₆	1	3	yes	—
28	C₄×C₂₁₁	—	2	6	no	—
29	C₄×C₂₁₁	C₄⋊C₈	1	12	yes	complemented
30	C₄×C₂₁₁	#127	1	1	yes	characteristic
31	C₄×C₂₁₁	C₄⋊Q₈	1	3	yes	—
32	C₂×C₂₁₁	#384	1	1	yes	characteristic
33	C₂×C₂₁₁	C₂×C₄⋊C₈	1	3	yes	—
34	C₂×C₂₁₁	#423	1	3	yes	—
35	C₂₁₁	#1842	1	1	yes	characteristic, direct product factor, Sylow
36	#1842	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₄⋊C₈	C₂×C₂₁₁	1	6	yes	—
38	C₄³	C₂×C₂₁₁	1	1	yes	characteristic
39	C₂²×C₈	C₂²×C₂₁₁	1	4	yes	—
40	C₂×C₄²	C₄×C₂₁₁	1	3	yes	—
41	C₂×C₄²	C₄×C₂₁₁	1	1	yes	characteristic
42	C₂×C₄²	C₂²×C₂₁₁	1	3	yes	—
43	C₄⋊C₈	—	2	12	no	—
44	C₂²×C₄	C₂×C₄×C₂₁₁	1	3	yes	—
45	C₂²×C₄	C₂×C₄×C₂₁₁	1	3	yes	—
46	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
47	C₂×C₈	—	2	12	no	—
48	C₄²	—	2	6	no	—
49	C₄²	C₈×C₂₁₁	1	4	yes	complemented
50	C₄²	D₄×C₂₁₁	1	6	yes	—
51	C₄²	Q₈×C₂₁₁	1	6	yes	—
52	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
53	C₂×C₄	—	2	6	no	—
54	C₂×C₄	C₄⋊C₄×C₂₁₁	1	6	yes	—
55	C₂×C₄	C₄⋊C₄×C₂₁₁	1	6	yes	—
56	C₂×C₄	C₂×C₈×C₂₁₁	1	6	yes	—
57	C₂×C₄	M₁₆×C₂₁₁	1	6	yes	—
58	C₂×C₄	C₂×D₄×C₂₁₁	1	3	yes	—
59	C₂×C₄	C₂×Q₈×C₂₁₁	1	3	yes	—
60	C₈	—	4	4	no	—
61	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	3	yes	—
62	C₂²	C₂²×C₈×C₂₁₁	1	1	yes	characteristic
63	C₂²	C₂×M₁₆×C₂₁₁	1	3	yes	—
64	C₄	—	2	6	no	—
65	C₄	C₄⋊C₈×C₂₁₁	1	12	yes	complemented
66	C₄	#127×C₂₁₁	1	1	yes	characteristic
67	C₄	C₄⋊Q₈×C₂₁₁	1	3	yes	—
68	C₂	#384×C₂₁₁	1	1	yes	characteristic
69	C₂	C₂×C₄⋊C₈×C₂₁₁	1	3	yes	—
70	C₂	#423×C₂₁₁	1	3	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂²×C₈×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₄²×C₂₁₁	C₈	semidirect
29. C₄×C₂₁₁	C₄⋊C₈	semidirect
35. C₂₁₁	#1842	direct (N × H)
36. #1842	C₂₁₁	direct (N × H)
49. C₄²	C₈×C₂₁₁	semidirect
65. C₄	C₄⋊C₈×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊C₈×C₂₁₁	2
C₄³×C₂₁₁	2
#1842	211

Subgroups — #1842×C211 #41009462

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1842×C₂₁₁ #41009462