Subgroups — #2183×C₂₁₁ #41009803

Order 27008.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₁₆×C₂₁₁	C₂	1	2	yes	maximal
4	#346×C₂₁₁	C₂	1	2	yes	maximal
5	C₄×C₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₂₁₁	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₁₆×C₂₁₁	C₂²	1	2	yes	—
9	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
10	C₄⋊C₈×C₂₁₁	C₄	1	2	yes	—
11	C₄⋊C₈×C₂₁₁	C₄	1	1	yes	characteristic
12	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₁₁	C₂²	1	2	yes	—
14	C₂×Q₈×C₂₁₁	C₈	1	1	yes	characteristic
15	C₂×C₈×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
18	C₄⋊C₄×C₂₁₁	C₈	1	2	yes	—
19	C₄⋊C₄×C₂₁₁	C₈	1	1	yes	characteristic
20	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₁₆×C₂₁₁	—	2	1	no	—
23	C₁₆×C₂₁₁	—	2	2	no	—
24	C₁₆×C₂₁₁	Q₈	1	2	yes	complemented
25	Q₈×C₂₁₁	—	2	2	no	—
26	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	C₂×C₈	1	2	yes	—
29	C₂×C₄×C₂₁₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
31	C₈×C₂₁₁	—	2	1	no	—
32	C₈×C₂₁₁	—	2	2	no	—
33	C₈×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₂₁₁	#41	1	1	yes	characteristic
35	C₂²×C₂₁₁	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₂₁₁	—	2	2	no	—
37	C₄×C₂₁₁	—	2	2	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	M₃₂	1	2	yes	—
40	C₄×C₂₁₁	C₄×Q₈	1	1	yes	characteristic
41	C₄×C₂₁₁	#131	1	1	yes	characteristic
42	C₂×C₂₁₁	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₂₁₁	C₂×M₃₂	1	1	yes	characteristic
44	C₂×C₂₁₁	#504	1	1	yes	characteristic
45	C₂₁₁	#2183	1	1	yes	characteristic, direct product factor, Sylow
46	#2183	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₈×Q₈	C₂×C₂₁₁	1	1	yes	characteristic
48	C₄⋊C₁₆	C₂×C₂₁₁	1	1	yes	characteristic
49	C₄⋊C₁₆	C₂×C₂₁₁	1	2	yes	—
50	#346	C₂×C₂₁₁	1	2	yes	—
51	C₄×C₁₆	C₂×C₂₁₁	1	1	yes	characteristic
52	C₄×Q₈	C₄×C₂₁₁	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
54	C₂×C₁₆	C₂²×C₂₁₁	1	2	yes	—
55	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
56	C₄⋊C₈	C₄×C₂₁₁	1	2	yes	—
57	C₄⋊C₈	C₄×C₂₁₁	1	1	yes	characteristic
58	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
59	C₄×C₈	C₂²×C₂₁₁	1	2	yes	—
60	C₂×Q₈	C₈×C₂₁₁	1	1	yes	characteristic
61	C₂×C₈	C₂×C₄×C₂₁₁	1	1	yes	characteristic
62	C₂×C₈	C₂×C₄×C₂₁₁	1	2	yes	—
63	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
64	C₄⋊C₄	C₈×C₂₁₁	1	2	yes	—
65	C₄⋊C₄	C₈×C₂₁₁	1	1	yes	characteristic
66	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
67	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	2	2	no	—
70	C₁₆	Q₈×C₂₁₁	1	2	yes	complemented
71	Q₈	—	2	2	no	—
72	C₂×C₄	C₂×C₈×C₂₁₁	1	2	yes	—
73	C₂×C₄	C₂×C₈×C₂₁₁	1	1	yes	characteristic
74	C₂×C₄	C₂×C₈×C₂₁₁	1	2	yes	—
75	C₂×C₄	C₂×C₈×C₂₁₁	1	1	yes	characteristic
76	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	2	2	no	—
79	C₈	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
80	C₈	#41×C₂₁₁	1	1	yes	characteristic
81	C₂²	C₂²×C₈×C₂₁₁	1	1	yes	characteristic
82	C₄	—	2	2	no	—
83	C₄	—	2	2	no	—
84	C₄	—	2	1	no	—
85	C₄	M₃₂×C₂₁₁	1	2	yes	—
86	C₄	C₄×Q₈×C₂₁₁	1	1	yes	characteristic
87	C₄	#131×C₂₁₁	1	1	yes	characteristic
88	C₂	C₈×Q₈×C₂₁₁	1	1	yes	characteristic
89	C₂	C₂×M₃₂×C₂₁₁	1	1	yes	characteristic
90	C₂	#504×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₂₁₁	C₂³
commutator	C₂²	C₂²×C₈×C₂₁₁
Frattini, cosocle	C₂×C₈	C₂³×C₂₁₁
socle, prime radical	C₂²×C₂₁₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₆×C₂₁₁	Q₈	semidirect
45. C₂₁₁	#2183	direct (N × H)
46. #2183	C₂₁₁	direct (N × H)
70. C₁₆	Q₈×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×Q₈×C₂₁₁	2
C₄⋊C₁₆×C₂₁₁	2
#346×C₂₁₁	2
C₄×C₁₆×C₂₁₁	2
#2183	211

Subgroups — #2183×C211 #41009803

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2183×C₂₁₁ #41009803