Subgroups — #1708×C₂₁₁ #41009328

Order 27008.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#498×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#492×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#487×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	#486×C₂₁₁	C₂	1	2	yes	maximal
5	C₈²×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₂₁₁	—	2	2	no	—
7	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
9	#127×C₂₁₁	C₂²	1	1	yes	characteristic
10	#106×C₂₁₁	—	2	2	no	—
11	#103×C₂₁₁	—	2	4	no	—
12	#102×C₂₁₁	—	2	4	no	—
13	C₄×C₈×C₂₁₁	C₂²	1	2	yes	—
14	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	—	2	1	no	—
17	C₂×D₄×C₂₁₁	—	2	2	no	—
18	C₂×D₄×C₂₁₁	—	2	1	no	—
19	SD₁₆×C₂₁₁	—	4	4	no	—
20	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
21	C₂×C₈×C₂₁₁	D₄	1	2	yes	—
22	C₄⋊C₄×C₂₁₁	—	2	2	no	—
23	C₄⋊C₄×C₂₁₁	—	2	2	no	—
24	C₄⋊C₄×C₂₁₁	—	2	2	no	—
25	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂³×C₂₁₁	—	4	1	no	—
27	Q₈×C₂₁₁	—	4	2	no	—
28	Q₈×C₂₁₁	—	4	4	no	—
29	D₄×C₂₁₁	—	4	2	no	—
30	D₄×C₂₁₁	—	4	4	no	—
31	C₂×C₄×C₂₁₁	—	4	1	no	—
32	C₂×C₄×C₂₁₁	—	4	2	no	—
33	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
35	C₈×C₂₁₁	—	2	4	no	—
36	C₈×C₂₁₁	SD₁₆	1	4	yes	—
37	C₂²×C₂₁₁	—	8	1	no	—
38	C₂²×C₂₁₁	—	8	2	no	—
39	C₂²×C₂₁₁	#127	1	1	yes	characteristic
40	C₄×C₂₁₁	—	8	1	no	—
41	C₄×C₂₁₁	—	8	2	no	—
42	C₄×C₂₁₁	C₂×SD₁₆	1	2	yes	—
43	C₄×C₂₁₁	#135	1	4	yes	—
44	C₂×C₂₁₁	—	16	1	no	—
45	C₂×C₂₁₁	#492	1	1	yes	characteristic
46	C₂×C₂₁₁	#495	1	2	yes	—
47	C₂₁₁	#1708	1	1	yes	characteristic, direct product factor, Sylow
48	#1708	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	#498	C₂×C₂₁₁	1	1	yes	characteristic, complemented
50	#492	C₂×C₂₁₁	1	1	yes	characteristic
51	#487	C₂×C₂₁₁	1	2	yes	complemented
52	#486	C₂×C₂₁₁	1	2	yes	—
53	C₈²	C₂×C₂₁₁	1	1	yes	characteristic, complemented
54	C₂×SD₁₆	—	2	2	no	—
55	C₄⋊Q₈	C₂²×C₂₁₁	1	2	yes	—
56	C₄⋊Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
57	#127	C₂²×C₂₁₁	1	1	yes	characteristic
58	#106	—	2	2	no	—
59	#103	—	2	4	no	—
60	#102	—	2	4	no	—
61	C₄×C₈	C₂²×C₂₁₁	1	2	yes	—
62	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	2	1	no	—
65	C₂×D₄	—	2	2	no	—
66	C₂×D₄	—	2	1	no	—
67	SD₁₆	—	4	4	no	—
68	C₂×C₈	D₄×C₂₁₁	1	4	yes	—
69	C₂×C₈	D₄×C₂₁₁	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
74	C₂³	—	4	1	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	4	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	4	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
82	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
83	C₈	—	2	4	no	—
84	C₈	SD₁₆×C₂₁₁	1	4	yes	—
85	C₂²	—	8	1	no	—
86	C₂²	—	8	2	no	—
87	C₂²	#127×C₂₁₁	1	1	yes	characteristic
88	C₄	—	8	1	no	—
89	C₄	—	8	2	no	—
90	C₄	C₂×SD₁₆×C₂₁₁	1	2	yes	—
91	C₄	#135×C₂₁₁	1	4	yes	—
92	C₂	—	16	1	no	—
93	C₂	#492×C₂₁₁	1	1	yes	characteristic
94	C₂	#495×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₂₁₁	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₁₁
prime radical	#127×C₂₁₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₄²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #498×C₂₁₁	C₂	semidirect
3. #487×C₂₁₁	C₂	semidirect
5. C₈²×C₂₁₁	C₂	semidirect
47. C₂₁₁	#1708	direct (N × H)
48. #1708	C₂₁₁	direct (N × H)
49. #498	C₂×C₂₁₁	semidirect
51. #487	C₂×C₂₁₁	semidirect
53. C₈²	C₂×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#498×C₂₁₁	2
#492×C₂₁₁	2
#487×C₂₁₁	2
#486×C₂₁₁	2
C₈²×C₂₁₁	2
#1708	211

Subgroups — #1708×C211 #41009328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1708×C₂₁₁ #41009328