Subgroups — #2179×C₂₁₁ #41009799

Order 27008.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#443×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	#368×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#359×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	#346×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₂₁₁	C₂²	1	2	yes	—
7	#131×C₂₁₁	—	2	2	no	—
8	Q₃₂×C₂₁₁	C₄	1	1	yes	characteristic, complemented
9	SD₃₂×C₂₁₁	C₄	1	2	yes	complemented
10	D₁₆×C₂₁₁	C₄	1	1	yes	characteristic, complemented
11	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
13	#108×C₂₁₁	C₂²	1	2	yes	—
14	#104×C₂₁₁	—	2	2	no	—
15	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
16	#41×C₂₁₁	—	2	2	no	—
17	Q₁₆×C₂₁₁	C₂×C₄	1	2	yes	—
18	SD₁₆×C₂₁₁	—	2	2	no	—
19	D₈×C₂₁₁	C₂×C₄	1	2	yes	—
20	M₁₆×C₂₁₁	—	2	2	no	—
21	M₁₆×C₂₁₁	—	4	2	no	—
22	C₂×C₈×C₂₁₁	—	4	2	no	—
23	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
26	C₁₆×C₂₁₁	—	2	1	no	—
27	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	Q₈×C₂₁₁	—	2	2	no	—
30	D₄×C₂₁₁	—	2	2	no	—
31	D₄×C₂₁₁	—	4	2	no	—
32	C₂×C₄×C₂₁₁	—	2	1	no	—
33	C₂×C₄×C₂₁₁	—	4	2	no	—
34	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₂₁₁	—	4	2	no	—
36	C₈×C₂₁₁	D₈	1	2	yes	—
37	C₈×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₂₁₁	#41	1	1	yes	characteristic
39	C₂²×C₂₁₁	—	4	2	no	—
40	C₂²×C₂₁₁	#135	1	1	yes	characteristic
41	C₄×C₂₁₁	—	4	1	no	—
42	C₄×C₂₁₁	—	4	2	no	—
43	C₄×C₂₁₁	C₄×D₄	1	1	yes	characteristic
44	C₄×C₂₁₁	C₂×D₈	1	1	yes	characteristic
45	C₂×C₂₁₁	—	2	1	no	—
46	C₂×C₂₁₁	—	8	2	no	—
47	C₂×C₂₁₁	C₄×D₈	1	1	yes	characteristic
48	C₂₁₁	#2179	1	1	yes	characteristic, direct product factor, Sylow
49	#2179	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	#508	C₂×C₂₁₁	1	1	yes	characteristic
51	#443	C₂×C₂₁₁	1	2	yes	complemented
52	#368	C₂×C₂₁₁	1	1	yes	characteristic, complemented
53	#359	C₂×C₂₁₁	1	2	yes	complemented
54	#346	C₂×C₂₁₁	1	1	yes	characteristic, complemented
55	#135	C₂²×C₂₁₁	1	2	yes	—
56	#131	—	2	2	no	—
57	Q₃₂	C₄×C₂₁₁	1	1	yes	characteristic, complemented
58	SD₃₂	C₄×C₂₁₁	1	2	yes	complemented
59	D₁₆	C₄×C₂₁₁	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
61	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
62	#108	C₂²×C₂₁₁	1	2	yes	—
63	#104	—	2	2	no	—
64	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
65	#41	—	2	2	no	—
66	Q₁₆	C₂×C₄×C₂₁₁	1	2	yes	—
67	SD₁₆	—	2	2	no	—
68	D₈	C₂×C₄×C₂₁₁	1	2	yes	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	2	no	—
71	C₂×C₈	—	4	2	no	—
72	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
73	C₂×C₈	C₂³×C₂₁₁	1	1	yes	characteristic
74	C₄²	D₄×C₂₁₁	1	1	yes	characteristic
75	C₁₆	—	2	1	no	—
76	C₁₆	C₂×C₄×C₂₁₁	1	1	yes	characteristic
77	C₁₆	C₂×C₄×C₂₁₁	1	1	yes	characteristic
78	Q₈	—	2	2	no	—
79	D₄	—	2	2	no	—
80	D₄	—	4	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	D₈×C₂₁₁	1	2	yes	—
86	C₈	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
87	C₈	#41×C₂₁₁	1	1	yes	characteristic
88	C₂²	—	4	2	no	—
89	C₂²	#135×C₂₁₁	1	1	yes	characteristic
90	C₄	—	4	1	no	—
91	C₄	—	4	2	no	—
92	C₄	C₄×D₄×C₂₁₁	1	1	yes	characteristic
93	C₄	C₂×D₈×C₂₁₁	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	8	2	no	—
96	C₂	C₄×D₈×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₂₁₁	C₂×D₈
commutator	C₈	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂×C₈	C₂³×C₂₁₁
socle	C₂×C₂₁₁	C₄×D₈
prime radical	#508×C₂₁₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₁₁ ◁ C₂×C₈×C₂₁₁ ◁ C₄×C₈×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #443×C₂₁₁	C₂	semidirect
3. #368×C₂₁₁	C₂	semidirect
4. #359×C₂₁₁	C₂	semidirect
5. #346×C₂₁₁	C₂	semidirect
8. Q₃₂×C₂₁₁	C₄	semidirect
9. SD₃₂×C₂₁₁	C₄	semidirect
10. D₁₆×C₂₁₁	C₄	semidirect
48. C₂₁₁	#2179	direct (N × H)
49. #2179	C₂₁₁	direct (N × H)
51. #443	C₂×C₂₁₁	semidirect
52. #368	C₂×C₂₁₁	semidirect
53. #359	C₂×C₂₁₁	semidirect
54. #346	C₂×C₂₁₁	semidirect
57. Q₃₂	C₄×C₂₁₁	semidirect
58. SD₃₂	C₄×C₂₁₁	semidirect
59. D₁₆	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#508×C₂₁₁	2
#443×C₂₁₁	2
#368×C₂₁₁	2
#359×C₂₁₁	2
#346×C₂₁₁	2
#2179	211

Subgroups — #2179×C211 #41009799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2179×C₂₁₁ #41009799