Subgroups — C₂²×#89166 #13183401

Order 1760.

472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₅₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89166	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×D₅₅	C₄	1	8	yes	complemented
7	C₂²×D₅₅	C₂²	1	6	yes	—
8	C₂×C₁₁×Dic₅	C₂²	1	6	yes	complemented
9	C₂×C₅×Dic₁₁	C₂²	1	6	yes	complemented
10	#89166	C₂²	1	16	yes	direct product factor
11	C₂²×C₄×D₁₁	—	5	1	no	maximal, Hall
12	C₂²×C₅×C₁₁	C₂×C₄	1	4	yes	complemented
13	C₂²×C₅×C₁₁	C₂³	1	3	yes	—
14	C₂×D₅₅	C₂×C₄	1	24	yes	complemented
15	C₂×D₅₅	C₂³	1	4	yes	—
16	C₅×Dic₁₁	C₂³	1	4	yes	complemented
17	C₁₁×Dic₅	C₂³	1	4	yes	complemented
18	C₂³×D₁₁	—	5	1	no	—
19	C₂²×C₄×C₁₁	—	5	1	no	—
20	C₂²×Dic₁₁	D₅	1	1	yes	characteristic, complemented
21	C₂×C₄×D₁₁	—	5	12	no	—
22	C₂²×C₄×D₅	—	11	1	no	maximal, Hall
23	C₂×C₅×C₁₁	C₂²×C₄	1	6	yes	complemented
24	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
25	D₅₅	C₂²×C₄	1	8	yes	complemented
26	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
27	C₂²×D₁₁	—	5	8	no	—
28	C₂²×D₁₁	—	5	6	no	—
29	C₂×C₄×C₁₁	—	5	6	no	—
30	C₂×Dic₁₁	C₂×D₅	1	6	yes	complemented
31	C₄×D₁₁	—	5	16	no	—
32	C₂³×D₅	—	11	1	no	—
33	C₂²×C₄×C₅	—	11	1	no	—
34	C₂²×Dic₅	D₁₁	1	1	yes	characteristic, complemented
35	C₂×C₄×D₅	—	11	12	no	—
36	C₅×C₁₁	C₂³×C₄	1	1	yes	characteristic, complemented, Hall
37	C₂²×C₁₁	C₄×D₅	1	4	yes	complemented
38	C₂²×C₁₁	C₂²×D₅	1	3	yes	—
39	C₂×D₁₁	—	5	24	no	—
40	C₂×D₁₁	—	5	4	no	—
41	C₄×C₁₁	—	5	4	no	—
42	Dic₁₁	C₂²×D₅	1	4	yes	complemented
43	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
44	C₂²×D₅	—	11	6	no	—
45	C₂²×D₅	—	11	8	no	—
46	C₂×C₄×C₅	—	11	6	no	—
47	C₂×Dic₅	C₂×D₁₁	1	6	yes	complemented
48	C₄×D₅	—	11	16	no	—
49	C₂³×C₄	—	55	1	no	Sylow
50	C₂×C₁₁	C₂×C₄×D₅	1	6	yes	complemented
51	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
52	D₁₁	—	5	8	no	—
53	C₂²×C₅	C₄×D₁₁	1	4	yes	complemented
54	C₂²×C₅	C₂²×D₁₁	1	3	yes	—
55	C₂×D₅	—	11	24	no	—
56	C₂×D₅	—	11	4	no	—
57	C₄×C₅	—	11	4	no	—
58	Dic₅	C₂²×D₁₁	1	4	yes	complemented
59	C₂⁴	—	55	1	no	—
60	C₂²×C₄	—	5	1	no	—
61	C₂²×C₄	—	11	1	no	—
62	C₂²×C₄	—	55	12	no	—
63	C₁₁	C₂²×C₄×D₅	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₅	C₂×C₄×D₁₁	1	6	yes	complemented
65	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
66	D₅	—	11	8	no	—
67	C₂³	—	55	8	no	—
68	C₂³	—	55	6	no	—
69	C₂³	D₅×D₁₁	1	1	yes	characteristic
70	C₂×C₄	—	5	6	no	—
71	C₂×C₄	—	11	6	no	—
72	C₂×C₄	—	55	16	no	—
73	C₅	C₂²×C₄×D₁₁	1	1	yes	characteristic, complemented, Sylow
74	C₂²	—	55	24	no	—
75	C₂²	—	55	4	no	—
76	C₂²	#89166	1	4	yes	direct product factor
77	C₂²	C₂×D₅×D₁₁	1	3	yes	—
78	C₄	—	5	4	no	—
79	C₄	—	11	4	no	—
80	C₂	—	55	8	no	—
81	C₂	C₂×#89166	1	6	yes	direct product factor
82	C₂	C₂²×D₅×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₁₁
commutator, hypocenter	C₅×C₁₁	C₂³×C₄
Frattini	C₂	C₂²×D₅×D₁₁
Fitting, socle	C₂³×C₅×C₁₁	C₂²
prime radical	C₂³×D₅₅	C₂
cosocle	C₂×C₅×C₁₁	C₂⁴
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₁×Dic₅	C₂	semidirect
3. C₂²×C₅×Dic₁₁	C₂	semidirect
4. C₂×#89166	C₂	direct (N × H)
6. C₂²×D₅₅	C₄	semidirect
8. C₂×C₁₁×Dic₅	C₂²	semidirect
9. C₂×C₅×Dic₁₁	C₂²	semidirect
10. #89166	C₂²	direct (N × H)
12. C₂²×C₅×C₁₁	C₂×C₄	semidirect
14. C₂×D₅₅	C₂×C₄	semidirect
16. C₅×Dic₁₁	C₂³	semidirect
17. C₁₁×Dic₅	C₂³	semidirect
20. C₂²×Dic₁₁	D₅	semidirect
23. C₂×C₅×C₁₁	C₂²×C₄	semidirect
25. D₅₅	C₂²×C₄	semidirect
30. C₂×Dic₁₁	C₂×D₅	semidirect
34. C₂²×Dic₅	D₁₁	semidirect
36. C₅×C₁₁	C₂³×C₄	semidirect
37. C₂²×C₁₁	C₄×D₅	semidirect
42. Dic₁₁	C₂²×D₅	semidirect
47. C₂×Dic₅	C₂×D₁₁	semidirect
50. C₂×C₁₁	C₂×C₄×D₅	semidirect
53. C₂²×C₅	C₄×D₁₁	semidirect
58. Dic₅	C₂²×D₁₁	semidirect
63. C₁₁	C₂²×C₄×D₅	semidirect
64. C₂×C₅	C₂×C₄×D₁₁	semidirect
73. C₅	C₂²×C₄×D₁₁	semidirect
76. C₂²	#89166	direct (N × H)
81. C₂	C₂×#89166	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₅₅	2
C₂²×C₁₁×Dic₅	2
C₂²×C₅×Dic₁₁	2
C₂×#89166	2
C₂²×C₄×D₁₁	5
C₂²×C₄×D₅	11

Subgroups — C22×#89166 #13183401

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89166 #13183401