Subgroups — C₂²×#89170 #13183405

Order 1760.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₅₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₁×Dic₅	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#89170	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂×Dic₅₅	C₂²	1	6	yes	—
7	C₂×C₁₁×Dic₅	C₂²	1	6	yes	—
8	C₂×C₅×Dic₁₁	C₂²	1	6	yes	—
9	#89170	C₂²	1	16	yes	direct product factor
10	C₂²×Dic₂₂	—	5	1	no	maximal, Hall
11	C₂²×C₅×C₁₁	Q₈	1	4	yes	complemented
12	C₂²×C₅×C₁₁	C₂³	1	3	yes	—
13	Dic₅₅	C₂³	1	4	yes	—
14	C₅×Dic₁₁	C₂³	1	4	yes	—
15	C₁₁×Dic₅	C₂³	1	4	yes	—
16	C₂²×C₄×C₁₁	—	5	1	no	—
17	C₂²×Dic₁₁	—	5	1	no	—
18	C₂²×Dic₁₁	D₅	1	1	yes	characteristic
19	C₂×Dic₂₂	—	5	12	no	—
20	C₂²×Dic₁₀	—	11	1	no	maximal, Hall
21	C₂×C₅×C₁₁	C₂×Q₈	1	6	yes	complemented
22	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
23	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
24	C₂×C₄×C₁₁	—	5	6	no	—
25	C₂×Dic₁₁	—	5	6	no	—
26	C₂×Dic₁₁	C₂×D₅	1	6	yes	—
27	Dic₂₂	—	5	16	no	—
28	C₂²×C₄×C₅	—	11	1	no	—
29	C₂²×Dic₅	—	11	1	no	—
30	C₂²×Dic₅	D₁₁	1	1	yes	characteristic
31	C₂×Dic₁₀	—	11	12	no	—
32	C₅×C₁₁	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
33	C₂²×C₁₁	Dic₁₀	1	4	yes	complemented
34	C₂²×C₁₁	C₂²×D₅	1	3	yes	—
35	C₄×C₁₁	—	5	4	no	—
36	Dic₁₁	—	5	4	no	—
37	Dic₁₁	C₂²×D₅	1	4	yes	—
38	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
39	C₂×C₄×C₅	—	11	6	no	—
40	C₂×Dic₅	—	11	6	no	—
41	C₂×Dic₅	C₂×D₁₁	1	6	yes	—
42	Dic₁₀	—	11	16	no	—
43	C₂²×Q₈	—	55	1	no	Sylow
44	C₂×C₁₁	C₂×Dic₁₀	1	6	yes	complemented
45	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
46	C₂²×C₅	Dic₂₂	1	4	yes	complemented
47	C₂²×C₅	C₂²×D₁₁	1	3	yes	—
48	C₄×C₅	—	11	4	no	—
49	Dic₅	—	11	4	no	—
50	Dic₅	C₂²×D₁₁	1	4	yes	—
51	C₂×Q₈	—	55	12	no	—
52	C₂²×C₄	—	5	1	no	—
53	C₂²×C₄	—	11	1	no	—
54	C₂²×C₄	—	55	1	no	—
55	C₁₁	C₂²×Dic₁₀	1	1	yes	characteristic, complemented, Sylow
56	C₂×C₅	C₂×Dic₂₂	1	6	yes	complemented
57	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
58	C₂³	D₅×D₁₁	1	1	yes	characteristic
59	Q₈	—	55	16	no	—
60	C₂×C₄	—	5	6	no	—
61	C₂×C₄	—	11	6	no	—
62	C₂×C₄	—	55	6	no	—
63	C₅	C₂²×Dic₂₂	1	1	yes	characteristic, complemented, Sylow
64	C₂²	#89170	1	4	yes	direct product factor
65	C₂²	C₂×D₅×D₁₁	1	3	yes	—
66	C₄	—	5	4	no	—
67	C₄	—	11	4	no	—
68	C₄	—	55	4	no	—
69	C₂	C₂×#89170	1	6	yes	direct product factor
70	C₂	C₂²×D₅×D₁₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	D₅×D₁₁
commutator, cosocle	C₂×C₅×C₁₁	C₂⁴
Frattini	C₂	C₂²×D₅×D₁₁
Fitting, socle, prime radical	C₂³×C₅×C₁₁	C₂²
hypocenter	C₅×C₁₁	C₂²×Q₈
LCS	G ▷ C₂×C₅×C₁₁ ▷ C₅×C₁₁
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
4. C₂×#89170	C₂	direct (N × H)
9. #89170	C₂²	direct (N × H)
11. C₂²×C₅×C₁₁	Q₈	semidirect
21. C₂×C₅×C₁₁	C₂×Q₈	semidirect
32. C₅×C₁₁	C₂²×Q₈	semidirect
33. C₂²×C₁₁	Dic₁₀	semidirect
44. C₂×C₁₁	C₂×Dic₁₀	semidirect
46. C₂²×C₅	Dic₂₂	semidirect
55. C₁₁	C₂²×Dic₁₀	semidirect
56. C₂×C₅	C₂×Dic₂₂	semidirect
63. C₅	C₂²×Dic₂₂	semidirect
64. C₂²	#89170	direct (N × H)
69. C₂	C₂×#89170	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Dic₅₅	2
C₂²×C₁₁×Dic₅	2
C₂²×C₅×Dic₁₁	2
C₂×#89170	2
C₂²×Dic₂₂	5
C₂²×Dic₁₀	11

Subgroups — C22×#89170 #13183405

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89170 #13183405