Subgroups — C₂²×#89169 #13183404

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, complemented
2	C₂³×D₅×C₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×Dic₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89169	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₅×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×D₅₅	—	2	4	no	—
7	C₂²×D₅₅	C₂²	1	6	yes	complemented
8	C₂²×D₅×C₁₁	—	2	4	no	—
9	C₂²×D₅×C₁₁	C₂²	1	6	yes	complemented
10	C₂×C₅×Dic₁₁	C₂²	1	6	yes	complemented
11	#89169	C₂²	1	16	yes	direct product factor
12	C₂²×C₂²⋊D₁₁	—	5	1	no	maximal, Hall
13	C₂²×C₅×C₁₁	D₄	1	4	yes	complemented
14	C₂²×C₅×C₁₁	C₂³	1	3	yes	—
15	C₂×D₅₅	—	2	12	no	—
16	C₂×D₅₅	C₂³	1	4	yes	complemented
17	C₂×D₅×C₁₁	—	2	12	no	—
18	C₂×D₅×C₁₁	C₂³	1	4	yes	complemented
19	C₅×Dic₁₁	C₂³	1	4	yes	complemented
20	C₂⁴×C₁₁	—	5	1	no	—
21	C₂³×D₁₁	—	5	1	no	—
22	C₂×C₂²⋊D₁₁	—	5	12	no	—
23	C₂²×Dic₁₁	D₅	1	1	yes	characteristic, complemented
24	C₂²×D₂₀	—	11	1	no	maximal, Hall
25	C₂×C₅×C₁₁	C₂×D₄	1	6	yes	complemented
26	C₂×C₅×C₁₁	C₂⁴	1	1	yes	characteristic
27	D₅₅	—	2	4	no	—
28	D₅×C₁₁	—	2	4	no	—
29	C₂³×C₁₁	—	5	6	no	—
30	C₂³×C₁₁	—	10	4	no	—
31	C₂³×C₁₁	C₂×D₅	1	1	yes	characteristic
32	C₂²×D₁₁	—	5	6	no	—
33	C₂²×D₁₁	—	10	4	no	—
34	C₂²⋊D₁₁	—	5	16	no	—
35	C₂×Dic₁₁	C₂×D₅	1	6	yes	complemented
36	C₂³×D₅	—	11	1	no	—
37	C₂³×D₅	D₁₁	1	1	yes	characteristic, complemented
38	C₂²×C₄×C₅	—	11	1	no	—
39	C₂×D₂₀	—	11	12	no	—
40	C₅×C₁₁	C₂²×D₄	1	1	yes	characteristic, complemented, Hall
41	C₂²×C₁₁	—	5	4	no	—
42	C₂²×C₁₁	—	10	12	no	—
43	C₂²×C₁₁	D₂₀	1	4	yes	complemented
44	C₂²×C₁₁	C₂²×D₅	1	3	yes	—
45	C₂×D₁₁	—	5	4	no	—
46	C₂×D₁₁	—	10	12	no	—
47	Dic₁₁	C₂²×D₅	1	4	yes	complemented
48	C₂³×C₅	C₂×D₁₁	1	1	yes	characteristic
49	C₂²×D₅	—	2	4	no	—
50	C₂²×D₅	—	11	6	no	—
51	C₂²×D₅	—	22	4	no	—
52	C₂²×D₅	C₂×D₁₁	1	6	yes	complemented
53	C₂×C₄×C₅	—	11	6	no	—
54	D₂₀	—	11	16	no	—
55	C₂²×D₄	—	55	1	no	Sylow
56	C₂×C₁₁	—	10	4	no	—
57	C₂×C₁₁	C₂×D₂₀	1	6	yes	complemented
58	C₂×C₁₁	C₂³×D₅	1	1	yes	characteristic
59	D₁₁	—	10	4	no	—
60	C₂²×C₅	C₂²⋊D₁₁	1	4	yes	complemented
61	C₂²×C₅	C₂²×D₁₁	1	3	yes	—
62	C₂×D₅	—	2	12	no	—
63	C₂×D₅	—	11	4	no	—
64	C₂×D₅	—	22	12	no	—
65	C₂×D₅	C₂²×D₁₁	1	4	yes	complemented
66	C₄×C₅	—	11	4	no	—
67	C₂⁴	—	5	1	no	—
68	C₂⁴	—	55	1	no	—
69	C₂×D₄	—	55	12	no	—
70	C₂²×C₄	—	11	1	no	—
71	C₁₁	C₂²×D₂₀	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₅	C₂×C₂²⋊D₁₁	1	6	yes	complemented
73	C₂×C₅	C₂³×D₁₁	1	1	yes	characteristic
74	D₅	—	2	4	no	—
75	D₅	—	22	4	no	—
76	C₂³	—	5	6	no	—
77	C₂³	—	10	4	no	—
78	C₂³	—	55	6	no	—
79	C₂³	—	110	4	no	—
80	C₂³	D₅×D₁₁	1	1	yes	characteristic
81	D₄	—	55	16	no	—
82	C₂×C₄	—	11	6	no	—
83	C₅	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic, complemented, Sylow
84	C₂²	—	5	4	no	—
85	C₂²	—	10	12	no	—
86	C₂²	—	55	4	no	—
87	C₂²	—	110	12	no	—
88	C₂²	#89169	1	4	yes	direct product factor
89	C₂²	C₂×D₅×D₁₁	1	3	yes	—
90	C₄	—	11	4	no	—
91	C₂	—	10	4	no	—
92	C₂	—	110	4	no	—
93	C₂	C₂×#89169	1	6	yes	direct product factor
94	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	C₂³×C₅×C₁₁	C₂²
hypocenter	C₅×C₁₁	C₂²×D₄
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
1. C₂³×D₅₅	C₂	semidirect
2. C₂³×D₅×C₁₁	C₂	semidirect
3. C₂²×C₅×Dic₁₁	C₂	semidirect
4. C₂×#89169	C₂	direct (N × H)
7. C₂²×D₅₅	C₂²	semidirect
9. C₂²×D₅×C₁₁	C₂²	semidirect
10. C₂×C₅×Dic₁₁	C₂²	semidirect
11. #89169	C₂²	direct (N × H)
13. C₂²×C₅×C₁₁	D₄	semidirect
16. C₂×D₅₅	C₂³	semidirect
18. C₂×D₅×C₁₁	C₂³	semidirect
19. C₅×Dic₁₁	C₂³	semidirect
23. C₂²×Dic₁₁	D₅	semidirect
25. C₂×C₅×C₁₁	C₂×D₄	semidirect
35. C₂×Dic₁₁	C₂×D₅	semidirect
37. C₂³×D₅	D₁₁	semidirect
40. C₅×C₁₁	C₂²×D₄	semidirect
43. C₂²×C₁₁	D₂₀	semidirect
47. Dic₁₁	C₂²×D₅	semidirect
52. C₂²×D₅	C₂×D₁₁	semidirect
57. C₂×C₁₁	C₂×D₂₀	semidirect
60. C₂²×C₅	C₂²⋊D₁₁	semidirect
65. C₂×D₅	C₂²×D₁₁	semidirect
71. C₁₁	C₂²×D₂₀	semidirect
72. C₂×C₅	C₂×C₂²⋊D₁₁	semidirect
83. C₅	C₂²×C₂²⋊D₁₁	semidirect
88. C₂²	#89169	direct (N × H)
93. C₂	C₂×#89169	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
C₂³×D₅×C₁₁	2
C₂²×C₅×Dic₁₁	2
C₂×#89169	2
C₂²×C₂²⋊D₁₁	5
C₂²×D₂₀	11

Subgroups — C22×#89169 #13183404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89169 #13183404