Subgroups — C₂×#504×C₁₀₉ #28360660

Order 13952.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#504×C₁₀₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×M₃₂×C₁₀₉	C₂	1	3	yes	maximal, complemented
4	C₂²×C₁₆×C₁₀₉	C₂	1	3	yes	maximal, complemented
5	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
6	#131×C₁₀₉	C₄	1	4	yes	—
7	#131×C₁₀₉	C₂²	1	4	yes	—
8	C₂×M₁₆×C₁₀₉	C₄	1	3	yes	—
9	C₂²×C₈×C₁₀₉	C₂²	1	3	yes	—
10	M₃₂×C₁₀₉	C₂²	1	12	yes	complemented
11	C₂×C₁₆×C₁₀₉	C₂²	1	3	yes	—
12	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×C₁₀₉	C₂²	1	12	yes	complemented
14	#41×C₁₀₉	C₈	1	4	yes	—
15	#41×C₁₀₉	C₂×C₄	1	4	yes	—
16	C₂×Q₈×C₁₀₉	C₈	1	1	yes	characteristic
17	C₂×D₄×C₁₀₉	C₈	1	3	yes	—
18	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
19	M₁₆×C₁₀₉	C₂×C₄	1	12	yes	—
20	C₂×C₈×C₁₀₉	C₂×C₄	1	3	yes	—
21	C₂×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
22	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₀₉	C₂³	1	6	yes	—
24	C₁₆×C₁₀₉	C₂³	1	6	yes	—
25	C₁₆×C₁₀₉	C₂³	1	2	yes	—
26	C₂³×C₁₀₉	C₂×C₈	1	3	yes	—
27	Q₈×C₁₀₉	C₂×C₈	1	4	yes	—
28	D₄×C₁₀₉	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₈	1	3	yes	—
30	C₂×C₄×C₁₀₉	C₂×C₈	1	6	yes	—
31	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₀₉	C₂²×C₄	1	6	yes	—
33	C₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
34	C₈×C₁₀₉	C₂²×C₄	1	6	yes	—
35	C₈×C₁₀₉	C₂⁴	1	1	yes	characteristic
36	C₂²×C₁₀₉	—	2	6	no	—
37	C₂²×C₁₀₉	C₂²×C₈	1	6	yes	—
38	C₂²×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₀₉	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₁₀₉	C₂²×C₈	1	6	yes	—
41	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
42	C₂×C₁₀₉	—	2	6	no	—
43	C₂×C₁₀₉	#504	1	2	yes	direct product factor
44	C₂×C₁₀₉	C₂³×C₈	1	1	yes	characteristic
45	C₂×#504	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₁₀₉	C₂×#504	1	1	yes	characteristic, direct product factor, Sylow
47	C₂×#131	C₂×C₁₀₉	1	1	yes	characteristic
48	#504	C₂×C₁₀₉	1	8	yes	direct product factor
49	C₂×M₃₂	C₂×C₁₀₉	1	3	yes	complemented
50	C₂²×C₁₆	C₂×C₁₀₉	1	3	yes	complemented
51	C₂×#41	C₄×C₁₀₉	1	1	yes	characteristic
52	#131	C₄×C₁₀₉	1	4	yes	—
53	#131	C₂²×C₁₀₉	1	4	yes	—
54	C₂×M₁₆	C₄×C₁₀₉	1	3	yes	—
55	C₂²×C₈	C₂²×C₁₀₉	1	3	yes	—
56	M₃₂	C₂²×C₁₀₉	1	12	yes	complemented
57	C₂×C₁₆	C₂²×C₁₀₉	1	3	yes	—
58	C₂×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
59	C₂×C₁₆	C₂²×C₁₀₉	1	12	yes	complemented
60	#41	C₈×C₁₀₉	1	4	yes	—
61	#41	C₂×C₄×C₁₀₉	1	4	yes	—
62	C₂×Q₈	C₈×C₁₀₉	1	1	yes	characteristic
63	C₂×D₄	C₈×C₁₀₉	1	3	yes	—
64	C₂²×C₄	C₂×C₄×C₁₀₉	1	3	yes	—
65	M₁₆	C₂×C₄×C₁₀₉	1	12	yes	—
66	C₂×C₈	C₂×C₄×C₁₀₉	1	3	yes	—
67	C₂×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
68	C₂×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
69	C₂×C₈	C₂³×C₁₀₉	1	6	yes	—
70	C₁₆	C₂³×C₁₀₉	1	2	yes	—
71	C₁₆	C₂³×C₁₀₉	1	6	yes	—
72	C₂³	C₂×C₈×C₁₀₉	1	3	yes	—
73	Q₈	C₂×C₈×C₁₀₉	1	4	yes	—
74	D₄	C₂×C₈×C₁₀₉	1	12	yes	—
75	C₂×C₄	C₂×C₈×C₁₀₉	1	3	yes	—
76	C₂×C₄	C₂×C₈×C₁₀₉	1	6	yes	—
77	C₂×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₄×C₁₀₉	1	6	yes	—
79	C₈	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
80	C₈	C₂²×C₄×C₁₀₉	1	6	yes	—
81	C₈	C₂⁴×C₁₀₉	1	1	yes	characteristic
82	C₂²	—	2	6	no	—
83	C₂²	C₂²×C₈×C₁₀₉	1	6	yes	—
84	C₂²	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
85	C₄	C₂²×C₈×C₁₀₉	1	1	yes	characteristic
86	C₄	C₂²×C₈×C₁₀₉	1	6	yes	—
87	C₄	C₂³×C₄×C₁₀₉	1	1	yes	characteristic
88	C₂	—	2	6	no	—
89	C₂	#504×C₁₀₉	1	2	yes	direct product factor
90	C₂	C₂³×C₈×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₁₀₉	C₂²
commutator	C₂	C₂³×C₈×C₁₀₉
Frattini, cosocle	C₈	C₂⁴×C₁₀₉
socle	C₂²×C₁₀₉	C₂²×C₈
prime radical	C₂×#41×C₁₀₉	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. #504×C₁₀₉	C₂	direct (N × H)
3. C₂×M₃₂×C₁₀₉	C₂	semidirect
4. C₂²×C₁₆×C₁₀₉	C₂	semidirect
10. M₃₂×C₁₀₉	C₂²	semidirect
13. C₂×C₁₆×C₁₀₉	C₂²	semidirect
43. C₂×C₁₀₉	#504	direct (N × H)
45. C₂×#504	C₁₀₉	direct (N × H)
46. C₁₀₉	C₂×#504	direct (N × H)
48. #504	C₂×C₁₀₉	direct (N × H)
49. C₂×M₃₂	C₂×C₁₀₉	semidirect
50. C₂²×C₁₆	C₂×C₁₀₉	semidirect
56. M₃₂	C₂²×C₁₀₉	semidirect
59. C₂×C₁₆	C₂²×C₁₀₉	semidirect
89. C₂	#504×C₁₀₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#131×C₁₀₉	2
#504×C₁₀₉	2
C₂×M₃₂×C₁₀₉	2
C₂²×C₁₆×C₁₀₉	2
C₂×#504	109

Subgroups — C2×#504×C109 #28360660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#504×C₁₀₉ #28360660