Subgroups — C₁₀₉×#12542 #42130536

Order 27904.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#2110	C₂	1	3	yes	maximal
2	C₁₀₉×#2108	C₂	1	3	yes	maximal
3	C₁₀₉×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×M₃₂×C₁₀₉	C₂²	1	3	yes	—
5	C₂²×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
6	#405×C₁₀₉	C₂²	1	3	yes	—
7	#347×C₁₀₉	C₄	1	12	yes	complemented
8	#346×C₁₀₉	C₄	1	6	yes	complemented
9	C₄×C₁₆×C₁₀₉	C₄	1	6	yes	complemented
10	#338×C₁₀₉	C₄	1	1	yes	characteristic
11	H(3,ℤ/4)×C₁₀₉	C₄	1	3	yes	—
12	C₂×M₁₆×C₁₀₉	C₂×C₄	1	3	yes	—
13	C₂²×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
14	#117×C₁₀₉	C₂×C₄	1	3	yes	—
15	M₃₂×C₁₀₉	C₂×C₄	1	12	yes	—
16	C₂×C₁₆×C₁₀₉	C₂×C₄	1	3	yes	—
17	C₂×C₁₆×C₁₀₉	C₂×C₄	1	6	yes	—
18	C₂×C₁₆×C₁₀₉	C₂×C₄	1	3	yes	—
19	#97×C₁₀₉	C₂×C₄	1	6	yes	—
20	C₄×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
21	C₂²×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
22	M₁₆×C₁₀₉	—	2	6	no	—
23	C₂×C₈×C₁₀₉	C₄²	1	6	yes	—
24	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
25	C₂×C₈×C₁₀₉	C₂²×C₄	1	3	yes	—
26	C₄⋊C₄×C₁₀₉	—	2	3	no	—
27	C₂²⋊C₄×C₁₀₉	—	2	3	no	—
28	C₄²×C₁₀₉	C₄²	1	6	yes	—
29	C₁₆×C₁₀₉	C₄²	1	3	yes	—
30	C₁₆×C₁₀₉	C₄²	1	12	yes	—
31	C₁₆×C₁₀₉	C₄²	1	1	yes	characteristic
32	C₂³×C₁₀₉	#131	1	1	yes	characteristic
33	C₂×C₄×C₁₀₉	—	2	6	no	—
34	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₁₀₉	#131	1	3	yes	—
36	C₈×C₁₀₉	—	2	6	no	—
37	C₈×C₁₀₉	C₂×C₄²	1	3	yes	—
38	C₈×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
39	C₂²×C₁₀₉	—	4	1	no	—
40	C₂²×C₁₀₉	#405	1	3	yes	—
41	C₄×C₁₀₉	—	4	6	no	—
42	C₄×C₁₀₉	C₄³	1	1	yes	characteristic
43	C₄×C₁₀₉	#405	1	3	yes	—
44	#12542	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₁₀₉	—	2	3	no	—
46	C₂×C₁₀₉	#1726	1	1	yes	characteristic
47	#2110	C₂×C₁₀₉	1	3	yes	—
48	#2108	C₂×C₁₀₉	1	3	yes	—
49	#1747	C₂×C₁₀₉	1	1	yes	characteristic
50	C₁₀₉	#12542	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×M₃₂	C₂²×C₁₀₉	1	3	yes	—
52	C₂²×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
53	#405	C₂²×C₁₀₉	1	3	yes	—
54	#347	C₄×C₁₀₉	1	12	yes	complemented
55	#346	C₄×C₁₀₉	1	6	yes	complemented
56	C₄×C₁₆	C₄×C₁₀₉	1	6	yes	complemented
57	#338	C₄×C₁₀₉	1	1	yes	characteristic
58	H(3,ℤ/4)	C₄×C₁₀₉	1	3	yes	—
59	C₂×M₁₆	C₂×C₄×C₁₀₉	1	3	yes	—
60	C₂²×C₈	C₂³×C₁₀₉	1	1	yes	characteristic
61	#117	C₂×C₄×C₁₀₉	1	3	yes	—
62	M₃₂	C₂×C₄×C₁₀₉	1	12	yes	—
63	C₂×C₁₆	C₂×C₄×C₁₀₉	1	3	yes	—
64	C₂×C₁₆	C₂×C₄×C₁₀₉	1	6	yes	—
65	C₂×C₁₆	C₂×C₄×C₁₀₉	1	3	yes	—
66	#97	C₂×C₄×C₁₀₉	1	6	yes	—
67	C₄×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
68	C₂²×C₄	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	C₄²×C₁₀₉	1	6	yes	—
71	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
72	C₂×C₈	C₂²×C₄×C₁₀₉	1	3	yes	—
73	C₄⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₄²	C₄²×C₁₀₉	1	6	yes	—
76	C₁₆	C₄²×C₁₀₉	1	3	yes	—
77	C₁₆	C₄²×C₁₀₉	1	12	yes	—
78	C₁₆	C₄²×C₁₀₉	1	1	yes	characteristic
79	C₂³	#131×C₁₀₉	1	1	yes	characteristic
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
82	C₂×C₄	#131×C₁₀₉	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
85	C₈	C₂×C₄²×C₁₀₉	1	3	yes	—
86	C₂²	—	4	1	no	—
87	C₂²	#405×C₁₀₉	1	3	yes	—
88	C₄	—	4	6	no	—
89	C₄	C₄³×C₁₀₉	1	1	yes	characteristic
90	C₄	#405×C₁₀₉	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	C₁₀₉×#1726	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₀₉	C₄²
commutator	C₄	C₄³×C₁₀₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₀₉
socle	C₂×C₁₀₉	#1726
prime radical	C₂³×C₁₀₉	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. #347×C₁₀₉	C₄	semidirect
8. #346×C₁₀₉	C₄	semidirect
9. C₄×C₁₆×C₁₀₉	C₄	semidirect
44. #12542	C₁₀₉	direct (N × H)
50. C₁₀₉	#12542	direct (N × H)
54. #347	C₄×C₁₀₉	semidirect
55. #346	C₄×C₁₀₉	semidirect
56. C₄×C₁₆	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
C₁₀₉×#2110	2
C₁₀₉×#2108	2
C₁₀₉×#1747	2
#12542	109

Subgroups — C109×#12542 #42130536

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#12542 #42130536