Subgroups — C₁₀₉×#7845 #42125839

Order 27904.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₀₉×#1744	C₂	1	3	yes	maximal
2	C₂×#322×C₁₀₉	C₂	1	3	yes	maximal
3	C₂×C₈²×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
5	C₂×C₄×C₈×C₁₀₉	C₂²	1	3	yes	—
6	C₄³×C₁₀₉	C₂²	1	1	yes	characteristic
7	#336×C₁₀₉	C₄	1	12	yes	—
8	#322×C₁₀₉	C₄	1	12	yes	complemented
9	C₈²×C₁₀₉	C₄	1	4	yes	complemented
10	C₂²×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
11	C₂²×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
12	C₂×C₄²×C₁₀₉	C₂×C₄	1	6	yes	—
13	C₂×C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
15	C₄×C₈×C₁₀₉	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₁₀₉	C₂×C₄	1	12	yes	—
17	C₂²×C₄×C₁₀₉	C₄²	1	4	yes	—
18	C₂²×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
19	C₂×C₈×C₁₀₉	—	2	12	no	—
20	C₂×C₈×C₁₀₉	—	2	12	no	—
21	C₂×C₈×C₁₀₉	C₄²	1	12	yes	—
22	C₂×C₈×C₁₀₉	C₄²	1	12	yes	—
23	C₄²×C₁₀₉	—	2	12	no	—
24	C₄²×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
25	C₄²×C₁₀₉	C₂²×C₄	1	3	yes	—
26	C₂³×C₁₀₉	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₀₉	—	2	12	no	—
28	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₁₀₉	C₂×C₄²	1	3	yes	—
30	C₂×C₄×C₁₀₉	#131	1	12	yes	—
31	C₈×C₁₀₉	—	2	12	no	—
32	C₈×C₁₀₉	—	2	12	no	—
33	C₂²×C₁₀₉	C₄³	1	1	yes	characteristic
34	C₂²×C₁₀₉	#405	1	6	yes	—
35	C₄×C₁₀₉	—	4	4	no	—
36	C₄×C₁₀₉	#405	1	6	yes	—
37	C₄×C₁₀₉	#405	1	6	yes	—
38	#7845	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₁₀₉	#1454	1	4	yes	—
40	C₂×C₁₀₉	#1726	1	3	yes	—
41	#1744	C₂×C₁₀₉	1	3	yes	—
42	C₂×#322	C₂×C₁₀₉	1	3	yes	—
43	C₂×C₈²	C₂×C₁₀₉	1	1	yes	characteristic
44	C₁₀₉	#7845	1	1	yes	characteristic, direct product factor, Sylow
45	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
46	C₂×C₄×C₈	C₂²×C₁₀₉	1	3	yes	—
47	C₄³	C₂²×C₁₀₉	1	1	yes	characteristic
48	#336	C₄×C₁₀₉	1	12	yes	—
49	#322	C₄×C₁₀₉	1	12	yes	complemented
50	C₈²	C₄×C₁₀₉	1	4	yes	complemented
51	C₂²×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
52	C₂²×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
53	C₂×C₄²	C₂×C₄×C₁₀₉	1	6	yes	—
54	C₂×C₄²	C₂³×C₁₀₉	1	1	yes	characteristic
55	C₄×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
56	C₄×C₈	C₂×C₄×C₁₀₉	1	6	yes	—
57	C₄×C₈	C₂×C₄×C₁₀₉	1	12	yes	—
58	C₂²×C₄	C₄²×C₁₀₉	1	4	yes	—
59	C₂²×C₄	C₂²×C₄×C₁₀₉	1	3	yes	—
60	C₂×C₈	—	2	12	no	—
61	C₂×C₈	—	2	12	no	—
62	C₂×C₈	C₄²×C₁₀₉	1	12	yes	—
63	C₂×C₈	C₄²×C₁₀₉	1	12	yes	—
64	C₄²	—	2	12	no	—
65	C₄²	C₂²×C₄×C₁₀₉	1	1	yes	characteristic
66	C₄²	C₂²×C₄×C₁₀₉	1	3	yes	—
67	C₂³	C₂×C₄²×C₁₀₉	1	1	yes	characteristic
68	C₂×C₄	—	2	12	no	—
69	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
70	C₂×C₄	C₂×C₄²×C₁₀₉	1	3	yes	—
71	C₂×C₄	#131×C₁₀₉	1	12	yes	—
72	C₈	—	2	12	no	—
73	C₈	—	2	12	no	—
74	C₂²	C₄³×C₁₀₉	1	1	yes	characteristic
75	C₂²	#405×C₁₀₉	1	6	yes	—
76	C₄	—	4	4	no	—
77	C₄	#405×C₁₀₉	1	6	yes	—
78	C₄	#405×C₁₀₉	1	6	yes	—
79	C₂	C₁₀₉×#1454	1	4	yes	—
80	C₂	C₁₀₉×#1726	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
8. #322×C₁₀₉	C₄	semidirect
9. C₈²×C₁₀₉	C₄	semidirect
38. #7845	C₁₀₉	direct (N × H)
44. C₁₀₉	#7845	direct (N × H)
49. #322	C₄×C₁₀₉	semidirect
50. C₈²	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×#1744	2
C₂×#322×C₁₀₉	2
C₂×C₈²×C₁₀₉	2
#7845	109

Subgroups — C109×#7845 #42125839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#7845 #42125839