Subgroups — C₁₀₉×#2070 #28359324

Order 13952.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#430×C₁₀₉	C₂	1	3	yes	maximal
2	C₂×#101×C₁₀₉	C₂	1	3	yes	maximal
3	#338×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₁₀₉	C₂²	1	3	yes	—
6	C₂×M₁₆×C₁₀₉	C₂²	1	3	yes	—
7	#108×C₁₀₉	—	2	6	no	—
8	#101×C₁₀₉	—	2	6	no	—
9	C₂×Q₈×C₁₀₉	—	2	3	no	—
10	C₂×Q₈×C₁₀₉	—	2	3	no	—
11	C₂²×C₄×C₁₀₉	Q₈	1	2	yes	—
12	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
13	M₁₆×C₁₀₉	—	2	3	no	—
14	M₁₆×C₁₀₉	—	2	3	no	—
15	M₁₆×C₁₀₉	—	2	3	no	—
16	M₁₆×C₁₀₉	D₄	1	6	yes	—
17	C₂×C₈×C₁₀₉	—	2	3	no	—
18	C₂×C₈×C₁₀₉	—	2	3	no	—
19	C₂³×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
20	Q₈×C₁₀₉	—	4	3	no	—
21	Q₈×C₁₀₉	—	4	1	no	—
22	C₂×C₄×C₁₀₉	—	2	6	no	—
23	C₂×C₄×C₁₀₉	C₂×D₄	1	3	yes	—
24	C₂×C₄×C₁₀₉	#41	1	3	yes	—
25	C₈×C₁₀₉	—	2	6	no	—
26	C₈×C₁₀₉	—	4	3	no	—
27	C₂²×C₁₀₉	—	4	1	no	—
28	C₂²×C₁₀₉	C₂²⋊Q₈	1	3	yes	—
29	C₄×C₁₀₉	—	4	2	no	—
30	C₄×C₁₀₉	#121	1	3	yes	—
31	C₄×C₁₀₉	#126	1	1	yes	characteristic
32	C₂×C₁₀₉	—	2	3	no	—
33	C₂×C₁₀₉	#396	1	1	yes	characteristic
34	#2070	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₁₀₉	#2070	1	1	yes	characteristic, direct product factor, Sylow
36	#430	C₂×C₁₀₉	1	3	yes	—
37	C₂×#101	C₂×C₁₀₉	1	3	yes	—
38	#338	C₂×C₁₀₉	1	1	yes	characteristic
39	C₂²×Q₈	C₂²×C₁₀₉	1	1	yes	characteristic
40	C₂×M₁₆	C₂²×C₁₀₉	1	3	yes	—
41	C₂×M₁₆	C₂²×C₁₀₉	1	3	yes	—
42	#108	—	2	6	no	—
43	#101	—	2	6	no	—
44	C₂×Q₈	—	2	3	no	—
45	C₂×Q₈	—	2	3	no	—
46	C₂²×C₄	Q₈×C₁₀₉	1	2	yes	—
47	C₂²×C₄	C₂³×C₁₀₉	1	1	yes	characteristic
48	M₁₆	—	2	3	no	—
49	M₁₆	—	2	3	no	—
50	M₁₆	—	2	3	no	—
51	M₁₆	D₄×C₁₀₉	1	6	yes	—
52	C₂×C₈	—	2	3	no	—
53	C₂×C₈	—	2	3	no	—
54	C₂³	C₂×Q₈×C₁₀₉	1	1	yes	characteristic
55	Q₈	—	4	3	no	—
56	Q₈	—	4	1	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	C₂×D₄×C₁₀₉	1	3	yes	—
59	C₂×C₄	#41×C₁₀₉	1	3	yes	—
60	C₈	—	2	6	no	—
61	C₈	—	4	3	no	—
62	C₂²	—	4	1	no	—
63	C₂²	C₂²⋊Q₈×C₁₀₉	1	3	yes	—
64	C₄	—	4	2	no	—
65	C₄	#121×C₁₀₉	1	3	yes	—
66	C₄	#126×C₁₀₉	1	1	yes	characteristic
67	C₂	—	2	3	no	—
68	C₂	#396×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₀₉	#396
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₀₉
prime radical	C₂³×C₁₀₉	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂²×C₄×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
34. #2070	C₁₀₉	direct (N × H)
35. C₁₀₉	#2070	direct (N × H)

Maximal subgroups

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

subgroup	index
#430×C₁₀₉	2
C₂×#101×C₁₀₉	2
#338×C₁₀₉	2
#2070	109

Subgroups — C109×#2070 #28359324

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#2070 #28359324