Subgroups — C₁₀₉×#1398 #28358652

Order 13952.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₆₄×C₁₀₉	C₂	1	2	yes	maximal
2	#346×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
5	C₄×C₈×C₁₀₉	C₄	1	1	yes	characteristic
6	C₃₂×C₁₀₉	C₄	1	4	yes	complemented
7	C₂×C₈×C₁₀₉	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₀₉	C₈	1	1	yes	characteristic
10	C₁₆×C₁₀₉	C₈	1	2	yes	—
11	C₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₁₀₉	—	2	1	no	—
13	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
14	C₈×C₁₀₉	—	2	1	no	—
15	C₈×C₁₀₉	C₄²	1	1	yes	characteristic
16	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
17	C₂²×C₁₀₉	M₃₂	1	1	yes	characteristic
18	C₄×C₁₀₉	—	4	1	no	—
19	C₄×C₁₀₉	C₄×C₈	1	1	yes	characteristic
20	C₄×C₁₀₉	M₃₂	1	1	yes	characteristic
21	C₂×C₁₀₉	—	2	1	no	—
22	C₂×C₁₀₉	#346	1	1	yes	characteristic
23	#1398	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₁₀₉	#1398	1	1	yes	characteristic, direct product factor, Sylow
25	M₆₄	C₂×C₁₀₉	1	2	yes	—
26	#346	C₂×C₁₀₉	1	1	yes	characteristic
27	C₂×C₁₆	C₄×C₁₀₉	1	1	yes	characteristic
28	C₂×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
29	C₄×C₈	C₄×C₁₀₉	1	1	yes	characteristic
30	C₃₂	C₄×C₁₀₉	1	4	yes	complemented
31	C₂×C₈	C₈×C₁₀₉	1	1	yes	characteristic
32	C₂×C₈	C₂×C₄×C₁₀₉	1	1	yes	characteristic
33	C₄²	C₈×C₁₀₉	1	1	yes	characteristic
34	C₁₆	C₈×C₁₀₉	1	2	yes	—
35	C₁₆	C₂×C₄×C₁₀₉	1	2	yes	—
36	C₂×C₄	—	2	1	no	—
37	C₂×C₄	C₂×C₈×C₁₀₉	1	1	yes	characteristic
38	C₈	—	2	1	no	—
39	C₈	C₄²×C₁₀₉	1	1	yes	characteristic
40	C₈	C₂×C₈×C₁₀₉	1	1	yes	characteristic
41	C₂²	M₃₂×C₁₀₉	1	1	yes	characteristic
42	C₄	—	4	1	no	—
43	C₄	C₄×C₈×C₁₀₉	1	1	yes	characteristic
44	C₄	M₃₂×C₁₀₉	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	#346×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₀₉	C₄²
commutator	C₄	C₄×C₈×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₀₉
socle	C₂×C₁₀₉	#346
prime radical	C₂²×C₁₀₉	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₈×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃₂×C₁₀₉	C₄	semidirect
23. #1398	C₁₀₉	direct (N × H)
24. C₁₀₉	#1398	direct (N × H)
30. C₃₂	C₄×C₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
M₆₄×C₁₀₉	2
#346×C₁₀₉	2
#1398	109

Subgroups — C109×#1398 #28358652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1398 #28358652