Subgroups — C₁₀₉×#1426 #28358680

Order 13952.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#368×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	#367×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
5	#108×C₁₀₉	—	2	1	no	—
6	#107×C₁₀₉	—	2	1	no	—
7	C₃₂×C₁₀₉	C₄	1	2	yes	complemented
8	M₁₆×C₁₀₉	—	4	1	no	—
9	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
10	C₄⋊C₄×C₁₀₉	—	4	1	no	—
11	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₀₉	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₀₉	—	8	1	no	—
14	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
15	C₈×C₁₀₉	—	8	1	no	—
16	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₀₉	SD₃₂	1	1	yes	characteristic
19	C₄×C₁₀₉	—	16	1	no	—
20	C₄×C₁₀₉	#107	1	1	yes	characteristic
21	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
22	C₂×C₁₀₉	—	2	1	no	—
23	C₂×C₁₀₉	#367	1	1	yes	characteristic
24	#1426	C₁₀₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₁₀₉	#1426	1	1	yes	characteristic, direct product factor, Sylow
26	M₆₄	C₂×C₁₀₉	1	1	yes	characteristic
27	#368	C₂×C₁₀₉	1	1	yes	characteristic
28	#367	C₂×C₁₀₉	1	1	yes	characteristic
29	C₂×C₁₆	C₂²×C₁₀₉	1	1	yes	characteristic
30	#108	—	2	1	no	—
31	#107	—	2	1	no	—
32	C₃₂	C₄×C₁₀₉	1	2	yes	complemented
33	M₁₆	—	4	1	no	—
34	C₂×C₈	D₄×C₁₀₉	1	1	yes	characteristic
35	C₄⋊C₄	—	4	1	no	—
36	C₁₆	C₂×C₄×C₁₀₉	1	1	yes	characteristic
37	C₁₆	Q₈×C₁₀₉	1	1	yes	characteristic
38	C₂×C₄	—	8	1	no	—
39	C₂×C₄	D₈×C₁₀₉	1	1	yes	characteristic
40	C₈	—	8	1	no	—
41	C₈	C₄⋊C₄×C₁₀₉	1	1	yes	characteristic
42	C₈	Q₁₆×C₁₀₉	1	1	yes	characteristic
43	C₂²	SD₃₂×C₁₀₉	1	1	yes	characteristic
44	C₄	—	16	1	no	—
45	C₄	#107×C₁₀₉	1	1	yes	characteristic
46	C₄	SD₃₂×C₁₀₉	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#367×C₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₀₉	#367
commutator	C₁₆	C₂×C₄×C₁₀₉
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₀₉
prime radical	C₂²×C₁₀₉	SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₀₉ ◁ C₂×C₄×C₁₀₉ ◁ C₂×C₈×C₁₀₉ ◁ C₂×C₁₆×C₁₀₉ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₃₂×C₁₀₉	C₄	semidirect
24. #1426	C₁₀₉	direct (N × H)
25. C₁₀₉	#1426	direct (N × H)
32. 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
#368×C₁₀₉	2
#367×C₁₀₉	2
#1426	109

Subgroups — C109×#1426 #28358680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₀₉×#1426 #28358680