Subgroups — #28361237

Order 13952.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20294860	C₂	1	1	yes	maximal, characteristic
2	#20294856	C₂	1	1	yes	maximal, characteristic
3	#367×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812635	—	2	1	no	—
5	#16812605	C₄	1	2	yes	complemented
6	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
7	#107×C₁₀₉	—	2	1	no	—
8	#13182013	—	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	C₄⋊C₄	1	1	yes	characteristic
16	C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
17	#11724784	—	8	1	no	—
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	—	109	1	no	maximal, Sylow
25	C₁₀₉	#1426	1	1	yes	characteristic, complemented, Sylow
26	M₆₄	—	109	1	no	—
27	#368	—	109	1	no	—
28	#367	D₁₀₉	1	1	yes	characteristic
29	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
30	#108	—	218	1	no	—
31	#107	—	2	1	no	—
32	C₃₂	—	109	2	no	—
33	M₁₆	—	436	1	no	—
34	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
35	C₄⋊C₄	—	4	1	no	—
36	C₁₆	Dic₂₁₈	1	1	yes	characteristic
37	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
38	C₂×C₄	—	8	1	no	—
39	C₂×C₄	#13182014	1	1	yes	characteristic
40	C₈	—	872	1	no	—
41	C₈	#13182008	1	1	yes	characteristic
42	C₈	#13182019	1	1	yes	characteristic
43	C₂²	#16812643	1	1	yes	characteristic
44	C₄	—	16	1	no	—
45	C₄	#16812632	1	1	yes	characteristic
46	C₄	#16812642	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20294853	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20294853
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	#367×C₁₀₉	C₂
socle	C₂×C₁₀₉	#367
prime radical	C₂²×C₁₀₉	SD₃₂
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	#367	D₁₀₉
hypocenter	C₁₀₉	#1426
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #367

Complements

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

N	H ≅ G/N	type
5. #16812605	C₄	semidirect
25. C₁₀₉	#1426	semidirect

Maximal subgroups

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

subgroup	index
#20294860	2
#20294856	2
#367×C₁₀₉	2
#1426	109