Subgroups — #28378198

Order 13952.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295955	C₂	1	1	yes	maximal, characteristic
2	#20295953	C₂	1	1	yes	maximal, characteristic
3	#20294859	C₂	1	1	yes	maximal, characteristic
4	#16812770	—	2	1	no	—
5	#16812768	—	2	1	no	—
6	C₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
7	#16812605	C₄	1	1	yes	characteristic, complemented
8	C₃₂×C₁₀₉	C₄	1	1	yes	characteristic, complemented
9	#13182039	—	4	1	no	—
10	C₄⋊#89137	—	4	1	no	—
11	C₈×D₁₀₉	D₄	1	1	yes	characteristic
12	#13182004	Q₈	1	1	yes	characteristic
13	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
14	C₂×#89137	—	8	1	no	—
15	C₄×D₁₀₉	D₈	1	1	yes	characteristic
16	#11724786	—	8	1	no	—
17	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
18	#11724784	Q₁₆	1	1	yes	characteristic
19	C₂×D₁₀₉	SD₃₂	1	1	yes	characteristic
20	#89137	—	16	1	no	—
21	C₄×C₁₀₉	#107	1	1	yes	characteristic
22	Dic₁₀₉	SD₃₂	1	1	yes	characteristic
23	C₂×C₁₀₉	#367	1	1	yes	characteristic
24	D₁₀₉	—	2	1	no	—
25	#1426	—	109	1	no	maximal, Sylow
26	C₁₀₉	#1426	1	1	yes	characteristic, complemented, Sylow
27	M₆₄	—	109	1	no	—
28	#368	—	109	1	no	—
29	#367	—	109	1	no	—
30	C₂×C₁₆	—	109	1	no	—
31	#108	—	218	1	no	—
32	#107	—	218	1	no	—
33	C₃₂	—	109	1	no	—
34	C₃₂	#89137	1	1	yes	characteristic, complemented
35	M₁₆	—	436	1	no	—
36	C₂×C₈	—	109	1	no	—
37	C₄⋊C₄	—	436	1	no	—
38	C₁₆	—	109	1	no	—
39	C₁₆	C₂×#89137	1	1	yes	characteristic
40	C₂×C₄	—	872	1	no	—
41	C₂×C₄	—	109	1	no	—
42	C₈	—	872	1	no	—
43	C₈	—	109	1	no	—
44	C₈	C₄⋊#89137	1	1	yes	characteristic
45	C₂²	—	109	1	no	—
46	C₄	—	109	1	no	—
47	C₄	—	1744	1	no	—
48	C₄	#16812768	1	1	yes	characteristic
49	C₂	—	218	1	no	—
50	C₂	#20295953	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20295953
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₁₆	C₂×#89137
Fitting	C₃₂×C₁₀₉	C₄
socle	C₂×C₁₀₉	#367
prime radical	C₂×D₁₀₉	SD₃₂
cosocle	C₁₆×D₁₀₉	C₂²
hypercenter	C₃₂	#89137
hypocenter	C₁₀₉	#1426
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

Complements

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

N	H ≅ G/N	type
7. #16812605	C₄	semidirect
8. C₃₂×C₁₀₉	C₄	semidirect
26. C₁₀₉	#1426	semidirect
34. C₃₂	#89137	semidirect

Maximal subgroups

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

subgroup	index
#20295955	2
#20295953	2
#20294859	2
#1426	109