Subgroups — #28361233

Order 13952.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812605	C₂	1	1	yes	maximal, characteristic
2	#20294852	C₂	1	1	yes	maximal, characteristic
3	#366×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	#16812633	—	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	C₄⋊Dic₁₀₉	—	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	C₂×Dic₁₀₉	—	8	1	no	—
16	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₀₉	Q₁₆	1	1	yes	characteristic
18	C₂²×C₁₀₉	D₁₆	1	1	yes	characteristic
19	C₄×C₁₀₉	—	16	1	no	—
20	C₄×C₁₀₉	#107	1	1	yes	characteristic
21	C₄×C₁₀₉	Q₃₂	1	1	yes	characteristic
22	Dic₁₀₉	—	16	1	no	—
23	C₂×C₁₀₉	#366	1	1	yes	characteristic
24	C₂×C₁₀₉	SD₆₄	1	1	yes	characteristic
25	C₂×C₁₀₉	SD₆₄	1	1	yes	characteristic
26	#1424	—	109	1	no	maximal, Sylow
27	C₁₀₉	#1424	1	1	yes	characteristic, complemented, Sylow
28	C₂×C₃₂	—	109	1	no	—
29	#366	—	109	1	no	—
30	#366	D₁₀₉	1	1	yes	characteristic
31	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
32	#107	—	2	1	no	—
33	#107	—	218	1	no	—
34	C₃₂	—	109	2	no	—
35	C₂×C₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
36	C₄⋊C₄	—	4	1	no	—
37	C₄⋊C₄	—	436	1	no	—
38	C₁₆	Dic₂₁₈	1	1	yes	characteristic
39	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	872	1	no	—
42	C₂×C₄	#13182014	1	1	yes	characteristic
43	C₈	#13182008	1	1	yes	characteristic
44	C₈	#13182019	1	1	yes	characteristic
45	C₂²	#16812640	1	1	yes	characteristic
46	C₄	—	16	1	no	—
47	C₄	—	1744	1	no	—
48	C₄	#16812632	1	1	yes	characteristic
49	C₄	#16812645	1	1	yes	characteristic
50	C₂	#20294851	1	1	yes	characteristic
51	C₂	#20294863	1	1	yes	characteristic
52	C₂	#20294864	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16812640
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	#366×C₁₀₉	C₂
socle, prime radical	C₂²×C₁₀₉	D₁₆
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	#366	D₁₀₉
hypocenter	C₁₀₉	#1424
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
5. #16812605	C₄	semidirect
27. C₁₀₉	#1424	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812605	2
#20294852	2
#366×C₁₀₉	2
#1424	109