Subgroups — #28361229

Order 13952.

58 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	C₁₆×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#13182004	C₄	1	1	yes	characteristic
5	C₈×Dic₁₀₉	C₄	1	1	yes	characteristic
6	#16812605	—	2	1	no	—
7	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
8	C₃₂×C₁₀₉	—	2	1	no	—
9	C₂×#11724784	C₈	1	1	yes	characteristic
10	C₄×Dic₁₀₉	C₈	1	1	yes	characteristic
11	#13182004	—	2	1	no	—
12	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
13	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
14	C₁₆×C₁₀₉	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	C₂×C₈	1	1	yes	characteristic
16	C₂×Dic₁₀₉	—	2	1	no	—
17	C₈×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
18	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
19	#11724784	C₁₆	1	2	yes	—
20	C₂²×C₁₀₉	M₃₂	1	1	yes	characteristic
21	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
22	C₄×C₁₀₉	C₂×C₁₆	1	1	yes	characteristic
23	Dic₁₀₉	—	2	2	no	—
24	C₂×C₁₀₉	—	2	1	no	—
25	C₂×C₁₀₉	C₄⋊C₁₆	1	1	yes	characteristic
26	#1422	—	109	1	no	maximal, Sylow
27	C₁₀₉	#1422	1	1	yes	characteristic, complemented, Sylow
28	M₆₄	—	109	1	no	—
29	M₆₄	D₁₀₉	1	1	yes	characteristic
30	C₄×C₁₆	—	109	1	no	—
31	C₂×C₁₆	—	109	1	no	—
32	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
33	C₄×C₈	—	109	1	no	—
34	C₃₂	—	2	1	no	—
35	C₃₂	—	218	1	no	—
36	C₂×C₈	—	109	1	no	—
37	C₂×C₈	C₄×D₁₀₉	1	1	yes	characteristic
38	C₄²	—	109	1	no	—
39	C₁₆	—	218	1	no	—
40	C₁₆	Dic₂₁₈	1	1	yes	characteristic
41	C₁₆	C₂²⋊D₁₀₉	1	1	yes	characteristic
42	C₂×C₄	—	218	1	no	—
43	C₂×C₄	C₈×D₁₀₉	1	1	yes	characteristic
44	C₈	—	109	2	no	—
45	C₈	#13182008	1	1	yes	characteristic
46	C₈	#13182012	1	1	yes	characteristic
47	C₂²	#16812638	1	1	yes	characteristic
48	C₄	—	218	2	no	—
49	C₄	#16812628	1	1	yes	characteristic
50	C₄	C₁₆×D₁₀₉	1	1	yes	characteristic
51	C₂	—	2	1	no	—
52	C₂	#20294844	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₂²⋊D₁₀₉
commutator	C₄×C₁₀₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	M₆₄×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₄⋊C₁₆
prime radical	C₂²×C₁₀₉	M₃₂
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	#1422
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
27. C₁₀₉	#1422	semidirect

Maximal subgroups

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

subgroup	index
#20294860	2
C₁₆×Dic₁₀₉	2
M₆₄×C₁₀₉	2
#1422	109