Subgroups — #28378060

Order 13952.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295948	C₂	1	2	yes	maximal
2	#20294801	C₂	1	1	yes	maximal, characteristic
3	#16812773	—	2	2	no	—
4	C₂×#13182004	C₄	1	1	yes	characteristic
5	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
6	C₂×C₁₆×C₁₀₉	C₄	1	1	yes	characteristic
7	#13182032	—	4	2	no	—
8	C₂×#11724784	D₄	1	1	yes	characteristic
9	C₄×Dic₁₀₉	Q₈	1	1	yes	characteristic
10	#13182004	C₈	1	2	yes	—
11	C₂×C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₁₆×C₁₀₉	—	2	1	no	—
13	C₂×C₄×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
14	C₂×Dic₁₀₉	—	2	1	no	—
15	C₈×C₁₀₉	C₂×C₈	1	1	yes	characteristic
16	C₈×C₁₀₉	M₁₆	1	1	yes	characteristic
17	#11724784	SD₁₆	1	2	yes	—
18	C₂²×C₁₀₉	#108	1	1	yes	characteristic
19	C₄×C₁₀₉	C₄⋊C₈	1	1	yes	characteristic
20	C₄×C₁₀₉	#106	1	1	yes	characteristic
21	Dic₁₀₉	—	4	1	no	—
22	C₂×C₁₀₉	—	2	1	no	—
23	C₂×C₁₀₉	#334	1	1	yes	characteristic
24	#1369	—	109	1	no	maximal, Sylow
25	C₁₀₉	#1369	1	1	yes	characteristic, complemented, Sylow
26	#364	—	109	2	no	—
27	#346	—	109	1	no	—
28	M₃₂	—	218	2	no	—
29	C₂×C₁₆	—	109	1	no	—
30	C₂×C₁₆	#89137	1	1	yes	characteristic
31	C₄×C₈	—	109	1	no	—
32	C₂×C₈	—	109	1	no	—
33	C₂×C₈	C₂×#89137	1	1	yes	characteristic
34	C₄²	—	109	1	no	—
35	C₁₆	—	2	1	no	—
36	C₁₆	—	109	2	no	—
37	C₁₆	—	436	2	no	—
38	C₂×C₄	—	218	1	no	—
39	C₂×C₄	C₄⋊#89137	1	1	yes	characteristic
40	C₈	—	109	2	no	—
41	C₈	D₁₀₉⋊C₈	1	1	yes	characteristic
42	C₈	#13182038	1	1	yes	characteristic
43	C₂²	#16812770	1	1	yes	characteristic
44	C₄	—	436	1	no	—
45	C₄	#16812766	1	1	yes	characteristic
46	C₄	#16812767	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	#20295888	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812767
commutator	C₈×C₁₀₉	C₂×C₈
Frattini	C₂×C₈	C₂×#89137
Fitting	C₂×C₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#334
prime radical	C₂²×C₁₀₉	#108
cosocle	C₈×Dic₁₀₉	C₂²
hypercenter	C₂×C₁₆	#89137
hypocenter	C₁₀₉	#1369
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
25. C₁₀₉	#1369	semidirect

Maximal subgroups

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

subgroup	index
#20295948	2
#20294801	2
#1369	109