Subgroups — #28361226

Order 13952.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₈₇₂	C₂	1	1	yes	maximal, characteristic
2	#20294856	C₂	1	1	yes	maximal, characteristic
3	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₄₃₆	—	2	1	no	—
5	Dic₈₇₂	—	2	1	no	—
6	Dic₈₇₂	C₄	1	2	yes	—
7	#16812635	—	2	1	no	—
8	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
9	C₃₂×C₁₀₉	—	2	1	no	—
10	C₂×Dic₂₁₈	—	4	1	no	—
11	Dic₄₃₆	—	2	2	no	—
12	Dic₄₃₆	—	4	1	no	—
13	#13182013	—	4	1	no	—
14	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
15	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
18	C₂×Dic₁₀₉	—	8	1	no	—
19	Dic₂₁₈	—	4	2	no	—
20	Dic₂₁₈	—	8	1	no	—
21	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
23	#11724784	—	8	1	no	—
24	C₂²×C₁₀₉	SD₃₂	1	1	yes	characteristic
25	C₄×C₁₀₉	#102	1	1	yes	characteristic
26	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
27	Dic₁₀₉	—	8	2	no	—
28	C₂×C₁₀₉	—	2	1	no	—
29	C₂×C₁₀₉	#357	1	1	yes	characteristic
30	#1420	—	109	1	no	maximal, Sylow
31	C₁₀₉	#1420	1	1	yes	characteristic, complemented, Sylow
32	C₂×Q₃₂	—	109	1	no	—
33	M₆₄	D₁₀₉	1	1	yes	characteristic
34	#368	—	109	1	no	—
35	C₂×Q₁₆	—	218	1	no	—
36	Q₃₂	—	109	2	no	—
37	Q₃₂	—	218	1	no	—
38	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
39	#108	—	218	1	no	—
40	C₃₂	—	2	1	no	—
41	C₂×Q₈	—	436	1	no	—
42	Q₁₆	—	436	1	no	—
43	Q₁₆	—	218	2	no	—
44	M₁₆	—	436	1	no	—
45	C₂×C₈	D₄₃₆	1	1	yes	characteristic
46	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
47	C₁₆	C₂²⋊D₁₀₉	1	1	yes	characteristic
48	Q₈	—	872	1	no	—
49	Q₈	—	436	2	no	—
50	C₂×C₄	—	872	1	no	—
51	C₂×C₄	D₈₇₂	1	1	yes	characteristic
52	C₈	—	872	1	no	—
53	C₈	#13182006	1	1	yes	characteristic
54	C₈	#13182018	1	1	yes	characteristic
55	C₂²	#16812644	1	1	yes	characteristic
56	C₄	—	872	2	no	—
57	C₄	#16812621	1	1	yes	characteristic
58	C₄	D₁₇₄₄	1	1	yes	characteristic
59	C₂	—	2	1	no	—
60	C₂	#20294828	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20294828
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	M₆₄×C₁₀₉	C₂
socle	C₂×C₁₀₉	#357
prime radical	C₂²×C₁₀₉	SD₃₂
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	#1420
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
31. C₁₀₉	#1420	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₈₇₂	2
#20294856	2
M₆₄×C₁₀₉	2
#1420	109