Subgroups — #28361217

Order 13952.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295458	C₂	1	1	yes	maximal, characteristic
2	#20294856	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812721	—	2	1	no	—
5	Dic₈₇₂	C₄	1	1	yes	characteristic
6	#16812644	—	2	1	no	—
7	D₁₇₄₄	C₄	1	1	yes	characteristic
8	#16812635	—	2	1	no	—
9	C₂×C₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₀₉	—	2	1	no	—
11	#13182029	—	4	1	no	—
12	Dic₄₃₆	—	2	1	no	—
13	#13182018	—	4	1	no	—
14	D₈₇₂	—	2	1	no	—
15	#13182013	—	4	1	no	—
16	C₂×C₈×C₁₀₉	D₄	1	1	yes	characteristic
17	C₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₀₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₀₉	D₈	1	1	yes	characteristic
20	C₂²⋊D₁₀₉	—	8	1	no	—
21	D₄₃₆	—	4	1	no	—
22	C₄×D₁₀₉	—	8	1	no	—
23	Dic₂₁₈	—	4	1	no	—
24	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
26	#11724784	—	8	1	no	—
27	C₂²×C₁₀₉	SD₃₂	1	1	yes	characteristic
28	C₂×D₁₀₉	—	8	1	no	—
29	C₄×C₁₀₉	#102	1	1	yes	characteristic
30	C₄×C₁₀₉	D₁₆	1	1	yes	characteristic
31	Dic₁₀₉	—	8	1	no	—
32	C₂×C₁₀₉	—	2	1	no	—
33	C₂×C₁₀₉	#357	1	1	yes	characteristic
34	D₁₀₉	—	16	1	no	—
35	#1417	—	109	1	no	maximal, Sylow
36	C₁₀₉	#1417	1	1	yes	characteristic, complemented, Sylow
37	#508	—	109	1	no	—
38	C₂×C₃₂	D₁₀₉	1	1	yes	characteristic, complemented
39	#368	—	109	1	no	—
40	#135	—	218	1	no	—
41	Q₃₂	—	109	1	no	—
42	SD₃₂	—	218	1	no	—
43	D₁₆	—	109	1	no	—
44	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
45	#108	—	218	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	436	1	no	—
48	Q₁₆	—	218	1	no	—
49	SD₁₆	—	436	1	no	—
50	D₈	—	218	1	no	—
51	M₁₆	—	436	1	no	—
52	C₂×C₈	D₄₃₆	1	1	yes	characteristic
53	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
54	C₁₆	C₂²⋊D₁₀₉	1	1	yes	characteristic
55	Q₈	—	436	1	no	—
56	D₄	—	872	1	no	—
57	D₄	—	436	1	no	—
58	C₂×C₄	—	872	1	no	—
59	C₂×C₄	D₈₇₂	1	1	yes	characteristic
60	C₈	—	872	1	no	—
61	C₈	#13182006	1	1	yes	characteristic
62	C₈	#13182018	1	1	yes	characteristic
63	C₂²	—	872	1	no	—
64	C₂²	#16812644	1	1	yes	characteristic
65	C₄	—	872	1	no	—
66	C₄	#16812621	1	1	yes	characteristic
67	C₄	D₁₇₄₄	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	1744	1	no	—
70	C₂	#20294828	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₁₇₄₄
commutator	C₁₆×C₁₀₉	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₀₉
Fitting	C₂×C₃₂×C₁₀₉	C₂
socle	C₂×C₁₀₉	#357
prime radical	#20295458	C₂
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	C₂×C₃₂	D₁₀₉
hypocenter	C₁₀₉	#1417
LCS	G ▷ C₁₆×C₁₀₉ ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #20294856	C₂	semidirect
3. C₂×C₃₂×C₁₀₉	C₂	semidirect
36. C₁₀₉	#1417	semidirect
38. C₂×C₃₂	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295458	2
#20294856	2
C₂×C₃₂×C₁₀₉	2
#1417	109