Subgroups — #28361220

Order 13952.

74 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	#20294854	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#16812721	—	2	1	no	—
5	Dic₈₇₂	C₄	1	1	yes	characteristic, complemented
6	#16812644	—	2	1	no	—
7	D₁₇₄₄	C₄	1	1	yes	characteristic, complemented
8	#16812633	—	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	C₄⋊Dic₁₀₉	—	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	C₂×Dic₁₀₉	—	8	1	no	—
22	D₄₃₆	—	4	1	no	—
23	C₄×D₁₀₉	—	8	1	no	—
24	Dic₂₁₈	—	4	1	no	—
25	C₈×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₀₉	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₀₉	D₁₆	1	1	yes	characteristic
28	C₂×D₁₀₉	—	8	1	no	—
29	C₄×C₁₀₉	#102	1	1	yes	characteristic
30	C₄×C₁₀₉	SD₃₂	1	1	yes	characteristic
31	Dic₁₀₉	—	8	1	no	—
32	Dic₁₀₉	—	16	1	no	—
33	C₂×C₁₀₉	—	2	1	no	—
34	C₂×C₁₀₉	#357	1	1	yes	characteristic
35	D₁₀₉	—	16	1	no	—
36	#1418	—	109	1	no	maximal, Sylow
37	C₁₀₉	#1418	1	1	yes	characteristic, complemented, Sylow
38	#508	—	109	1	no	—
39	M₆₄	D₁₀₉	1	1	yes	characteristic, complemented
40	#367	—	109	1	no	—
41	#135	—	218	1	no	—
42	Q₃₂	—	109	1	no	—
43	SD₃₂	—	218	1	no	—
44	D₁₆	—	109	1	no	—
45	C₂×C₁₆	C₂×D₁₀₉	1	1	yes	characteristic
46	#107	—	218	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	436	1	no	—
49	Q₁₆	—	218	1	no	—
50	SD₁₆	—	436	1	no	—
51	D₈	—	218	1	no	—
52	C₂×C₈	D₄₃₆	1	1	yes	characteristic
53	C₄⋊C₄	—	436	1	no	—
54	C₁₆	C₄×D₁₀₉	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₀₉	1	1	yes	characteristic
56	Q₈	—	436	1	no	—
57	D₄	—	872	1	no	—
58	D₄	—	436	1	no	—
59	C₂×C₄	—	872	1	no	—
60	C₂×C₄	—	872	1	no	—
61	C₂×C₄	D₈₇₂	1	1	yes	characteristic
62	C₈	#13182006	1	1	yes	characteristic
63	C₈	#13182018	1	1	yes	characteristic
64	C₂²	—	872	1	no	—
65	C₂²	D₁₇₄₄	1	1	yes	characteristic
66	C₄	—	872	1	no	—
67	C₄	—	1744	1	no	—
68	C₄	#16812621	1	1	yes	characteristic
69	C₄	#16812644	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	1744	1	no	—
72	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	#20295458	C₂
cosocle	C₂×C₁₆×C₁₀₉	C₂²
hypercenter	M₆₄	D₁₀₉
hypocenter	C₁₀₉	#1418
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
2. #20294854	C₂	semidirect
3. M₆₄×C₁₀₉	C₂	semidirect
5. Dic₈₇₂	C₄	semidirect
7. D₁₇₄₄	C₄	semidirect
37. C₁₀₉	#1418	semidirect
39. M₆₄	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
#20294854	2
M₆₄×C₁₀₉	2
#1418	109