Subgroups — #20295939

Order 6976.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#16812768	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₁₀₉	C₂	1	1	yes	maximal, characteristic
4	C₄⋊#89137	—	2	1	no	—
5	#13182032	—	2	1	no	—
6	Q₈×D₁₀₉	—	2	1	no	—
7	Dic₄₃₆	C₄	1	1	yes	characteristic, complemented
8	#13182019	—	2	1	no	—
9	C₈×D₁₀₉	C₂²	1	1	yes	characteristic
10	Q₁₆×C₁₀₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#89137	—	4	1	no	—
12	Q₈×C₁₀₉	—	2	1	no	—
13	C₄×D₁₀₉	—	4	1	no	—
14	C₄×D₁₀₉	D₄	1	1	yes	characteristic
15	Dic₂₁₈	—	2	1	no	—
16	Dic₂₁₈	—	4	1	no	—
17	C₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
18	#11724784	D₄	1	1	yes	characteristic
19	C₂×D₁₀₉	D₈	1	1	yes	characteristic
20	#89137	—	8	1	no	—
21	C₄×C₁₀₉	—	4	1	no	—
22	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₁₀₉	—	4	1	no	—
24	Dic₁₀₉	SD₁₆	1	1	yes	characteristic
25	C₂×C₁₀₉	#102	1	1	yes	characteristic
26	D₁₀₉	SD₃₂	1	1	yes	characteristic
27	D₁₀₉	Q₃₂	1	1	yes	characteristic
28	C₁₀₉	#358	1	1	yes	characteristic, complemented, Sylow
29	#358	—	109	1	no	maximal, Sylow
30	C₂×Q₁₆	—	109	1	no	—
31	C₂×C₁₆	—	109	1	no	—
32	#107	—	109	1	no	—
33	C₂×Q₈	—	218	1	no	—
34	Q₁₆	—	109	1	no	—
35	Q₁₆	—	218	1	no	—
36	Q₁₆	#89137	1	1	yes	characteristic, complemented
37	C₂×C₈	—	109	1	no	—
38	C₄⋊C₄	—	218	1	no	—
39	C₁₆	—	218	1	no	—
40	Q₈	—	2	1	no	—
41	Q₈	—	436	1	no	—
42	Q₈	—	218	1	no	—
43	C₂×C₄	—	109	1	no	—
44	C₂×C₄	—	436	1	no	—
45	C₂×C₄	—	436	1	no	—
46	C₈	—	109	1	no	—
47	C₈	C₂×#89137	1	1	yes	characteristic
48	C₂²	—	109	1	no	—
49	C₄	—	4	1	no	—
50	C₄	—	872	1	no	—
51	C₄	—	109	1	no	—
52	C₄	—	436	1	no	—
53	C₄	C₂²⋊#89137	1	1	yes	characteristic
54	C₂	—	109	1	no	—
55	C₂	—	109	1	no	—
56	C₂	#16812761	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16812761
commutator	C₈×C₁₀₉	C₂×C₄
Frattini	C₈	C₂×#89137
Fitting	Q₁₆×C₁₀₉	C₄
socle	C₂×C₁₀₉	#102
hypercenter	Q₁₆	#89137
hypocenter	C₁₀₉	#358
LCS	G ▷ C₈×C₁₀₉ ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
7. Dic₄₃₆	C₄	semidirect
10. Q₁₆×C₁₀₉	C₄	semidirect
28. C₁₀₉	#358	semidirect
36. Q₁₆	#89137	semidirect