Subgroups — #20294789

Order 6976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊Dic₁₀₉	C₂	1	2	yes	maximal
2	C₂²×C₈×C₁₀₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×Dic₁₀₉	—	2	2	no	—
4	C₄⋊Dic₁₀₉	—	2	2	no	—
5	C₄⋊Dic₁₀₉	C₄	1	4	yes	complemented
6	C₂²×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×C₈×C₁₀₉	—	2	2	no	—
8	C₂×C₈×C₁₀₉	C₄	1	2	yes	complemented
9	C₂³×C₁₀₉	D₄	1	1	yes	characteristic
10	C₂×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
11	C₂×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₁₀₉	D₄	1	2	yes	—
13	C₂×C₄×C₁₀₉	Q₈	1	1	yes	characteristic
14	C₂×Dic₁₀₉	—	2	4	no	—
15	C₂×Dic₁₀₉	—	4	2	no	—
16	C₂×Dic₁₀₉	—	4	2	no	—
17	C₈×C₁₀₉	—	2	2	no	—
18	C₂²×C₁₀₉	C₂²⋊C₄	1	2	yes	—
19	C₂²×C₁₀₉	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₁₀₉	D₈	1	1	yes	characteristic
21	C₂²×C₁₀₉	SD₁₆	1	2	yes	—
22	C₂²×C₁₀₉	Q₁₆	1	1	yes	characteristic
23	C₄×C₁₀₉	C₄²	1	1	yes	characteristic
24	C₄×C₁₀₉	C₂²⋊C₄	1	1	yes	characteristic
25	C₄×C₁₀₉	C₄⋊C₄	1	2	yes	—
26	Dic₁₀₉	—	4	4	no	—
27	C₂×C₁₀₉	#95	1	1	yes	characteristic
28	C₂×C₁₀₉	#102	1	2	yes	—
29	C₂×C₁₀₉	#103	1	2	yes	—
30	C₂×C₁₀₉	#106	1	1	yes	characteristic
31	C₂×C₁₀₉	#107	1	1	yes	characteristic
32	C₁₀₉	#340	1	1	yes	characteristic, complemented, Sylow
33	#340	—	109	1	no	maximal, Sylow
34	C₂²×C₈	D₁₀₉	1	1	yes	characteristic
35	C₂×C₄⋊C₄	—	109	2	no	—
36	C₂²×C₄	—	218	2	no	—
37	C₂²×C₄	C₂×D₁₀₉	1	1	yes	characteristic
38	C₂×C₈	—	2	2	no	—
39	C₂×C₈	Dic₁₀₉	1	2	yes	complemented
40	C₄⋊C₄	—	109	4	no	—
41	C₄⋊C₄	—	218	2	no	—
42	C₂³	D₄₃₆	1	1	yes	characteristic
43	C₂×C₄	—	436	2	no	—
44	C₂×C₄	—	436	2	no	—
45	C₂×C₄	—	218	4	no	—
46	C₂×C₄	Dic₂₁₈	1	1	yes	characteristic
47	C₂×C₄	C₄×D₁₀₉	1	2	yes	—
48	C₂×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₀₉	1	2	yes	—
50	C₈	—	2	2	no	—
51	C₂²	#13182006	1	2	yes	—
52	C₂²	C₄⋊Dic₁₀₉	1	1	yes	characteristic
53	C₂²	D₈₇₂	1	1	yes	characteristic
54	C₂²	#13182018	1	2	yes	—
55	C₂²	Dic₄₃₆	1	1	yes	characteristic
56	C₄	—	436	4	no	—
57	C₄	C₄×Dic₁₀₉	1	1	yes	characteristic
58	C₄	C₂²⋊Dic₁₀₉	1	1	yes	characteristic
59	C₄	#13182008	1	2	yes	—
60	C₂	#16812606	1	1	yes	characteristic
61	C₂	#16812621	1	2	yes	—
62	C₂	#16812624	1	2	yes	—
63	C₂	#16812631	1	1	yes	characteristic
64	C₂	#16812633	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄₃₆
commutator	C₄×C₁₀₉	C₄²
Frattini	C₂²×C₄	C₂×D₁₀₉
Fitting	C₂²×C₈×C₁₀₉	C₂
socle	C₂³×C₁₀₉	D₄
hypercenter	C₂²×C₈	D₁₀₉
hypocenter	C₁₀₉	#340
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊Dic₁₀₉	C₄	semidirect
8. C₂×C₈×C₁₀₉	C₄	semidirect
32. C₁₀₉	#340	semidirect
39. C₂×C₈	Dic₁₀₉	semidirect