Subgroups — #20295427

Order 6976.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812690	C₂	1	2	yes	maximal
2	#16812633	C₂	1	1	yes	maximal, characteristic
3	#16812630	C₂	1	2	yes	maximal
4	C₈×Dic₁₀₉	C₂	1	1	yes	maximal, characteristic
5	#107×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#11724784	C₂²	1	1	yes	characteristic
7	C₄⋊Dic₁₀₉	—	2	2	no	—
8	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
9	#13182008	—	2	2	no	—
10	C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
13	C₂×C₄×C₁₀₉	—	2	2	no	—
14	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
15	C₂×Dic₁₀₉	—	2	2	no	—
16	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
17	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
18	C₈×C₁₀₉	Q₈	1	2	yes	—
19	#11724784	Q₈	1	2	yes	—
20	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₀₉	—	4	2	no	—
22	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₀₉	—	2	1	no	—
25	Dic₁₀₉	—	2	1	no	—
26	Dic₁₀₉	—	4	2	no	—
27	C₂×C₁₀₉	C₄⋊Q₈	1	1	yes	characteristic
28	C₂×C₁₀₉	#135	1	1	yes	characteristic
29	C₂×C₁₀₉	#135	1	1	yes	characteristic
30	C₁₀₉	#499	1	1	yes	characteristic, complemented, Sylow
31	#499	—	109	1	no	maximal, Sylow
32	#125	—	109	2	no	—
33	#107	—	109	1	no	—
34	#107	D₁₀₉	1	1	yes	characteristic
35	#106	—	109	2	no	—
36	C₄×C₈	—	109	1	no	—
37	C₂×C₈	—	109	1	no	—
38	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
39	C₄⋊C₄	—	109	2	no	—
40	C₄⋊C₄	—	218	2	no	—
41	C₄⋊C₄	—	218	2	no	—
42	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
43	C₄²	—	109	1	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	109	1	no	—
46	C₂×C₄	—	109	1	no	—
47	C₂×C₄	—	218	2	no	—
48	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
49	C₈	—	109	2	no	—
50	C₈	Dic₂₁₈	1	2	yes	—
51	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
52	C₄	—	4	2	no	—
53	C₄	—	436	2	no	—
54	C₄	—	218	1	no	—
55	C₄	—	218	1	no	—
56	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
57	C₄	C₂×Dic₂₁₈	1	1	yes	characteristic
58	C₂	#16812698	1	1	yes	characteristic
59	C₂	#16812718	1	1	yes	characteristic
60	C₂	#16812722	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₂×C₄×C₁₀₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#107×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#107	D₁₀₉
hypocenter	C₁₀₉	#499
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
30. C₁₀₉	#499	semidirect