Subgroups — #20295061

Order 6976.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₁₀₉	C₂	1	1	yes	maximal, characteristic
2	#16812657	C₂	1	2	yes	maximal
3	#16812632	C₂	1	2	yes	maximal, complemented
4	#16812631	C₂	1	1	yes	maximal, characteristic, complemented
5	#106×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
7	C₂×#11724784	C₂²	1	1	yes	characteristic
8	C₈×D₁₀₉	C₄	1	4	yes	complemented
9	C₄⋊Dic₁₀₉	C₂²	1	2	yes	—
10	#13182006	—	2	2	no	—
11	C₄×Dic₁₀₉	—	2	2	no	—
12	C₂×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
14	C₂²×D₁₀₉	D₄	1	1	yes	characteristic
15	C₂×C₄×C₁₀₉	—	2	2	no	—
16	C₂×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
17	C₂×Dic₁₀₉	—	2	2	no	—
18	C₂×Dic₁₀₉	D₄	1	1	yes	characteristic
19	C₄×D₁₀₉	C₂×C₄	1	2	yes	—
20	C₄×D₁₀₉	Q₈	1	2	yes	—
21	C₈×C₁₀₉	C₂×C₄	1	2	yes	—
22	#11724784	C₂×C₄	1	2	yes	—
23	C₂²×C₁₀₉	C₂×D₄	1	1	yes	characteristic
24	C₂×D₁₀₉	—	2	2	no	—
25	C₂×D₁₀₉	C₄⋊C₄	1	2	yes	—
26	C₄×C₁₀₉	—	4	2	no	—
27	C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₀₉	C₂×Q₈	1	1	yes	characteristic
29	Dic₁₀₉	—	4	2	no	—
30	Dic₁₀₉	C₄⋊C₄	1	2	yes	—
31	C₂×C₁₀₉	C₂×C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₁₀₉	#135	1	2	yes	—
33	D₁₀₉	—	2	2	no	—
34	C₁₀₉	#427	1	1	yes	characteristic, complemented, Sylow
35	#427	—	109	1	no	maximal, Sylow
36	C₂²×C₈	—	109	1	no	—
37	#117	—	109	2	no	—
38	#107	—	109	2	no	—
39	#106	—	109	1	no	—
40	#106	D₁₀₉	1	1	yes	characteristic, complemented
41	C₂²×C₄	—	109	1	no	—
42	C₂×C₈	—	109	1	no	—
43	C₂×C₈	—	109	4	no	—
44	C₂×C₈	C₂×D₁₀₉	1	1	yes	characteristic
45	C₄⋊C₄	—	109	2	no	—
46	C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
47	C₂²⋊C₄	—	218	2	no	—
48	C₄²	—	218	2	no	—
49	C₂³	—	109	1	no	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	109	1	no	—
52	C₂×C₄	—	109	2	no	—
53	C₂×C₄	—	109	2	no	—
54	C₂×C₄	—	218	2	no	—
55	C₂×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
56	C₈	—	109	2	no	—
57	C₈	C₄×D₁₀₉	1	2	yes	—
58	C₂²	—	109	2	no	—
59	C₂²	—	218	2	no	—
60	C₂²	D₄×D₁₀₉	1	1	yes	characteristic
61	C₄	—	4	2	no	—
62	C₄	—	109	2	no	—
63	C₄	—	436	2	no	—
64	C₄	C₂×C₄×D₁₀₉	1	1	yes	characteristic
65	C₄	Q₈×D₁₀₉	1	1	yes	characteristic
66	C₂	—	218	2	no	—
67	C₂	C₄⋊C₄×D₁₀₉	1	1	yes	characteristic
68	C₂	#16812720	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₀₉
commutator	C₄×C₁₀₉	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₀₉
Fitting	#106×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂×D₄
hypercenter	#106	D₁₀₉
hypocenter	C₁₀₉	#427
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

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

N	H ≅ G/N	type
3. #16812632	C₂	semidirect
4. #16812631	C₂	semidirect
5. #106×C₁₀₉	C₂	semidirect
8. C₈×D₁₀₉	C₄	semidirect
34. C₁₀₉	#427	semidirect
40. #106	D₁₀₉	semidirect