Subgroups — #16812745

Order 3488.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13182030	C₂	1	2	yes	maximal, complemented
2	#13182029	C₂	1	3	yes	maximal, complemented
3	C₂×D₄₃₆	C₂	1	3	yes	maximal, complemented
4	D₄×D₁₀₉	C₂	1	6	yes	maximal, complemented
5	#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₀₉	C₂²	1	6	yes	complemented
7	Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₁₀₉	C₂²	1	3	yes	complemented
9	C₂×C₄×C₁₀₉	C₂²	1	3	yes	complemented
10	C₂²⋊D₁₀₉	C₂²	1	6	yes	complemented
11	D₄₃₆	C₂²	1	6	yes	complemented
12	D₄₃₆	C₂²	1	3	yes	complemented
13	C₄×D₁₀₉	C₂²	1	6	yes	complemented
14	Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₁₀₉	C₂³	1	3	yes	—
16	C₂×D₁₀₉	—	4	6	no	—
17	C₂×D₁₀₉	C₂³	1	6	yes	—
18	C₄×C₁₀₉	C₂³	1	1	yes	characteristic
19	C₄×C₁₀₉	C₂³	1	3	yes	—
20	Dic₁₀₉	C₂³	1	2	yes	—
21	C₂×C₁₀₉	—	2	3	no	—
22	C₂×C₁₀₉	C₂⁴	1	1	yes	characteristic
23	D₁₀₉	—	2	6	no	—
24	C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	109	1	no	maximal, Sylow
26	#41	—	109	2	no	—
27	#41	—	109	3	no	—
28	#41	D₁₀₉	1	1	yes	characteristic, complemented
29	C₂×D₄	—	109	6	no	—
30	C₂×D₄	—	109	3	no	—
31	C₂³	—	109	6	no	—
32	Q₈	—	109	1	no	—
33	Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
34	D₄	—	109	6	no	—
35	D₄	—	109	6	no	—
36	D₄	—	109	3	no	—
37	D₄	C₂×D₁₀₉	1	3	yes	complemented
38	C₂×C₄	—	109	6	no	—
39	C₂×C₄	C₂×D₁₀₉	1	3	yes	complemented
40	C₂²	—	109	6	no	—
41	C₂²	—	436	6	no	—
42	C₂²	C₂²×D₁₀₉	1	3	yes	—
43	C₄	—	109	2	no	—
44	C₄	C₂²×D₁₀₉	1	3	yes	—
45	C₄	C₂²×D₁₀₉	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	218	6	no	—
48	C₂	C₂³×D₁₀₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₀₉
commutator, socle, cosocle	C₂×C₁₀₉	C₂⁴
Fitting	#41×C₁₀₉	C₂
hypercenter	#41	D₁₀₉
hypocenter	C₁₀₉	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #13182030	C₂	semidirect
2. #13182029	C₂	semidirect
3. C₂×D₄₃₆	C₂	semidirect
4. D₄×D₁₀₉	C₂	semidirect
5. #41×C₁₀₉	C₂	semidirect
6. C₂²×D₁₀₉	C₂²	semidirect
7. Q₈×C₁₀₉	C₂²	semidirect
8. D₄×C₁₀₉	C₂²	semidirect
9. C₂×C₄×C₁₀₉	C₂²	semidirect
10. C₂²⋊D₁₀₉	C₂²	semidirect
11. D₄₃₆	C₂²	semidirect
12. D₄₃₆	C₂²	semidirect
13. C₄×D₁₀₉	C₂²	semidirect
14. Dic₂₁₈	C₂²	semidirect
24. C₁₀₉	H(5,𝔽₂)	semidirect
28. #41	D₁₀₉	semidirect
33. Q₈	C₂×D₁₀₉	semidirect
37. D₄	C₂×D₁₀₉	semidirect
39. C₂×C₄	C₂×D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13182030	2
#13182029	2
C₂×D₄₃₆	2
D₄×D₁₀₉	2
#41×C₁₀₉	2
H(5,𝔽₂)	109