Subgroups — #88256

Order 416.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	#6131	C₂	1	2	yes	maximal, complemented
3	D₄×D₁₃	C₂	1	6	yes	maximal, complemented
4	#6127	C₂	1	3	yes	maximal, complemented
5	C₂×D₅₂	C₂	1	3	yes	maximal, complemented
6	C₂²×D₁₃	C₂²	1	6	yes	complemented
7	Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
8	D₄×C₁₃	C₂²	1	3	yes	complemented
9	C₂×C₄×C₁₃	C₂²	1	3	yes	complemented
10	C₂²⋊D₁₃	C₂²	1	6	yes	complemented
11	D₅₂	C₂²	1	3	yes	complemented
12	D₅₂	C₂²	1	6	yes	complemented
13	C₄×D₁₃	C₂²	1	6	yes	complemented
14	Dic₂₆	C₂²	1	1	yes	characteristic, complemented
15	C₂²×C₁₃	C₂³	1	3	yes	—
16	C₂×D₁₃	—	4	6	no	—
17	C₂×D₁₃	C₂³	1	6	yes	—
18	C₄×C₁₃	C₂³	1	1	yes	characteristic
19	C₄×C₁₃	C₂³	1	3	yes	—
20	Dic₁₃	C₂³	1	2	yes	—
21	H(5,𝔽₂)	—	13	1	no	maximal, Sylow
22	C₂×C₁₃	—	2	3	no	—
23	C₂×C₁₃	C₂⁴	1	1	yes	characteristic
24	D₁₃	—	2	6	no	—
25	#41	—	13	2	no	—
26	#41	—	13	3	no	—
27	#41	D₁₃	1	1	yes	characteristic, complemented
28	C₂×D₄	—	13	6	no	—
29	C₂×D₄	—	13	3	no	—
30	C₁₃	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
31	C₂³	—	13	6	no	—
32	Q₈	—	13	1	no	—
33	Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
34	D₄	—	13	6	no	—
35	D₄	—	13	6	no	—
36	D₄	—	13	3	no	—
37	D₄	C₂×D₁₃	1	3	yes	complemented
38	C₂×C₄	—	13	6	no	—
39	C₂×C₄	C₂×D₁₃	1	3	yes	complemented
40	C₂²	—	13	6	no	—
41	C₂²	—	52	6	no	—
42	C₂²	C₂²×D₁₃	1	3	yes	—
43	C₄	—	13	2	no	—
44	C₄	C₂²×D₁₃	1	3	yes	—
45	C₄	C₂²×D₁₃	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	26	6	no	—
48	C₂	C₂³×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₃
commutator, socle, cosocle	C₂×C₁₃	C₂⁴
Fitting	C₁₃×#41	C₂
hypercenter	#41	D₁₃
hypocenter	C₁₃	H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ #41

Complements

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

N	H ≅ G/N	type
1. C₁₃×#41	C₂	semidirect
2. #6131	C₂	semidirect
3. D₄×D₁₃	C₂	semidirect
4. #6127	C₂	semidirect
5. C₂×D₅₂	C₂	semidirect
6. C₂²×D₁₃	C₂²	semidirect
7. Q₈×C₁₃	C₂²	semidirect
8. D₄×C₁₃	C₂²	semidirect
9. C₂×C₄×C₁₃	C₂²	semidirect
10. C₂²⋊D₁₃	C₂²	semidirect
11. D₅₂	C₂²	semidirect
12. D₅₂	C₂²	semidirect
13. C₄×D₁₃	C₂²	semidirect
14. Dic₂₆	C₂²	semidirect
27. #41	D₁₃	semidirect
30. C₁₃	H(5,𝔽₂)	semidirect
33. Q₈	C₂×D₁₃	semidirect
37. D₄	C₂×D₁₃	semidirect
39. C₂×C₄	C₂×D₁₃	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#41	2
#6131	2
D₄×D₁₃	2
#6127	2
C₂×D₅₂	2
H(5,𝔽₂)	13