Subgroups — #88237

Order 416.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊#261	C₂	1	2	yes	maximal
3	C₄⋊#261	C₂	1	2	yes	maximal, complemented
4	C₄×#261	C₂	1	2	yes	maximal, complemented
5	C₂²×D₁₃	C₂²	1	1	yes	characteristic
6	C₂×#261	C₂²	1	4	yes	—
7	C₂×C₄×C₁₃	C₄	1	1	yes	characteristic, complemented
8	C₂×Dic₁₃	C₄	1	1	yes	characteristic, complemented
9	C₄×D₁₃	C₄	1	1	yes	characteristic, complemented
10	C₄×D₁₃	C₄	1	1	yes	characteristic, complemented
11	C₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₄×D₁₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂×D₁₃	—	2	2	no	—
15	C₂×D₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×D₁₃	C₂³	1	1	yes	characteristic
17	#261	—	2	4	no	—
18	C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	Dic₁₃	C₂×C₄	1	1	yes	characteristic
21	Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	#117	—	13	1	no	maximal, Sylow
23	C₂×C₁₃	—	2	1	no	—
24	C₂×C₁₃	C₂²×C₄	1	1	yes	characteristic
25	D₁₃	—	2	1	no	—
26	D₁₃	#41	1	2	yes	—
27	C₂²×C₄	—	13	1	no	—
28	C₄⋊C₄	—	13	2	no	—
29	C₂²⋊C₄	—	13	2	no	—
30	C₄²	—	13	2	no	—
31	C₁₃	#117	1	1	yes	characteristic, complemented, Sylow
32	C₂³	—	13	1	no	—
33	C₂×C₄	—	13	1	no	—
34	C₂×C₄	—	13	1	no	—
35	C₂×C₄	—	13	1	no	—
36	C₂×C₄	—	13	1	no	—
37	C₂×C₄	—	13	1	no	—
38	C₂×C₄	—	13	4	no	—
39	C₂×C₄	#261	1	1	yes	characteristic, complemented
40	C₂²	—	13	1	no	—
41	C₂²	—	13	1	no	—
42	C₂²	—	26	2	no	—
43	C₂²	C₂×#261	1	1	yes	characteristic
44	C₄	—	13	1	no	—
45	C₄	—	13	1	no	—
46	C₄	—	26	4	no	—
47	C₄	C₂×#261	1	1	yes	characteristic
48	C₄	C₂×#261	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	13	2	no	—
51	C₂	—	26	1	no	—
52	C₂	C₂²×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#261
commutator, socle	C₂×C₁₃	C₂²×C₄
Frattini	C₂	C₂²×#261
Fitting	C₂×C₄×C₁₃	C₄
prime radical	C₂²×D₁₃	C₂²
cosocle	C₂×D₁₃	C₂³
hypercenter	C₂×C₄	#261
hypocenter	C₁₃	#117
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
3. C₄⋊#261	C₂	semidirect
4. C₄×#261	C₂	semidirect
7. C₂×C₄×C₁₃	C₄	semidirect
8. C₂×Dic₁₃	C₄	semidirect
9. C₄×D₁₃	C₄	semidirect
10. C₄×D₁₃	C₄	semidirect
31. C₁₃	#117	semidirect
39. C₂×C₄	#261	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₁₃	2
C₂²⋊#261	2
C₄⋊#261	2
C₄×#261	2
#117	13