Subgroups — #88046

Order 416.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×C₁₃	C₂²	1	1	yes	characteristic
5	C₂²×D₁₃	C₄	1	1	yes	characteristic, complemented
6	C₂×C₄×C₁₃	—	2	1	no	—
7	C₂²⋊D₁₃	—	2	1	no	—
8	C₂²⋊D₁₃	—	2	1	no	—
9	C₂×Dic₁₃	—	2	1	no	—
10	C₂×Dic₁₃	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₁₃	—	4	1	no	—
12	C₂²×C₁₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₁₃	D₄	1	1	yes	characteristic
14	C₂²×C₁₃	D₄	1	1	yes	characteristic
15	C₂×D₁₃	—	2	1	no	—
16	C₂×D₁₃	—	4	1	no	—
17	C₄×C₁₃	—	4	1	no	—
18	Dic₁₃	—	2	1	no	—
19	Dic₁₃	—	4	1	no	—
20	#99	—	13	1	no	maximal, Sylow
21	C₂×C₁₃	—	2	1	no	—
22	C₂×C₁₃	—	2	1	no	—
23	C₂×C₁₃	—	2	1	no	—
24	C₂×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
25	D₁₃	—	4	1	no	—
26	C₂×D₄	—	13	1	no	—
27	C₂²⋊C₄	—	13	1	no	—
28	C₂²⋊C₄	D₁₃	1	1	yes	characteristic, complemented
29	C₁₃	#99	1	1	yes	characteristic, complemented, Sylow
30	C₂³	—	13	1	no	—
31	C₂³	C₂×D₁₃	1	1	yes	characteristic
32	D₄	—	26	1	no	—
33	D₄	—	26	1	no	—
34	C₂×C₄	—	2	1	no	—
35	C₂×C₄	—	13	1	no	—
36	C₂×C₄	—	26	1	no	—
37	C₂²	—	4	1	no	—
38	C₂²	—	26	1	no	—
39	C₂²	—	52	1	no	—
40	C₂²	C₄×D₁₃	1	1	yes	characteristic
41	C₂²	D₅₂	1	1	yes	characteristic
42	C₂²	C₂²⋊D₁₃	1	1	yes	characteristic
43	C₄	—	4	1	no	—
44	C₄	—	26	1	no	—
45	C₄	—	52	1	no	—
46	C₂	—	2	1	no	—
47	C₂	—	2	1	no	—
48	C₂	—	2	1	no	—
49	C₂	—	52	1	no	—
50	C₂	#6103	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6103
commutator	C₂²×C₁₃	C₂×C₄
Frattini	C₂³	C₂×D₁₃
Fitting	C₁₃×C₂²⋊C₄	C₂
socle	C₂×C₁₃	C₂²⋊C₄
prime radical	C₂×C₂²⋊D₁₃	C₂
cosocle	C₂³×C₁₃	C₂²
hypercenter	C₂²⋊C₄	D₁₃
hypocenter	C₁₃	#99
LCS	G ▷ C₂²×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₁₃×C₂²⋊C₄	C₂	semidirect
3. C₂²⋊Dic₁₃	C₂	semidirect
5. C₂²×D₁₃	C₄	semidirect
10. C₂×Dic₁₃	C₄	semidirect
28. C₂²⋊C₄	D₁₃	semidirect
29. C₁₃	#99	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₁₃	2
C₁₃×C₂²⋊C₄	2
C₂²⋊Dic₁₃	2
#99	13