Subgroups — #88172

Order 416.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6131	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#6107	C₂	1	1	yes	maximal, characteristic, complemented
5	#6106	C₂	1	1	yes	maximal, characteristic, complemented
6	#6095	C₂	1	1	yes	maximal, characteristic, complemented
7	#6094	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	D₅₂	—	2	1	no	—
11	D₅₂	C₂²	1	1	yes	characteristic
12	C₄×D₁₃	—	2	1	no	—
13	C₄×D₁₃	—	2	1	no	—
14	C₄×D₁₃	C₂²	1	1	yes	characteristic
15	Dic₂₆	—	2	1	no	—
16	Dic₂₆	C₂²	1	1	yes	characteristic
17	C₈×C₁₃	C₂²	1	1	yes	characteristic
18	#1055	C₂²	1	1	yes	characteristic
19	C₂×D₁₃	—	2	1	no	—
20	C₂×D₁₃	D₄	1	1	yes	characteristic
21	C₄×C₁₃	—	2	1	no	—
22	C₄×C₁₃	—	2	1	no	—
23	C₄×C₁₃	C₂³	1	1	yes	characteristic
24	Dic₁₃	—	2	1	no	—
25	Dic₁₃	D₄	1	1	yes	characteristic
26	#137	—	13	1	no	maximal, Sylow
27	C₂×C₁₃	C₂×D₄	1	1	yes	characteristic
28	D₁₃	—	2	1	no	—
29	D₁₃	—	4	1	no	—
30	#41	—	13	1	no	—
31	C₂×Q₈	—	13	1	no	—
32	Q₁₆	—	13	1	no	—
33	Q₁₆	D₁₃	1	1	yes	characteristic, complemented
34	SD₁₆	—	13	1	no	—
35	SD₁₆	—	13	1	no	—
36	M₁₆	—	13	1	no	—
37	C₁₃	#137	1	1	yes	characteristic, complemented, Sylow
38	Q₈	—	13	1	no	—
39	Q₈	—	26	1	no	—
40	Q₈	C₂×D₁₃	1	1	yes	characteristic
41	Q₈	C₂×D₁₃	1	1	yes	characteristic
42	D₄	—	13	1	no	—
43	D₄	—	26	1	no	—
44	C₂×C₄	—	13	1	no	—
45	C₂×C₄	—	26	1	no	—
46	C₂×C₄	—	26	1	no	—
47	C₈	—	13	1	no	—
48	C₈	C₂×D₁₃	1	1	yes	characteristic
49	C₂²	—	13	1	no	—
50	C₂²	—	26	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	2	1	no	—
53	C₄	—	13	1	no	—
54	C₄	—	26	1	no	—
55	C₄	C₂²×D₁₃	1	1	yes	characteristic
56	C₂	—	26	1	no	—
57	C₂	—	52	1	no	—
58	C₂	D₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₃
commutator, cosocle	C₄×C₁₃	C₂³
Frattini	C₄	C₂²×D₁₃
Fitting	C₁₃×Q₁₆	C₂
socle	C₂×C₁₃	C₂×D₄
prime radical	#6131	C₂
hypercenter	Q₁₆	D₁₃
hypocenter	C₁₃	#137
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×D₁₃	C₂	semidirect
3. C₁₃×Q₁₆	C₂	semidirect
4. #6107	C₂	semidirect
5. #6106	C₂	semidirect
6. #6095	C₂	semidirect
7. #6094	C₂	semidirect
33. Q₁₆	D₁₃	semidirect
37. C₁₃	#137	semidirect

Maximal subgroups

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

subgroup	index
#6131	2
Q₈×D₁₃	2
C₁₃×Q₁₆	2
#6107	2
#6106	2
#6095	2
#6094	2
#137	13