Subgroups — #11929125

Order 1248.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#41⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₁₃⋊#226	C₂	1	2	yes	maximal, complemented
3	#88205	—	3	1	no	maximal, Hall
4	Dic₇₈	—	4	1	no	maximal
5	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₁₃×#41	D₃	1	1	yes	characteristic
7	C₂×Dic₂₆	—	3	1	no	—
8	#6107	—	3	2	no	—
9	#6105	—	3	2	no	—
10	#6099	—	3	1	no	—
11	C₃×C₄×C₁₃	—	4	1	no	—
12	Dic₃₉	—	4	2	no	—
13	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic
14	D₄×C₁₃	—	3	1	no	—
15	C₂×C₄×C₁₃	—	3	1	no	—
16	C₂×Dic₁₃	—	6	1	no	—
17	Dic₂₆	—	3	2	no	—
18	Dic₂₆	—	6	1	no	—
19	#1055	—	3	2	no	—
20	#984	—	13	1	no	maximal, Hall
21	C₂×C₃×C₁₃	—	4	1	no	—
22	C₂²×C₁₃	—	3	1	no	—
23	C₄×C₁₃	—	3	1	no	—
24	C₄×C₁₃	S₄	1	1	yes	characteristic
25	Dic₁₃	—	6	2	no	—
26	#41⋊C₃	D₁₃	1	1	yes	characteristic
27	#226	—	13	2	no	—
28	C₃×C₁₃	—	4	1	no	Hall
29	#137	—	39	1	no	Sylow
30	C₂×C₁₃	—	6	1	no	—
31	C₂×C₁₃	C₂×S₄	1	1	yes	characteristic
32	Dic₆	—	52	1	no	—
33	SL(2,𝔽₃)	C₂×D₁₃	1	1	yes	characteristic
34	#41	D₃₉	1	1	yes	characteristic
35	C₂×Q₈	—	39	1	no	—
36	Q₁₆	—	39	2	no	—
37	SD₁₆	—	39	2	no	—
38	M₁₆	—	39	1	no	—
39	C₁₃	#984	1	1	yes	characteristic, complemented, Sylow
40	C₃×C₄	—	4	1	no	—
41	Dic₃	—	52	2	no	—
42	Q₈	—	39	2	no	—
43	Q₈	—	78	1	no	—
44	Q₈	C₂×D₃₉	1	1	yes	characteristic
45	D₄	—	3	1	no	—
46	C₂×C₄	—	3	1	no	—
47	C₂×C₄	—	78	1	no	—
48	C₈	—	39	2	no	—
49	C₂×C₃	—	4	1	no	—
50	C₂²	—	3	1	no	—
51	C₄	—	3	1	no	—
52	C₄	—	78	2	no	—
53	C₄	A₄⋊D₁₃	1	1	yes	characteristic
54	C₃	—	4	1	no	Sylow
55	C₂	—	6	1	no	—
56	C₂	C₂×A₄⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×A₄⋊D₁₃
commutator, cosocle, hypocenter	C₁₃×SL(2,𝔽₃)	C₂²
Fitting	C₁₃×#41	D₃
socle	C₂×C₁₃	C₂×S₄
prime radical	C₁₃×#41⋊C₃	C₂
hypercenter	C₄	A₄⋊D₁₃
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₁₃⋊#226	C₂	semidirect
39. C₁₃	#984	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#41⋊C₃	2
C₁₃⋊#226	2
#88205	3
Dic₇₈	4
#984	13