Subgroups — #11929264

Order 1248.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41⋊(C₁₃⋊C₃)	C₂	1	3	yes	maximal, complemented
2	C₁₃×H(5,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	Q₈⋊(C₁₃⋊C₃)	C₂²	1	1	yes	characteristic, complemented
4	Q₈×C₁₃⋊C₃	—	4	1	no	maximal
5	C₁₃×#41	—	3	1	no	—
6	C₁₃×#41	C₂×C₃	1	3	yes	—
7	C₂×D₄×C₁₃	—	3	3	no	—
8	C₄×C₁₃⋊C₃	—	4	3	no	—
9	C₂³×C₁₃	—	3	2	no	—
10	Q₈×C₁₃	A₄	1	1	yes	characteristic
11	Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
12	D₄×C₁₃	—	3	3	no	—
13	D₄×C₁₃	—	3	3	no	—
14	C₂×C₄×C₁₃	—	3	3	no	—
15	#994	—	13	1	no	maximal, Hall
16	C₂×C₁₃⋊C₃	—	4	1	no	—
17	C₂²×C₁₃	—	3	3	no	—
18	C₂²×C₁₃	—	12	2	no	—
19	C₄×C₁₃	—	3	1	no	—
20	C₄×C₁₃	C₂×A₄	1	3	yes	—
21	#41⋊C₃	—	13	3	no	—
22	C₁₃⋊C₃	—	4	1	no	Hall
23	H(5,𝔽₂)	C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
24	C₂×C₁₃	—	6	3	no	—
25	C₂×C₁₃	C₂²×A₄	1	1	yes	characteristic
26	C₃×Q₈	—	52	1	no	—
27	SL(2,𝔽₃)	—	13	1	no	—
28	#41	—	3	1	no	—
29	#41	C₂×C₁₃⋊C₃	1	3	yes	—
30	C₂×D₄	—	3	3	no	—
31	C₁₃	#994	1	1	yes	characteristic, complemented, Sylow
32	C₃×C₄	—	52	3	no	—
33	C₂³	—	3	2	no	—
34	Q₈	C₂²×C₁₃⋊C₃	1	1	yes	characteristic
35	Q₈	C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic
36	D₄	—	3	3	no	—
37	D₄	—	3	3	no	—
38	C₂×C₄	—	3	3	no	—
39	C₂×C₃	—	52	1	no	—
40	C₂²	—	3	3	no	—
41	C₂²	—	12	2	no	—
42	C₄	—	3	1	no	—
43	C₄	C₂×C₂²⋊(C₁₃⋊C₃)	1	3	yes	—
44	C₃	—	52	1	no	Sylow
45	C₂	—	6	3	no	—
46	C₂	C₂²×C₂²⋊(C₁₃⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₂²⋊(C₁₃⋊C₃)
commutator, cosocle, hypocenter	Q₈×C₁₃	C₂²×C₃
Fitting	C₁₃×H(5,𝔽₂)	C₃
socle	C₂×C₁₃	C₂²×A₄
hypercenter	Q₈	C₂²⋊(C₁₃⋊C₃)
LCS	G ▷ Q₈×C₁₃
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. #41⋊(C₁₃⋊C₃)	C₂	semidirect
2. C₁₃×H(5,𝔽₂)	C₃	semidirect
3. Q₈⋊(C₁₃⋊C₃)	C₂²	semidirect
23. H(5,𝔽₂)	C₁₃⋊C₃	semidirect
31. C₁₃	#994	semidirect

Maximal subgroups

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

subgroup	index
#41⋊(C₁₃⋊C₃)	2
C₁₃×H(5,𝔽₂)	3
Q₈×C₁₃⋊C₃	4
#994	13