Subgroups — C₁₁₃×#13737 #43281757

Order 28928.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁₃×#2258	C₂	1	1	yes	maximal, characteristic
2	C₁₁₃×M₁₂₈	C₂	1	3	yes	maximal, complemented
3	C₂×C₆₄×C₁₁₃	C₂	1	3	yes	maximal, complemented
4	#504×C₁₁₃	C₄	1	1	yes	characteristic
5	M₆₄×C₁₁₃	C₄	1	3	yes	—
6	C₂×C₃₂×C₁₁₃	C₂²	1	3	yes	—
7	C₆₄×C₁₁₃	C₂²	1	3	yes	—
8	C₆₄×C₁₁₃	C₂²	1	1	yes	characteristic
9	#131×C₁₁₃	C₈	1	1	yes	characteristic
10	M₃₂×C₁₁₃	C₈	1	3	yes	—
11	C₂×C₁₆×C₁₁₃	C₂×C₄	1	3	yes	—
12	C₃₂×C₁₁₃	C₂×C₄	1	3	yes	—
13	C₃₂×C₁₁₃	C₂³	1	1	yes	characteristic
14	#41×C₁₁₃	C₁₆	1	1	yes	characteristic
15	M₁₆×C₁₁₃	C₁₆	1	3	yes	—
16	C₂×C₈×C₁₁₃	C₂×C₈	1	3	yes	—
17	C₁₆×C₁₁₃	C₂×C₈	1	3	yes	—
18	C₁₆×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
19	Q₈×C₁₁₃	C₃₂	1	1	yes	characteristic
20	D₄×C₁₁₃	C₃₂	1	3	yes	—
21	C₂×C₄×C₁₁₃	C₂×C₁₆	1	3	yes	—
22	C₈×C₁₁₃	C₂×C₁₆	1	3	yes	—
23	C₈×C₁₁₃	C₂²×C₈	1	1	yes	characteristic
24	C₂²×C₁₁₃	C₂×C₃₂	1	3	yes	—
25	C₄×C₁₁₃	C₂×C₃₂	1	3	yes	—
26	C₄×C₁₁₃	C₂²×C₁₆	1	1	yes	characteristic
27	#13737	C₁₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂×C₁₁₃	—	2	3	no	—
29	C₂×C₁₁₃	C₂²×C₃₂	1	1	yes	characteristic
30	#2258	C₂×C₁₁₃	1	1	yes	characteristic
31	M₁₂₈	C₂×C₁₁₃	1	3	yes	complemented
32	C₂×C₆₄	C₂×C₁₁₃	1	3	yes	complemented
33	C₁₁₃	#13737	1	1	yes	characteristic, direct product factor, Sylow
34	#504	C₄×C₁₁₃	1	1	yes	characteristic
35	M₆₄	C₄×C₁₁₃	1	3	yes	—
36	C₂×C₃₂	C₂²×C₁₁₃	1	3	yes	—
37	C₆₄	C₂²×C₁₁₃	1	3	yes	—
38	C₆₄	C₂²×C₁₁₃	1	1	yes	characteristic
39	#131	C₈×C₁₁₃	1	1	yes	characteristic
40	M₃₂	C₈×C₁₁₃	1	3	yes	—
41	C₂×C₁₆	C₂×C₄×C₁₁₃	1	3	yes	—
42	C₃₂	C₂×C₄×C₁₁₃	1	3	yes	—
43	C₃₂	C₂³×C₁₁₃	1	1	yes	characteristic
44	#41	C₁₆×C₁₁₃	1	1	yes	characteristic
45	M₁₆	C₁₆×C₁₁₃	1	3	yes	—
46	C₂×C₈	C₂×C₈×C₁₁₃	1	3	yes	—
47	C₁₆	C₂×C₈×C₁₁₃	1	3	yes	—
48	C₁₆	C₂²×C₄×C₁₁₃	1	1	yes	characteristic
49	Q₈	C₃₂×C₁₁₃	1	1	yes	characteristic
50	D₄	C₃₂×C₁₁₃	1	3	yes	—
51	C₂×C₄	C₂×C₁₆×C₁₁₃	1	3	yes	—
52	C₈	C₂×C₁₆×C₁₁₃	1	3	yes	—
53	C₈	C₂²×C₈×C₁₁₃	1	1	yes	characteristic
54	C₂²	C₂×C₃₂×C₁₁₃	1	3	yes	—
55	C₄	C₂×C₃₂×C₁₁₃	1	3	yes	—
56	C₄	C₂²×C₁₆×C₁₁₃	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₃₂×C₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄×C₁₁₃	C₂²
commutator	C₂	C₂²×C₃₂×C₁₁₃
Frattini, cosocle	C₃₂	C₂³×C₁₁₃
socle	C₂×C₁₁₃	C₂²×C₃₂
prime radical	#41×C₁₁₃	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₆₄×C₁₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₁₃×M₁₂₈	C₂	semidirect
3. C₂×C₆₄×C₁₁₃	C₂	semidirect
27. #13737	C₁₁₃	direct (N × H)
31. M₁₂₈	C₂×C₁₁₃	semidirect
32. C₂×C₆₄	C₂×C₁₁₃	semidirect
33. C₁₁₃	#13737	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₁₃×#2258	2
C₁₁₃×M₁₂₈	2
C₂×C₆₄×C₁₁₃	2
#13737	113

Subgroups — C113×#13737 #43281757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁₃×#13737 #43281757