Subgroups — #11927990

Order 1248.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#1055	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#88054	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×Dic₁₃	C₄	1	1	yes	characteristic
7	C₂×C₃×Dic₁₃	C₄	1	1	yes	characteristic
8	C₃⋊#1055	—	2	1	no	—
9	C₁₃×C₃⋊C₈	—	2	1	no	—
10	C₂×C₈×C₁₃	—	3	1	no	—
11	C₄×Dic₁₃	D₃	1	1	yes	characteristic
12	C₂×#1055	—	3	1	no	—
13	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₃×C₄×C₁₃	D₄	1	1	yes	characteristic
15	C₃×C₄×C₁₃	Q₈	1	1	yes	characteristic
16	C₃×Dic₁₃	—	2	1	no	—
17	C₃×Dic₁₃	C₈	1	2	yes	complemented
18	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
19	C₂×Dic₁₃	Dic₃	1	1	yes	characteristic
20	C₂×Dic₁₃	Dic₃	1	1	yes	characteristic
21	C₈×C₁₃	—	6	1	no	—
22	#1055	—	6	1	no	—
23	C₄⋊(C₃⋊C₈)	—	13	1	no	maximal, Hall
24	C₂×C₃×C₁₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂×C₃×C₁₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃×C₁₃	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₃	C₂×Dic₃	1	1	yes	characteristic
28	C₄×C₁₃	Dic₆	1	1	yes	characteristic
29	C₄×C₁₃	D₁₂	1	1	yes	characteristic
30	Dic₁₃	—	2	1	no	—
31	Dic₁₃	C₃⋊C₈	1	2	yes	complemented
32	C₃×C₄²	—	13	1	no	—
33	C₂×C₃⋊C₈	—	13	1	no	—
34	C₂×C₃⋊C₈	D₁₃	1	1	yes	characteristic
35	C₃×C₁₃	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
36	C₄⋊C₈	—	39	1	no	Sylow
37	C₂×C₁₃	C₂×C₃⋊C₈	1	1	yes	characteristic
38	C₂×C₁₃	#208	1	1	yes	characteristic
39	C₂×C₁₃	C₄⋊Dic₃	1	1	yes	characteristic
40	C₂×C₃×C₄	—	13	1	no	—
41	C₂×C₃×C₄	—	13	1	no	—
42	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
43	C₃⋊C₈	—	2	1	no	—
44	C₃⋊C₈	—	26	1	no	—
45	C₂×C₈	—	3	1	no	—
46	C₂×C₈	—	39	1	no	—
47	C₄²	—	13	1	no	—
48	C₁₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₃	C₄×D₁₃	1	1	yes	characteristic
50	C₃×C₄	—	13	2	no	—
51	C₃×C₄	—	26	1	no	—
52	C₃×C₄	Dic₂₆	1	1	yes	characteristic
53	C₃×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
54	C₂×C₄	—	13	1	no	—
55	C₂×C₄	—	13	1	no	—
56	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
57	C₈	—	6	1	no	—
58	C₈	—	78	1	no	—
59	C₂×C₃	C₈×D₁₃	1	1	yes	characteristic
60	C₂×C₃	#6094	1	1	yes	characteristic
61	C₂×C₃	#6101	1	1	yes	characteristic
62	C₂²	Dic₃×D₁₃	1	1	yes	characteristic
63	C₄	—	13	2	no	—
64	C₄	—	26	1	no	—
65	C₄	#64604	1	1	yes	characteristic
66	C₄	#64605	1	1	yes	characteristic
67	C₃	#88054	1	1	yes	characteristic, complemented, Sylow
68	C₂	C₃⋊C₈×D₁₃	1	1	yes	characteristic
69	C₂	#10597592	1	1	yes	characteristic
70	C₂	#10597611	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	D₃×D₁₃
commutator	C₂×C₃×C₁₃	C₂×C₈
Fitting, cosocle	C₂×C₃×C₄×C₁₃	C₂²
socle, prime radical	C₂²×C₃×C₁₃	C₂×C₄
hypocenter	C₃×C₁₃	C₄⋊C₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
17. C₃×Dic₁₃	C₈	semidirect
31. Dic₁₃	C₃⋊C₈	semidirect
35. C₃×C₁₃	C₄⋊C₈	semidirect
48. C₁₃	C₄⋊(C₃⋊C₈)	semidirect
67. C₃	#88054	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#1055	2
C₂×C₁₃×C₃⋊C₈	2
C₃×C₄×Dic₁₃	2
#88054	3
C₄⋊(C₃⋊C₈)	13