Subgroups — #11928374

Order 1248.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#1057	C₂	1	2	yes	maximal
2	C₃×C₄×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	#88112	—	3	1	no	maximal, Hall
4	C₂×C₃×C₄×C₁₃	C₄	1	1	yes	characteristic
5	C₂×C₃×Dic₁₃	C₄	1	1	yes	characteristic
6	C₂×C₃×Dic₁₃	C₂²	1	1	yes	characteristic
7	C₃⋊#1057	—	2	2	no	—
8	C₂×#1057	—	3	2	no	—
9	C₄×Dic₁₃	D₃	1	1	yes	characteristic
10	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
11	C₃×C₄×C₁₃	—	2	1	no	—
12	C₃×Dic₁₃	C₈	1	2	yes	complemented
13	C₃×Dic₁₃	D₄	1	1	yes	characteristic
14	C₃×Dic₁₃	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₁₃	Dic₃	1	1	yes	characteristic
16	C₂×Dic₁₃	Dic₃	1	1	yes	characteristic
17	C₂×Dic₁₃	C₂×D₃	1	1	yes	characteristic
18	#1057	—	6	2	no	—
19	C₄⋊(C₃⋊C₈)	—	13	1	no	maximal, Hall
20	C₂×C₃×C₁₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₃×C₁₃	C₂×C₈	1	1	yes	characteristic
22	C₂×C₃×C₁₃	M₁₆	1	1	yes	characteristic
23	C₂²×C₁₃	C₂×Dic₃	1	1	yes	characteristic
24	C₄×C₁₃	—	2	1	no	—
25	Dic₁₃	C₃⋊C₈	1	2	yes	complemented
26	Dic₁₃	Dic₆	1	1	yes	characteristic
27	Dic₁₃	D₁₂	1	1	yes	characteristic
28	C₃×C₄²	—	13	1	no	—
29	C₂×C₃⋊C₈	—	13	2	no	—
30	C₃×C₁₃	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
31	C₄⋊C₈	—	39	1	no	Sylow
32	C₂×C₁₃	C₂×C₃⋊C₈	1	1	yes	characteristic
33	C₂×C₁₃	#208	1	1	yes	characteristic
34	C₂×C₁₃	C₄⋊Dic₃	1	1	yes	characteristic
35	C₂×C₃×C₄	—	13	1	no	—
36	C₂×C₃×C₄	—	13	1	no	—
37	C₂×C₃×C₄	#261	1	1	yes	characteristic
38	C₃⋊C₈	—	26	2	no	—
39	C₂×C₈	—	39	2	no	—
40	C₄²	—	13	1	no	—
41	C₁₃	C₄⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₃	C₂×#261	1	1	yes	characteristic
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	13	2	no	—
45	C₃×C₄	—	13	1	no	—
46	C₃×C₄	—	13	1	no	—
47	C₂×C₄	—	13	1	no	—
48	C₂×C₄	—	13	1	no	—
49	C₂×C₄	C₃⋊#261	1	1	yes	characteristic
50	C₈	—	78	2	no	—
51	C₂×C₃	D₁₃⋊C₈	1	1	yes	characteristic
52	C₂×C₃	C₄⋊#261	1	1	yes	characteristic
53	C₂×C₃	#6122	1	1	yes	characteristic
54	C₂²	C₂×C₃⋊#261	1	1	yes	characteristic
55	C₄	—	2	1	no	—
56	C₄	—	13	2	no	—
57	C₄	—	13	1	no	—
58	C₄	—	13	1	no	—
59	C₃	#88112	1	1	yes	characteristic, complemented, Sylow
60	C₂	D₁₃⋊(C₃⋊C₈)	1	1	yes	characteristic
61	C₂	C₄⋊(C₃⋊#261)	1	1	yes	characteristic
62	C₂	#10597728	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊#261
commutator	C₂×C₃×C₁₃	C₂×C₈
Fitting	C₂×C₃×C₄×C₁₃	C₄
socle, prime radical	C₂²×C₃×C₁₃	C₂×C₄
cosocle	C₂×C₃×Dic₁₃	C₂²
hypercenter	C₂×C₄	C₃⋊#261
hypocenter	C₃×C₁₃	C₄⋊C₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
12. C₃×Dic₁₃	C₈	semidirect
25. Dic₁₃	C₃⋊C₈	semidirect
30. C₃×C₁₃	C₄⋊C₈	semidirect
41. C₁₃	C₄⋊(C₃⋊C₈)	semidirect
59. C₃	#88112	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#1057	2
C₃×C₄×Dic₁₃	2
#88112	3
C₄⋊(C₃⋊C₈)	13