Subgroups — #11928126

Order 1248.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₃₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊#1055	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×Dic₃₉	C₄	1	1	yes	characteristic
7	C₂×Dic₃₉	C₄	1	1	yes	characteristic
8	C₃×C₈×C₁₃	—	2	1	no	—
9	C₃⋊#1055	—	2	1	no	—
10	C₂×C₈×C₁₃	D₃	1	1	yes	characteristic
11	C₄×Dic₁₃	—	3	1	no	—
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	Dic₃₉	—	2	1	no	—
17	Dic₃₉	C₈	1	2	yes	complemented
18	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
19	C₂×Dic₁₃	—	3	1	no	—
20	C₂×Dic₁₃	—	3	1	no	—
21	C₈×C₁₃	—	2	1	no	—
22	#1055	—	6	1	no	—
23	Dic₃⋊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	C₄×D₃	1	1	yes	characteristic
28	C₄×C₁₃	Dic₆	1	1	yes	characteristic
29	C₄×C₁₃	#67	1	1	yes	characteristic
30	Dic₁₃	—	3	2	no	—
31	Dic₁₃	—	6	1	no	—
32	C₂×C₃×C₈	D₁₃	1	1	yes	characteristic
33	C₄×Dic₃	—	13	1	no	—
34	C₂×C₃⋊C₈	—	13	1	no	—
35	C₃×C₁₃	C₄⋊C₈	1	1	yes	characteristic, complemented, Hall
36	C₄⋊C₈	—	39	1	no	Sylow
37	C₂×C₁₃	D₃×C₈	1	1	yes	characteristic
38	C₂×C₁₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
39	C₂×C₁₃	Dic₃⋊C₄	1	1	yes	characteristic
40	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
41	C₂×Dic₃	—	13	1	no	—
42	C₂×Dic₃	—	13	1	no	—
43	C₃×C₈	—	2	1	no	—
44	C₃⋊C₈	—	26	1	no	—
45	C₂×C₈	—	39	1	no	—
46	C₂×C₈	D₃₉	1	1	yes	characteristic
47	C₄²	—	39	1	no	—
48	C₁₃	Dic₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
49	C₂²×C₃	C₄×D₁₃	1	1	yes	characteristic
50	C₃×C₄	Dic₂₆	1	1	yes	characteristic
51	C₃×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
52	Dic₃	—	13	2	no	—
53	Dic₃	—	26	1	no	—
54	C₂×C₄	—	39	1	no	—
55	C₂×C₄	—	39	1	no	—
56	C₂×C₄	C₂×D₃₉	1	1	yes	characteristic
57	C₈	—	2	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	C₄×D₃₉	1	1	yes	characteristic
63	C₄	—	39	2	no	—
64	C₄	—	78	1	no	—
65	C₄	Dic₇₈	1	1	yes	characteristic
66	C₄	#64625	1	1	yes	characteristic
67	C₃	#88054	1	1	yes	characteristic, complemented, Sylow
68	C₂	C₈×D₃₉	1	1	yes	characteristic
69	C₂	#10597657	1	1	yes	characteristic
70	C₂	#10597664	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃₉
commutator	C₂×C₃×C₁₃	C₂×C₈
Fitting	C₂×C₃×C₈×C₁₃	C₂
socle, prime radical	C₂²×C₃×C₁₃	C₂×C₄
cosocle	C₂×C₃×C₄×C₁₃	C₂²
hypercenter	C₂×C₈	D₃₉
hypocenter	C₃×C₁₃	C₄⋊C₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
17. Dic₃₉	C₈	semidirect
35. C₃×C₁₃	C₄⋊C₈	semidirect
48. C₁₃	Dic₃⋊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×C₈×C₁₃	2
C₄×Dic₃₉	2
C₂×C₃⋊#1055	2
#88054	3
Dic₃⋊C₈	13