Subgroups — #11928480

Order 1248.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#235	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6099	C₂	1	1	yes	maximal, characteristic, complemented
4	#10597593	C₂	1	2	yes	maximal, complemented
5	D₃×#1055	C₂	1	2	yes	maximal, complemented
6	#88202	—	3	1	no	maximal, Hall
7	C₂×C₃×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#67	C₄	1	2	yes	—
9	C₁₃×D₁₂	C₄	1	1	yes	characteristic
10	C₄×D₃×C₁₃	C₂²	1	2	yes	—
11	C₁₃×Dic₆	C₄	1	1	yes	characteristic
12	C₃⋊#1055	C₂²	1	1	yes	characteristic
13	C₃⋊#1055	C₂²	1	1	yes	characteristic
14	C₃×#1055	C₂²	1	2	yes	—
15	C₁₃×#41	—	3	1	no	—
16	#6099	—	3	2	no	—
17	#6099	D₃	1	1	yes	characteristic, complemented
18	C₂×#1055	—	3	1	no	—
19	C₂×#1055	—	3	2	no	—
20	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₃×C₁₃	C₂×C₄	1	2	yes	—
22	C₃×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₃×C₄×C₁₃	C₂³	1	1	yes	characteristic
24	Dic₃×C₁₃	C₂×C₄	1	2	yes	—
25	Q₈×C₁₃	—	3	1	no	—
26	D₄×C₁₃	—	3	2	no	—
27	D₄×C₁₃	—	3	1	no	—
28	C₂×C₄×C₁₃	—	3	2	no	—
29	C₂×C₄×C₁₃	C₂×D₃	1	1	yes	characteristic
30	#1055	—	3	1	no	—
31	#1055	—	3	1	no	—
32	#1055	C₂×D₃	1	2	yes	—
33	#907	—	13	1	no	maximal, Hall
34	C₂×C₃×C₁₃	—	2	1	no	—
35	C₂×C₃×C₁₃	C₂²×C₄	1	1	yes	characteristic
36	D₃×C₁₃	—	2	2	no	—
37	C₂²×C₁₃	—	3	2	no	—
38	C₂²×C₁₃	C₄×D₃	1	1	yes	characteristic
39	C₄×C₁₃	—	3	2	no	—
40	C₄×C₁₃	C₄×D₃	1	1	yes	characteristic
41	C₄×C₁₃	C₂²×D₃	1	1	yes	characteristic
42	#235	D₁₃	1	1	yes	characteristic
43	C₃×M₁₆	—	13	1	no	—
44	C₂×C₃⋊C₈	—	13	1	no	—
45	ΓL(1,𝔽₂₅)	—	13	2	no	—
46	D₃×C₈	—	13	2	no	—
47	C₃×C₁₃	#131	1	1	yes	characteristic, complemented, Hall
48	#131	—	39	1	no	Sylow
49	C₂×C₁₃	—	2	1	no	—
50	C₂×C₁₃	—	6	2	no	—
51	C₂×C₁₃	C₂×C₄×D₃	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₁₃	1	1	yes	characteristic
53	#67	Dic₁₃	1	2	yes	—
54	D₁₂	Dic₁₃	1	1	yes	characteristic
55	C₄×D₃	C₂×D₁₃	1	2	yes	—
56	Dic₆	Dic₁₃	1	1	yes	characteristic
57	C₃×C₈	—	13	2	no	—
58	C₃⋊C₈	—	13	1	no	—
59	C₃⋊C₈	—	13	1	no	—
60	#41	—	3	1	no	—
61	M₁₆	—	13	1	no	—
62	M₁₆	—	39	2	no	—
63	C₂×C₈	—	39	2	no	—
64	C₂×C₈	—	39	1	no	—
65	C₁₃	#907	1	1	yes	characteristic, complemented, Sylow
66	C₂²×C₃	C₂×Dic₁₃	1	1	yes	characteristic
67	C₂×D₃	C₂×Dic₁₃	1	2	yes	—
68	C₃×C₄	C₂×Dic₁₃	1	1	yes	characteristic
69	C₃×C₄	C₂²×D₁₃	1	1	yes	characteristic
70	Dic₃	C₂×Dic₁₃	1	2	yes	—
71	Q₈	—	3	1	no	—
72	D₄	—	3	2	no	—
73	D₄	—	3	1	no	—
74	C₂×C₄	—	3	2	no	—
75	C₂×C₄	D₃×D₁₃	1	1	yes	characteristic
76	C₈	—	13	2	no	—
77	C₈	—	39	1	no	—
78	C₈	—	39	1	no	—
79	C₂×C₃	—	2	1	no	—
80	C₂×C₃	C₂²×Dic₁₃	1	1	yes	characteristic
81	D₃	—	2	2	no	—
82	C₂²	—	3	2	no	—
83	C₂²	D₃×Dic₁₃	1	1	yes	characteristic
84	C₄	—	3	2	no	—
85	C₄	D₃×Dic₁₃	1	1	yes	characteristic
86	C₄	C₂×D₃×D₁₃	1	1	yes	characteristic
87	C₃	#88202	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	6	2	no	—
90	C₂	C₂×D₃×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×D₃×D₁₃
commutator, socle	C₂×C₃×C₁₃	C₂²×C₄
Fitting	C₂×C₃×C₄×C₁₃	C₂²
prime radical	C₁₃×#235	C₂
cosocle	C₃×C₄×C₁₃	C₂³
hypercenter	C₂×C₄	D₃×D₁₃
hypocenter	C₃×C₁₃	#131
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊#1055	C₂	semidirect
3. C₃×#6099	C₂	semidirect
4. #10597593	C₂	semidirect
5. D₃×#1055	C₂	semidirect
17. #6099	D₃	semidirect
47. C₃×C₁₃	#131	semidirect
65. C₁₃	#907	semidirect
87. C₃	#88202	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#235	2
C₂×C₃⋊#1055	2
C₃×#6099	2
#10597593	2
D₃×#1055	2
#88202	3
#907	13