Subgroups — #11929126

Order 1248.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#41⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃⋊GL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃⋊#226	C₂	1	1	yes	maximal, characteristic, complemented
4	#88204	—	3	1	no	maximal, Hall
5	C₄×D₃₉	—	4	1	no	maximal
6	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₁₃×#41	D₃	1	1	yes	characteristic, complemented
8	#6127	—	3	1	no	—
9	#6107	—	3	1	no	—
10	#6106	—	3	1	no	—
11	#6105	—	3	1	no	—
12	#6104	—	3	1	no	—
13	C₂×#1055	—	3	1	no	—
14	C₂×D₃₉	—	4	1	no	—
15	C₃×C₄×C₁₃	—	4	1	no	—
16	Dic₃₉	—	4	1	no	—
17	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic
18	D₄×C₁₃	—	3	1	no	—
19	C₂×C₄×C₁₃	—	3	1	no	—
20	C₂²⋊D₁₃	—	6	1	no	—
21	D₅₂	—	3	1	no	—
22	C₄×D₁₃	—	6	1	no	—
23	Dic₂₆	—	3	1	no	—
24	#1055	—	3	1	no	—
25	#1055	—	3	1	no	—
26	#985	—	13	1	no	maximal, Hall
27	C₂×C₃×C₁₃	—	4	1	no	—
28	D₃₉	—	4	2	no	—
29	C₂²×C₁₃	—	3	1	no	—
30	C₂×D₁₃	—	6	1	no	—
31	C₄×C₁₃	—	3	1	no	—
32	C₄×C₁₃	S₄	1	1	yes	characteristic
33	Dic₁₃	—	6	1	no	—
34	#41⋊C₃	D₁₃	1	1	yes	characteristic, complemented
35	GL(2,𝔽₃)	—	13	1	no	—
36	#226	—	13	1	no	—
37	C₃×C₁₃	—	4	1	no	Hall
38	#135	—	39	1	no	Sylow
39	C₂×C₁₃	—	6	1	no	—
40	C₂×C₁₃	C₂×S₄	1	1	yes	characteristic
41	D₁₃	—	12	1	no	—
42	C₄×D₃	—	52	1	no	—
43	SL(2,𝔽₃)	C₂×D₁₃	1	1	yes	characteristic
44	#41	—	39	1	no	—
45	#41	D₃₉	1	1	yes	characteristic, complemented
46	Q₁₆	—	39	1	no	—
47	SD₁₆	—	39	1	no	—
48	SD₁₆	—	39	1	no	—
49	D₈	—	39	1	no	—
50	C₂×C₈	—	39	1	no	—
51	C₁₃	#985	1	1	yes	characteristic, complemented, Sylow
52	C₂×D₃	—	52	1	no	—
53	C₃×C₄	—	4	1	no	—
54	Dic₃	—	52	1	no	—
55	Q₈	—	39	1	no	—
56	Q₈	C₂×D₃₉	1	1	yes	characteristic
57	D₄	—	3	1	no	—
58	D₄	—	39	1	no	—
59	D₄	—	78	1	no	—
60	C₂×C₄	—	3	1	no	—
61	C₂×C₄	—	78	1	no	—
62	C₈	—	39	1	no	—
63	C₈	—	39	1	no	—
64	C₂×C₃	—	4	1	no	—
65	D₃	—	52	2	no	—
66	C₂²	—	3	1	no	—
67	C₂²	—	78	1	no	—
68	C₄	—	3	1	no	—
69	C₄	—	78	1	no	—
70	C₄	A₄⋊D₁₃	1	1	yes	characteristic
71	C₃	—	4	1	no	Sylow
72	C₂	—	6	1	no	—
73	C₂	—	156	1	no	—
74	C₂	C₂×A₄⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	A₄⋊D₁₃
commutator, cosocle, hypocenter	C₁₃×SL(2,𝔽₃)	C₂²
Frattini	C₂	C₂×A₄⋊D₁₃
Fitting	C₁₃×#41	D₃
socle	C₂×C₁₃	C₂×S₄
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₁₃×#41⋊C₃	C₂	semidirect
2. C₁₃⋊GL(2,𝔽₃)	C₂	semidirect
3. C₁₃⋊#226	C₂	semidirect
7. C₁₃×#41	D₃	semidirect
34. #41⋊C₃	D₁₃	semidirect
45. #41	D₃₉	semidirect
51. C₁₃	#985	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#41⋊C₃	2
C₁₃⋊GL(2,𝔽₃)	2
C₁₃⋊#226	2
#88204	3
C₄×D₃₉	4
#985	13