Subgroups — #11929068

Order 1248.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	SL(2,𝔽₃)×D₁₃	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	#88172	—	3	1	no	maximal, Hall
5	C₁₃×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic
6	#64603	—	4	1	no	maximal
7	#6131	—	3	1	no	—
8	Q₈×D₁₃	D₃	1	1	yes	characteristic, complemented
9	C₁₃×Q₁₆	—	3	1	no	—
10	#6107	—	3	1	no	—
11	#6106	—	3	1	no	—
12	#6095	—	3	1	no	—
13	#6094	—	3	1	no	—
14	C₂×D₃₉	—	4	1	no	—
15	C₂×C₃×D₁₃	—	4	1	no	—
16	Dic₃×C₁₃	—	4	1	no	—
17	Q₈×C₁₃	—	3	1	no	—
18	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic
19	D₅₂	—	3	1	no	—
20	D₅₂	—	6	1	no	—
21	C₄×D₁₃	—	3	1	no	—
22	C₄×D₁₃	—	6	1	no	—
23	Dic₂₆	—	3	1	no	—
24	C₈×C₁₃	—	3	1	no	—
25	#1055	—	3	1	no	—
26	#983	—	13	1	no	maximal, Hall
27	C₂×C₃×C₁₃	—	4	1	no	—
28	D₃₉	—	8	1	no	—
29	C₃×D₁₃	—	8	1	no	—
30	C₂×D₁₃	—	6	1	no	—
31	C₂×D₁₃	S₄	1	1	yes	characteristic
32	C₄×C₁₃	—	3	1	no	—
33	C₄×C₁₃	—	6	1	no	—
34	Dic₁₃	—	3	1	no	—
35	C₂×SL(2,𝔽₃)	—	13	1	no	—
36	GL(2,𝔽₃)	—	13	1	no	—
37	#226	D₁₃	1	1	yes	characteristic, complemented
38	C₃×C₁₃	—	4	1	no	Hall
39	#137	—	39	1	no	Sylow
40	C₂×C₁₃	C₂×S₄	1	1	yes	characteristic
41	D₁₃	—	2	1	no	—
42	D₁₃	—	12	1	no	—
43	#67	—	52	1	no	—
44	SL(2,𝔽₃)	C₂×D₁₃	1	1	yes	characteristic
45	#41	—	39	1	no	—
46	C₂×Q₈	—	13	1	no	—
47	Q₁₆	—	3	1	no	—
48	Q₁₆	—	39	1	no	—
49	SD₁₆	—	39	1	no	—
50	SD₁₆	—	39	1	no	—
51	M₁₆	—	39	1	no	—
52	C₁₃	#983	1	1	yes	characteristic, complemented, Sylow
53	C₂²×C₃	—	52	1	no	—
54	C₂×D₃	—	52	1	no	—
55	Dic₃	—	4	1	no	—
56	Q₈	—	3	1	no	—
57	Q₈	—	39	1	no	—
58	Q₈	D₃×D₁₃	1	1	yes	characteristic
59	D₄	—	39	1	no	—
60	D₄	—	78	1	no	—
61	C₂×C₄	—	39	1	no	—
62	C₂×C₄	—	78	1	no	—
63	C₈	—	3	1	no	—
64	C₈	—	39	1	no	—
65	C₂×C₃	—	4	1	no	—
66	C₂×C₃	—	104	1	no	—
67	D₃	—	104	1	no	—
68	C₂²	—	13	1	no	—
69	C₂²	—	78	1	no	—
70	C₄	—	3	1	no	—
71	C₄	—	6	1	no	—
72	C₄	—	39	1	no	—
73	C₃	—	4	1	no	Sylow
74	C₂	—	26	1	no	—
75	C₂	—	156	1	no	—
76	C₂	S₄×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄×D₁₃
commutator, cosocle, hypocenter	C₁₃×SL(2,𝔽₃)	C₂²
Fitting	Q₈×C₁₃	C₂×D₃
socle	C₂×C₁₃	C₂×S₄
LCS	G ▷ C₁₃×SL(2,𝔽₃)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. SL(2,𝔽₃)×D₁₃	C₂	semidirect
2. C₁₃⋊GL(2,𝔽₃)	C₂	semidirect
3. C₁₃×#226	C₂	semidirect
8. Q₈×D₁₃	D₃	semidirect
37. #226	D₁₃	semidirect
52. C₁₃	#983	semidirect

Maximal subgroups

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

subgroup	index
SL(2,𝔽₃)×D₁₃	2
C₁₃⋊GL(2,𝔽₃)	2
C₁₃×#226	2
#88172	3
#64603	4
#983	13