Subgroups — #11929162

Order 1248.

92 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41⋊(C₁₃⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	#10597713	C₂	1	2	yes	maximal, complemented
3	#88256	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	Q₈⋊(C₁₃⋊C₃)	C₂²	1	1	yes	characteristic, complemented
5	(C₁₃⋊C₃)⋊Q₈	—	4	1	no	maximal
6	C₁₃×#41	C₂×C₃	1	1	yes	characteristic
7	#6131	C₂×C₃	1	2	yes	—
8	D₄×D₁₃	—	3	2	no	—
9	#6127	—	3	1	no	—
10	C₂×D₅₂	—	3	1	no	—
11	C₄×C₁₃⋊C₃	—	4	1	no	—
12	Dic₁₃⋊C₃	—	4	2	no	—
13	C₂²×D₁₃	—	3	2	no	—
14	Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic
15	D₄×C₁₃	—	3	1	no	—
16	C₂×C₄×C₁₃	—	3	1	no	—
17	C₂²⋊D₁₃	—	3	2	no	—
18	D₅₂	—	3	2	no	—
19	D₅₂	—	3	1	no	—
20	C₄×D₁₃	—	3	2	no	—
21	Dic₂₆	A₄	1	1	yes	characteristic
22	#994	—	13	1	no	maximal, Hall
23	C₂×C₁₃⋊C₃	—	4	1	no	—
24	C₂²×C₁₃	—	3	1	no	—
25	C₂×D₁₃	—	3	2	no	—
26	C₂×D₁₃	—	12	2	no	—
27	C₄×C₁₃	—	3	1	no	—
28	C₄×C₁₃	C₂×A₄	1	1	yes	characteristic
29	Dic₁₃	C₂×A₄	1	2	yes	—
30	#41⋊C₃	—	13	2	no	—
31	#41⋊C₃	—	13	1	no	—
32	C₁₃⋊C₃	—	4	1	no	Hall
33	H(5,𝔽₂)	—	13	1	no	Sylow
34	C₂×C₁₃	—	6	1	no	—
35	C₂×C₁₃	C₂²×A₄	1	1	yes	characteristic
36	D₁₃	—	6	2	no	—
37	C₃×Q₈	—	52	1	no	—
38	SL(2,𝔽₃)	—	13	1	no	—
39	#41	—	13	2	no	—
40	#41	—	39	1	no	—
41	#41	D₁₃⋊C₃	1	1	yes	characteristic
42	C₂×D₄	—	39	1	no	—
43	C₂×D₄	—	39	2	no	—
44	C₁₃	#994	1	1	yes	characteristic, complemented, Sylow
45	C₃×C₄	—	52	2	no	—
46	C₃×C₄	—	52	1	no	—
47	C₂³	—	39	2	no	—
48	Q₈	—	13	1	no	—
49	Q₈	C₂×D₁₃⋊C₃	1	1	yes	characteristic
50	D₄	—	3	1	no	—
51	D₄	—	39	2	no	—
52	D₄	—	39	2	no	—
53	D₄	—	39	1	no	—
54	C₂×C₄	—	3	1	no	—
55	C₂×C₄	—	39	2	no	—
56	C₂×C₃	—	52	1	no	—
57	C₂²	—	3	1	no	—
58	C₂²	—	39	2	no	—
59	C₂²	—	156	2	no	—
60	C₄	—	3	1	no	—
61	C₄	—	13	2	no	—
62	C₄	D₁₃⋊A₄	1	1	yes	characteristic
63	C₃	—	52	1	no	Sylow
64	C₂	—	6	1	no	—
65	C₂	—	78	2	no	—
66	C₂	C₂×D₁₃⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃⋊A₄
commutator, cosocle, hypocenter	Q₈×C₁₃	C₂²×C₃
Fitting	C₁₃×#41	C₂×C₃
socle	C₂×C₁₃	C₂²×A₄
hypercenter	C₄	D₁₃⋊A₄
LCS	G ▷ Q₈×C₁₃
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. #41⋊(C₁₃⋊C₃)	C₂	semidirect
2. #10597713	C₂	semidirect
3. #88256	C₃	semidirect
4. Q₈⋊(C₁₃⋊C₃)	C₂²	semidirect
44. C₁₃	#994	semidirect

Maximal subgroups

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

subgroup	index
#41⋊(C₁₃⋊C₃)	2
#10597713	2
#88256	3
(C₁₃⋊C₃)⋊Q₈	4
#994	13