Subgroups — #11929139

Order 1248.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10597707	C₂	1	4	yes	maximal, complemented
3	Q₈×D₁₃⋊C₃	C₂	1	4	yes	maximal, complemented
4	#6127⋊C₃	C₂	1	6	yes	maximal, complemented
5	#88254	C₃	1	1	yes	maximal, characteristic, complemented, Hall
6	Q₈×C₁₃⋊C₃	C₂²	1	4	yes	—
7	C₂×C₄×C₁₃⋊C₃	C₂²	1	3	yes	—
8	(C₂²⋊D₁₃)⋊C₃	C₂²	1	4	yes	—
9	C₄⋊(D₁₃⋊C₃)	C₂²	1	6	yes	—
10	C₄×D₁₃⋊C₃	C₂²	1	12	yes	—
11	(C₁₃⋊C₃)⋊Q₈	C₂²	1	6	yes	—
12	C₂×Q₈×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
13	#6131	C₂×C₃	1	4	yes	complemented
14	Q₈×D₁₃	C₂×C₃	1	4	yes	complemented
15	#6127	C₂×C₃	1	6	yes	complemented
16	C₂²×C₁₃⋊C₃	C₂³	1	1	yes	characteristic
17	C₂×D₁₃⋊C₃	C₂³	1	4	yes	—
18	C₄×C₁₃⋊C₃	C₂³	1	6	yes	—
19	Dic₁₃⋊C₃	C₂³	1	4	yes	—
20	Q₈×C₁₃	C₂²×C₃	1	4	yes	—
21	C₂×C₄×C₁₃	C₂²×C₃	1	3	yes	—
22	C₂²⋊D₁₃	C₂²×C₃	1	4	yes	—
23	D₅₂	C₂²×C₃	1	6	yes	—
24	C₄×D₁₃	C₂²×C₃	1	12	yes	—
25	Dic₂₆	C₂²×C₃	1	6	yes	—
26	C₃×2₋¹⁺⁴	—	13	1	no	maximal, Hall
27	C₂×C₁₃⋊C₃	—	2	1	no	—
28	C₂×C₁₃⋊C₃	C₂⁴	1	1	yes	characteristic
29	D₁₃⋊C₃	—	2	4	no	—
30	C₂²×C₁₃	C₂³×C₃	1	1	yes	characteristic
31	C₂×D₁₃	C₂³×C₃	1	4	yes	—
32	C₄×C₁₃	C₂³×C₃	1	6	yes	—
33	Dic₁₃	C₂³×C₃	1	4	yes	—
34	C₃×#41	—	13	6	no	—
35	C₃×#41	—	13	4	no	—
36	C₂×C₃×Q₈	—	13	4	no	—
37	C₂×C₃×Q₈	—	13	1	no	—
38	C₁₃⋊C₃	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
39	2₋¹⁺⁴	—	13	1	no	Sylow
40	C₂×C₁₃	—	2	1	no	—
41	C₂×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
42	D₁₃	—	2	4	no	—
43	C₃×Q₈	—	13	6	no	—
44	C₃×Q₈	—	13	4	no	—
45	C₃×D₄	—	13	6	no	—
46	C₃×D₄	—	13	4	no	—
47	C₂×C₃×C₄	—	13	12	no	—
48	C₂×C₃×C₄	—	13	3	no	—
49	#41	—	13	4	no	—
50	#41	—	13	6	no	—
51	C₂×Q₈	—	13	4	no	—
52	C₂×Q₈	D₁₃⋊C₃	1	1	yes	characteristic, complemented
53	C₁₃	C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₃	—	13	4	no	—
55	C₂²×C₃	—	13	1	no	—
56	C₃×C₄	—	13	4	no	—
57	C₃×C₄	—	13	6	no	—
58	Q₈	—	13	6	no	—
59	Q₈	C₂×D₁₃⋊C₃	1	4	yes	—
60	D₄	—	13	4	no	—
61	D₄	—	13	6	no	—
62	C₂×C₄	—	13	12	no	—
63	C₂×C₄	C₂×D₁₃⋊C₃	1	3	yes	—
64	C₂×C₃	—	13	1	no	—
65	C₂×C₃	—	26	1	no	—
66	C₂×C₃	—	26	4	no	—
67	C₂²	—	13	4	no	—
68	C₂²	C₂²×D₁₃⋊C₃	1	1	yes	characteristic
69	C₄	—	13	4	no	—
70	C₄	C₂²×D₁₃⋊C₃	1	6	yes	—
71	C₃	—	13	1	no	Sylow
72	C₂	—	2	1	no	—
73	C₂	—	26	4	no	—
74	C₂	C₂³×D₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₁₃⋊C₃
commutator, socle, cosocle	C₂×C₁₃	C₂⁴×C₃
Fitting	C₂×Q₈×C₁₃	C₂×C₃
hypercenter	C₂×Q₈	D₁₃⋊C₃
hypocenter	C₁₃	C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₁₃⋊C₃	C₂	semidirect
2. #10597707	C₂	semidirect
3. Q₈×D₁₃⋊C₃	C₂	semidirect
4. #6127⋊C₃	C₂	semidirect
5. #88254	C₃	semidirect
12. C₂×Q₈×C₁₃	C₂×C₃	semidirect
13. #6131	C₂×C₃	semidirect
14. Q₈×D₁₃	C₂×C₃	semidirect
15. #6127	C₂×C₃	semidirect
38. C₁₃⋊C₃	2₋¹⁺⁴	semidirect
52. C₂×Q₈	D₁₃⋊C₃	semidirect
53. C₁₃	C₃×2₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₃⋊C₃	2
#10597707	2
Q₈×D₁₃⋊C₃	2
#6127⋊C₃	2
#88254	3
C₃×2₋¹⁺⁴	13