Subgroups — #11929309

Order 1248.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#10597794	C₂	1	4	yes	maximal, complemented
3	Q₈×D₃₉	C₂	1	4	yes	maximal, complemented
4	#10597790	C₂	1	6	yes	maximal, complemented
5	#88254	—	3	1	no	maximal, Hall
6	C₃×Q₈×C₁₃	C₂²	1	4	yes	—
7	C₂×C₃×C₄×C₁₃	C₂²	1	3	yes	—
8	#64625	C₂²	1	4	yes	—
9	D₁₅₆	C₂²	1	6	yes	—
10	C₄×D₃₉	C₂²	1	12	yes	—
11	Dic₇₈	C₂²	1	6	yes	—
12	C₂×Q₈×C₁₃	D₃	1	1	yes	characteristic, complemented
13	#6131	—	3	4	no	—
14	Q₈×D₁₃	—	3	4	no	—
15	#6127	—	3	6	no	—
16	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×D₃₉	C₂³	1	4	yes	—
18	C₃×C₄×C₁₃	C₂³	1	6	yes	—
19	Dic₃₉	C₂³	1	4	yes	—
20	Q₈×C₁₃	C₂×D₃	1	4	yes	—
21	C₂×C₄×C₁₃	C₂×D₃	1	3	yes	—
22	C₂²⋊D₁₃	—	3	4	no	—
23	D₅₂	—	3	6	no	—
24	C₄×D₁₃	—	3	12	no	—
25	Dic₂₆	—	3	6	no	—
26	#1007	—	13	1	no	maximal, Hall
27	C₂×C₃×C₁₃	—	2	1	no	—
28	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
29	D₃₉	—	2	4	no	—
30	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
31	C₂×D₁₃	—	3	4	no	—
32	C₄×C₁₃	C₂²×D₃	1	6	yes	—
33	Dic₁₃	—	3	4	no	—
34	C₂×C₃×Q₈	D₁₃	1	1	yes	characteristic, complemented
35	#239	—	13	4	no	—
36	D₃×Q₈	—	13	4	no	—
37	#235	—	13	6	no	—
38	C₃×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
39	2₋¹⁺⁴	—	39	1	no	Sylow
40	C₂×C₁₃	—	2	1	no	—
41	C₂×C₁₃	C₂³×D₃	1	1	yes	characteristic
42	D₁₃	—	6	4	no	—
43	C₃×Q₈	C₂×D₁₃	1	4	yes	—
44	C₂×C₃×C₄	C₂×D₁₃	1	3	yes	—
45	#67	—	13	4	no	—
46	D₁₂	—	13	6	no	—
47	C₄×D₃	—	13	12	no	—
48	Dic₆	—	13	6	no	—
49	#41	—	39	4	no	—
50	#41	—	39	6	no	—
51	C₂×Q₈	—	39	4	no	—
52	C₂×Q₈	D₃₉	1	1	yes	characteristic, complemented
53	C₁₃	#1007	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
55	C₂×D₃	—	13	4	no	—
56	C₃×C₄	C₂²×D₁₃	1	6	yes	—
57	Dic₃	—	13	4	no	—
58	Q₈	—	39	6	no	—
59	Q₈	C₂×D₃₉	1	4	yes	—
60	D₄	—	39	4	no	—
61	D₄	—	39	6	no	—
62	C₂×C₄	—	39	12	no	—
63	C₂×C₄	C₂×D₃₉	1	3	yes	—
64	C₂×C₃	—	2	1	no	—
65	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
66	D₃	—	26	4	no	—
67	C₂²	—	39	4	no	—
68	C₂²	C₂²×D₃₉	1	1	yes	characteristic
69	C₄	—	39	4	no	—
70	C₄	C₂²×D₃₉	1	6	yes	—
71	C₃	#88254	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	1	no	—
73	C₂	—	78	4	no	—
74	C₂	C₂³×D₃₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₉
commutator, socle, cosocle	C₂×C₃×C₁₃	C₂⁴
Fitting	C₂×C₃×Q₈×C₁₃	C₂
hypercenter	C₂×Q₈	D₃₉
hypocenter	C₃×C₁₃	2₋¹⁺⁴
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×C₁₃	C₂	semidirect
2. #10597794	C₂	semidirect
3. Q₈×D₃₉	C₂	semidirect
4. #10597790	C₂	semidirect
12. C₂×Q₈×C₁₃	D₃	semidirect
34. C₂×C₃×Q₈	D₁₃	semidirect
38. C₃×C₁₃	2₋¹⁺⁴	semidirect
52. C₂×Q₈	D₃₉	semidirect
53. C₁₃	#1007	semidirect
71. C₃	#88254	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈×C₁₃	2
#10597794	2
Q₈×D₃₉	2
#10597790	2
#88254	3
#1007	13