Subgroups — #11929015

Order 1248.

260 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#64625	C₂	1	3	yes	maximal, complemented
3	#10597671	C₂	1	3	yes	maximal, complemented
4	#88207	—	3	1	no	maximal, Hall
5	C₂³×C₃×C₁₃	—	2	6	no	—
6	C₂³×C₃×C₁₃	C₂²	1	3	yes	—
7	C₂²×D₃₉	C₂²	1	1	yes	characteristic, complemented
8	#64625	—	2	6	no	—
9	C₂×Dic₃₉	C₂²	1	3	yes	complemented
10	C₂⁴×C₁₃	D₃	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₁₃	—	3	3	no	—
12	C₂²⋊Dic₁₃	—	3	3	no	—
13	C₂²×C₃×C₁₃	—	2	8	no	—
14	C₂²×C₃×C₁₃	—	2	6	no	—
15	C₂²×C₃×C₁₃	D₄	1	6	yes	complemented
16	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×D₃₉	—	2	3	no	—
18	Dic₃₉	—	2	3	no	—
19	C₂³×C₁₃	—	2	6	no	—
20	C₂³×C₁₃	C₂×D₃	1	3	yes	—
21	C₂²×D₁₃	—	3	1	no	—
22	C₂²⋊D₁₃	—	6	6	no	—
23	C₂×Dic₁₃	—	3	3	no	—
24	#953	—	13	1	no	maximal, Hall
25	C₂×C₃×C₁₃	—	2	6	no	—
26	C₂×C₃×C₁₃	C₂×D₄	1	3	yes	—
27	D₃₉	—	4	1	no	—
28	C₂²×C₁₃	—	2	8	no	—
29	C₂²×C₁₃	—	2	6	no	—
30	C₂²×C₁₃	#67	1	6	yes	complemented
31	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic
32	C₂×D₁₃	—	6	3	no	—
33	Dic₁₃	—	6	3	no	—
34	C₂⁴×C₃	D₁₃	1	1	yes	characteristic, complemented
35	C₂×#67	—	13	3	no	—
36	#217	—	13	3	no	—
37	C₃×C₁₃	#120	1	1	yes	characteristic, complemented, Hall
38	#120	—	39	1	no	Sylow
39	C₂×C₁₃	—	2	6	no	—
40	C₂×C₁₃	C₂×#67	1	3	yes	—
41	D₁₃	—	12	1	no	—
42	C₂³×C₃	—	2	6	no	—
43	C₂³×C₃	C₂×D₁₃	1	3	yes	—
44	C₂²×D₃	—	13	1	no	—
45	#67	—	26	6	no	—
46	C₂×Dic₃	—	13	3	no	—
47	C₂⁴	D₃₉	1	1	yes	characteristic, complemented
48	C₂×D₄	—	39	3	no	—
49	C₂²⋊C₄	—	39	3	no	—
50	C₁₃	#953	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₃	—	2	8	no	—
52	C₂²×C₃	—	2	6	no	—
53	C₂²×C₃	C₂²⋊D₁₃	1	6	yes	complemented
54	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
55	C₂×D₃	—	26	3	no	—
56	Dic₃	—	26	3	no	—
57	C₂³	—	2	6	no	—
58	C₂³	—	39	1	no	—
59	C₂³	C₂×D₃₉	1	3	yes	—
60	D₄	—	78	6	no	—
61	C₂×C₄	—	39	3	no	—
62	C₂×C₃	—	2	6	no	—
63	C₂×C₃	C₂×C₂²⋊D₁₃	1	3	yes	—
64	D₃	—	52	1	no	—
65	C₂²	—	2	8	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	78	3	no	—
68	C₂²	#64625	1	6	yes	complemented
69	C₂²	C₂²×D₃₉	1	1	yes	characteristic
70	C₄	—	78	3	no	—
71	C₃	#88207	1	1	yes	characteristic, complemented, Sylow
72	C₂	—	2	6	no	—
73	C₂	—	156	1	no	—
74	C₂	C₂×#64625	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₉
commutator, socle, cosocle	C₂²×C₃×C₁₃	C₂³
Fitting	C₂⁴×C₃×C₁₃	C₂
hypercenter	C₂⁴	D₃₉
hypocenter	C₃×C₁₃	#120
LCS	G ▷ C₂²×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂² ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₁₃	C₂	semidirect
2. C₂×#64625	C₂	semidirect
3. #10597671	C₂	semidirect
7. C₂²×D₃₉	C₂²	semidirect
9. C₂×Dic₃₉	C₂²	semidirect
10. C₂⁴×C₁₃	D₃	semidirect
15. C₂²×C₃×C₁₃	D₄	semidirect
30. C₂²×C₁₃	#67	semidirect
34. C₂⁴×C₃	D₁₃	semidirect
37. C₃×C₁₃	#120	semidirect
47. C₂⁴	D₃₉	semidirect
50. C₁₃	#953	semidirect
53. C₂²×C₃	C₂²⋊D₁₃	semidirect
68. C₂²	#64625	semidirect
71. C₃	#88207	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₁₃	2
C₂×#64625	2
#10597671	2
#88207	3
#953	13