Subgroups — C₂²×C₁₃×S₄ #11929328

Order 1248.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×A₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×S₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×D₄×C₁₃	—	3	1	no	maximal, Hall
4	C₂²×D₃×C₁₃	—	4	1	no	maximal
5	C₂×A₄×C₁₃	C₂²	1	3	yes	complemented
6	C₁₃×S₄	C₂²	1	4	yes	direct product factor
7	C₂⁴×C₁₃	—	3	1	no	—
8	C₂⁴×C₁₃	D₃	1	1	yes	characteristic, complemented
9	C₂×D₄×C₁₃	—	3	6	no	—
10	C₂×D₄×C₁₃	—	3	6	no	—
11	C₂²×C₄×C₁₃	—	3	1	no	—
12	C₂²×C₃×C₁₃	—	4	1	no	—
13	C₂×D₃×C₁₃	—	4	6	no	—
14	A₄×C₁₃	C₂³	1	1	yes	characteristic, complemented
15	C₂³×C₁₃	—	3	1	no	—
16	C₂³×C₁₃	—	3	3	no	—
17	C₂³×C₁₃	—	3	6	no	—
18	C₂³×C₁₃	—	6	3	no	—
19	C₂³×C₁₃	—	6	1	no	—
20	C₂³×C₁₃	C₂×D₃	1	3	yes	complemented
21	D₄×C₁₃	—	3	12	no	—
22	D₄×C₁₃	—	3	4	no	—
23	C₂×C₄×C₁₃	—	3	6	no	—
24	C₂²×S₄	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₃×C₁₃	—	4	3	no	—
26	D₃×C₁₃	—	4	4	no	—
27	C₂²×C₁₃	—	3	3	no	—
28	C₂²×C₁₃	—	3	3	no	—
29	C₂²×C₁₃	—	3	3	no	—
30	C₂²×C₁₃	—	3	4	no	—
31	C₂²×C₁₃	—	6	1	no	—
32	C₂²×C₁₃	—	6	6	no	—
33	C₂²×C₁₃	—	6	6	no	—
34	C₂²×C₁₃	S₄	1	1	yes	characteristic, direct product factor
35	C₂²×C₁₃	C₂²×D₃	1	1	yes	characteristic, complemented
36	C₄×C₁₃	—	3	4	no	—
37	C₂²×A₄	C₂×C₁₃	1	1	yes	characteristic, complemented
38	C₂×S₄	C₂×C₁₃	1	6	yes	direct product factor
39	C₃×C₁₃	—	4	1	no	Hall
40	C₂²×D₄	—	3	1	no	Sylow
41	C₂×C₁₃	—	3	1	no	—
42	C₂×C₁₃	—	3	3	no	—
43	C₂×C₁₃	—	6	4	no	—
44	C₂×C₁₃	C₂×S₄	1	3	yes	direct product factor
45	C₂²×D₃	—	4	1	no	—
46	C₂×A₄	C₂²×C₁₃	1	3	yes	complemented
47	S₄	C₂²×C₁₃	1	4	yes	direct product factor
48	C₂⁴	—	3	1	no	—
49	C₂⁴	D₃×C₁₃	1	1	yes	characteristic, complemented
50	C₂×D₄	—	3	6	no	—
51	C₂×D₄	—	3	6	no	—
52	C₂²×C₄	—	3	1	no	—
53	C₁₃	C₂²×S₄	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²×C₃	—	4	1	no	—
55	C₂×D₃	—	4	6	no	—
56	A₄	C₂³×C₁₃	1	1	yes	characteristic, complemented
57	C₂³	—	3	1	no	—
58	C₂³	—	3	3	no	—
59	C₂³	—	3	6	no	—
60	C₂³	—	6	3	no	—
61	C₂³	—	6	1	no	—
62	C₂³	C₂×D₃×C₁₃	1	3	yes	complemented
63	D₄	—	3	12	no	—
64	D₄	—	3	4	no	—
65	C₂×C₄	—	3	6	no	—
66	C₂×C₃	—	4	3	no	—
67	D₃	—	4	4	no	—
68	C₂²	—	3	3	no	—
69	C₂²	—	3	3	no	—
70	C₂²	—	3	3	no	—
71	C₂²	—	3	4	no	—
72	C₂²	—	6	1	no	—
73	C₂²	—	6	6	no	—
74	C₂²	—	6	6	no	—
75	C₂²	C₁₃×S₄	1	1	yes	characteristic, direct product factor
76	C₂²	C₂²×D₃×C₁₃	1	1	yes	characteristic, complemented
77	C₄	—	3	4	no	—
78	C₃	—	4	1	no	Sylow
79	C₂	—	3	1	no	—
80	C₂	—	3	3	no	—
81	C₂	—	6	4	no	—
82	C₂	C₂×C₁₃×S₄	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₁₃	S₄
commutator, cosocle, hypocenter	A₄	C₂³×C₁₃
Frattini	C₁	G
Fitting, socle	C₂⁴×C₁₃	D₃
LCS	G ▷ A₄
UCS	C₁ ◁ C₂²×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₂²×A₄×C₁₃	C₂	semidirect
2. C₂×C₁₃×S₄	C₂	direct (N × H)
5. C₂×A₄×C₁₃	C₂²	semidirect
6. C₁₃×S₄	C₂²	direct (N × H)
8. C₂⁴×C₁₃	D₃	semidirect
14. A₄×C₁₃	C₂³	semidirect
20. C₂³×C₁₃	C₂×D₃	semidirect
24. C₂²×S₄	C₁₃	direct (N × H)
34. C₂²×C₁₃	S₄	direct (N × H)
35. C₂²×C₁₃	C₂²×D₃	semidirect
37. C₂²×A₄	C₂×C₁₃	semidirect
38. C₂×S₄	C₂×C₁₃	direct (N × H)
44. C₂×C₁₃	C₂×S₄	direct (N × H)
46. C₂×A₄	C₂²×C₁₃	semidirect
47. S₄	C₂²×C₁₃	direct (N × H)
49. C₂⁴	D₃×C₁₃	semidirect
53. C₁₃	C₂²×S₄	direct (N × H)
56. A₄	C₂³×C₁₃	semidirect
62. C₂³	C₂×D₃×C₁₃	semidirect
75. C₂²	C₁₃×S₄	direct (N × H)
76. C₂²	C₂²×D₃×C₁₃	semidirect
82. C₂	C₂×C₁₃×S₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×A₄×C₁₃	2
C₂×C₁₃×S₄	2
C₂²×D₄×C₁₃	3
C₂²×D₃×C₁₃	4
C₂²×S₄	13

Subgroups — C22×C13×S4 #11929328

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₁₃×S₄ #11929328