Subgroups — C₂³×#6122 #13131821

Order 1664.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6122	C₂	1	28	yes	maximal, direct product factor
3	C₂³×#1057	C₂	1	2	yes	maximal, complemented
4	C₂⁵×C₁₃	C₄	1	1	yes	characteristic
5	C₂³×Dic₁₃	C₄	1	14	yes	—
6	C₂³×Dic₁₃	C₄	1	1	yes	characteristic
7	C₂³×Dic₁₃	C₂²	1	1	yes	characteristic
8	C₂³×Dic₁₃	C₂²	1	14	yes	—
9	C₂×#6122	C₂²	1	112	yes	direct product factor
10	C₂²×#1057	C₂²	1	28	yes	complemented
11	C₂⁴×C₁₃	—	2	8	no	—
12	C₂⁴×C₁₃	C₂×C₄	1	14	yes	—
13	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₃	C₂×C₄	1	84	yes	—
15	C₂²×Dic₁₃	C₂×C₄	1	7	yes	—
16	C₂²×Dic₁₃	C₂×C₄	1	14	yes	—
17	C₂²×Dic₁₃	C₂³	1	7	yes	—
18	C₂²×Dic₁₃	C₂³	1	28	yes	—
19	#6122	C₂³	1	64	yes	direct product factor
20	C₂×#1057	C₂³	1	56	yes	complemented
21	C₂³×M₁₆	—	13	1	no	maximal, Sylow
22	C₂³×C₁₃	—	2	56	no	—
23	C₂³×C₁₃	M₁₆	1	8	yes	complemented
24	C₂³×C₁₃	C₂²×C₄	1	28	yes	—
25	C₂³×C₁₃	C₂²×C₄	1	7	yes	—
26	C₂×Dic₁₃	C₂²×C₄	1	56	yes	—
27	C₂×Dic₁₃	C₂²×C₄	1	21	yes	—
28	C₂×Dic₁₃	C₂²×C₄	1	28	yes	—
29	C₂×Dic₁₃	C₂⁴	1	7	yes	—
30	C₂×Dic₁₃	C₂⁴	1	8	yes	—
31	#1057	C₂⁴	1	16	yes	complemented
32	C₂⁴×C₄	—	13	1	no	—
33	C₂²×M₁₆	—	13	28	no	—
34	C₂³×C₈	—	13	2	no	—
35	C₂²×C₁₃	—	2	56	no	—
36	C₂²×C₁₃	C₂×M₁₆	1	28	yes	complemented
37	C₂²×C₁₃	C₂³×C₄	1	8	yes	—
38	C₂²×C₁₃	C₂³×C₄	1	7	yes	—
39	Dic₁₃	C₂³×C₄	1	7	yes	—
40	Dic₁₃	C₂³×C₄	1	8	yes	—
41	Dic₁₃	C₂⁵	1	1	yes	characteristic
42	C₂⁵	#261	1	1	yes	characteristic
43	C₂³×C₄	—	13	14	no	—
44	C₂³×C₄	—	13	14	no	—
45	C₂³×C₄	—	13	1	no	—
46	C₂³×C₄	—	13	1	no	—
47	C₂×M₁₆	—	13	112	no	—
48	C₂²×C₈	—	13	28	no	—
49	C₂×C₁₃	—	2	8	no	—
50	C₂×C₁₃	C₂²×M₁₆	1	14	yes	complemented
51	C₂×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
52	C₂⁴	—	2	8	no	—
53	C₂⁴	C₂×#261	1	14	yes	—
54	C₂⁴	C₂×#261	1	1	yes	characteristic
55	C₂²×C₄	—	13	7	no	—
56	C₂²×C₄	—	13	28	no	—
57	C₂²×C₄	—	13	84	no	—
58	C₂²×C₄	—	13	14	no	—
59	C₂²×C₄	—	13	7	no	—
60	M₁₆	—	13	64	no	—
61	C₂×C₈	—	13	56	no	—
62	C₁₃	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂³	—	2	56	no	—
64	C₂³	#6122	1	8	yes	direct product factor
65	C₂³	C₂²×#261	1	28	yes	—
66	C₂³	C₂²×#261	1	7	yes	—
67	C₂×C₄	—	13	7	no	—
68	C₂×C₄	—	13	8	no	—
69	C₂×C₄	—	13	56	no	—
70	C₂×C₄	—	13	21	no	—
71	C₂×C₄	—	13	28	no	—
72	C₈	—	13	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#6122	1	28	yes	direct product factor
75	C₂²	C₂³×#261	1	8	yes	—
76	C₂²	C₂³×#261	1	7	yes	—
77	C₄	—	13	7	no	—
78	C₄	—	13	8	no	—
79	C₄	—	13	1	no	—
80	C₂	—	2	8	no	—
81	C₂	C₂²×#6122	1	14	yes	direct product factor
82	C₂	C₂⁴×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×#261
commutator	C₂×C₁₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#261
Fitting, prime radical	C₂⁵×C₁₃	C₄
socle	C₂⁴×C₁₃	C₂×C₄
cosocle	Dic₁₃	C₂⁵
hypercenter	C₂⁵	#261
hypocenter	C₁₃	C₂³×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁵

Complements

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

N	H ≅ G/N	type
2. C₂²×#6122	C₂	direct (N × H)
3. C₂³×#1057	C₂	semidirect
9. C₂×#6122	C₂²	direct (N × H)
10. C₂²×#1057	C₂²	semidirect
19. #6122	C₂³	direct (N × H)
20. C₂×#1057	C₂³	semidirect
23. C₂³×C₁₃	M₁₆	semidirect
31. #1057	C₂⁴	semidirect
36. C₂²×C₁₃	C₂×M₁₆	semidirect
50. C₂×C₁₃	C₂²×M₁₆	semidirect
62. C₁₃	C₂³×M₁₆	semidirect
64. C₂³	#6122	direct (N × H)
74. C₂²	C₂×#6122	direct (N × H)
81. C₂	C₂²×#6122	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₃	2
C₂²×#6122	2
C₂³×#1057	2
C₂³×M₁₆	13

Subgroups — C23×#6122 #13131821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#6122 #13131821