Subgroups — C₂×C₃×C₄×#261 #11929170

Order 1248.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×#261	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×#261	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×#261	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂²×C₃×D₁₃	C₂²	1	1	yes	characteristic
6	C₂×C₃×#261	C₄	1	8	yes	direct product factor
7	C₂×C₃×#261	C₂²	1	4	yes	—
8	C₂×C₃×C₄×C₁₃	C₄	1	1	yes	characteristic, complemented
9	C₂×C₃×Dic₁₃	C₄	1	1	yes	characteristic, complemented
10	C₃×C₄×D₁₃	C₄	1	2	yes	complemented
11	C₃×C₄×D₁₃	C₂²	1	2	yes	—
12	C₂²×#261	C₂×C₃	1	2	yes	—
13	C₂×C₄×D₁₃	C₂×C₃	1	1	yes	characteristic
14	C₄×#261	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×D₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×D₁₃	C₂×C₄	1	4	yes	—
18	C₂×C₃×D₁₃	C₂³	1	1	yes	characteristic
19	C₃×#261	C₂×C₄	1	8	yes	direct product factor
20	C₃×C₄×C₁₃	C₂×C₄	1	2	yes	complemented
21	C₃×Dic₁₃	C₂×C₄	1	2	yes	complemented
22	C₂²×D₁₃	C₂²×C₃	1	1	yes	characteristic
23	C₂×#261	C₃×C₄	1	8	yes	direct product factor
24	C₂×#261	C₂²×C₃	1	4	yes	—
25	C₂×C₄×C₁₃	C₃×C₄	1	1	yes	characteristic, complemented
26	C₂×Dic₁₃	C₃×C₄	1	1	yes	characteristic, complemented
27	C₄×D₁₃	C₃×C₄	1	2	yes	complemented
28	C₄×D₁₃	C₂²×C₃	1	2	yes	—
29	C₂×C₃×C₄²	—	13	1	no	maximal, Hall
30	C₂×C₃×C₁₃	C₄²	1	2	yes	complemented
31	C₂×C₃×C₁₃	C₂²×C₄	1	1	yes	characteristic
32	C₃×D₁₃	C₄²	1	2	yes	complemented
33	C₃×D₁₃	C₂²×C₄	1	2	yes	—
34	C₂²×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
35	C₂×D₁₃	C₂×C₃×C₄	1	1	yes	characteristic
36	C₂×D₁₃	C₂×C₃×C₄	1	4	yes	—
37	C₂×D₁₃	C₂³×C₃	1	1	yes	characteristic
38	#261	C₂×C₃×C₄	1	8	yes	direct product factor
39	C₄×C₁₃	C₂×C₃×C₄	1	2	yes	complemented
40	Dic₁₃	C₂×C₃×C₄	1	2	yes	complemented
41	C₂²×C₃×C₄	—	13	2	no	—
42	C₂²×C₃×C₄	—	13	1	no	—
43	C₃×C₄²	—	13	4	no	—
44	C₃×C₁₃	C₂×C₄²	1	1	yes	characteristic, complemented, Hall
45	C₂×C₄²	—	13	1	no	Sylow
46	C₂×C₁₃	C₃×C₄²	1	2	yes	complemented
47	C₂×C₁₃	C₂²×C₃×C₄	1	1	yes	characteristic
48	D₁₃	C₃×C₄²	1	2	yes	complemented
49	D₁₃	C₂²×C₃×C₄	1	2	yes	—
50	C₂³×C₃	—	13	1	no	—
51	C₂×C₃×C₄	—	13	8	no	—
52	C₂×C₃×C₄	—	13	4	no	—
53	C₂×C₃×C₄	—	13	1	no	—
54	C₂×C₃×C₄	—	13	2	no	—
55	C₂×C₃×C₄	—	13	2	no	—
56	C₂×C₃×C₄	#261	1	1	yes	characteristic, direct product factor
57	C₂²×C₄	—	13	2	no	—
58	C₂²×C₄	—	13	1	no	—
59	C₄²	—	13	4	no	—
60	C₁₃	C₂×C₃×C₄²	1	1	yes	characteristic, complemented, Sylow
61	C₂²×C₃	—	13	4	no	—
62	C₂²×C₃	—	13	1	no	—
63	C₂²×C₃	—	13	1	no	—
64	C₂²×C₃	C₂×#261	1	1	yes	characteristic
65	C₃×C₄	—	13	8	no	—
66	C₃×C₄	—	13	2	no	—
67	C₃×C₄	C₂×#261	1	2	yes	direct product factor
68	C₂³	—	13	1	no	—
69	C₂×C₄	—	13	4	no	—
70	C₂×C₄	—	13	8	no	—
71	C₂×C₄	—	13	1	no	—
72	C₂×C₄	—	13	2	no	—
73	C₂×C₄	—	13	2	no	—
74	C₂×C₄	C₃×#261	1	1	yes	characteristic, direct product factor
75	C₂×C₃	—	13	2	no	—
76	C₂×C₃	—	13	2	no	—
77	C₂×C₃	C₄×#261	1	2	yes	direct product factor
78	C₂×C₃	C₂²×#261	1	1	yes	characteristic
79	C₂²	—	13	4	no	—
80	C₂²	—	13	1	no	—
81	C₂²	—	13	1	no	—
82	C₂²	C₂×C₃×#261	1	1	yes	characteristic
83	C₄	—	13	8	no	—
84	C₄	—	13	2	no	—
85	C₄	C₂×C₃×#261	1	2	yes	direct product factor
86	C₃	C₂×C₄×#261	1	1	yes	characteristic, direct product factor, Sylow
87	C₂	—	13	2	no	—
88	C₂	—	13	2	no	—
89	C₂	C₃×C₄×#261	1	2	yes	direct product factor
90	C₂	C₂²×C₃×#261	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₄	#261
commutator, hypocenter	C₁₃	C₂×C₃×C₄²
Frattini	C₂	C₂²×C₃×#261
Fitting	C₂×C₃×C₄×C₁₃	C₄
socle	C₂²×C₃×C₁₃	C₂×C₄
prime radical	C₂²×C₃×D₁₃	C₂²
cosocle	C₂×D₁₃	C₂³×C₃
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×#261	C₂	direct (N × H)
4. C₂×C₄×#261	C₃	direct (N × H)
6. C₂×C₃×#261	C₄	direct (N × H)
8. C₂×C₃×C₄×C₁₃	C₄	semidirect
9. C₂×C₃×Dic₁₃	C₄	semidirect
10. C₃×C₄×D₁₃	C₄	semidirect
14. C₄×#261	C₂×C₃	direct (N × H)
19. C₃×#261	C₂×C₄	direct (N × H)
20. C₃×C₄×C₁₃	C₂×C₄	semidirect
21. C₃×Dic₁₃	C₂×C₄	semidirect
23. C₂×#261	C₃×C₄	direct (N × H)
25. C₂×C₄×C₁₃	C₃×C₄	semidirect
26. C₂×Dic₁₃	C₃×C₄	semidirect
27. C₄×D₁₃	C₃×C₄	semidirect
30. C₂×C₃×C₁₃	C₄²	semidirect
32. C₃×D₁₃	C₄²	semidirect
38. #261	C₂×C₃×C₄	direct (N × H)
39. C₄×C₁₃	C₂×C₃×C₄	semidirect
40. Dic₁₃	C₂×C₃×C₄	semidirect
44. C₃×C₁₃	C₂×C₄²	semidirect
46. C₂×C₁₃	C₃×C₄²	semidirect
48. D₁₃	C₃×C₄²	semidirect
56. C₂×C₃×C₄	#261	direct (N × H)
60. C₁₃	C₂×C₃×C₄²	semidirect
67. C₃×C₄	C₂×#261	direct (N × H)
74. C₂×C₄	C₃×#261	direct (N × H)
77. C₂×C₃	C₄×#261	direct (N × H)
85. C₄	C₂×C₃×#261	direct (N × H)
86. C₃	C₂×C₄×#261	direct (N × H)
89. C₂	C₃×C₄×#261	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×C₃×#261	2
C₂×C₃×C₄×D₁₃	2
C₃×C₄×#261	2
C₂×C₄×#261	3
C₂×C₃×C₄²	13

Subgroups — C2×C3×C4×#261 #11929170

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×#261 #11929170