Subgroups — C₂×C₃×#6122 #11929178

Order 1248.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	C₃×#6122	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃×#1057	C₂	1	2	yes	maximal, complemented
4	C₂×#6122	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₁₃	C₄	1	1	yes	characteristic
6	C₂×C₃×Dic₁₃	C₄	1	2	yes	—
7	C₂×C₃×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×C₃×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
10	C₃×#1057	C₂²	1	4	yes	complemented
11	C₂²×Dic₁₃	C₂×C₃	1	1	yes	characteristic
12	#6122	C₂×C₃	1	4	yes	direct product factor
13	C₂×#1057	C₂×C₃	1	2	yes	complemented
14	C₂²×C₃×C₁₃	—	2	2	no	—
15	C₂²×C₃×C₁₃	C₂×C₄	1	2	yes	—
16	C₂²×C₃×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₃×Dic₁₃	C₂×C₄	1	1	yes	characteristic
18	C₃×Dic₁₃	C₂×C₄	1	2	yes	—
19	C₃×Dic₁₃	C₂³	1	1	yes	characteristic
20	C₂³×C₁₃	C₃×C₄	1	1	yes	characteristic
21	C₂×Dic₁₃	C₃×C₄	1	2	yes	—
22	C₂×Dic₁₃	C₃×C₄	1	1	yes	characteristic
23	C₂×Dic₁₃	C₂²×C₃	1	1	yes	characteristic
24	C₂×Dic₁₃	C₂²×C₃	1	2	yes	—
25	#1057	C₂²×C₃	1	4	yes	complemented
26	C₂×C₃×M₁₆	—	13	1	no	maximal, Hall
27	C₂×C₃×C₁₃	—	2	2	no	—
28	C₂×C₃×C₁₃	M₁₆	1	2	yes	complemented
29	C₂×C₃×C₁₃	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₁₃	—	2	2	no	—
31	C₂²×C₁₃	C₂×C₃×C₄	1	2	yes	—
32	C₂²×C₁₃	C₂×C₃×C₄	1	1	yes	characteristic
33	Dic₁₃	C₂×C₃×C₄	1	1	yes	characteristic
34	Dic₁₃	C₂×C₃×C₄	1	2	yes	—
35	Dic₁₃	C₂³×C₃	1	1	yes	characteristic
36	C₂²×C₃×C₄	—	13	1	no	—
37	C₃×M₁₆	—	13	4	no	—
38	C₂×C₃×C₈	—	13	2	no	—
39	C₃×C₁₃	C₂×M₁₆	1	1	yes	characteristic, complemented, Hall
40	C₂×M₁₆	—	13	1	no	Sylow
41	C₂×C₁₃	—	2	2	no	—
42	C₂×C₁₃	C₃×M₁₆	1	2	yes	complemented
43	C₂×C₁₃	C₂²×C₃×C₄	1	1	yes	characteristic
44	C₂³×C₃	#261	1	1	yes	characteristic
45	C₂×C₃×C₄	—	13	2	no	—
46	C₂×C₃×C₄	—	13	2	no	—
47	C₂×C₃×C₄	—	13	1	no	—
48	C₂×C₃×C₄	—	13	1	no	—
49	C₃×C₈	—	13	4	no	—
50	C₂²×C₄	—	13	1	no	—
51	M₁₆	—	13	4	no	—
52	C₂×C₈	—	13	2	no	—
53	C₁₃	C₂×C₃×M₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	C₂×#261	1	2	yes	—
56	C₂²×C₃	C₂×#261	1	1	yes	characteristic
57	C₃×C₄	—	13	1	no	—
58	C₃×C₄	—	13	1	no	—
59	C₃×C₄	—	13	2	no	—
60	C₂³	C₃×#261	1	1	yes	characteristic
61	C₂×C₄	—	13	1	no	—
62	C₂×C₄	—	13	2	no	—
63	C₂×C₄	—	13	2	no	—
64	C₂×C₄	—	13	1	no	—
65	C₈	—	13	4	no	—
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	#6122	1	2	yes	direct product factor
68	C₂×C₃	C₂²×#261	1	1	yes	characteristic
69	C₂²	—	2	2	no	—
70	C₂²	C₂×C₃×#261	1	2	yes	—
71	C₂²	C₂×C₃×#261	1	1	yes	characteristic
72	C₄	—	13	1	no	—
73	C₄	—	13	2	no	—
74	C₄	—	13	1	no	—
75	C₃	C₂×#6122	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	2	2	no	—
77	C₂	C₃×#6122	1	2	yes	direct product factor
78	C₂	C₂²×C₃×#261	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×#261
commutator	C₂×C₁₃	C₂²×C₃×C₄
Frattini	C₂	C₂²×C₃×#261
Fitting, prime radical	C₂³×C₃×C₁₃	C₄
socle	C₂²×C₃×C₁₃	C₂×C₄
cosocle	Dic₁₃	C₂³×C₃
hypercenter	C₂³×C₃	#261
hypocenter	C₁₃	C₂×C₃×M₁₆
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₃×#6122	C₂	direct (N × H)
3. C₂×C₃×#1057	C₂	semidirect
4. C₂×#6122	C₃	direct (N × H)
10. C₃×#1057	C₂²	semidirect
12. #6122	C₂×C₃	direct (N × H)
13. C₂×#1057	C₂×C₃	semidirect
25. #1057	C₂²×C₃	semidirect
28. C₂×C₃×C₁₃	M₁₆	semidirect
39. C₃×C₁₃	C₂×M₁₆	semidirect
42. C₂×C₁₃	C₃×M₁₆	semidirect
53. C₁₃	C₂×C₃×M₁₆	semidirect
67. C₂×C₃	#6122	direct (N × H)
75. C₃	C₂×#6122	direct (N × H)
77. C₂	C₃×#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
C₂²×C₃×Dic₁₃	2
C₃×#6122	2
C₂×C₃×#1057	2
C₂×#6122	3
C₂×C₃×M₁₆	13

Subgroups — C2×C3×#6122 #11929178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6122 #11929178