Subgroups — C₂×C₃×#6131 #11929278

Order 1248.

328 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6131	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃×D₅₂	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×C₄×D₁₃	C₂	1	3	yes	maximal, complemented
5	C₂×#6131	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×D₁₃	C₂²	1	3	yes	—
7	C₃×Q₈×C₁₃	C₂²	1	4	yes	complemented
8	C₂×C₃×C₄×C₁₃	C₂²	1	3	yes	—
9	C₂×C₃×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₃×D₅₂	C₂²	1	12	yes	complemented
11	C₃×C₄×D₁₃	C₂²	1	12	yes	complemented
12	C₂×Q₈×C₁₃	C₂×C₃	1	1	yes	characteristic, complemented
13	#6131	C₂×C₃	1	8	yes	direct product factor
14	C₂×D₅₂	C₂×C₃	1	3	yes	complemented
15	C₂×C₄×D₁₃	C₂×C₃	1	3	yes	complemented
16	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
17	C₂×C₃×D₁₃	—	2	6	no	—
18	C₂×C₃×D₁₃	C₂³	1	6	yes	—
19	C₃×C₄×C₁₃	C₂³	1	6	yes	—
20	C₃×Dic₁₃	C₂³	1	2	yes	—
21	C₂²×D₁₃	C₂²×C₃	1	3	yes	—
22	Q₈×C₁₃	C₂²×C₃	1	4	yes	complemented
23	C₂×C₄×C₁₃	C₂²×C₃	1	3	yes	—
24	C₂×Dic₁₃	C₂²×C₃	1	1	yes	characteristic
25	D₅₂	C₂²×C₃	1	12	yes	complemented
26	C₄×D₁₃	C₂²×C₃	1	12	yes	complemented
27	C₂×C₃×#41	—	13	1	no	maximal, Hall
28	C₂×C₃×C₁₃	#41	1	2	yes	complemented
29	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
30	C₃×D₁₃	—	2	6	no	—
31	C₂²×C₁₃	C₂³×C₃	1	1	yes	characteristic
32	C₂×D₁₃	—	2	6	no	—
33	C₂×D₁₃	C₂³×C₃	1	6	yes	—
34	C₄×C₁₃	C₂³×C₃	1	6	yes	—
35	Dic₁₃	C₂³×C₃	1	2	yes	—
36	C₃×#41	—	13	8	no	—
37	C₂×C₃×Q₈	D₁₃	1	1	yes	characteristic, complemented
38	C₂×C₃×D₄	—	13	3	no	—
39	C₂²×C₃×C₄	—	13	3	no	—
40	C₃×C₁₃	C₂×#41	1	1	yes	characteristic, complemented, Hall
41	C₂×#41	—	13	1	no	Sylow
42	C₂×C₁₃	C₃×#41	1	2	yes	complemented
43	C₂×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
44	D₁₃	—	2	6	no	—
45	C₂³×C₃	—	13	3	no	—
46	C₃×Q₈	C₂×D₁₃	1	4	yes	complemented
47	C₃×D₄	—	13	12	no	—
48	C₂×C₃×C₄	—	13	1	no	—
49	C₂×C₃×C₄	—	13	12	no	—
50	C₂×C₃×C₄	C₂×D₁₃	1	3	yes	—
51	#41	—	13	8	no	—
52	C₂×Q₈	C₃×D₁₃	1	1	yes	characteristic, complemented
53	C₂×D₄	—	13	3	no	—
54	C₂²×C₄	—	13	3	no	—
55	C₁₃	C₂×C₃×#41	1	1	yes	characteristic, complemented, Sylow
56	C₂²×C₃	—	13	6	no	—
57	C₂²×C₃	—	26	6	no	—
58	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
59	C₃×C₄	—	13	2	no	—
60	C₃×C₄	C₂²×D₁₃	1	6	yes	—
61	C₂³	—	13	3	no	—
62	Q₈	C₂×C₃×D₁₃	1	4	yes	complemented
63	D₄	—	13	12	no	—
64	C₂×C₄	—	13	1	no	—
65	C₂×C₄	—	13	12	no	—
66	C₂×C₄	C₂×C₃×D₁₃	1	3	yes	—
67	C₂×C₃	—	26	6	no	—
68	C₂×C₃	#6131	1	2	yes	direct product factor
69	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
70	C₂²	—	13	6	no	—
71	C₂²	—	26	6	no	—
72	C₂²	C₂²×C₃×D₁₃	1	1	yes	characteristic
73	C₄	—	13	2	no	—
74	C₄	C₂²×C₃×D₁₃	1	6	yes	—
75	C₃	C₂×#6131	1	1	yes	characteristic, direct product factor, Sylow
76	C₂	—	26	6	no	—
77	C₂	C₃×#6131	1	2	yes	direct product factor
78	C₂	C₂³×C₃×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁴×C₃
Frattini	C₂	C₂³×C₃×D₁₃
Fitting	C₂×C₃×Q₈×C₁₃	C₂
socle	C₂²×C₃×C₁₃	C₂³
hypercenter	C₂×C₃×Q₈	D₁₃
hypocenter	C₁₃	C₂×C₃×#41
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×Q₈×C₁₃	C₂	semidirect
2. C₃×#6131	C₂	direct (N × H)
3. C₂×C₃×D₅₂	C₂	semidirect
4. C₂×C₃×C₄×D₁₃	C₂	semidirect
5. C₂×#6131	C₃	direct (N × H)
7. C₃×Q₈×C₁₃	C₂²	semidirect
10. C₃×D₅₂	C₂²	semidirect
11. C₃×C₄×D₁₃	C₂²	semidirect
12. C₂×Q₈×C₁₃	C₂×C₃	semidirect
13. #6131	C₂×C₃	direct (N × H)
14. C₂×D₅₂	C₂×C₃	semidirect
15. C₂×C₄×D₁₃	C₂×C₃	semidirect
22. Q₈×C₁₃	C₂²×C₃	semidirect
25. D₅₂	C₂²×C₃	semidirect
26. C₄×D₁₃	C₂²×C₃	semidirect
28. C₂×C₃×C₁₃	#41	semidirect
37. C₂×C₃×Q₈	D₁₃	semidirect
40. C₃×C₁₃	C₂×#41	semidirect
42. C₂×C₁₃	C₃×#41	semidirect
46. C₃×Q₈	C₂×D₁₃	semidirect
52. C₂×Q₈	C₃×D₁₃	semidirect
55. C₁₃	C₂×C₃×#41	semidirect
62. Q₈	C₂×C₃×D₁₃	semidirect
68. C₂×C₃	#6131	direct (N × H)
75. C₃	C₂×#6131	direct (N × H)
77. C₂	C₃×#6131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×Q₈×C₁₃	2
C₃×#6131	2
C₂×C₃×D₅₂	2
C₂×C₃×C₄×D₁₃	2
C₂×#6131	3
C₂×C₃×#41	13

Subgroups — C2×C3×#6131 #11929278

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6131 #11929278