Subgroups — C₂×C₃×#6101 #11928813

Order 1248.

184 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#6101	C₂	1	4	yes	maximal, direct product factor
5	C₂×#6101	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂³×C₃×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₁₃	—	2	2	no	—
8	C₂×C₃×C₄×C₁₃	C₂²	1	2	yes	—
9	C₂×C₃×Dic₁₃	—	2	2	no	—
10	C₂×C₃×Dic₁₃	C₄	1	4	yes	complemented
11	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
12	C₂×C₃×Dic₁₃	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₃	C₂×C₃	1	1	yes	characteristic
14	C₂²×Dic₁₃	C₂×C₃	1	1	yes	characteristic
15	C₂²×Dic₁₃	C₂×C₃	1	1	yes	characteristic
16	#6101	C₂×C₃	1	4	yes	direct product factor
17	C₂²×C₃×C₁₃	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₁₃	D₄	1	2	yes	—
19	C₂²×C₃×C₁₃	Q₈	1	2	yes	—
20	C₂²×C₃×C₁₃	C₂³	1	1	yes	characteristic
21	C₃×C₄×C₁₃	—	2	2	no	—
22	C₃×Dic₁₃	—	2	2	no	—
23	C₃×Dic₁₃	C₂×C₄	1	4	yes	complemented
24	C₂³×C₁₃	C₂²×C₃	1	1	yes	characteristic
25	C₂×C₄×C₁₃	—	2	2	no	—
26	C₂×C₄×C₁₃	C₂²×C₃	1	2	yes	—
27	C₂×Dic₁₃	—	2	2	no	—
28	C₂×Dic₁₃	C₃×C₄	1	4	yes	complemented
29	C₂×Dic₁₃	C₂²×C₃	1	2	yes	—
30	C₂×Dic₁₃	C₂²×C₃	1	2	yes	—
31	C₂×C₃×C₄⋊C₄	—	13	1	no	maximal, Hall
32	C₂×C₃×C₁₃	C₄⋊C₄	1	4	yes	complemented
33	C₂×C₃×C₁₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₃×C₁₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃×C₁₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₃	C₂×C₃×C₄	1	2	yes	—
37	C₂²×C₁₃	C₃×D₄	1	2	yes	—
38	C₂²×C₁₃	C₃×Q₈	1	2	yes	—
39	C₂²×C₁₃	C₂³×C₃	1	1	yes	characteristic
40	C₄×C₁₃	—	2	2	no	—
41	Dic₁₃	—	2	2	no	—
42	Dic₁₃	C₂×C₃×C₄	1	4	yes	complemented
43	C₂²×C₃×C₄	—	13	1	no	—
44	C₂²×C₃×C₄	—	13	1	no	—
45	C₂²×C₃×C₄	D₁₃	1	1	yes	characteristic
46	C₃×C₄⋊C₄	—	13	4	no	—
47	C₃×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
48	C₂×C₄⋊C₄	—	13	1	no	Sylow
49	C₂×C₁₃	C₃×C₄⋊C₄	1	4	yes	complemented
50	C₂×C₁₃	C₂²×C₃×C₄	1	1	yes	characteristic
51	C₂×C₁₃	C₂×C₃×D₄	1	1	yes	characteristic
52	C₂×C₁₃	C₂×C₃×Q₈	1	1	yes	characteristic
53	C₂³×C₃	C₂×D₁₃	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	13	4	no	—
56	C₂×C₃×C₄	—	13	2	no	—
57	C₂×C₃×C₄	—	13	2	no	—
58	C₂×C₃×C₄	—	26	2	no	—
59	C₂×C₃×C₄	C₂×D₁₃	1	2	yes	—
60	C₂²×C₄	—	13	1	no	—
61	C₂²×C₄	—	13	1	no	—
62	C₂²×C₄	C₃×D₁₃	1	1	yes	characteristic
63	C₄⋊C₄	—	13	4	no	—
64	C₁₃	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₃	Dic₂₆	1	2	yes	—
66	C₂²×C₃	C₄×D₁₃	1	2	yes	—
67	C₂²×C₃	C₂²⋊D₁₃	1	2	yes	—
68	C₂²×C₃	C₂²×D₁₃	1	1	yes	characteristic
69	C₃×C₄	—	2	2	no	—
70	C₃×C₄	—	13	4	no	—
71	C₃×C₄	—	26	2	no	—
72	C₂³	C₂×C₃×D₁₃	1	1	yes	characteristic
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	13	4	no	—
75	C₂×C₄	—	13	2	no	—
76	C₂×C₄	—	13	2	no	—
77	C₂×C₄	—	26	2	no	—
78	C₂×C₄	C₂×C₃×D₁₃	1	2	yes	—
79	C₂×C₃	#6101	1	4	yes	direct product factor
80	C₂×C₃	C₂×Dic₂₆	1	1	yes	characteristic
81	C₂×C₃	C₂×C₄×D₁₃	1	1	yes	characteristic
82	C₂×C₃	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
83	C₂²	C₃×Dic₂₆	1	2	yes	—
84	C₂²	C₃×C₄×D₁₃	1	2	yes	—
85	C₂²	C₃×C₂²⋊D₁₃	1	2	yes	—
86	C₂²	C₂²×C₃×D₁₃	1	1	yes	characteristic
87	C₄	—	2	2	no	—
88	C₄	—	13	4	no	—
89	C₄	—	26	2	no	—
90	C₃	C₂×#6101	1	1	yes	characteristic, direct product factor, Sylow
91	C₂	C₃×#6101	1	4	yes	direct product factor
92	C₂	C₂×C₃×Dic₂₆	1	1	yes	characteristic
93	C₂	C₂×C₃×C₄×D₁₃	1	1	yes	characteristic
94	C₂	C₂×C₃×C₂²⋊D₁₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃×#6101	C₂	direct (N × H)
5. C₂×#6101	C₃	direct (N × H)
10. C₂×C₃×Dic₁₃	C₄	semidirect
16. #6101	C₂×C₃	direct (N × H)
23. C₃×Dic₁₃	C₂×C₄	semidirect
28. C₂×Dic₁₃	C₃×C₄	semidirect
32. C₂×C₃×C₁₃	C₄⋊C₄	semidirect
42. Dic₁₃	C₂×C₃×C₄	semidirect
47. C₃×C₁₃	C₂×C₄⋊C₄	semidirect
49. C₂×C₁₃	C₃×C₄⋊C₄	semidirect
64. C₁₃	C₂×C₃×C₄⋊C₄	semidirect
79. C₂×C₃	#6101	direct (N × H)
90. C₃	C₂×#6101	direct (N × H)
91. C₂	C₃×#6101	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×C₄×C₁₃	2
C₂²×C₃×Dic₁₃	2
C₃×#6101	2
C₂×#6101	3
C₂×C₃×C₄⋊C₄	13

Subgroups — C2×C3×#6101 #11928813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#6101 #11928813