Subgroups — D₃×#126 #5703

Order 192.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#126	C₂	1	1	yes	maximal, characteristic, complemented
2	#898	C₂	1	3	yes	maximal, complemented
3	D₃×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
4	D₃×C₂²⋊C₄	C₂	1	3	yes	maximal
5	#879	C₂	1	3	yes	maximal, complemented
6	#876	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄²×D₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#126	—	3	1	no	maximal, Sylow
9	C₂³×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃	C₂²	1	3	yes	—
11	C₂×C₄×D₃	C₂²	1	3	yes	—
12	C₃×C₄⋊C₄	C₂²	1	3	yes	complemented
13	C₃×C₂²⋊C₄	C₂²	1	3	yes	—
14	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
15	#217	C₂²	1	3	yes	—
16	(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
17	(C₂×D₃)⋊C₄	C₂²	1	3	yes	—
18	C₄⋊Dic₃	C₂²	1	3	yes	complemented
19	Dic₃⋊C₄	C₂²	1	3	yes	complemented
20	Dic₃⋊C₄	C₂²	1	3	yes	complemented
21	C₄×Dic₃	C₂²	1	3	yes	complemented
22	#126	—	3	3	no	—
23	#126	—	3	3	no	—
24	#126	—	3	1	no	—
25	#126	D₃	1	1	yes	characteristic, direct product factor
26	C₂×C₄⋊C₄	—	3	3	no	—
27	C₂×C₂²⋊C₄	—	3	3	no	—
28	C₂×C₄²	—	3	1	no	—
29	C₂³×C₃	C₂³	1	1	yes	characteristic
30	C₂²×D₃	—	2	6	no	—
31	C₂²×D₃	C₂³	1	1	yes	characteristic
32	C₂²×D₃	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂³	1	3	yes	—
34	C₂×C₃×C₄	C₂³	1	3	yes	—
35	C₂×Dic₃	C₂³	1	3	yes	—
36	C₂×Dic₃	C₂³	1	3	yes	—
37	C₄×D₃	—	2	6	no	—
38	C₄×D₃	—	2	6	no	—
39	C₂⁴	—	3	1	no	—
40	C₂²×C₄	—	3	3	no	—
41	C₂²×C₄	—	3	3	no	—
42	C₄⋊C₄	—	3	3	no	—
43	C₄⋊C₄	—	3	3	no	—
44	C₄⋊C₄	—	3	3	no	—
45	C₄⋊C₄	C₂×D₃	1	3	yes	complemented
46	C₂²⋊C₄	—	3	3	no	—
47	C₂²⋊C₄	—	3	3	no	—
48	C₂²⋊C₄	—	3	3	no	—
49	C₂²⋊C₄	C₂×D₃	1	3	yes	—
50	C₄²	—	3	3	no	—
51	C₄²	C₂×D₃	1	1	yes	characteristic, complemented
52	C₂²×C₃	—	2	3	no	—
53	C₂²×C₃	C₂⁴	1	1	yes	characteristic
54	C₂×D₃	—	2	3	no	—
55	C₂×D₃	—	4	4	no	—
56	C₂×D₃	#41	1	6	yes	—
57	C₃×C₄	—	2	3	no	—
58	C₃×C₄	—	2	3	no	—
59	Dic₃	—	2	3	no	—
60	Dic₃	—	2	3	no	—
61	C₂³	—	3	1	no	—
62	C₂³	—	3	1	no	—
63	C₂³	—	6	6	no	—
64	C₂³	C₂²×D₃	1	1	yes	characteristic
65	C₂×C₄	—	3	3	no	—
66	C₂×C₄	—	3	3	no	—
67	C₂×C₄	—	6	6	no	—
68	C₂×C₄	—	6	6	no	—
69	C₂×C₄	C₂²×D₃	1	3	yes	—
70	C₂×C₄	C₂²×D₃	1	3	yes	—
71	C₂×C₃	—	4	1	no	—
72	C₂×C₃	C₂×#41	1	3	yes	—
73	D₃	—	4	1	no	—
74	D₃	#126	1	4	yes	direct product factor
75	C₂²	—	2	3	no	—
76	C₂²	—	3	6	no	—
77	C₂²	—	6	3	no	—
78	C₂²	—	12	4	no	—
79	C₂²	C₂³×D₃	1	1	yes	characteristic
80	C₄	—	2	3	no	—
81	C₄	—	2	3	no	—
82	C₄	—	6	3	no	—
83	C₄	—	6	3	no	—
84	C₃	C₂×#126	1	1	yes	characteristic, complemented, Sylow
85	C₂	—	3	4	no	—
86	C₂	—	4	1	no	—
87	C₂	—	12	1	no	—
88	C₂	D₃×#41	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#126	C₂
prime radical	C₂³×D₃	C₂²
hypercenter	#126	D₃
hypocenter	C₃	C₂×#126
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #126

Complements

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

N	H ≅ G/N	type
1. C₃×#126	C₂	semidirect
2. #898	C₂	semidirect
3. D₃×C₄⋊C₄	C₂	semidirect
5. #879	C₂	semidirect
6. #876	C₂	semidirect
7. C₄²×D₃	C₂	semidirect
12. C₃×C₄⋊C₄	C₂²	semidirect
14. C₃×C₄²	C₂²	semidirect
18. C₄⋊Dic₃	C₂²	semidirect
19. Dic₃⋊C₄	C₂²	semidirect
20. Dic₃⋊C₄	C₂²	semidirect
21. C₄×Dic₃	C₂²	semidirect
25. #126	D₃	direct (N × H)
45. C₄⋊C₄	C₂×D₃	semidirect
51. C₄²	C₂×D₃	semidirect
74. D₃	#126	direct (N × H)
84. C₃	C₂×#126	semidirect

Maximal subgroups

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

subgroup	index
C₃×#126	2
#898	2
D₃×C₄⋊C₄	2
D₃×C₂²⋊C₄	2
#879	2
#876	2
C₄²×D₃	2
C₂×#126	3

Subgroups — D3×#126 #5703

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×#126 #5703