Subgroups — C₂×C₃³×#131 #13180022

Order 1728.

3724 conjugacy classes of subgroups.

The subgroup lattice has 132 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×#41	C₂	1	1	yes	maximal, characteristic
2	C₃³×#131	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃³×M₁₆	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃³×C₈	C₂	1	3	yes	maximal, complemented
5	C₂×C₃²×#131	C₃	1	13	yes	maximal, direct product factor
6	C₃³×#41	C₄	1	4	yes	—
7	C₃³×#41	C₂²	1	4	yes	—
8	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
9	C₂×C₃³×D₄	C₄	1	3	yes	—
10	C₂²×C₃³×C₄	C₂²	1	3	yes	—
11	C₃³×M₁₆	C₂²	1	12	yes	complemented
12	C₂×C₃³×C₈	C₂²	1	3	yes	—
13	C₂×C₃³×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃³×C₈	C₂²	1	12	yes	complemented
15	C₂×C₃²×#41	C₂×C₃	1	13	yes	—
16	C₃²×#131	C₂×C₃	1	104	yes	direct product factor
17	C₂×C₃²×M₁₆	C₂×C₃	1	39	yes	complemented
18	C₂²×C₃²×C₈	C₂×C₃	1	39	yes	complemented
19	C₂³×C₃³	C₂×C₄	1	3	yes	—
20	C₃³×Q₈	C₂×C₄	1	4	yes	—
21	C₃³×D₄	C₂×C₄	1	12	yes	—
22	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
23	C₂×C₃³×C₄	C₂×C₄	1	6	yes	—
24	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
25	C₂×C₃³×C₄	C₂³	1	6	yes	—
26	C₃³×C₈	C₂³	1	6	yes	—
27	C₃³×C₈	C₂³	1	2	yes	—
28	C₂×C₃×#131	C₃²	1	13	yes	direct product factor
29	C₃²×#41	C₃×C₄	1	52	yes	—
30	C₃²×#41	C₂²×C₃	1	52	yes	—
31	C₂×C₃²×Q₈	C₃×C₄	1	13	yes	—
32	C₂×C₃²×D₄	C₃×C₄	1	39	yes	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	39	yes	—
34	C₃²×M₁₆	C₂²×C₃	1	156	yes	complemented
35	C₂×C₃²×C₈	C₂²×C₃	1	39	yes	—
36	C₂×C₃²×C₈	C₂²×C₃	1	13	yes	—
37	C₂×C₃²×C₈	C₂²×C₃	1	156	yes	complemented
38	C₂²×C₃³	—	2	6	no	—
39	C₂²×C₃³	C₂²×C₄	1	6	yes	—
40	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
41	C₃³×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₃³×C₄	C₂²×C₄	1	6	yes	—
43	C₃³×C₄	C₂⁴	1	1	yes	characteristic
44	C₂×C₃×#41	C₂×C₃²	1	13	yes	—
45	C₃×#131	C₂×C₃²	1	104	yes	direct product factor
46	C₂×C₃×M₁₆	C₂×C₃²	1	39	yes	complemented
47	C₂²×C₃×C₈	C₂×C₃²	1	39	yes	complemented
48	C₂³×C₃²	C₂×C₃×C₄	1	39	yes	—
49	C₃²×Q₈	C₂×C₃×C₄	1	52	yes	—
50	C₃²×D₄	C₂×C₃×C₄	1	156	yes	—
51	C₂×C₃²×C₄	C₂×C₃×C₄	1	39	yes	—
52	C₂×C₃²×C₄	C₂×C₃×C₄	1	78	yes	—
53	C₂×C₃²×C₄	C₂³×C₃	1	13	yes	—
54	C₂×C₃²×C₄	C₂³×C₃	1	78	yes	—
55	C₃²×C₈	C₂³×C₃	1	26	yes	—
56	C₃²×C₈	C₂³×C₃	1	78	yes	—
57	C₂×#131	C₃³	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×C₃³	—	2	6	no	—
59	C₂×C₃³	#131	1	2	yes	direct product factor
60	C₂×C₃³	C₂³×C₄	1	1	yes	characteristic
61	C₃×#41	C₃²×C₄	1	52	yes	—
62	C₃×#41	C₂²×C₃²	1	52	yes	—
63	C₂×C₃×Q₈	C₃²×C₄	1	13	yes	—
64	C₂×C₃×D₄	C₃²×C₄	1	39	yes	—
65	C₂²×C₃×C₄	C₂²×C₃²	1	39	yes	—
66	C₃×M₁₆	C₂²×C₃²	1	156	yes	complemented
67	C₂×C₃×C₈	C₂²×C₃²	1	13	yes	—
68	C₂×C₃×C₈	C₂²×C₃²	1	39	yes	—
69	C₂×C₃×C₈	C₂²×C₃²	1	156	yes	complemented
70	C₂²×C₃²	—	2	78	no	—
71	C₂²×C₃²	C₂²×C₃×C₄	1	78	yes	—
72	C₂²×C₃²	C₂²×C₃×C₄	1	13	yes	—
73	C₃²×C₄	C₂²×C₃×C₄	1	13	yes	—
74	C₃²×C₄	C₂²×C₃×C₄	1	78	yes	—
75	C₃²×C₄	C₂⁴×C₃	1	13	yes	—
76	C₂×#41	C₂×C₃³	1	1	yes	characteristic
77	#131	C₂×C₃³	1	8	yes	direct product factor
78	C₂×M₁₆	C₂×C₃³	1	3	yes	complemented
79	C₂²×C₈	C₂×C₃³	1	3	yes	complemented
80	C₃³	C₂×#131	1	1	yes	characteristic, direct product factor, Sylow
81	C₂³×C₃	C₂×C₃²×C₄	1	39	yes	—
82	C₃×Q₈	C₂×C₃²×C₄	1	52	yes	—
83	C₃×D₄	C₂×C₃²×C₄	1	156	yes	—
84	C₂×C₃×C₄	C₂×C₃²×C₄	1	39	yes	—
85	C₂×C₃×C₄	C₂×C₃²×C₄	1	78	yes	—
86	C₂×C₃×C₄	C₂³×C₃²	1	13	yes	—
87	C₂×C₃×C₄	C₂³×C₃²	1	78	yes	—
88	C₃×C₈	C₂³×C₃²	1	26	yes	—
89	C₃×C₈	C₂³×C₃²	1	78	yes	—
90	C₂×C₃²	—	2	78	no	—
91	C₂×C₃²	C₃×#131	1	26	yes	direct product factor
92	C₂×C₃²	C₂³×C₃×C₄	1	13	yes	—
93	#41	C₃³×C₄	1	4	yes	—
94	#41	C₂²×C₃³	1	4	yes	—
95	C₂×Q₈	C₃³×C₄	1	1	yes	characteristic
96	C₂×D₄	C₃³×C₄	1	3	yes	—
97	C₂²×C₄	C₂²×C₃³	1	3	yes	—
98	M₁₆	C₂²×C₃³	1	12	yes	complemented
99	C₂×C₈	C₂²×C₃³	1	1	yes	characteristic
100	C₂×C₈	C₂²×C₃³	1	3	yes	—
101	C₂×C₈	C₂²×C₃³	1	12	yes	complemented
102	C₂²×C₃	—	2	78	no	—
103	C₂²×C₃	C₂²×C₃²×C₄	1	78	yes	—
104	C₂²×C₃	C₂²×C₃²×C₄	1	13	yes	—
105	C₃×C₄	C₂²×C₃²×C₄	1	13	yes	—
106	C₃×C₄	C₂²×C₃²×C₄	1	78	yes	—
107	C₃×C₄	C₂⁴×C₃²	1	13	yes	—
108	C₃²	C₂×C₃×#131	1	13	yes	direct product factor
109	C₂³	C₂×C₃³×C₄	1	3	yes	—
110	Q₈	C₂×C₃³×C₄	1	4	yes	—
111	D₄	C₂×C₃³×C₄	1	12	yes	—
112	C₂×C₄	C₂×C₃³×C₄	1	3	yes	—
113	C₂×C₄	C₂×C₃³×C₄	1	6	yes	—
114	C₂×C₄	C₂³×C₃³	1	1	yes	characteristic
115	C₂×C₄	C₂³×C₃³	1	6	yes	—
116	C₈	C₂³×C₃³	1	2	yes	—
117	C₈	C₂³×C₃³	1	6	yes	—
118	C₂×C₃	—	2	78	no	—
119	C₂×C₃	C₃²×#131	1	26	yes	direct product factor
120	C₂×C₃	C₂³×C₃²×C₄	1	13	yes	—
121	C₂²	—	2	6	no	—
122	C₂²	C₂²×C₃³×C₄	1	6	yes	—
123	C₂²	C₂²×C₃³×C₄	1	1	yes	characteristic
124	C₄	C₂²×C₃³×C₄	1	1	yes	characteristic
125	C₄	C₂²×C₃³×C₄	1	6	yes	—
126	C₄	C₂⁴×C₃³	1	1	yes	characteristic
127	C₃	C₂×C₃²×#131	1	13	yes	direct product factor
128	C₂	—	2	6	no	—
129	C₂	C₃³×#131	1	2	yes	direct product factor
130	C₂	C₂³×C₃³×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃³×C₈	C₂²
commutator	C₂	C₂³×C₃³×C₄
Frattini, cosocle	C₄	C₂⁴×C₃³
socle	C₂²×C₃³	C₂²×C₄
prime radical	C₂×C₃³×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃³×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃³×#131	C₂	direct (N × H)
3. C₂×C₃³×M₁₆	C₂	semidirect
4. C₂²×C₃³×C₈	C₂	semidirect
5. C₂×C₃²×#131	C₃	direct (N × H)
11. C₃³×M₁₆	C₂²	semidirect
14. C₂×C₃³×C₈	C₂²	semidirect
16. C₃²×#131	C₂×C₃	direct (N × H)
17. C₂×C₃²×M₁₆	C₂×C₃	semidirect
18. C₂²×C₃²×C₈	C₂×C₃	semidirect
28. C₂×C₃×#131	C₃²	direct (N × H)
34. C₃²×M₁₆	C₂²×C₃	semidirect
37. C₂×C₃²×C₈	C₂²×C₃	semidirect
45. C₃×#131	C₂×C₃²	direct (N × H)
46. C₂×C₃×M₁₆	C₂×C₃²	semidirect
47. C₂²×C₃×C₈	C₂×C₃²	semidirect
57. C₂×#131	C₃³	direct (N × H)
59. C₂×C₃³	#131	direct (N × H)
66. C₃×M₁₆	C₂²×C₃²	semidirect
69. C₂×C₃×C₈	C₂²×C₃²	semidirect
77. #131	C₂×C₃³	direct (N × H)
78. C₂×M₁₆	C₂×C₃³	semidirect
79. C₂²×C₈	C₂×C₃³	semidirect
80. C₃³	C₂×#131	direct (N × H)
91. C₂×C₃²	C₃×#131	direct (N × H)
98. M₁₆	C₂²×C₃³	semidirect
101. C₂×C₈	C₂²×C₃³	semidirect
108. C₃²	C₂×C₃×#131	direct (N × H)
119. C₂×C₃	C₃²×#131	direct (N × H)
127. C₃	C₂×C₃²×#131	direct (N × H)
129. C₂	C₃³×#131	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×C₃³×#41	2
C₃³×#131	2
C₂×C₃³×M₁₆	2
C₂²×C₃³×C₈	2
C₂×C₃²×#131	3

Subgroups — C2×C33×#131 #13180022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃³×#131 #13180022