Subgroups — #13144235

Order 1728.

890 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊Dic₉	C₂	1	2	yes	maximal
3	#11720572	C₂	1	4	yes	maximal
4	#10592631	—	3	1	no	maximal
5	#10588612	—	3	3	no	maximal
6	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
7	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₃⋊Dic₉	C₂²	1	4	yes	—
9	C₃×C₄²×C₉	C₄	1	4	yes	complemented
10	C₄×C₃⋊Dic₉	C₄	1	8	yes	complemented
11	C₂×C₃²×C₄²	D₃	1	1	yes	characteristic
12	C₂×C₄×C₃⋊Dic₃	—	3	2	no	—
13	#63658	—	3	4	no	—
14	C₂×C₄²×C₉	D₃	1	3	yes	—
15	C₂×C₄×Dic₉	—	3	6	no	—
16	C₉⋊#95	—	3	12	no	—
17	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
18	C₂×C₃×C₄×C₉	—	2	4	no	—
19	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
20	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
22	C₂×C₃⋊Dic₉	—	2	8	no	—
23	C₂×C₃⋊Dic₉	C₂×C₄	1	8	yes	—
24	#4932	—	9	3	no	—
25	#4932	—	9	1	no	—
26	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
27	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₃⋊Dic₃	—	3	4	no	—
29	C₃²×C₄²	Dic₃	1	4	yes	—
30	C₄×C₃⋊Dic₃	—	3	8	no	—
31	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
32	C₂²×C₄×C₉	C₂×D₃	1	3	yes	—
33	C₂²×Dic₉	—	3	12	no	—
34	C₄²×C₉	Dic₃	1	12	yes	complemented
35	C₄×Dic₉	—	3	24	no	—
36	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
37	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₃×C₉	#41	1	4	yes	—
39	C₃×C₄×C₉	—	2	4	no	—
40	C₃×C₄×C₉	C₄²	1	4	yes	—
41	C₃⋊Dic₉	—	2	8	no	—
42	C₂×C₃×C₄²	D₉	1	3	yes	—
43	C₂×C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
44	C₂×C₄×Dic₃	—	9	6	no	—
45	C₂×C₄×Dic₃	—	9	2	no	—
46	C₃⋊#95	—	9	12	no	—
47	C₃⋊#95	—	9	4	no	—
48	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
49	C₂×C₃²×C₄	—	2	4	no	—
50	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
51	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
52	C₂×C₃²×C₄	C₂×Dic₃	1	2	yes	—
53	C₂×C₃⋊Dic₃	—	3	8	no	—
54	C₂×C₃⋊Dic₃	—	6	8	no	—
55	C₂³×C₉	C₂²×D₃	1	3	yes	—
56	C₂×C₄×C₉	—	2	12	no	—
57	C₂×C₄×C₉	C₄×D₃	1	12	yes	—
58	C₂×C₄×C₉	C₂×Dic₃	1	12	yes	—
59	C₂×C₄×C₉	C₂×Dic₃	1	6	yes	—
60	C₂×Dic₉	—	3	24	no	—
61	C₂×Dic₉	—	6	24	no	—
62	#376	—	27	1	no	Sylow
63	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
64	C₂×C₃×C₉	#117	1	4	yes	—
65	C₂×C₃×C₉	#117	1	2	yes	—
66	C₂²×C₃×C₄	C₂×D₉	1	6	yes	—
67	C₂²×C₃×C₄	C₂×D₉	1	3	yes	—
68	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
69	C₂²×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
70	C₂²×Dic₃	—	9	12	no	—
71	C₂²×Dic₃	—	9	4	no	—
72	C₃×C₄²	Dic₉	1	12	yes	complemented
73	C₃×C₄²	C₃⋊Dic₃	1	4	yes	—
74	C₄×Dic₃	—	9	24	no	—
75	C₄×Dic₃	—	9	8	no	—
76	C₂²×C₃²	C₂×C₄×D₃	1	2	yes	—
77	C₂²×C₃²	#235	1	4	yes	—
78	C₂²×C₃²	C₂²×Dic₃	1	1	yes	characteristic
79	C₃²×C₄	—	2	4	no	—
80	C₃²×C₄	C₄×Dic₃	1	4	yes	—
81	C₃⋊Dic₃	—	6	8	no	—
82	C₂²×C₉	C₂×C₄×D₃	1	6	yes	—
83	C₂²×C₉	#235	1	12	yes	—
84	C₂²×C₉	C₂²×Dic₃	1	3	yes	—
85	C₄×C₉	—	2	12	no	—
86	C₄×C₉	C₄×Dic₃	1	12	yes	—
87	Dic₉	—	6	24	no	—
88	C₂×C₄²	—	27	2	no	—
89	C₂×C₄²	C₃⋊D₉	1	1	yes	characteristic
90	#95	—	27	4	no	—
91	C₃×C₉	#376	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₃	C₂²×D₉	1	3	yes	—
93	C₂³×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
94	C₂×C₃×C₄	—	2	12	no	—
95	C₂×C₃×C₄	—	2	4	no	—
96	C₂×C₃×C₄	C₄×D₉	1	12	yes	—
97	C₂×C₃×C₄	C₂×Dic₉	1	12	yes	—
98	C₂×C₃×C₄	C₂×Dic₉	1	6	yes	—
99	C₂×C₃×C₄	C₄×C₃⋊D₃	1	4	yes	—
100	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	4	yes	—
101	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	2	yes	—
102	C₂×Dic₃	—	9	24	no	—
103	C₂×Dic₃	—	9	8	no	—
104	C₂×Dic₃	—	18	24	no	—
105	C₂×Dic₃	—	18	8	no	—
106	C₂×C₃²	#872	1	4	yes	—
107	C₂×C₃²	C₂×C₄×Dic₃	1	1	yes	characteristic
108	C₂×C₃²	#926	1	2	yes	—
109	C₂×C₉	#872	1	12	yes	—
110	C₂×C₉	C₂×C₄×Dic₃	1	3	yes	—
111	C₂×C₉	#926	1	6	yes	—
112	C₂²×C₄	—	27	4	no	—
113	C₂²×C₄	C₂×C₃⋊D₉	1	2	yes	—
114	C₂²×C₄	C₂×C₃⋊D₉	1	1	yes	characteristic
115	C₄²	—	27	8	no	—
116	C₄²	C₃⋊Dic₉	1	4	yes	complemented
117	C₂²×C₃	C₂×C₄×D₉	1	6	yes	—
118	C₂²×C₃	#3697	1	12	yes	—
119	C₂²×C₃	C₂²×Dic₉	1	3	yes	—
120	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	2	yes	—
121	C₂²×C₃	#3828	1	4	yes	—
122	C₂²×C₃	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
123	C₃×C₄	—	2	12	no	—
124	C₃×C₄	—	2	4	no	—
125	C₃×C₄	C₄×Dic₉	1	12	yes	—
126	C₃×C₄	C₄×C₃⋊Dic₃	1	4	yes	—
127	Dic₃	—	18	8	no	—
128	Dic₃	—	18	24	no	—
129	C₃²	#4932	1	1	yes	characteristic
130	C₉	#4932	1	3	yes	complemented
131	C₂³	C₂²×C₃⋊D₉	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	27	8	no	—
134	C₂×C₄	—	54	8	no	—
135	C₂×C₄	C₄×C₃⋊D₉	1	4	yes	—
136	C₂×C₄	C₂×C₃⋊Dic₉	1	4	yes	—
137	C₂×C₄	C₂×C₃⋊Dic₉	1	2	yes	—
138	C₂×C₃	#63434	1	12	yes	—
139	C₂×C₃	C₂×C₄×Dic₉	1	3	yes	—
140	C₂×C₃	#63488	1	6	yes	—
141	C₂×C₃	#64081	1	4	yes	—
142	C₂×C₃	C₂×C₄×C₃⋊Dic₃	1	1	yes	characteristic
143	C₂×C₃	#64135	1	2	yes	—
144	C₂²	C₂×C₄×C₃⋊D₉	1	2	yes	—
145	C₂²	#88736	1	4	yes	—
146	C₂²	C₂²×C₃⋊Dic₉	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	54	8	no	—
149	C₄	C₄×C₃⋊Dic₉	1	4	yes	—
150	C₃	#10588612	1	3	yes	complemented
151	C₃	#10592631	1	1	yes	characteristic
152	C₂	#11721709	1	4	yes	—
153	C₂	C₂×C₄×C₃⋊Dic₉	1	1	yes	characteristic
154	C₂	#11721815	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂³×C₃	C₂²×C₃⋊D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle, prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂³×C₃×C₉	C₂³
hypercenter	C₂×C₄²	C₃⋊D₉
hypocenter	C₃×C₉	#376
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
9. C₃×C₄²×C₉	C₄	semidirect
10. C₄×C₃⋊Dic₉	C₄	semidirect
34. C₄²×C₉	Dic₃	semidirect
72. C₃×C₄²	Dic₉	semidirect
91. C₃×C₉	#376	semidirect
116. C₄²	C₃⋊Dic₉	semidirect
130. C₉	#4932	semidirect
150. C₃	#10588612	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄²×C₉	2
C₂×C₄×C₃⋊Dic₉	2
#11720572	2
#10592631	3
#10588612	3