Subgroups — C₂²×C₄×C₉×Dic₃ #13153532

Order 1728.

1730 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₉×Dic₃	C₂	1	2	yes	maximal
3	C₂×C₄×C₉×Dic₃	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×C₄×Dic₃	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₉	—	3	1	no	maximal
6	C₂⁴×C₃×C₉	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₄×C₉	C₄	1	8	yes	complemented
8	C₂²×C₃×C₄×C₉	C₂²	1	6	yes	—
9	C₂²×C₉×Dic₃	C₄	1	16	yes	direct product factor
10	C₂²×C₉×Dic₃	C₂²	1	12	yes	—
11	C₄×C₉×Dic₃	C₂²	1	16	yes	direct product factor
12	C₂³×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
13	C₂³×C₃×Dic₃	C₂×C₃	1	2	yes	—
14	C₂×C₃×C₄×Dic₃	C₂×C₃	1	12	yes	—
15	C₂³×C₄×C₉	—	2	1	no	—
16	C₂³×C₄×C₉	—	3	2	no	—
17	C₂³×C₄×C₉	D₃	1	1	yes	characteristic
18	C₂×C₄²×C₉	—	3	12	no	—
19	C₂³×C₃×C₉	C₂×C₄	1	8	yes	—
20	C₂³×C₃×C₉	C₂×C₄	1	4	yes	—
21	C₂³×C₃×C₉	C₂³	1	3	yes	—
22	C₂×C₃×C₄×C₉	C₂×C₄	1	24	yes	complemented
23	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
24	C₂×C₉×Dic₃	C₂×C₄	1	48	yes	direct product factor
25	C₂×C₉×Dic₃	C₂³	1	8	yes	—
26	C₂²×C₃×C₄²	—	3	1	no	—
27	C₂²×C₄×Dic₃	C₉	1	1	yes	characteristic, direct product factor
28	C₂⁴×C₃²	C₂²×C₃	1	1	yes	characteristic
29	C₂²×C₃²×C₄	C₃×C₄	1	8	yes	—
30	C₂²×C₃²×C₄	C₂²×C₃	1	6	yes	—
31	C₂²×C₃×Dic₃	C₃×C₄	1	16	yes	—
32	C₂²×C₃×Dic₃	C₂²×C₃	1	12	yes	—
33	C₂⁴×C₉	—	2	1	no	—
34	C₂⁴×C₉	C₂×D₃	1	1	yes	characteristic
35	C₃×C₄×Dic₃	C₂²×C₃	1	16	yes	—
36	C₂²×C₄×C₉	—	2	8	no	—
37	C₂²×C₄×C₉	—	2	6	no	—
38	C₂²×C₄×C₉	—	3	16	no	—
39	C₂²×C₄×C₉	—	3	12	no	—
40	C₂²×C₄×C₉	Dic₃	1	8	yes	direct product factor
41	C₂²×C₄×C₉	C₂×D₃	1	6	yes	—
42	C₄²×C₉	—	3	16	no	—
43	C₂²×C₃×C₉	C₄²	1	16	yes	complemented
44	C₂²×C₃×C₉	C₂²×C₄	1	12	yes	—
45	C₂²×C₃×C₉	C₂²×C₄	1	6	yes	—
46	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
47	C₃×C₄×C₉	C₂²×C₄	1	8	yes	complemented
48	C₉×Dic₃	C₂²×C₄	1	16	yes	direct product factor
49	C₂³×C₃×C₄	—	2	1	no	—
50	C₂³×C₃×C₄	—	3	2	no	—
51	C₂³×C₃×C₄	C₂×C₉	1	1	yes	characteristic
52	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
53	C₂³×Dic₃	C₂×C₉	1	2	yes	—
54	C₂×C₃×C₄²	—	3	12	no	—
55	C₂×C₄×Dic₃	C₂×C₉	1	12	yes	direct product factor
56	C₂³×C₃²	C₂×C₃×C₄	1	8	yes	—
57	C₂³×C₃²	C₂×C₃×C₄	1	4	yes	—
58	C₂³×C₃²	C₂³×C₃	1	3	yes	—
59	C₂×C₃²×C₄	C₂×C₃×C₄	1	24	yes	—
60	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
61	C₂×C₃×Dic₃	C₂×C₃×C₄	1	48	yes	—
62	C₂×C₃×Dic₃	C₂³×C₃	1	8	yes	—
63	C₂³×C₉	—	2	8	no	—
64	C₂³×C₉	—	2	4	no	—
65	C₂³×C₉	—	2	3	no	—
66	C₂³×C₉	C₄×D₃	1	8	yes	—
67	C₂³×C₉	C₂×Dic₃	1	4	yes	—
68	C₂³×C₉	C₂²×D₃	1	3	yes	—
69	C₂×C₄×C₉	—	2	24	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	3	48	no	—
72	C₂×C₄×C₉	—	3	8	no	—
73	C₂×C₄×C₉	C₂×Dic₃	1	24	yes	direct product factor
74	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
75	C₂²×C₄²	—	3	1	no	Sylow
76	C₂×C₃×C₉	C₂×C₄²	1	12	yes	complemented
77	C₂×C₃×C₉	C₂³×C₄	1	2	yes	—
78	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
79	C₂⁴×C₃	—	2	1	no	—
80	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
81	C₂⁴×C₃	C₂×C₃×D₃	1	1	yes	characteristic
82	C₂²×C₃×C₄	—	2	8	no	—
83	C₂²×C₃×C₄	—	2	6	no	—
84	C₂²×C₃×C₄	—	3	16	no	—
85	C₂²×C₃×C₄	—	3	12	no	—
86	C₂²×C₃×C₄	C₄×C₉	1	8	yes	complemented
87	C₂²×C₃×C₄	C₂²×C₉	1	6	yes	—
88	C₂²×C₃×C₄	C₃×Dic₃	1	8	yes	—
89	C₂²×C₃×C₄	C₂×C₃×D₃	1	6	yes	—
90	C₂²×Dic₃	C₄×C₉	1	16	yes	direct product factor
91	C₂²×Dic₃	C₂²×C₉	1	12	yes	—
92	C₃×C₄²	—	3	16	no	—
93	C₄×Dic₃	C₂²×C₉	1	16	yes	direct product factor
94	C₂²×C₃²	C₃×C₄²	1	16	yes	—
95	C₂²×C₃²	C₂²×C₃×C₄	1	12	yes	—
96	C₂²×C₃²	C₂²×C₃×C₄	1	6	yes	—
97	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
98	C₃²×C₄	C₂²×C₃×C₄	1	8	yes	—
99	C₃×Dic₃	C₂²×C₃×C₄	1	16	yes	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	16	no	—
102	C₂²×C₉	—	2	12	no	—
103	C₂²×C₉	—	2	6	no	—
104	C₂²×C₉	C₄×Dic₃	1	16	yes	direct product factor
105	C₂²×C₉	C₂×C₄×D₃	1	12	yes	—
106	C₂²×C₉	C₂²×Dic₃	1	6	yes	—
107	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
108	C₄×C₉	—	2	8	no	—
109	C₄×C₉	—	3	16	no	—
110	C₄×C₉	C₂²×Dic₃	1	8	yes	direct product factor
111	C₂³×C₄	—	3	2	no	—
112	C₂³×C₄	D₃×C₉	1	1	yes	characteristic
113	C₂×C₄²	—	3	12	no	—
114	C₃×C₉	C₂²×C₄²	1	1	yes	characteristic, complemented, Sylow
115	C₂³×C₃	—	2	8	no	—
116	C₂³×C₃	—	2	4	no	—
117	C₂³×C₃	—	2	3	no	—
118	C₂³×C₃	C₂×C₄×C₉	1	8	yes	—
119	C₂³×C₃	C₂×C₄×C₉	1	4	yes	—
120	C₂³×C₃	C₂³×C₉	1	3	yes	—
121	C₂³×C₃	C₃×C₄×D₃	1	8	yes	—
122	C₂³×C₃	C₂×C₃×Dic₃	1	4	yes	—
123	C₂³×C₃	C₂²×C₃×D₃	1	3	yes	—
124	C₂×C₃×C₄	—	2	24	no	—
125	C₂×C₃×C₄	—	2	4	no	—
126	C₂×C₃×C₄	—	3	8	no	—
127	C₂×C₃×C₄	—	3	48	no	—
128	C₂×C₃×C₄	C₂×C₄×C₉	1	24	yes	complemented
129	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
130	C₂×C₃×C₄	C₂×C₃×Dic₃	1	24	yes	—
131	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
132	C₂×Dic₃	C₂×C₄×C₉	1	48	yes	direct product factor
133	C₂×Dic₃	C₂³×C₉	1	8	yes	—
134	C₂×C₃²	C₂×C₃×C₄²	1	12	yes	—
135	C₂×C₃²	C₂³×C₃×C₄	1	2	yes	—
136	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
137	C₂×C₉	—	2	12	no	—
138	C₂×C₉	—	2	2	no	—
139	C₂×C₉	—	2	1	no	—
140	C₂×C₉	C₂×C₄×Dic₃	1	12	yes	direct product factor
141	C₂×C₉	C₂²×C₄×D₃	1	2	yes	—
142	C₂×C₉	C₂³×Dic₃	1	1	yes	characteristic
143	C₂⁴	C₂×D₃×C₉	1	1	yes	characteristic
144	C₂²×C₄	—	3	16	no	—
145	C₂²×C₄	—	3	12	no	—
146	C₂²×C₄	C₉×Dic₃	1	8	yes	direct product factor
147	C₂²×C₄	C₂×D₃×C₉	1	6	yes	—
148	C₄²	—	3	16	no	—
149	C₂²×C₃	—	2	1	no	—
150	C₂²×C₃	—	2	12	no	—
151	C₂²×C₃	—	2	6	no	—
152	C₂²×C₃	—	2	16	no	—
153	C₂²×C₃	C₄²×C₉	1	16	yes	complemented
154	C₂²×C₃	C₂²×C₄×C₉	1	12	yes	—
155	C₂²×C₃	C₂²×C₄×C₉	1	6	yes	—
156	C₂²×C₃	C₃×C₄×Dic₃	1	16	yes	—
157	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
158	C₂²×C₃	C₂×C₃×C₄×D₃	1	12	yes	—
159	C₂²×C₃	C₂²×C₃×Dic₃	1	6	yes	—
160	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
161	C₃×C₄	—	2	8	no	—
162	C₃×C₄	—	3	16	no	—
163	C₃×C₄	C₂²×C₄×C₉	1	8	yes	complemented
164	C₃×C₄	C₂²×C₃×Dic₃	1	8	yes	—
165	Dic₃	C₂²×C₄×C₉	1	16	yes	direct product factor
166	C₃²	C₂²×C₃×C₄²	1	1	yes	characteristic
167	C₉	—	2	1	no	—
168	C₉	C₂²×C₄×Dic₃	1	1	yes	characteristic, direct product factor
169	C₂³	C₄×D₃×C₉	1	8	yes	—
170	C₂³	C₂×C₉×Dic₃	1	4	yes	—
171	C₂³	C₂²×D₃×C₉	1	3	yes	—
172	C₂×C₄	—	3	8	no	—
173	C₂×C₄	—	3	48	no	—
174	C₂×C₄	C₂×C₉×Dic₃	1	24	yes	direct product factor
175	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
176	C₂×C₃	—	2	2	no	—
177	C₂×C₃	—	2	12	no	—
178	C₂×C₃	—	2	1	no	—
179	C₂×C₃	C₂×C₄²×C₉	1	12	yes	complemented
180	C₂×C₃	C₂³×C₄×C₉	1	2	yes	—
181	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
182	C₂×C₃	C₂×C₃×C₄×Dic₃	1	12	yes	—
183	C₂×C₃	C₂²×C₃×C₄×D₃	1	2	yes	—
184	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
185	C₂²	C₄×C₉×Dic₃	1	16	yes	direct product factor
186	C₂²	C₂×C₄×D₃×C₉	1	12	yes	—
187	C₂²	C₂²×C₉×Dic₃	1	6	yes	—
188	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
189	C₄	—	3	16	no	—
190	C₄	C₂²×C₉×Dic₃	1	8	yes	direct product factor
191	C₃	—	2	1	no	—
192	C₃	C₂²×C₄²×C₉	1	1	yes	characteristic, complemented
193	C₃	C₂²×C₃×C₄×Dic₃	1	1	yes	characteristic
194	C₂	C₂×C₄×C₉×Dic₃	1	12	yes	direct product factor
195	C₂	C₂²×C₄×D₃×C₉	1	2	yes	—
196	C₂	C₂³×C₉×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄×C₉	D₃
commutator, hypocenter	C₃	C₂²×C₄²×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, prime radical	C₂⁴×C₃²	C₂²×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₄×C₉

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₉×Dic₃	C₂	direct (N × H)
7. C₂²×C₃×C₄×C₉	C₄	semidirect
9. C₂²×C₉×Dic₃	C₄	direct (N × H)
11. C₄×C₉×Dic₃	C₂²	direct (N × H)
22. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
24. C₂×C₉×Dic₃	C₂×C₄	direct (N × H)
27. C₂²×C₄×Dic₃	C₉	direct (N × H)
40. C₂²×C₄×C₉	Dic₃	direct (N × H)
43. C₂²×C₃×C₉	C₄²	semidirect
47. C₃×C₄×C₉	C₂²×C₄	semidirect
48. C₉×Dic₃	C₂²×C₄	direct (N × H)
55. C₂×C₄×Dic₃	C₂×C₉	direct (N × H)
73. C₂×C₄×C₉	C₂×Dic₃	direct (N × H)
76. C₂×C₃×C₉	C₂×C₄²	semidirect
86. C₂²×C₃×C₄	C₄×C₉	semidirect
90. C₂²×Dic₃	C₄×C₉	direct (N × H)
93. C₄×Dic₃	C₂²×C₉	direct (N × H)
104. C₂²×C₉	C₄×Dic₃	direct (N × H)
110. C₄×C₉	C₂²×Dic₃	direct (N × H)
114. C₃×C₉	C₂²×C₄²	semidirect
128. C₂×C₃×C₄	C₂×C₄×C₉	semidirect
132. C₂×Dic₃	C₂×C₄×C₉	direct (N × H)
140. C₂×C₉	C₂×C₄×Dic₃	direct (N × H)
146. C₂²×C₄	C₉×Dic₃	direct (N × H)
153. C₂²×C₃	C₄²×C₉	semidirect
163. C₃×C₄	C₂²×C₄×C₉	semidirect
165. Dic₃	C₂²×C₄×C₉	direct (N × H)
168. C₉	C₂²×C₄×Dic₃	direct (N × H)
174. C₂×C₄	C₂×C₉×Dic₃	direct (N × H)
179. C₂×C₃	C₂×C₄²×C₉	semidirect
185. C₂²	C₄×C₉×Dic₃	direct (N × H)
190. C₄	C₂²×C₉×Dic₃	direct (N × H)
192. C₃	C₂²×C₄²×C₉	semidirect
194. C₂	C₂×C₄×C₉×Dic₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₉	2
C₂³×C₉×Dic₃	2
C₂×C₄×C₉×Dic₃	2
C₂²×C₃×C₄×Dic₃	3
C₂²×C₄²×C₉	3

Subgroups — C22×C4×C9×Dic3 #13153532

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×C₉×Dic₃ #13153532