Subgroups — C₂²×C₃×C₄×Dic₉ #13153527

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×C₃×Dic₉	C₂	1	2	yes	maximal
3	C₂×C₃×C₄×Dic₉	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₃×C₄×Dic₃	—	3	1	no	maximal
5	C₂²×C₄×Dic₉	C₃	1	1	yes	maximal, characteristic, direct product factor
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×C₃×Dic₉	C₄	1	16	yes	direct product factor
10	C₂²×C₃×Dic₉	C₂²	1	12	yes	—
11	C₃×C₄×Dic₉	C₂²	1	16	yes	direct product factor
12	C₂³×C₃²×C₄	D₃	1	1	yes	characteristic
13	C₂³×C₃×Dic₃	—	3	2	no	—
14	C₂×C₃×C₄×Dic₃	—	3	12	no	—
15	C₂³×C₄×C₉	—	2	1	no	—
16	C₂³×C₄×C₉	C₂×C₃	1	1	yes	characteristic
17	C₂³×Dic₉	C₂×C₃	1	2	yes	—
18	C₂×C₄×Dic₉	C₂×C₃	1	12	yes	direct product factor
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×C₃×Dic₉	C₂×C₄	1	48	yes	direct product factor
25	C₂×C₃×Dic₉	C₂³	1	8	yes	—
26	C₂²×C₃×C₄²	—	9	1	no	—
27	C₂²×C₄×Dic₃	—	3	1	no	—
28	C₂⁴×C₃²	C₂×D₃	1	1	yes	characteristic
29	C₂²×C₃²×C₄	Dic₃	1	8	yes	—
30	C₂²×C₃²×C₄	C₂×D₃	1	6	yes	—
31	C₂²×C₃×Dic₃	—	3	16	no	—
32	C₂²×C₃×Dic₃	—	3	12	no	—
33	C₂⁴×C₉	—	2	1	no	—
34	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
35	C₃×C₄×Dic₃	—	3	16	no	—
36	C₂²×C₄×C₉	—	2	8	no	—
37	C₂²×C₄×C₉	—	2	6	no	—
38	C₂²×C₄×C₉	C₃×C₄	1	8	yes	complemented
39	C₂²×C₄×C₉	C₂²×C₃	1	6	yes	—
40	C₂²×Dic₉	C₃×C₄	1	16	yes	direct product factor
41	C₂²×Dic₉	C₂²×C₃	1	12	yes	—
42	C₄×Dic₉	C₂²×C₃	1	16	yes	direct product factor
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	C₃×Dic₉	C₂²×C₄	1	16	yes	direct product factor
49	C₂³×C₃×C₄	—	2	1	no	—
50	C₂³×C₃×C₄	—	9	2	no	—
51	C₂³×C₃×C₄	D₉	1	1	yes	characteristic
52	C₂³×C₃×C₄	C₃×D₃	1	1	yes	characteristic
53	C₂³×Dic₃	—	3	2	no	—
54	C₂×C₃×C₄²	—	9	12	no	—
55	C₂×C₄×Dic₃	—	3	12	no	—
56	C₂³×C₃²	C₄×D₃	1	8	yes	—
57	C₂³×C₃²	C₂×Dic₃	1	4	yes	—
58	C₂³×C₃²	C₂²×D₃	1	3	yes	—
59	C₂×C₃²×C₄	C₂×Dic₃	1	24	yes	—
60	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
61	C₂×C₃×Dic₃	—	3	48	no	—
62	C₂×C₃×Dic₃	—	3	8	no	—
63	C₂³×C₉	—	2	8	no	—
64	C₂³×C₉	—	2	4	no	—
65	C₂³×C₉	—	2	3	no	—
66	C₂³×C₉	C₂×C₃×C₄	1	8	yes	—
67	C₂³×C₉	C₂×C₃×C₄	1	4	yes	—
68	C₂³×C₉	C₂³×C₃	1	3	yes	—
69	C₂×C₄×C₉	—	2	24	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	C₂×C₃×C₄	1	24	yes	complemented
72	C₂×C₄×C₉	C₂³×C₃	1	4	yes	—
73	C₂×Dic₉	C₂×C₃×C₄	1	48	yes	direct product factor
74	C₂×Dic₉	C₂³×C₃	1	8	yes	—
75	C₂²×C₄²	—	9	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	C₂×D₉	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	—	9	16	no	—
85	C₂²×C₃×C₄	—	9	12	no	—
86	C₂²×C₃×C₄	Dic₉	1	8	yes	direct product factor
87	C₂²×C₃×C₄	C₂×D₉	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	—	3	16	no	—
91	C₂²×Dic₃	—	3	12	no	—
92	C₃×C₄²	—	9	16	no	—
93	C₄×Dic₃	—	3	16	no	—
94	C₂²×C₃²	C₄×Dic₃	1	16	yes	—
95	C₂²×C₃²	C₂×C₄×D₃	1	12	yes	—
96	C₂²×C₃²	C₂²×Dic₃	1	6	yes	—
97	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
98	C₃²×C₄	C₂²×Dic₃	1	8	yes	—
99	C₃×Dic₃	—	3	16	no	—
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	C₃×C₄²	1	16	yes	complemented
105	C₂²×C₉	C₂²×C₃×C₄	1	12	yes	—
106	C₂²×C₉	C₂²×C₃×C₄	1	6	yes	—
107	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
108	C₄×C₉	—	2	8	no	—
109	C₄×C₉	C₂²×C₃×C₄	1	8	yes	complemented
110	Dic₉	C₂²×C₃×C₄	1	16	yes	direct product factor
111	C₂³×C₄	—	9	2	no	—
112	C₂³×C₄	C₃×D₉	1	1	yes	characteristic
113	C₂×C₄²	—	9	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	C₄×D₉	1	8	yes	—
119	C₂³×C₃	C₂×Dic₉	1	4	yes	—
120	C₂³×C₃	C₂²×D₉	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	—	9	48	no	—
127	C₂×C₃×C₄	—	9	8	no	—
128	C₂×C₃×C₄	C₂×Dic₉	1	24	yes	direct product factor
129	C₂×C₃×C₄	C₂²×D₉	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	—	3	8	no	—
133	C₂×Dic₃	—	3	48	no	—
134	C₂×C₃²	C₂×C₄×Dic₃	1	12	yes	—
135	C₂×C₃²	C₂²×C₄×D₃	1	2	yes	—
136	C₂×C₃²	C₂³×Dic₃	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×C₃×C₄²	1	12	yes	complemented
141	C₂×C₉	C₂³×C₃×C₄	1	2	yes	—
142	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
143	C₂⁴	C₂×C₃×D₉	1	1	yes	characteristic
144	C₂²×C₄	—	9	16	no	—
145	C₂²×C₄	—	9	12	no	—
146	C₂²×C₄	C₃×Dic₉	1	8	yes	direct product factor
147	C₂²×C₄	C₂×C₃×D₉	1	6	yes	—
148	C₄²	—	9	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	C₄×Dic₉	1	16	yes	direct product factor
154	C₂²×C₃	C₂×C₄×D₉	1	12	yes	—
155	C₂²×C₃	C₂²×Dic₉	1	6	yes	—
156	C₂²×C₃	C₃×C₄×Dic₃	1	16	yes	—
157	C₂²×C₃	C₂³×D₉	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	—	9	16	no	—
163	C₃×C₄	C₂²×Dic₉	1	8	yes	direct product factor
164	C₃×C₄	C₂²×C₃×Dic₃	1	8	yes	—
165	Dic₃	—	3	16	no	—
166	C₃²	C₂²×C₄×Dic₃	1	1	yes	characteristic
167	C₉	—	2	1	no	—
168	C₉	C₂²×C₃×C₄²	1	1	yes	characteristic, complemented
169	C₂³	C₃×C₄×D₉	1	8	yes	—
170	C₂³	C₂×C₃×Dic₉	1	4	yes	—
171	C₂³	C₂²×C₃×D₉	1	3	yes	—
172	C₂×C₄	—	9	8	no	—
173	C₂×C₄	—	9	48	no	—
174	C₂×C₄	C₂×C₃×Dic₉	1	24	yes	direct product factor
175	C₂×C₄	C₂²×C₃×D₉	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×C₄×Dic₉	1	12	yes	direct product factor
180	C₂×C₃	C₂²×C₄×D₉	1	2	yes	—
181	C₂×C₃	C₂³×Dic₉	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	C₃×C₄×Dic₉	1	16	yes	direct product factor
186	C₂²	C₂×C₃×C₄×D₉	1	12	yes	—
187	C₂²	C₂²×C₃×Dic₉	1	6	yes	—
188	C₂²	C₂³×C₃×D₉	1	1	yes	characteristic
189	C₄	—	9	16	no	—
190	C₄	C₂²×C₃×Dic₉	1	8	yes	direct product factor
191	C₃	—	2	1	no	—
192	C₃	C₂²×C₄×Dic₉	1	1	yes	characteristic, direct product factor
193	C₃	C₂²×C₃×C₄×Dic₃	1	1	yes	characteristic
194	C₂	C₂×C₃×C₄×Dic₉	1	12	yes	direct product factor
195	C₂	C₂²×C₃×C₄×D₉	1	2	yes	—
196	C₂	C₂³×C₃×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₃×C₄	D₉
commutator, hypocenter	C₉	C₂²×C₃×C₄²
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂³×C₃×C₄×C₉	C₂
socle, prime radical	C₂⁴×C₃²	C₂×D₃
cosocle	C₂²×C₉	C₂⁴×C₃
LCS	G ▷ C₉
UCS	C₁ ◁ C₂³×C₃×C₄

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄×Dic₉	C₂	direct (N × H)
5. C₂²×C₄×Dic₉	C₃	direct (N × H)
7. C₂²×C₃×C₄×C₉	C₄	semidirect
9. C₂²×C₃×Dic₉	C₄	direct (N × H)
11. C₃×C₄×Dic₉	C₂²	direct (N × H)
18. C₂×C₄×Dic₉	C₂×C₃	direct (N × H)
22. C₂×C₃×C₄×C₉	C₂×C₄	semidirect
24. C₂×C₃×Dic₉	C₂×C₄	direct (N × H)
38. C₂²×C₄×C₉	C₃×C₄	semidirect
40. C₂²×Dic₉	C₃×C₄	direct (N × H)
42. C₄×Dic₉	C₂²×C₃	direct (N × H)
43. C₂²×C₃×C₉	C₄²	semidirect
47. C₃×C₄×C₉	C₂²×C₄	semidirect
48. C₃×Dic₉	C₂²×C₄	direct (N × H)
71. C₂×C₄×C₉	C₂×C₃×C₄	semidirect
73. C₂×Dic₉	C₂×C₃×C₄	direct (N × H)
76. C₂×C₃×C₉	C₂×C₄²	semidirect
86. C₂²×C₃×C₄	Dic₉	direct (N × H)
104. C₂²×C₉	C₃×C₄²	semidirect
109. C₄×C₉	C₂²×C₃×C₄	semidirect
110. Dic₉	C₂²×C₃×C₄	direct (N × H)
114. C₃×C₉	C₂²×C₄²	semidirect
128. C₂×C₃×C₄	C₂×Dic₉	direct (N × H)
140. C₂×C₉	C₂×C₃×C₄²	semidirect
146. C₂²×C₄	C₃×Dic₉	direct (N × H)
153. C₂²×C₃	C₄×Dic₉	direct (N × H)
163. C₃×C₄	C₂²×Dic₉	direct (N × H)
168. C₉	C₂²×C₃×C₄²	semidirect
174. C₂×C₄	C₂×C₃×Dic₉	direct (N × H)
179. C₂×C₃	C₂×C₄×Dic₉	direct (N × H)
185. C₂²	C₃×C₄×Dic₉	direct (N × H)
190. C₄	C₂²×C₃×Dic₉	direct (N × H)
192. C₃	C₂²×C₄×Dic₉	direct (N × H)
194. C₂	C₂×C₃×C₄×Dic₉	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×C₃×Dic₉	2
C₂×C₃×C₄×Dic₉	2
C₂²×C₃×C₄×Dic₃	3
C₂²×C₄×Dic₉	3

Subgroups — C22×C3×C4×Dic9 #13153527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₄×Dic₉ #13153527