Subgroups — C₂×C₉×#953 #13153762

Order 1728.

2734 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₉×#67	C₂	1	3	yes	maximal, complemented
3	C₉×#953	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₉×#217	C₂	1	3	yes	maximal, complemented
5	C₂×C₃×#953	C₃	1	1	yes	maximal, characteristic
6	C₂×C₉×#120	—	3	1	no	maximal
7	C₂⁴×C₃×C₉	—	2	12	no	—
8	C₂⁴×C₃×C₉	C₂²	1	3	yes	—
9	C₂⁴×C₃×C₉	C₂²	1	4	yes	complemented
10	C₂³×D₃×C₉	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₉×#67	—	2	12	no	—
12	C₂×C₉×#67	C₂²	1	12	yes	complemented
13	C₂²×C₉×Dic₃	C₂²	1	3	yes	complemented
14	C₉×#217	C₂²	1	12	yes	complemented
15	C₂⁵×C₃²	C₂×C₃	1	1	yes	characteristic
16	C₂²×C₃×#67	C₂×C₃	1	3	yes	—
17	C₂⁵×C₉	—	2	1	no	—
18	C₂⁵×C₉	D₃	1	1	yes	characteristic, complemented
19	C₃×#953	C₂×C₃	1	8	yes	—
20	C₂×C₃×#217	C₂×C₃	1	3	yes	—
21	C₂²×D₄×C₉	—	3	3	no	—
22	C₉×#120	—	3	8	no	—
23	C₂×C₉×C₂²⋊C₄	—	3	3	no	—
24	C₂³×C₃×C₉	—	2	32	no	—
25	C₂³×C₃×C₉	—	2	12	no	—
26	C₂³×C₃×C₉	—	2	24	no	—
27	C₂³×C₃×C₉	D₄	1	12	yes	complemented
28	C₂³×C₃×C₉	C₂³	1	6	yes	—
29	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
30	C₂²×D₃×C₉	—	2	6	no	—
31	C₂²×D₃×C₉	C₂³	1	2	yes	complemented
32	C₉×#67	—	2	24	no	—
33	C₂×C₉×Dic₃	—	2	6	no	—
34	C₂×C₉×Dic₃	C₂³	1	6	yes	complemented
35	C₂×C₃×#120	—	3	1	no	—
36	C₂×#953	C₉	1	1	yes	characteristic, direct product factor
37	C₂⁴×C₃²	—	2	12	no	—
38	C₂⁴×C₃²	C₂²×C₃	1	4	yes	—
39	C₂⁴×C₃²	C₂²×C₃	1	3	yes	—
40	C₂³×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₃×#67	—	2	12	no	—
42	C₂×C₃×#67	C₂²×C₃	1	12	yes	—
43	C₂²×C₃×Dic₃	C₂²×C₃	1	3	yes	—
44	C₂⁴×C₉	—	2	24	no	—
45	C₂⁴×C₉	—	2	3	no	—
46	C₂⁴×C₉	—	2	4	no	—
47	C₂⁴×C₉	—	2	12	no	—
48	C₂⁴×C₉	—	3	1	no	—
49	C₂⁴×C₉	C₂×D₃	1	3	yes	—
50	C₂⁴×C₉	C₂×D₃	1	4	yes	complemented
51	C₃×#217	C₂²×C₃	1	12	yes	—
52	C₂×D₄×C₉	—	3	12	no	—
53	C₂×D₄×C₉	—	6	12	no	—
54	C₂²×C₄×C₉	—	3	3	no	—
55	C₉×C₂²⋊C₄	—	3	12	no	—
56	C₂²×C₃×C₉	—	2	32	no	—
57	C₂²×C₃×C₉	—	2	24	no	—
58	C₂²×C₃×C₉	—	2	12	no	—
59	C₂²×C₃×C₉	C₂×D₄	1	6	yes	—
60	C₂²×C₃×C₉	C₂×D₄	1	12	yes	complemented
61	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
62	C₂×D₃×C₉	—	2	6	no	—
63	C₂×D₃×C₉	—	4	4	no	—
64	C₉×Dic₃	—	2	6	no	—
65	C₂⁵×C₃	—	2	1	no	—
66	C₂⁵×C₃	C₂×C₉	1	1	yes	characteristic, complemented
67	C₂⁵×C₃	C₃×D₃	1	1	yes	characteristic
68	C₂²×C₃×D₄	—	3	3	no	—
69	C₂²×#67	C₂×C₉	1	3	yes	complemented
70	C₃×#120	—	3	8	no	—
71	C₂×C₃×C₂²⋊C₄	—	3	3	no	—
72	#953	C₂×C₉	1	8	yes	direct product factor
73	C₂×#217	C₂×C₉	1	3	yes	complemented
74	C₂³×C₃²	—	2	32	no	—
75	C₂³×C₃²	—	2	12	no	—
76	C₂³×C₃²	—	2	24	no	—
77	C₂³×C₃²	C₃×D₄	1	12	yes	—
78	C₂³×C₃²	C₂³×C₃	1	6	yes	—
79	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
80	C₂²×C₃×D₃	—	2	6	no	—
81	C₂²×C₃×D₃	C₂³×C₃	1	2	yes	—
82	C₃×#67	—	2	24	no	—
83	C₂×C₃×Dic₃	—	2	6	no	—
84	C₂×C₃×Dic₃	C₂³×C₃	1	6	yes	—
85	C₂³×C₉	—	2	64	no	—
86	C₂³×C₉	—	2	32	no	—
87	C₂³×C₉	—	2	24	no	—
88	C₂³×C₉	—	2	48	no	—
89	C₂³×C₉	—	2	12	no	—
90	C₂³×C₉	—	2	24	no	—
91	C₂³×C₉	—	2	12	no	—
92	C₂³×C₉	—	2	6	no	—
93	C₂³×C₉	—	2	1	no	—
94	C₂³×C₉	—	3	2	no	—
95	C₂³×C₉	—	6	6	no	—
96	C₂³×C₉	#67	1	12	yes	complemented
97	C₂³×C₉	C₂²×D₃	1	6	yes	—
98	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
99	D₄×C₉	—	6	24	no	—
100	C₂×C₄×C₉	—	3	6	no	—
101	C₂×C₄×C₉	—	6	6	no	—
102	C₂×#120	—	3	1	no	Sylow
103	C₂×C₃×C₉	—	2	12	no	—
104	C₂×C₃×C₉	#120	1	4	yes	complemented
105	C₂×C₃×C₉	C₂²×D₄	1	3	yes	—
106	D₃×C₉	—	4	2	no	—
107	C₂⁴×C₃	—	2	24	no	—
108	C₂⁴×C₃	—	2	12	no	—
109	C₂⁴×C₃	—	2	12	no	—
110	C₂⁴×C₃	—	2	3	no	—
111	C₂⁴×C₃	—	2	4	no	—
112	C₂⁴×C₃	—	3	1	no	—
113	C₂⁴×C₃	C₂²×C₉	1	3	yes	—
114	C₂⁴×C₃	C₂²×C₉	1	4	yes	complemented
115	C₂⁴×C₃	C₂×C₃×D₃	1	3	yes	—
116	C₂⁴×C₃	C₂×C₃×D₃	1	4	yes	—
117	C₂³×D₃	C₂²×C₉	1	1	yes	characteristic, complemented
118	C₂×C₃×D₄	—	3	12	no	—
119	C₂×C₃×D₄	—	6	12	no	—
120	C₂²×C₃×C₄	—	3	3	no	—
121	C₂×#67	—	2	12	no	—
122	C₂×#67	C₂²×C₉	1	12	yes	complemented
123	C₂²×Dic₃	C₂²×C₉	1	3	yes	complemented
124	C₃×C₂²⋊C₄	—	3	12	no	—
125	#217	C₂²×C₉	1	12	yes	complemented
126	C₂²×C₃²	—	2	32	no	—
127	C₂²×C₃²	—	2	24	no	—
128	C₂²×C₃²	—	2	12	no	—
129	C₂²×C₃²	C₂×C₃×D₄	1	6	yes	—
130	C₂²×C₃²	C₂×C₃×D₄	1	12	yes	—
131	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
132	C₂×C₃×D₃	—	2	6	no	—
133	C₂×C₃×D₃	—	4	4	no	—
134	C₃×Dic₃	—	2	6	no	—
135	C₂²×C₉	—	2	64	no	—
136	C₂²×C₉	—	2	32	no	—
137	C₂²×C₉	—	2	48	no	—
138	C₂²×C₉	—	2	24	no	—
139	C₂²×C₉	—	2	24	no	—
140	C₂²×C₉	—	2	12	no	—
141	C₂²×C₉	—	2	1	no	—
142	C₂²×C₉	—	2	6	no	—
143	C₂²×C₉	—	2	12	no	—
144	C₂²×C₉	—	6	6	no	—
145	C₂²×C₉	—	12	4	no	—
146	C₂²×C₉	C₂×#67	1	6	yes	—
147	C₂²×C₉	C₂×#67	1	12	yes	complemented
148	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
149	C₄×C₉	—	6	6	no	—
150	C₂⁵	D₃×C₉	1	1	yes	characteristic, complemented
151	C₂²×D₄	—	3	3	no	—
152	#120	—	3	8	no	—
153	C₂×C₂²⋊C₄	—	3	3	no	—
154	C₃×C₉	C₂×#120	1	1	yes	characteristic, complemented, Sylow
155	C₂³×C₃	—	2	32	no	—
156	C₂³×C₃	—	2	64	no	—
157	C₂³×C₃	—	2	32	no	—
158	C₂³×C₃	—	2	24	no	—
159	C₂³×C₃	—	2	48	no	—
160	C₂³×C₃	—	2	12	no	—
161	C₂³×C₃	—	2	24	no	—
162	C₂³×C₃	—	2	12	no	—
163	C₂³×C₃	—	2	24	no	—
164	C₂³×C₃	—	2	12	no	—
165	C₂³×C₃	—	2	6	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	—	3	2	no	—
168	C₂³×C₃	—	6	6	no	—
169	C₂³×C₃	D₄×C₉	1	12	yes	complemented
170	C₂³×C₃	C₂³×C₉	1	6	yes	—
171	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
172	C₂³×C₃	C₃×#67	1	12	yes	—
173	C₂³×C₃	C₂²×C₃×D₃	1	6	yes	—
174	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
175	C₂²×D₃	—	2	6	no	—
176	C₂²×D₃	C₂³×C₉	1	2	yes	complemented
177	C₃×D₄	—	6	24	no	—
178	C₂×C₃×C₄	—	3	6	no	—
179	C₂×C₃×C₄	—	6	6	no	—
180	#67	—	2	24	no	—
181	C₂×Dic₃	—	2	6	no	—
182	C₂×Dic₃	C₂³×C₉	1	6	yes	complemented
183	C₂×C₃²	—	2	12	no	—
184	C₂×C₃²	C₃×#120	1	4	yes	—
185	C₂×C₃²	C₂²×C₃×D₄	1	3	yes	—
186	C₃×D₃	—	4	2	no	—
187	C₂×C₉	—	2	24	no	—
188	C₂×C₉	—	2	12	no	—
189	C₂×C₉	—	2	4	no	—
190	C₂×C₉	—	2	3	no	—
191	C₂×C₉	—	12	2	no	—
192	C₂×C₉	#953	1	4	yes	direct product factor
193	C₂×C₉	C₂²×#67	1	3	yes	—
194	C₂⁴	—	2	12	no	—
195	C₂⁴	—	3	1	no	—
196	C₂⁴	C₂×D₃×C₉	1	3	yes	—
197	C₂⁴	C₂×D₃×C₉	1	4	yes	complemented
198	C₂×D₄	—	3	12	no	—
199	C₂×D₄	—	6	12	no	—
200	C₂²×C₄	—	3	3	no	—
201	C₂²⋊C₄	—	3	12	no	—
202	C₂²×C₃	—	2	32	no	—
203	C₂²×C₃	—	2	64	no	—
204	C₂²×C₃	—	2	32	no	—
205	C₂²×C₃	—	2	24	no	—
206	C₂²×C₃	—	2	48	no	—
207	C₂²×C₃	—	2	24	no	—
208	C₂²×C₃	—	2	24	no	—
209	C₂²×C₃	—	2	12	no	—
210	C₂²×C₃	—	2	12	no	—
211	C₂²×C₃	—	2	1	no	—
212	C₂²×C₃	—	2	6	no	—
213	C₂²×C₃	—	2	12	no	—
214	C₂²×C₃	—	6	6	no	—
215	C₂²×C₃	—	12	4	no	—
216	C₂²×C₃	C₂×D₄×C₉	1	6	yes	—
217	C₂²×C₃	C₂×D₄×C₉	1	12	yes	complemented
218	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
219	C₂²×C₃	C₂×C₃×#67	1	6	yes	—
220	C₂²×C₃	C₂×C₃×#67	1	12	yes	—
221	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
222	C₂×D₃	—	2	6	no	—
223	C₂×D₃	—	4	4	no	—
224	C₃×C₄	—	6	6	no	—
225	Dic₃	—	2	6	no	—
226	C₃²	C₂×C₃×#120	1	1	yes	characteristic
227	C₉	—	2	1	no	—
228	C₉	C₂×#953	1	1	yes	characteristic, direct product factor
229	C₂³	—	2	32	no	—
230	C₂³	—	2	12	no	—
231	C₂³	—	2	24	no	—
232	C₂³	—	3	2	no	—
233	C₂³	—	6	6	no	—
234	C₂³	C₉×#67	1	12	yes	complemented
235	C₂³	C₂²×D₃×C₉	1	6	yes	—
236	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
237	D₄	—	6	24	no	—
238	C₂×C₄	—	3	6	no	—
239	C₂×C₄	—	6	6	no	—
240	C₂×C₃	—	2	24	no	—
241	C₂×C₃	—	2	12	no	—
242	C₂×C₃	—	2	12	no	—
243	C₂×C₃	—	2	3	no	—
244	C₂×C₃	—	2	4	no	—
245	C₂×C₃	—	12	2	no	—
246	C₂×C₃	C₉×#120	1	4	yes	complemented
247	C₂×C₃	C₂²×D₄×C₉	1	3	yes	—
248	C₂×C₃	C₃×#953	1	4	yes	—
249	C₂×C₃	C₂²×C₃×#67	1	3	yes	—
250	D₃	—	4	2	no	—
251	C₂²	—	2	32	no	—
252	C₂²	—	2	12	no	—
253	C₂²	—	2	24	no	—
254	C₂²	—	6	6	no	—
255	C₂²	—	12	4	no	—
256	C₂²	C₂×C₉×#67	1	6	yes	—
257	C₂²	C₂×C₉×#67	1	12	yes	complemented
258	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
259	C₄	—	6	6	no	—
260	C₃	—	2	1	no	—
261	C₃	C₂×C₉×#120	1	1	yes	characteristic, complemented
262	C₃	C₂×C₃×#953	1	1	yes	characteristic
263	C₂	—	2	12	no	—
264	C₂	—	12	2	no	—
265	C₂	C₉×#953	1	4	yes	direct product factor
266	C₂	C₂²×C₉×#67	1	3	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂²×D₃
commutator	C₂²×C₃	C₂⁴×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂⁵×C₃×C₉	C₂
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂×C₃×#953	C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂⁵×C₉	D₃
hypocenter	C₃	C₂×C₉×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₉ ◁ C₂⁵×C₉

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₃×C₉	C₂	semidirect
2. C₂²×C₉×#67	C₂	semidirect
3. C₉×#953	C₂	direct (N × H)
4. C₂×C₉×#217	C₂	semidirect
9. C₂⁴×C₃×C₉	C₂²	semidirect
10. C₂³×D₃×C₉	C₂²	semidirect
12. C₂×C₉×#67	C₂²	semidirect
13. C₂²×C₉×Dic₃	C₂²	semidirect
14. C₉×#217	C₂²	semidirect
18. C₂⁵×C₉	D₃	semidirect
27. C₂³×C₃×C₉	D₄	semidirect
31. C₂²×D₃×C₉	C₂³	semidirect
34. C₂×C₉×Dic₃	C₂³	semidirect
36. C₂×#953	C₉	direct (N × H)
50. C₂⁴×C₉	C₂×D₃	semidirect
60. C₂²×C₃×C₉	C₂×D₄	semidirect
66. C₂⁵×C₃	C₂×C₉	semidirect
69. C₂²×#67	C₂×C₉	semidirect
72. #953	C₂×C₉	direct (N × H)
73. C₂×#217	C₂×C₉	semidirect
96. C₂³×C₉	#67	semidirect
104. C₂×C₃×C₉	#120	semidirect
114. C₂⁴×C₃	C₂²×C₉	semidirect
117. C₂³×D₃	C₂²×C₉	semidirect
122. C₂×#67	C₂²×C₉	semidirect
123. C₂²×Dic₃	C₂²×C₉	semidirect
125. #217	C₂²×C₉	semidirect
147. C₂²×C₉	C₂×#67	semidirect
150. C₂⁵	D₃×C₉	semidirect
154. C₃×C₉	C₂×#120	semidirect
169. C₂³×C₃	D₄×C₉	semidirect
176. C₂²×D₃	C₂³×C₉	semidirect
182. C₂×Dic₃	C₂³×C₉	semidirect
192. C₂×C₉	#953	direct (N × H)
197. C₂⁴	C₂×D₃×C₉	semidirect
217. C₂²×C₃	C₂×D₄×C₉	semidirect
228. C₉	C₂×#953	direct (N × H)
234. C₂³	C₉×#67	semidirect
246. C₂×C₃	C₉×#120	semidirect
257. C₂²	C₂×C₉×#67	semidirect
261. C₃	C₂×C₉×#120	semidirect
265. C₂	C₉×#953	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁵×C₃×C₉	2
C₂²×C₉×#67	2
C₉×#953	2
C₂×C₉×#217	2
C₂×C₃×#953	3
C₂×C₉×#120	3

Subgroups — C2×C9×#953 #13153762

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#953 #13153762