Subgroups — C₂×C₉×#540 #11768173

Order 1152.

1890 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#540	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₉×#123	C₂	1	4	yes	maximal, complemented
5	C₂×C₉×#121	C₂	1	4	yes	maximal, complemented
6	C₂×C₉×#121	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×D₄×C₉	C₂	1	2	yes	maximal, complemented
8	C₂×C₉×#117	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃×#540	C₃	1	1	yes	maximal, characteristic
10	C₂²×D₄×C₉	C₂²	1	2	yes	—
11	C₂²×D₄×C₉	C₂²	1	4	yes	—
12	C₂²×D₄×C₉	C₂²	1	4	yes	—
13	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₉	C₂²	1	4	yes	—
15	C₉×#127	C₂²	1	8	yes	complemented
16	C₉×#125	C₂²	1	8	yes	complemented
17	C₉×#123	C₂²	1	32	yes	complemented
18	C₉×#121	C₂²	1	32	yes	complemented
19	C₉×#121	C₂²	1	16	yes	complemented
20	C₄×D₄×C₉	C₂²	1	16	yes	complemented
21	C₉×#117	C₂²	1	8	yes	complemented
22	C₂×C₉×C₄⋊C₄	C₂²	1	4	yes	—
23	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
24	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
25	C₂×C₉×C₂²⋊C₄	C₂²	1	2	yes	—
26	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
27	C₂×C₉×C₂²⋊C₄	C₂²	1	4	yes	—
28	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
29	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
30	C₃×#540	C₂×C₃	1	16	yes	—
31	C₂×C₃×#127	C₂×C₃	1	1	yes	characteristic
32	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
33	C₂×C₃×#123	C₂×C₃	1	4	yes	—
34	C₂×C₃×#121	C₂×C₃	1	4	yes	—
35	C₂×C₃×#121	C₂×C₃	1	2	yes	—
36	C₂×C₃×C₄×D₄	C₂×C₃	1	2	yes	—
37	C₂×C₃×#117	C₂×C₃	1	1	yes	characteristic
38	C₂⁴×C₉	C₂³	1	1	yes	characteristic
39	C₂⁴×C₉	C₂³	1	4	yes	—
40	C₂×D₄×C₉	—	2	8	no	—
41	C₂×D₄×C₉	—	4	8	no	—
42	C₂×D₄×C₉	—	4	8	no	—
43	C₂×D₄×C₉	C₂³	1	8	yes	—
44	C₂×D₄×C₉	C₂³	1	16	yes	—
45	C₂×D₄×C₉	C₂³	1	16	yes	—
46	C₂²×C₄×C₉	—	2	4	no	—
47	C₂²×C₄×C₉	—	4	8	no	—
48	C₂²×C₄×C₉	C₂³	1	4	yes	—
49	C₂²×C₄×C₉	C₂³	1	2	yes	—
50	C₂²×C₄×C₉	C₂³	1	2	yes	—
51	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₉	C₂³	1	4	yes	—
53	C₂²×C₄×C₉	C₂³	1	16	yes	—
54	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
55	C₉×C₄⋊C₄	C₂³	1	8	yes	—
56	C₉×C₄⋊C₄	C₂³	1	8	yes	—
57	C₉×C₄⋊C₄	C₂³	1	16	yes	—
58	C₉×C₂²⋊C₄	C₂³	1	8	yes	—
59	C₉×C₂²⋊C₄	C₂³	1	16	yes	—
60	C₉×C₂²⋊C₄	C₂³	1	16	yes	—
61	C₄²×C₉	C₂³	1	4	yes	—
62	C₄²×C₉	C₂³	1	4	yes	—
63	C₂×#540	C₉	1	1	yes	characteristic, direct product factor, Sylow
64	C₂²×C₃×D₄	C₂²×C₃	1	2	yes	—
65	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
66	C₂²×C₃×D₄	C₂²×C₃	1	4	yes	—
67	C₂³×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
68	C₂³×C₃×C₄	C₂²×C₃	1	4	yes	—
69	C₃×#127	C₂²×C₃	1	8	yes	—
70	C₃×#125	C₂²×C₃	1	8	yes	—
71	C₃×#123	C₂²×C₃	1	32	yes	—
72	C₃×#121	C₂²×C₃	1	32	yes	—
73	C₃×#121	C₂²×C₃	1	16	yes	—
74	C₃×C₄×D₄	C₂²×C₃	1	16	yes	—
75	C₃×#117	C₂²×C₃	1	8	yes	—
76	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
77	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
79	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
80	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
81	C₂×C₃×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
82	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
83	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
84	C₂³×C₉	—	2	2	no	—
85	C₂³×C₉	—	2	8	no	—
86	C₂³×C₉	—	2	8	no	—
87	C₂³×C₉	—	2	4	no	—
88	C₂³×C₉	—	2	8	no	—
89	C₂³×C₉	C₂⁴	1	2	yes	—
90	C₂³×C₉	C₂⁴	1	8	yes	—
91	C₂³×C₉	C₂⁴	1	1	yes	characteristic
92	D₄×C₉	—	2	16	no	—
93	D₄×C₉	—	4	16	no	—
94	D₄×C₉	—	4	16	no	—
95	C₂×C₄×C₉	—	2	2	no	—
96	C₂×C₄×C₉	—	2	4	no	—
97	C₂×C₄×C₉	—	2	8	no	—
98	C₂×C₄×C₉	—	2	4	no	—
99	C₂×C₄×C₉	—	2	8	no	—
100	C₂×C₄×C₉	—	4	16	no	—
101	C₂×C₄×C₉	#41	1	4	yes	—
102	C₂×C₄×C₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₉	C₂⁴	1	4	yes	—
104	C₂×C₄×C₉	C₂⁴	1	2	yes	—
105	C₂×C₄×C₉	C₂⁴	1	8	yes	—
106	C₂×C₄×C₉	C₂⁴	1	2	yes	—
107	#540	C₂×C₉	1	16	yes	direct product factor
108	C₂×#127	C₂×C₉	1	1	yes	characteristic, complemented
109	C₂×#125	C₂×C₉	1	1	yes	characteristic, complemented
110	C₂×#123	C₂×C₉	1	4	yes	complemented
111	C₂×#121	C₂×C₉	1	4	yes	complemented
112	C₂×#121	C₂×C₉	1	2	yes	complemented
113	C₂×C₄×D₄	C₂×C₉	1	2	yes	complemented
114	C₂×#117	C₂×C₉	1	1	yes	characteristic, complemented
115	C₂⁴×C₃	C₂³×C₃	1	1	yes	characteristic
116	C₂⁴×C₃	C₂³×C₃	1	4	yes	—
117	C₂×C₃×D₄	—	2	8	no	—
118	C₂×C₃×D₄	—	4	8	no	—
119	C₂×C₃×D₄	—	4	8	no	—
120	C₂×C₃×D₄	C₂³×C₃	1	8	yes	—
121	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
122	C₂×C₃×D₄	C₂³×C₃	1	16	yes	—
123	C₂²×C₃×C₄	—	2	4	no	—
124	C₂²×C₃×C₄	—	4	8	no	—
125	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
126	C₂²×C₃×C₄	C₂³×C₃	1	2	yes	—
127	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
128	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
129	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
130	C₂²×C₃×C₄	C₂³×C₃	1	16	yes	—
131	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
132	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
133	C₃×C₄⋊C₄	C₂³×C₃	1	8	yes	—
134	C₃×C₄⋊C₄	C₂³×C₃	1	16	yes	—
135	C₃×C₂²⋊C₄	C₂³×C₃	1	8	yes	—
136	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
137	C₃×C₂²⋊C₄	C₂³×C₃	1	16	yes	—
138	C₃×C₄²	C₂³×C₃	1	4	yes	—
139	C₃×C₄²	C₂³×C₃	1	4	yes	—
140	C₂²×C₉	—	2	2	no	—
141	C₂²×C₉	—	2	8	no	—
142	C₂²×C₉	—	2	8	no	—
143	C₂²×C₉	—	2	4	no	—
144	C₂²×C₉	—	2	8	no	—
145	C₂²×C₉	—	4	4	no	—
146	C₂²×C₉	—	4	16	no	—
147	C₂²×C₉	C₂×#41	1	2	yes	—
148	C₂²×C₉	H(5,𝔽₂)	1	4	yes	—
149	C₂²×C₉	C₂⁵	1	1	yes	characteristic
150	C₄×C₉	—	2	2	no	—
151	C₄×C₉	—	2	4	no	—
152	C₄×C₉	—	2	8	no	—
153	C₄×C₉	—	2	4	no	—
154	C₄×C₉	C₂×#41	1	4	yes	—
155	C₂²×D₄	C₂²×C₉	1	2	yes	—
156	C₂²×D₄	C₂²×C₉	1	4	yes	—
157	C₂²×D₄	C₂²×C₉	1	4	yes	—
158	C₂³×C₄	C₂²×C₉	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₉	1	4	yes	—
160	#127	C₂²×C₉	1	8	yes	complemented
161	#125	C₂²×C₉	1	8	yes	complemented
162	#123	C₂²×C₉	1	32	yes	complemented
163	#121	C₂²×C₉	1	32	yes	complemented
164	#121	C₂²×C₉	1	16	yes	complemented
165	C₄×D₄	C₂²×C₉	1	16	yes	complemented
166	#117	C₂²×C₉	1	8	yes	complemented
167	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
168	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
169	C₂×C₄⋊C₄	C₂²×C₉	1	4	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₉	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₉	1	4	yes	—
173	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
174	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
175	C₂³×C₃	—	2	2	no	—
176	C₂³×C₃	—	2	8	no	—
177	C₂³×C₃	—	2	8	no	—
178	C₂³×C₃	—	2	4	no	—
179	C₂³×C₃	—	2	8	no	—
180	C₂³×C₃	C₂⁴×C₃	1	2	yes	—
181	C₂³×C₃	C₂⁴×C₃	1	8	yes	—
182	C₂³×C₃	C₂⁴×C₃	1	1	yes	characteristic
183	C₃×D₄	—	2	16	no	—
184	C₃×D₄	—	4	16	no	—
185	C₃×D₄	—	4	16	no	—
186	C₂×C₃×C₄	—	2	2	no	—
187	C₂×C₃×C₄	—	2	4	no	—
188	C₂×C₃×C₄	—	2	8	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	8	no	—
191	C₂×C₃×C₄	—	4	16	no	—
192	C₂×C₃×C₄	C₃×#41	1	4	yes	—
193	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
194	C₂×C₃×C₄	C₂⁴×C₃	1	4	yes	—
195	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
196	C₂×C₃×C₄	C₂⁴×C₃	1	8	yes	—
197	C₂×C₃×C₄	C₂⁴×C₃	1	2	yes	—
198	C₂×C₉	—	4	2	no	—
199	C₂×C₉	—	4	8	no	—
200	C₂×C₉	#540	1	4	yes	direct product factor
201	C₂×C₉	C₂²×#41	1	1	yes	characteristic
202	C₂×C₉	C₂×H(5,𝔽₂)	1	2	yes	—
203	C₂⁴	C₂³×C₉	1	1	yes	characteristic
204	C₂⁴	C₂³×C₉	1	4	yes	—
205	C₂×D₄	—	2	8	no	—
206	C₂×D₄	—	4	8	no	—
207	C₂×D₄	—	4	8	no	—
208	C₂×D₄	C₂³×C₉	1	8	yes	—
209	C₂×D₄	C₂³×C₉	1	16	yes	—
210	C₂×D₄	C₂³×C₉	1	16	yes	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	4	8	no	—
213	C₂²×C₄	C₂³×C₉	1	2	yes	—
214	C₂²×C₄	C₂³×C₉	1	2	yes	—
215	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₉	1	4	yes	—
217	C₂²×C₄	C₂³×C₉	1	4	yes	—
218	C₂²×C₄	C₂³×C₉	1	16	yes	—
219	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
220	C₄⋊C₄	C₂³×C₉	1	8	yes	—
221	C₄⋊C₄	C₂³×C₉	1	8	yes	—
222	C₄⋊C₄	C₂³×C₉	1	16	yes	—
223	C₂²⋊C₄	C₂³×C₉	1	8	yes	—
224	C₂²⋊C₄	C₂³×C₉	1	16	yes	—
225	C₂²⋊C₄	C₂³×C₉	1	16	yes	—
226	C₄²	C₂³×C₉	1	4	yes	—
227	C₄²	C₂³×C₉	1	4	yes	—
228	C₂²×C₃	—	2	2	no	—
229	C₂²×C₃	—	2	8	no	—
230	C₂²×C₃	—	2	8	no	—
231	C₂²×C₃	—	2	4	no	—
232	C₂²×C₃	—	2	8	no	—
233	C₂²×C₃	—	4	4	no	—
234	C₂²×C₃	—	4	16	no	—
235	C₂²×C₃	C₂×C₃×#41	1	2	yes	—
236	C₂²×C₃	C₃×H(5,𝔽₂)	1	4	yes	—
237	C₂²×C₃	C₂⁵×C₃	1	1	yes	characteristic
238	C₃×C₄	—	2	2	no	—
239	C₃×C₄	—	2	4	no	—
240	C₃×C₄	—	2	8	no	—
241	C₃×C₄	—	2	4	no	—
242	C₃×C₄	C₂×C₃×#41	1	4	yes	—
243	C₉	C₂×#540	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	2	2	no	—
245	C₂³	—	2	8	no	—
246	C₂³	—	2	8	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	2	8	no	—
249	C₂³	C₂⁴×C₉	1	2	yes	—
250	C₂³	C₂⁴×C₉	1	8	yes	—
251	C₂³	C₂⁴×C₉	1	1	yes	characteristic
252	D₄	—	2	16	no	—
253	D₄	—	4	16	no	—
254	D₄	—	4	16	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	8	no	—
260	C₂×C₄	—	4	16	no	—
261	C₂×C₄	C₉×#41	1	4	yes	—
262	C₂×C₄	C₂⁴×C₉	1	4	yes	—
263	C₂×C₄	C₂⁴×C₉	1	4	yes	—
264	C₂×C₄	C₂⁴×C₉	1	2	yes	—
265	C₂×C₄	C₂⁴×C₉	1	8	yes	—
266	C₂×C₄	C₂⁴×C₉	1	2	yes	—
267	C₂×C₃	—	4	2	no	—
268	C₂×C₃	—	4	8	no	—
269	C₂×C₃	C₃×#540	1	4	yes	—
270	C₂×C₃	C₂²×C₃×#41	1	1	yes	characteristic
271	C₂×C₃	C₂×C₃×H(5,𝔽₂)	1	2	yes	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	8	no	—
274	C₂²	—	2	8	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	8	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	16	no	—
279	C₂²	C₂×C₉×#41	1	2	yes	—
280	C₂²	C₉×H(5,𝔽₂)	1	4	yes	—
281	C₂²	C₂⁵×C₉	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	—	2	8	no	—
285	C₄	—	2	4	no	—
286	C₄	C₂×C₉×#41	1	4	yes	—
287	C₃	C₂×C₃×#540	1	1	yes	characteristic
288	C₂	—	4	2	no	—
289	C₂	—	4	8	no	—
290	C₂	C₉×#540	1	4	yes	direct product factor
291	C₂	C₂²×C₉×#41	1	1	yes	characteristic
292	C₂	C₂×C₉×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center	C₂³×C₉	C₂⁴
commutator	C₂²	C₂⁵×C₉
Frattini, cosocle	C₂²×C₃	C₂⁵×C₃
socle	C₂³×C₃	C₂⁴×C₃
prime radical	C₂×C₃×#540	C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₉×#540	C₂	direct (N × H)
2. C₂×C₉×#127	C₂	semidirect
3. C₂×C₉×#125	C₂	semidirect
4. C₂×C₉×#123	C₂	semidirect
5. C₂×C₉×#121	C₂	semidirect
6. C₂×C₉×#121	C₂	semidirect
7. C₂×C₄×D₄×C₉	C₂	semidirect
8. C₂×C₉×#117	C₂	semidirect
15. C₉×#127	C₂²	semidirect
16. C₉×#125	C₂²	semidirect
17. C₉×#123	C₂²	semidirect
18. C₉×#121	C₂²	semidirect
19. C₉×#121	C₂²	semidirect
20. C₄×D₄×C₉	C₂²	semidirect
21. C₉×#117	C₂²	semidirect
63. C₂×#540	C₉	direct (N × H)
107. #540	C₂×C₉	direct (N × H)
108. C₂×#127	C₂×C₉	semidirect
109. C₂×#125	C₂×C₉	semidirect
110. C₂×#123	C₂×C₉	semidirect
111. C₂×#121	C₂×C₉	semidirect
112. C₂×#121	C₂×C₉	semidirect
113. C₂×C₄×D₄	C₂×C₉	semidirect
114. C₂×#117	C₂×C₉	semidirect
160. #127	C₂²×C₉	semidirect
161. #125	C₂²×C₉	semidirect
162. #123	C₂²×C₉	semidirect
163. #121	C₂²×C₉	semidirect
164. #121	C₂²×C₉	semidirect
165. C₄×D₄	C₂²×C₉	semidirect
166. #117	C₂²×C₉	semidirect
200. C₂×C₉	#540	direct (N × H)
243. C₉	C₂×#540	direct (N × H)
290. C₂	C₉×#540	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#540	2
C₂×C₉×#127	2
C₂×C₉×#125	2
C₂×C₉×#123	2
C₂×C₉×#121	2
C₂×C₄×D₄×C₉	2
C₂×C₉×#117	2
C₂×C₃×#540	3

Subgroups — C2×C9×#540 #11768173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#540 #11768173