Subgroups — C₂×C₉×#455 #11767736

Order 1152.

1140 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#137	C₂	1	2	yes	maximal, complemented
3	C₂×C₉×#124	C₂	1	1	yes	maximal, characteristic, complemented
4	C₉×#455	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₉×#104	C₂	1	2	yes	maximal, complemented
6	C₂×C₉×#100	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×#455	C₃	1	1	yes	maximal, characteristic
8	C₉×H(5,𝔽₂)	—	2	4	no	—
9	C₉×H(5,𝔽₂)	—	2	2	no	—
10	C₉×H(5,𝔽₂)	C₂²	1	4	yes	—
11	C₂×C₉×#41	—	2	2	no	—
12	C₂×C₉×#41	C₂²	1	2	yes	—
13	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic, complemented
14	C₂²×D₄×C₉	—	2	2	no	—
15	C₂²×D₄×C₉	—	2	2	no	—
16	C₂²×D₄×C₉	C₂²	1	1	yes	characteristic
17	C₉×#137	—	2	4	no	—
18	C₉×#137	C₂²	1	8	yes	complemented
19	C₂×C₉×Q₁₆	—	2	2	no	—
20	C₂×C₉×SD₁₆	—	2	2	no	—
21	C₂×C₉×M₁₆	C₂²	1	2	yes	complemented
22	C₉×#124	—	2	2	no	—
23	C₉×#124	C₂²	1	4	yes	complemented
24	C₂×C₉×C₂²⋊C₄	—	2	2	no	—
25	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic, complemented
26	C₉×#104	C₂²	1	8	yes	complemented
27	C₉×#100	C₂²	1	4	yes	complemented
28	C₂×C₃×H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic
29	C₂×C₃×#137	C₂×C₃	1	2	yes	—
30	C₂×C₃×#124	C₂×C₃	1	1	yes	characteristic
31	C₃×#455	C₂×C₃	1	8	yes	—
32	C₂×C₃×#104	C₂×C₃	1	2	yes	—
33	C₂×C₃×#100	C₂×C₃	1	1	yes	characteristic
34	C₂⁴×C₉	—	2	2	no	—
35	C₂⁴×C₉	D₄	1	2	yes	—
36	C₉×#41	—	2	8	no	—
37	C₉×#41	—	2	8	no	—
38	C₉×#41	—	2	2	no	—
39	C₉×#41	—	2	2	no	—
40	C₉×#41	D₄	1	4	yes	—
41	C₉×#41	C₂³	1	4	yes	—
42	C₂×Q₈×C₉	—	2	1	no	—
43	C₂×Q₈×C₉	—	2	2	no	—
44	C₂×Q₈×C₉	—	2	2	no	—
45	C₂×Q₈×C₉	D₄	1	2	yes	—
46	C₂×Q₈×C₉	C₂³	1	2	yes	complemented
47	C₂×D₄×C₉	—	2	4	no	—
48	C₂×D₄×C₉	—	2	8	no	—
49	C₂×D₄×C₉	—	2	8	no	—
50	C₂×D₄×C₉	—	2	2	no	—
51	C₂×D₄×C₉	—	2	8	no	—
52	C₂×D₄×C₉	—	2	8	no	—
53	C₂×D₄×C₉	—	2	2	no	—
54	C₂×D₄×C₉	—	2	2	no	—
55	C₂×D₄×C₉	D₄	1	2	yes	—
56	C₂×D₄×C₉	D₄	1	2	yes	—
57	C₂×D₄×C₉	C₂³	1	2	yes	—
58	C₂²×C₄×C₉	—	2	2	no	—
59	C₂²×C₄×C₉	—	2	2	no	—
60	C₂²×C₄×C₉	—	2	1	no	—
61	C₂²×C₄×C₉	—	2	1	no	—
62	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
63	C₉×Q₁₆	—	2	8	no	—
64	C₉×SD₁₆	—	2	8	no	—
65	C₉×M₁₆	—	2	2	no	—
66	C₉×M₁₆	C₂³	1	4	yes	complemented
67	C₂×C₈×C₉	—	2	2	no	—
68	C₉×C₂²⋊C₄	—	2	8	no	—
69	C₄²×C₉	—	2	1	no	—
70	C₄²×C₉	C₂³	1	2	yes	complemented
71	C₂×#455	C₉	1	1	yes	characteristic, direct product factor, Sylow
72	C₃×H(5,𝔽₂)	—	2	4	no	—
73	C₃×H(5,𝔽₂)	—	2	2	no	—
74	C₃×H(5,𝔽₂)	C₂²×C₃	1	4	yes	—
75	C₂×C₃×#41	—	2	2	no	—
76	C₂×C₃×#41	C₂²×C₃	1	2	yes	—
77	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
78	C₂²×C₃×D₄	—	2	2	no	—
79	C₂²×C₃×D₄	—	2	2	no	—
80	C₂²×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
81	C₃×#137	—	2	4	no	—
82	C₃×#137	C₂²×C₃	1	8	yes	—
83	C₂×C₃×Q₁₆	—	2	2	no	—
84	C₂×C₃×SD₁₆	—	2	2	no	—
85	C₂×C₃×M₁₆	C₂²×C₃	1	2	yes	—
86	C₃×#124	—	2	2	no	—
87	C₃×#124	C₂²×C₃	1	4	yes	—
88	C₂×C₃×C₂²⋊C₄	—	2	2	no	—
89	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
90	C₃×#104	C₂²×C₃	1	8	yes	—
91	C₃×#100	C₂²×C₃	1	4	yes	—
92	C₂³×C₉	—	2	8	no	—
93	C₂³×C₉	—	2	2	no	—
94	C₂³×C₉	—	2	2	no	—
95	C₂³×C₉	—	2	2	no	—
96	C₂³×C₉	—	4	4	no	—
97	C₂³×C₉	—	8	4	no	—
98	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₉	C₂×D₄	1	4	yes	—
100	Q₈×C₉	—	2	2	no	—
101	Q₈×C₉	—	2	4	no	—
102	Q₈×C₉	—	4	2	no	—
103	Q₈×C₉	C₂×D₄	1	4	yes	—
104	D₄×C₉	—	2	16	no	—
105	D₄×C₉	—	2	4	no	—
106	D₄×C₉	—	2	4	no	—
107	D₄×C₉	—	2	4	no	—
108	D₄×C₉	—	2	2	no	—
109	D₄×C₉	—	2	4	no	—
110	D₄×C₉	C₂×D₄	1	4	yes	—
111	C₂×C₄×C₉	—	2	8	no	—
112	C₂×C₄×C₉	—	2	2	no	—
113	C₂×C₄×C₉	—	2	4	no	—
114	C₂×C₄×C₉	—	2	4	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	4	2	no	—
118	C₂×C₄×C₉	—	4	1	no	—
119	C₂×C₄×C₉	—	4	1	no	—
120	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
123	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
124	C₈×C₉	—	2	4	no	—
125	C₂×H(5,𝔽₂)	C₂×C₉	1	1	yes	characteristic
126	C₂×#137	C₂×C₉	1	2	yes	complemented
127	C₂×#124	C₂×C₉	1	1	yes	characteristic, complemented
128	#455	C₂×C₉	1	8	yes	direct product factor
129	C₂×#104	C₂×C₉	1	2	yes	complemented
130	C₂×#100	C₂×C₉	1	1	yes	characteristic, complemented
131	C₂⁴×C₃	—	2	2	no	—
132	C₂⁴×C₃	C₃×D₄	1	2	yes	—
133	C₃×#41	—	2	8	no	—
134	C₃×#41	—	2	8	no	—
135	C₃×#41	—	2	2	no	—
136	C₃×#41	—	2	2	no	—
137	C₃×#41	C₃×D₄	1	4	yes	—
138	C₃×#41	C₂³×C₃	1	4	yes	—
139	C₂×C₃×Q₈	—	2	1	no	—
140	C₂×C₃×Q₈	—	2	2	no	—
141	C₂×C₃×Q₈	—	2	2	no	—
142	C₂×C₃×Q₈	C₃×D₄	1	2	yes	—
143	C₂×C₃×Q₈	C₂³×C₃	1	2	yes	—
144	C₂×C₃×D₄	—	2	4	no	—
145	C₂×C₃×D₄	—	2	8	no	—
146	C₂×C₃×D₄	—	2	8	no	—
147	C₂×C₃×D₄	—	2	2	no	—
148	C₂×C₃×D₄	—	2	8	no	—
149	C₂×C₃×D₄	—	2	8	no	—
150	C₂×C₃×D₄	—	2	2	no	—
151	C₂×C₃×D₄	—	2	2	no	—
152	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
153	C₂×C₃×D₄	C₃×D₄	1	2	yes	—
154	C₂×C₃×D₄	C₂³×C₃	1	2	yes	—
155	C₂²×C₃×C₄	—	2	2	no	—
156	C₂²×C₃×C₄	—	2	2	no	—
157	C₂²×C₃×C₄	—	2	1	no	—
158	C₂²×C₃×C₄	—	2	1	no	—
159	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
160	C₃×Q₁₆	—	2	8	no	—
161	C₃×SD₁₆	—	2	8	no	—
162	C₃×M₁₆	—	2	2	no	—
163	C₃×M₁₆	C₂³×C₃	1	4	yes	—
164	C₂×C₃×C₈	—	2	2	no	—
165	C₃×C₂²⋊C₄	—	2	8	no	—
166	C₃×C₄²	—	2	1	no	—
167	C₃×C₄²	C₂³×C₃	1	2	yes	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	4	no	—
170	C₂²×C₉	—	2	4	no	—
171	C₂²×C₉	—	4	4	no	—
172	C₂²×C₉	—	4	4	no	—
173	C₂²×C₉	—	4	4	no	—
174	C₂²×C₉	—	8	2	no	—
175	C₂²×C₉	—	8	8	no	—
176	C₂²×C₉	#120	1	1	yes	characteristic
177	C₂²×C₉	#120	1	1	yes	characteristic
178	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
179	C₄×C₉	—	2	4	no	—
180	C₄×C₉	—	4	2	no	—
181	C₄×C₉	—	4	2	no	—
182	C₄×C₉	#120	1	2	yes	—
183	C₄×C₉	C₂²×D₄	1	2	yes	—
184	H(5,𝔽₂)	—	2	4	no	—
185	H(5,𝔽₂)	—	2	2	no	—
186	H(5,𝔽₂)	C₂²×C₉	1	4	yes	—
187	C₂×#41	—	2	2	no	—
188	C₂×#41	C₂²×C₉	1	2	yes	—
189	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic, complemented
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂²×D₄	C₂²×C₉	1	1	yes	characteristic
193	#137	—	2	4	no	—
194	#137	C₂²×C₉	1	8	yes	complemented
195	C₂×Q₁₆	—	2	2	no	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×M₁₆	C₂²×C₉	1	2	yes	complemented
198	#124	—	2	2	no	—
199	#124	C₂²×C₉	1	4	yes	complemented
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic, complemented
202	#104	C₂²×C₉	1	8	yes	complemented
203	#100	C₂²×C₉	1	4	yes	complemented
204	C₂³×C₃	—	2	8	no	—
205	C₂³×C₃	—	2	2	no	—
206	C₂³×C₃	—	2	2	no	—
207	C₂³×C₃	—	2	2	no	—
208	C₂³×C₃	—	4	4	no	—
209	C₂³×C₃	—	8	4	no	—
210	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂³×C₃	C₂×C₃×D₄	1	4	yes	—
212	C₃×Q₈	—	2	2	no	—
213	C₃×Q₈	—	2	4	no	—
214	C₃×Q₈	—	4	2	no	—
215	C₃×Q₈	C₂×C₃×D₄	1	4	yes	—
216	C₃×D₄	—	2	16	no	—
217	C₃×D₄	—	2	4	no	—
218	C₃×D₄	—	2	4	no	—
219	C₃×D₄	—	2	4	no	—
220	C₃×D₄	—	2	2	no	—
221	C₃×D₄	—	2	4	no	—
222	C₃×D₄	C₂×C₃×D₄	1	4	yes	—
223	C₂×C₃×C₄	—	2	8	no	—
224	C₂×C₃×C₄	—	2	2	no	—
225	C₂×C₃×C₄	—	2	4	no	—
226	C₂×C₃×C₄	—	2	4	no	—
227	C₂×C₃×C₄	—	2	2	no	—
228	C₂×C₃×C₄	—	2	2	no	—
229	C₂×C₃×C₄	—	4	2	no	—
230	C₂×C₃×C₄	—	4	1	no	—
231	C₂×C₃×C₄	—	4	1	no	—
232	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
234	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
235	C₂×C₃×C₄	C₂⁴×C₃	1	1	yes	characteristic
236	C₃×C₈	—	2	4	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	4	4	no	—
240	C₂×C₉	—	4	4	no	—
241	C₂×C₉	#455	1	2	yes	direct product factor
242	C₂×C₉	C₂×#120	1	1	yes	characteristic
243	C₂⁴	—	2	2	no	—
244	C₂⁴	D₄×C₉	1	2	yes	—
245	#41	—	2	8	no	—
246	#41	—	2	8	no	—
247	#41	—	2	2	no	—
248	#41	—	2	2	no	—
249	#41	D₄×C₉	1	4	yes	—
250	#41	C₂³×C₉	1	4	yes	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	D₄×C₉	1	2	yes	—
255	C₂×Q₈	C₂³×C₉	1	2	yes	complemented
256	C₂×D₄	—	2	4	no	—
257	C₂×D₄	—	2	8	no	—
258	C₂×D₄	—	2	8	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	8	no	—
261	C₂×D₄	—	2	8	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	D₄×C₉	1	2	yes	—
265	C₂×D₄	D₄×C₉	1	2	yes	—
266	C₂×D₄	C₂³×C₉	1	2	yes	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
272	Q₁₆	—	2	8	no	—
273	SD₁₆	—	2	8	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	C₂³×C₉	1	4	yes	complemented
276	C₂×C₈	—	2	2	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₄²	—	2	1	no	—
279	C₄²	C₂³×C₉	1	2	yes	complemented
280	C₂²×C₃	—	2	2	no	—
281	C₂²×C₃	—	2	4	no	—
282	C₂²×C₃	—	2	4	no	—
283	C₂²×C₃	—	4	4	no	—
284	C₂²×C₃	—	4	4	no	—
285	C₂²×C₃	—	4	4	no	—
286	C₂²×C₃	—	8	2	no	—
287	C₂²×C₃	—	8	8	no	—
288	C₂²×C₃	C₃×#120	1	1	yes	characteristic
289	C₂²×C₃	C₃×#120	1	1	yes	characteristic
290	C₂²×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
291	C₃×C₄	—	2	4	no	—
292	C₃×C₄	—	4	2	no	—
293	C₃×C₄	—	4	2	no	—
294	C₃×C₄	C₃×#120	1	2	yes	—
295	C₃×C₄	C₂²×C₃×D₄	1	2	yes	—
296	C₉	C₂×#455	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³	—	2	8	no	—
298	C₂³	—	2	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	8	4	no	—
303	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
304	C₂³	C₂×D₄×C₉	1	4	yes	—
305	Q₈	—	2	2	no	—
306	Q₈	—	2	4	no	—
307	Q₈	—	4	2	no	—
308	Q₈	C₂×D₄×C₉	1	4	yes	—
309	D₄	—	2	16	no	—
310	D₄	—	2	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	4	no	—
313	D₄	—	2	2	no	—
314	D₄	—	2	4	no	—
315	D₄	C₂×D₄×C₉	1	4	yes	—
316	C₂×C₄	—	2	8	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	4	no	—
319	C₂×C₄	—	2	4	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	C₂×D₄×C₉	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
327	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
328	C₂×C₄	C₂⁴×C₉	1	1	yes	characteristic
329	C₈	—	2	4	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	4	4	no	—
333	C₂×C₃	—	4	4	no	—
334	C₂×C₃	C₃×#455	1	2	yes	—
335	C₂×C₃	C₂×C₃×#120	1	1	yes	characteristic
336	C₂²	—	2	2	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	2	4	no	—
339	C₂²	—	4	4	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	8	no	—
344	C₂²	C₉×#120	1	1	yes	characteristic
345	C₂²	C₉×#120	1	1	yes	characteristic
346	C₂²	C₂²×D₄×C₉	1	1	yes	characteristic
347	C₄	—	2	4	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	C₉×#120	1	2	yes	—
351	C₄	C₂²×D₄×C₉	1	2	yes	—
352	C₃	C₂×C₃×#455	1	1	yes	characteristic
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	—	4	4	no	—
356	C₂	—	4	4	no	—
357	C₂	C₉×#455	1	2	yes	direct product factor
358	C₂	C₂×C₉×#120	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	#120
commutator	C₂×C₄	C₂⁴×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂⁴×C₃
socle	C₂²×C₃	C₃×#120
prime radical	C₂×C₃×H(5,𝔽₂)	C₂×C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₉×#137	C₂	semidirect
3. C₂×C₉×#124	C₂	semidirect
4. C₉×#455	C₂	direct (N × H)
5. C₂×C₉×#104	C₂	semidirect
6. C₂×C₉×#100	C₂	semidirect
13. C₂²×Q₈×C₉	C₂²	semidirect
18. C₉×#137	C₂²	semidirect
21. C₂×C₉×M₁₆	C₂²	semidirect
23. C₉×#124	C₂²	semidirect
25. C₂×C₄²×C₉	C₂²	semidirect
26. C₉×#104	C₂²	semidirect
27. C₉×#100	C₂²	semidirect
46. C₂×Q₈×C₉	C₂³	semidirect
66. C₉×M₁₆	C₂³	semidirect
70. C₄²×C₉	C₂³	semidirect
71. C₂×#455	C₉	direct (N × H)
126. C₂×#137	C₂×C₉	semidirect
127. C₂×#124	C₂×C₉	semidirect
128. #455	C₂×C₉	direct (N × H)
129. C₂×#104	C₂×C₉	semidirect
130. C₂×#100	C₂×C₉	semidirect
189. C₂²×Q₈	C₂²×C₉	semidirect
194. #137	C₂²×C₉	semidirect
197. C₂×M₁₆	C₂²×C₉	semidirect
199. #124	C₂²×C₉	semidirect
201. C₂×C₄²	C₂²×C₉	semidirect
202. #104	C₂²×C₉	semidirect
203. #100	C₂²×C₉	semidirect
241. C₂×C₉	#455	direct (N × H)
255. C₂×Q₈	C₂³×C₉	semidirect
275. M₁₆	C₂³×C₉	semidirect
279. C₄²	C₂³×C₉	semidirect
296. C₉	C₂×#455	direct (N × H)
357. C₂	C₉×#455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₉×H(5,𝔽₂)	2
C₂×C₉×#137	2
C₂×C₉×#124	2
C₉×#455	2
C₂×C₉×#104	2
C₂×C₉×#100	2
C₂×C₃×#455	3

Subgroups — C2×C9×#455 #11767736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#455 #11767736