Subgroups — #1450×D₈₉ #36520004

Order 22784.

982 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#431×D₈₉	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26018537	C₂	1	2	yes	maximal, complemented
4	#26018536	C₂	1	2	yes	maximal, complemented
5	#26018535	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018534	C₂	1	2	yes	maximal, complemented
7	#322×D₈₉	C₂	1	2	yes	maximal, complemented
8	C₈²×D₈₉	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₈₉	C₄	1	2	yes	—
11	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
12	#19151651	C₂²	1	2	yes	—
13	#19151650	C₄	1	4	yes	—
14	#19151649	C₂²	1	4	yes	—
15	#19151648	C₄	1	4	yes	—
16	C₄⋊C₈×D₈₉	C₄	1	4	yes	—
17	C₂²⋊C₈×D₈₉	C₄	1	4	yes	—
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
20	C₄×C₈×D₈₉	C₄	1	2	yes	—
21	C₄×C₈×D₈₉	C₄	1	2	yes	—
22	C₄×C₈×D₈₉	C₂²	1	4	yes	—
23	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
27	#19151320	C₂²	1	4	yes	complemented
28	#19151319	C₂²	1	2	yes	complemented
29	C₈×#10623494	C₂²	1	2	yes	complemented
30	C₈×#10623494	C₂²	1	4	yes	complemented
31	#431×C₈₉	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	#322×C₈₉	C₂²	1	2	yes	complemented
34	C₈²×C₈₉	C₂²	1	2	yes	complemented
35	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂²×#10623494	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₈	1	16	yes	complemented
39	C₂×C₈×D₈₉	C₂×C₄	1	8	yes	—
40	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
46	C₄²×D₈₉	C₂×C₄	1	2	yes	—
47	C₄²×D₈₉	C₂×C₄	1	2	yes	—
48	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
49	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₈₉	C₂³	1	1	yes	characteristic
51	C₄²×D₈₉	C₂³	1	1	yes	characteristic
52	C₄⋊#10623494	C₂×C₄	1	4	yes	—
53	#15667941	C₂×C₄	1	8	yes	—
54	C₂²⋊#10623494	C₂×C₄	1	4	yes	—
55	#15667924	C₂×C₄	1	8	yes	—
56	C₄×#10623494	C₂×C₄	1	2	yes	—
57	C₄×#10623494	C₂³	1	4	yes	—
58	C₄×#10623494	C₂³	1	1	yes	characteristic
59	C₄×#10623494	C₂³	1	1	yes	characteristic
60	C₈×Dic₈₉	C₂×C₄	1	8	yes	—
61	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
62	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
63	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
68	C₂²⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
70	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	4	yes	—
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂³×D₈₉	C₄²	1	1	yes	characteristic
74	C₂²×Dic₈₉	C₄²	1	2	yes	—
75	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
76	C₂×C₄×D₈₉	—	2	8	no	—
77	C₂×C₄×D₈₉	C₄²	1	2	yes	—
78	C₂×C₄×D₈₉	C₄²	1	2	yes	—
79	C₂×C₄×D₈₉	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₈₉	C₄²	1	1	yes	characteristic
81	C₂×C₄×D₈₉	C₂×C₈	1	8	yes	—
82	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
86	C₂×#10623494	C₂×C₈	1	8	yes	complemented
87	C₂×#10623494	C₂²×C₄	1	8	yes	—
88	C₂×#10623494	C₂²×C₄	1	2	yes	—
89	C₂×#10623494	C₂²×C₄	1	2	yes	—
90	C₈×D₈₉	—	2	16	no	—
91	C₈×D₈₉	C₂×C₈	1	8	yes	—
92	C₈×D₈₉	C₂×C₈	1	8	yes	—
93	C₈×D₈₉	C₂×C₈	1	8	yes	complemented
94	C₈×D₈₉	C₂×C₈	1	8	yes	complemented
95	C₄×Dic₈₉	C₄²	1	2	yes	—
96	C₄×Dic₈₉	C₄²	1	2	yes	—
97	C₄×Dic₈₉	C₄²	1	2	yes	—
98	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
100	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
101	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×C₈	1	8	yes	complemented
105	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
112	C₂²×D₈₉	—	2	4	no	—
113	C₂²×D₈₉	C₄×C₈	1	4	yes	—
114	C₂²×D₈₉	C₂×C₄²	1	1	yes	characteristic
115	C₂²×D₈₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₈₉	—	2	4	no	—
117	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
120	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
122	C₂×Dic₈₉	—	2	4	no	—
123	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
124	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
125	C₂×Dic₈₉	C₂×C₄²	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×C₄²	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
128	C₄×D₈₉	—	2	4	no	—
129	C₄×D₈₉	—	2	4	no	—
130	C₄×D₈₉	—	2	4	no	—
131	C₄×D₈₉	C₄×C₈	1	4	yes	—
132	C₄×D₈₉	C₄×C₈	1	4	yes	—
133	C₄×D₈₉	C₂²×C₈	1	4	yes	—
134	C₄×D₈₉	C₂²×C₈	1	4	yes	—
135	C₄×D₈₉	#131	1	8	yes	—
136	C₈×C₈₉	—	2	8	no	—
137	C₈×C₈₉	C₂²×C₈	1	4	yes	—
138	C₈×C₈₉	C₂²×C₈	1	4	yes	—
139	#10623494	—	2	8	no	—
140	#10623494	C₂²×C₈	1	4	yes	—
141	#10623494	C₂²×C₈	1	4	yes	—
142	C₂²×C₈₉	—	2	2	no	—
143	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
144	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
145	C₂×D₈₉	—	2	2	no	—
146	C₂×D₈₉	—	2	8	no	—
147	C₂×D₈₉	C₂×C₄×C₈	1	2	yes	—
148	C₂×D₈₉	C₂×C₄×C₈	1	2	yes	—
149	C₂×D₈₉	#405	1	4	yes	—
150	C₄×C₈₉	—	2	2	no	—
151	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
152	C₄×C₈₉	C₂³×C₈	1	2	yes	—
153	C₄×C₈₉	C₂×#131	1	4	yes	—
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	C₂×C₄×C₈	1	2	yes	—
156	Dic₈₉	C₂×C₄×C₈	1	2	yes	—
157	Dic₈₉	#405	1	4	yes	—
158	C₂×#1450	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#405	1	2	yes	—
162	D₈₉	—	2	2	no	—
163	D₈₉	#1450	1	4	yes	direct product factor
164	C₂×#431	—	89	2	no	—
165	C₂²×C₄×C₈	—	89	1	no	—
166	#1450	—	89	2	no	—
167	#1450	—	89	1	no	—
168	#1450	—	89	2	no	—
169	#1450	—	89	2	no	—
170	#1450	D₈₉	1	1	yes	characteristic, direct product factor
171	C₂×#322	—	89	2	no	—
172	C₂×C₈²	—	89	2	no	—
173	C₈₉	C₂×#1450	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₈	—	89	2	no	—
175	C₂²×C₄²	—	89	1	no	—
176	#431	—	89	2	no	—
177	#431	—	89	4	no	—
178	#431	—	89	4	no	—
179	#431	—	89	4	no	—
180	#431	C₂×D₈₉	1	2	yes	—
181	C₂×C₄⋊C₈	—	89	4	no	—
182	C₂×C₂²⋊C₈	—	89	4	no	—
183	C₂×C₄×C₈	—	89	2	no	—
184	C₂×C₄×C₈	—	89	1	no	—
185	C₂×C₄×C₈	—	89	2	no	—
186	C₂×C₄×C₈	—	89	2	no	—
187	C₂×C₄×C₈	—	89	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	C₂×C₄×C₈	—	89	4	no	—
190	C₂×C₄×C₈	—	89	1	no	—
191	C₂×C₄×C₈	—	89	1	no	—
192	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
193	#322	—	89	2	no	—
194	#322	—	89	4	no	—
195	#322	C₂×D₈₉	1	2	yes	complemented
196	C₈²	—	89	4	no	—
197	C₈²	—	89	2	no	—
198	C₈²	C₂×D₈₉	1	2	yes	complemented
199	C₂³×C₄	—	89	1	no	—
200	C₂³×C₄	—	89	2	no	—
201	C₂²×C₈	—	89	2	no	—
202	C₂²×C₈	—	89	16	no	—
203	C₂²×C₈	—	89	2	no	—
204	C₂²×C₈	—	89	2	no	—
205	C₂²×C₈	—	89	2	no	—
206	C₂²×C₈	—	89	2	no	—
207	C₂²×C₈	—	89	8	no	—
208	C₂²×C₈	C₄×D₈₉	1	2	yes	—
209	C₂×C₄²	—	89	1	no	—
210	C₂×C₄²	—	89	2	no	—
211	C₂×C₄²	—	89	1	no	—
212	C₂×C₄²	—	89	1	no	—
213	C₂×C₄²	—	89	2	no	—
214	C₂×C₄²	—	89	2	no	—
215	C₂×C₄²	—	89	1	no	—
216	C₂×C₄²	—	89	1	no	—
217	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
218	C₄⋊C₈	—	89	8	no	—
219	C₄⋊C₈	—	89	4	no	—
220	C₄⋊C₈	C₄×D₈₉	1	4	yes	—
221	C₂²⋊C₈	—	89	4	no	—
222	C₂²⋊C₈	—	89	8	no	—
223	C₂²⋊C₈	C₄×D₈₉	1	4	yes	—
224	C₄×C₈	—	89	2	no	—
225	C₄×C₈	—	89	2	no	—
226	C₄×C₈	—	89	2	no	—
227	C₄×C₈	—	89	2	no	—
228	C₄×C₈	—	89	8	no	—
229	C₄×C₈	—	89	2	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	—	89	4	no	—
232	C₄×C₈	—	89	1	no	—
233	C₄×C₈	C₄×D₈₉	1	2	yes	—
234	C₄×C₈	C₂²×D₈₉	1	4	yes	—
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂⁴	—	89	1	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	89	8	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	89	2	no	—
243	C₂²×C₄	—	89	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	2	no	—
246	C₂²×C₄	—	89	2	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	—	89	2	no	—
249	C₂²×C₄	—	178	8	no	—
250	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂×C₈	—	89	8	no	—
253	C₂×C₈	—	89	2	no	—
254	C₂×C₈	—	89	2	no	—
255	C₂×C₈	—	89	8	no	—
256	C₂×C₈	—	89	8	no	—
257	C₂×C₈	—	89	8	no	—
258	C₂×C₈	—	89	8	no	—
259	C₂×C₈	—	89	8	no	—
260	C₂×C₈	—	178	16	no	—
261	C₂×C₈	C₈×D₈₉	1	8	yes	complemented
262	C₂×C₈	C₂×C₄×D₈₉	1	8	yes	—
263	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
264	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
265	C₄²	—	89	1	no	—
266	C₄²	—	89	2	no	—
267	C₄²	—	89	1	no	—
268	C₄²	—	89	2	no	—
269	C₄²	—	89	2	no	—
270	C₄²	—	89	2	no	—
271	C₄²	—	89	2	no	—
272	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₄²	C₂×C₄×D₈₉	1	2	yes	—
274	C₄²	C₂³×D₈₉	1	1	yes	characteristic
275	C₂³	—	89	1	no	—
276	C₂³	—	89	1	no	—
277	C₂³	—	89	4	no	—
278	C₂³	—	178	4	no	—
279	C₂³	C₄²×D₈₉	1	1	yes	characteristic
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	89	1	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	89	4	no	—
285	C₂×C₄	—	89	4	no	—
286	C₂×C₄	—	89	4	no	—
287	C₂×C₄	—	89	4	no	—
288	C₂×C₄	—	89	4	no	—
289	C₂×C₄	—	89	8	no	—
290	C₂×C₄	—	89	2	no	—
291	C₂×C₄	—	178	4	no	—
292	C₂×C₄	—	178	4	no	—
293	C₂×C₄	—	178	4	no	—
294	C₂×C₄	—	178	4	no	—
295	C₂×C₄	C₄²×D₈₉	1	2	yes	—
296	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
298	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
300	C₈	—	2	8	no	—
301	C₈	—	89	4	no	—
302	C₈	—	89	4	no	—
303	C₈	—	178	8	no	—
304	C₈	C₂×C₈×D₈₉	1	4	yes	—
305	C₈	C₂×C₈×D₈₉	1	4	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	89	2	no	—
308	C₂²	—	89	4	no	—
309	C₂²	—	89	2	no	—
310	C₂²	—	178	8	no	—
311	C₂²	—	178	2	no	—
312	C₂²	C₄×C₈×D₈₉	1	2	yes	—
313	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	89	2	no	—
316	C₄	—	89	2	no	—
317	C₄	—	89	4	no	—
318	C₄	—	178	2	no	—
319	C₄	C₄×C₈×D₈₉	1	2	yes	—
320	C₄	C₂²×C₈×D₈₉	1	2	yes	—
321	C₄	#131×D₈₉	1	4	yes	—
322	C₂	—	2	2	no	—
323	C₂	—	89	4	no	—
324	C₂	—	178	2	no	—
325	C₂	C₂×C₄×C₈×D₈₉	1	1	yes	characteristic
326	C₂	#405×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1450	C₂
socle	C₂²×C₈₉	C₂²×C₄²
prime radical	C₂³×D₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1450	D₈₉
hypocenter	C₈₉	C₂×#1450
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #26018537	C₂	semidirect
4. #26018536	C₂	semidirect
5. #26018535	C₂	semidirect
6. #26018534	C₂	semidirect
7. #322×D₈₉	C₂	semidirect
8. C₈²×D₈₉	C₂	semidirect
9. C₈₉×#1450	C₂	semidirect
27. #19151320	C₂²	semidirect
28. #19151319	C₂²	semidirect
29. C₈×#10623494	C₂²	semidirect
30. C₈×#10623494	C₂²	semidirect
33. #322×C₈₉	C₂²	semidirect
34. C₈²×C₈₉	C₂²	semidirect
38. C₂×C₈×D₈₉	C₈	semidirect
86. C₂×#10623494	C₂×C₈	semidirect
93. C₈×D₈₉	C₂×C₈	semidirect
94. C₈×D₈₉	C₂×C₈	semidirect
104. C₂×C₈×C₈₉	C₂×C₈	semidirect
163. D₈₉	#1450	direct (N × H)
170. #1450	D₈₉	direct (N × H)
173. C₈₉	C₂×#1450	semidirect
195. #322	C₂×D₈₉	semidirect
198. C₈²	C₂×D₈₉	semidirect
261. C₂×C₈	C₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#431×D₈₉	2
C₂×C₄×C₈×D₈₉	2
#26018537	2
#26018536	2
#26018535	2
#26018534	2
#322×D₈₉	2
C₈²×D₈₉	2
C₈₉×#1450	2
C₂×#1450	89

Subgroups — #1450×D89 #36520004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1450×D₈₉ #36520004