Subgroups — #2110×D₈₉ #36707083

Order 22784.

738 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₈₉	C₂	1	2	yes	maximal
2	#405×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021857	C₂	1	1	yes	maximal, characteristic, complemented
4	#26021856	C₂	1	2	yes	maximal, complemented
5	#26021855	C₂	1	4	yes	maximal, complemented
6	#347×D₈₉	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	#117×D₈₉	C₄	1	1	yes	characteristic
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₂×#15667951	C₄	1	4	yes	complemented
13	M₃₂×D₈₉	C₄	1	4	yes	complemented
14	M₃₂×D₈₉	C₄	1	4	yes	complemented
15	M₃₂×D₈₉	C₂²	1	4	yes	—
16	M₃₂×D₈₉	C₂²	1	4	yes	—
17	C₂×C₁₆×D₈₉	C₄	1	4	yes	complemented
18	#19151556	C₄	1	1	yes	characteristic
19	#19151555	C₂²	1	2	yes	—
20	#19151554	C₄	1	1	yes	characteristic
21	#19151553	C₂²	1	1	yes	characteristic
22	#19151552	C₄	1	2	yes	—
23	#97×D₈₉	C₂²	1	2	yes	—
24	C₄×C₈×D₈₉	C₄	1	2	yes	—
25	#19151376	C₂²	1	4	yes	complemented
26	#19151375	C₂²	1	8	yes	complemented
27	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
28	#405×C₈₉	C₂²	1	1	yes	characteristic
29	#347×C₈₉	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083557	C₂×C₄	1	2	yes	—
33	M₁₆×D₈₉	—	2	2	no	—
34	M₁₆×D₈₉	—	2	2	no	—
35	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₈₉	C₂×C₄	1	2	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	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15667971	C₂×C₄	1	1	yes	characteristic
46	#15667970	—	2	1	no	—
47	#15667969	C₂×C₄	1	2	yes	—
48	#15667968	—	2	1	no	—
49	C₄⋊C₄×D₈₉	—	2	1	no	—
50	C₂²⋊C₄×D₈₉	—	2	1	no	—
51	C₄²×D₈₉	C₂×C₄	1	2	yes	—
52	#15667952	C₂×C₄	1	4	yes	complemented
53	#15667952	C₂³	1	4	yes	—
54	#15667951	C₂×C₄	1	8	yes	—
55	#15667951	C₂×C₄	1	8	yes	complemented
56	C₂×#13083549	C₂×C₄	1	4	yes	complemented
57	C₁₆×D₈₉	C₂×C₄	1	8	yes	—
58	C₁₆×D₈₉	C₂×C₄	1	8	yes	complemented
59	#15667923	C₂×C₄	1	2	yes	—
60	#15667922	C₂×C₄	1	2	yes	—
61	#15667922	C₂³	1	2	yes	—
62	C₄×#10623494	C₂×C₄	1	2	yes	—
63	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
69	M₃₂×C₈₉	C₂³	1	4	yes	—
70	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
71	#97×C₈₉	C₂³	1	2	yes	—
72	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
73	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
74	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	C₄²	1	2	yes	—
77	C₂×C₄×D₈₉	C₄²	1	2	yes	—
78	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
79	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₈₉	M₁₆	1	2	yes	—
81	C₂×C₄×D₈₉	M₁₆	1	2	yes	—
82	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
84	#13083558	—	2	2	no	—
85	#13083557	—	2	4	no	—
86	C₂×#10623494	C₄²	1	2	yes	—
87	C₂×#10623494	C₄²	1	2	yes	—
88	C₂×#10623494	C₂²×C₄	1	2	yes	—
89	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
91	C₈×D₈₉	—	2	2	no	—
92	C₈×D₈₉	—	2	2	no	—
93	C₈×D₈₉	C₄²	1	1	yes	characteristic
94	C₈×D₈₉	C₄²	1	1	yes	characteristic
95	C₈×D₈₉	C₄²	1	2	yes	—
96	C₈×D₈₉	C₄²	1	1	yes	characteristic
97	C₈×D₈₉	C₄²	1	1	yes	characteristic
98	C₈×D₈₉	C₄²	1	2	yes	—
99	C₈×D₈₉	C₂²×C₄	1	2	yes	—
100	C₈×D₈₉	C₂²×C₄	1	2	yes	—
101	C₈×D₈₉	C₂²×C₄	1	2	yes	—
102	C₈×D₈₉	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₈₉	—	2	1	no	—
104	#13083553	—	2	2	no	—
105	C₂²⋊Dic₈₉	—	2	1	no	—
106	#13083551	—	2	2	no	—
107	C₄×Dic₈₉	—	2	2	no	—
108	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
109	#13083549	C₂²×C₄	1	8	yes	—
110	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₂²⋊C₄×C₈₉	—	2	1	no	—
119	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
120	C₁₆×C₈₉	C₂²×C₄	1	8	yes	—
121	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₈₉	—	4	2	no	—
123	C₂²×D₈₉	#97	1	2	yes	—
124	C₂²×D₈₉	#97	1	2	yes	—
125	C₂²×D₈₉	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₈₉	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	#97	1	2	yes	—
134	C₂×Dic₈₉	#97	1	2	yes	—
135	C₂×Dic₈₉	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₈₉	—	4	4	no	—
138	C₄×D₈₉	#97	1	2	yes	—
139	C₄×D₈₉	#97	1	2	yes	—
140	C₄×D₈₉	#97	1	2	yes	—
141	C₄×D₈₉	#97	1	2	yes	—
142	C₄×D₈₉	C₂×C₄²	1	2	yes	—
143	C₄×D₈₉	C₂×C₄²	1	1	yes	characteristic
144	C₄×D₈₉	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₈₉	C₂×M₁₆	1	1	yes	characteristic
146	C₄×D₈₉	C₂×M₁₆	1	1	yes	characteristic
147	C₄×D₈₉	C₂×M₁₆	1	2	yes	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
150	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
151	C₈×C₈₉	C₂³×C₄	1	2	yes	—
152	#10623494	—	2	2	no	—
153	#10623494	C₂×C₄²	1	2	yes	—
154	#10623494	C₂×C₄²	1	1	yes	characteristic
155	#10623494	C₂×C₄²	1	1	yes	characteristic
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	C₂×#97	1	2	yes	—
158	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	2	4	no	—
161	C₂×D₈₉	—	4	2	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	C₂×#97	1	1	yes	characteristic
164	C₂×D₈₉	C₂×#97	1	2	yes	—
165	C₂×D₈₉	C₂×#97	1	1	yes	characteristic
166	C₄×C₈₉	—	4	2	no	—
167	C₄×C₈₉	C₂×#97	1	2	yes	—
168	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
169	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	Dic₈₉	C₂×#97	1	1	yes	characteristic
172	Dic₈₉	C₂×#97	1	1	yes	characteristic
173	Dic₈₉	C₂×#97	1	2	yes	—
174	C₂×#2110	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	2	1	no	—
176	C₂×C₈₉	—	2	2	no	—
177	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
178	D₈₉	—	2	1	no	—
179	D₈₉	—	2	2	no	—
180	D₈₉	#2110	1	2	yes	direct product factor
181	C₂²×M₃₂	—	89	2	no	—
182	C₂×#405	—	89	1	no	—
183	#2110	—	89	2	no	—
184	#2110	—	89	1	no	—
185	#2110	—	89	4	no	—
186	#2110	D₈₉	1	1	yes	characteristic, direct product factor
187	C₂×#347	—	89	4	no	—
188	C₈₉	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
189	C₂²×M₁₆	—	89	1	no	—
190	C₂³×C₈	—	89	1	no	—
191	C₂×#117	—	89	1	no	—
192	C₂×M₃₂	—	89	2	no	—
193	C₂×M₃₂	—	89	4	no	—
194	C₂×M₃₂	—	89	4	no	—
195	C₂×M₃₂	—	89	4	no	—
196	C₂×M₃₂	—	89	4	no	—
197	C₂×M₃₂	—	89	4	no	—
198	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
199	C₂²×C₁₆	—	89	4	no	—
200	#405	—	89	1	no	—
201	#405	—	89	1	no	—
202	#405	—	89	1	no	—
203	#405	—	89	2	no	—
204	#405	—	89	2	no	—
205	#405	C₂×D₈₉	1	1	yes	characteristic
206	C₂×#97	—	89	2	no	—
207	C₂×C₄×C₈	—	89	2	no	—
208	#347	—	89	8	no	—
209	#347	—	89	4	no	—
210	#347	C₂×D₈₉	1	4	yes	complemented
211	C₂³×C₄	—	89	1	no	—
212	C₂×M₁₆	—	89	1	no	—
213	C₂×M₁₆	—	89	2	no	—
214	C₂×M₁₆	—	178	2	no	—
215	C₂×M₁₆	—	178	2	no	—
216	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	89	2	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	—	89	2	no	—
223	C₂²×C₈	—	89	2	no	—
224	C₂²×C₈	—	89	1	no	—
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
228	#117	—	89	1	no	—
229	#117	—	89	2	no	—
230	#117	—	178	1	no	—
231	#117	—	178	1	no	—
232	#117	C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	178	1	no	—
234	C₂×C₂²⋊C₄	—	178	1	no	—
235	C₂×C₄²	—	89	2	no	—
236	M₃₂	—	89	4	no	—
237	M₃₂	—	89	4	no	—
238	M₃₂	—	89	8	no	—
239	M₃₂	—	89	8	no	—
240	M₃₂	C₄×D₈₉	1	4	yes	complemented
241	M₃₂	C₂²×D₈₉	1	4	yes	—
242	C₂×C₁₆	—	89	4	no	—
243	C₂×C₁₆	—	89	8	no	—
244	C₂×C₁₆	—	89	8	no	—
245	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
246	#97	—	89	2	no	—
247	#97	—	89	2	no	—
248	#97	—	89	2	no	—
249	#97	C₂²×D₈₉	1	2	yes	—
250	C₄×C₈	—	89	2	no	—
251	C₄×C₈	—	89	2	no	—
252	C₄×C₈	—	89	2	no	—
253	C₄×C₈	C₄×D₈₉	1	2	yes	—
254	C₂⁴	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	89	1	no	—
258	C₂²×C₄	—	89	2	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	2	no	—
264	C₂²×C₄	—	178	2	no	—
265	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
266	M₁₆	—	2	2	no	—
267	M₁₆	—	178	4	no	—
268	M₁₆	—	178	2	no	—
269	C₂×C₈	—	89	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	2	no	—
272	C₂×C₈	—	89	2	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	89	2	no	—
276	C₂×C₈	—	89	2	no	—
277	C₂×C₈	—	89	1	no	—
278	C₂×C₈	—	89	1	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	1	no	—
281	C₂×C₈	—	89	1	no	—
282	C₂×C₈	—	89	2	no	—
283	C₂×C₈	—	89	2	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	178	2	no	—
293	C₄⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	178	2	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₄²	—	89	2	no	—
298	C₄²	—	178	2	no	—
299	C₄²	C₂×C₄×D₈₉	1	2	yes	—
300	C₁₆	—	89	8	no	—
301	C₁₆	C₂×C₄×D₈₉	1	8	yes	—
302	C₂³	—	89	2	no	—
303	C₂³	—	89	1	no	—
304	C₂³	—	89	2	no	—
305	C₂³	—	89	1	no	—
306	C₂³	—	356	2	no	—
307	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	89	2	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	89	1	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	2	no	—
317	C₂×C₄	—	89	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	89	2	no	—
323	C₂×C₄	—	356	4	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	C₄²×D₈₉	1	2	yes	—
326	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
328	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	89	1	no	—
331	C₈	—	89	1	no	—
332	C₈	—	89	2	no	—
333	C₈	—	178	2	no	—
334	C₈	C₄²×D₈₉	1	1	yes	characteristic
335	C₈	C₄²×D₈₉	1	1	yes	characteristic
336	C₈	C₂²×C₄×D₈₉	1	2	yes	—
337	C₂²	—	4	1	no	—
338	C₂²	—	89	1	no	—
339	C₂²	—	89	1	no	—
340	C₂²	—	89	2	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	2	no	—
343	C₂²	—	178	2	no	—
344	C₂²	—	178	4	no	—
345	C₂²	#97×D₈₉	1	2	yes	—
346	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	89	1	no	—
349	C₄	—	89	1	no	—
350	C₄	—	89	2	no	—
351	C₄	—	356	2	no	—
352	C₄	#97×D₈₉	1	2	yes	—
353	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
354	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	89	2	no	—
358	C₂	—	178	1	no	—
359	C₂	—	178	2	no	—
360	C₂	C₂×#97×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2110	C₂
socle	C₂×C₈₉	C₂²×#97
prime radical	C₂³×D₈₉	M₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2110	D₈₉
hypocenter	C₈₉	C₂×#2110
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #26021857	C₂	semidirect
4. #26021856	C₂	semidirect
5. #26021855	C₂	semidirect
6. #347×D₈₉	C₂	semidirect
7. C₈₉×#2110	C₂	semidirect
12. C₂×#15667951	C₄	semidirect
13. M₃₂×D₈₉	C₄	semidirect
14. M₃₂×D₈₉	C₄	semidirect
17. C₂×C₁₆×D₈₉	C₄	semidirect
25. #19151376	C₂²	semidirect
26. #19151375	C₂²	semidirect
29. #347×C₈₉	C₂²	semidirect
52. #15667952	C₂×C₄	semidirect
55. #15667951	C₂×C₄	semidirect
56. C₂×#13083549	C₂×C₄	semidirect
58. C₁₆×D₈₉	C₂×C₄	semidirect
68. M₃₂×C₈₉	C₂×C₄	semidirect
70. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
180. D₈₉	#2110	direct (N × H)
186. #2110	D₈₉	direct (N × H)
188. C₈₉	C₂×#2110	semidirect
210. #347	C₂×D₈₉	semidirect
240. M₃₂	C₄×D₈₉	semidirect
245. C₂×C₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₈₉	2
#405×D₈₉	2
#26021857	2
#26021856	2
#26021855	2
#347×D₈₉	2
C₈₉×#2110	2
C₂×#2110	89

Subgroups — #2110×D89 #36707083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2110×D₈₉ #36707083