Subgroups — #2151×D₈₉ #36707408

Order 22784.

734 conjugacy classes of subgroups.

The subgroup lattice has 336 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	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#405×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#26022012	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022011	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022010	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022009	C₂	1	4	yes	maximal, complemented
8	C₄⋊C₁₆×D₈₉	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₈₉	C₄	1	1	yes	characteristic
11	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
12	#117×D₈₉	C₄	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂×#15667951	C₂²	1	2	yes	—
15	M₃₂×D₈₉	—	2	4	no	—
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×D₈₉	—	2	4	no	—
18	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
19	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
20	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
21	#19151556	C₄	1	1	yes	characteristic
22	#19151555	C₄	1	2	yes	—
23	#19151554	C₄	1	1	yes	characteristic
24	#19151553	C₂²	1	1	yes	characteristic
25	#19151552	C₂²	1	2	yes	—
26	#97×D₈₉	C₄	1	2	yes	—
27	C₄×C₈×D₈₉	C₂²	1	2	yes	—
28	C₄⋊#13083549	C₂²	1	4	yes	complemented
29	#19151416	C₂²	1	4	yes	complemented
30	#19151416	C₂²	1	4	yes	complemented
31	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	#405×C₈₉	C₂²	1	1	yes	characteristic
34	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	complemented
35	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	M₁₆×D₈₉	—	2	4	no	—
39	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	D₄	1	4	yes	—
44	C₂×C₈×D₈₉	Q₈	1	4	yes	—
45	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
47	#15667971	C₂×C₄	1	1	yes	characteristic
48	#15667970	C₈	1	2	yes	—
49	#15667969	C₂×C₄	1	2	yes	—
50	#15667968	C₈	1	2	yes	—
51	C₄⋊C₄×D₈₉	C₈	1	2	yes	—
52	C₂²⋊C₄×D₈₉	C₈	1	2	yes	—
53	C₄²×D₈₉	C₂×C₄	1	2	yes	—
54	#15667952	—	2	2	no	—
55	#15667951	—	2	4	no	—
56	C₂×#13083549	—	2	2	no	—
57	C₂×#13083549	C₂³	1	2	yes	—
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₁₆×D₈₉	—	2	4	no	—
60	C₁₆×D₈₉	—	2	4	no	—
61	C₁₆×D₈₉	—	2	4	no	—
62	#15667923	C₂×C₄	1	2	yes	—
63	#15667922	C₂×C₄	1	2	yes	—
64	#15667922	C₂×C₄	1	2	yes	—
65	C₄×#10623494	C₂×C₄	1	2	yes	—
66	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
67	C₈×Dic₈₉	C₂³	1	2	yes	—
68	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
71	M₃₂×C₈₉	—	2	2	no	—
72	C₂×C₁₆×C₈₉	—	2	2	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	#97×C₈₉	C₂×C₄	1	2	yes	—
76	C₄×C₈×C₈₉	C₂³	1	2	yes	—
77	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
78	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₄⋊C₄	1	4	yes	—
80	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
81	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	M₁₆	1	4	yes	—
84	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
86	#13083558	—	2	2	no	—
87	#13083557	—	2	4	no	—
88	C₂×#10623494	C₄⋊C₄	1	4	yes	—
89	C₂×#10623494	C₂²×C₄	1	2	yes	—
90	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
91	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
92	C₈×D₈₉	—	2	4	no	—
93	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
94	C₈×D₈₉	C₄⋊C₄	1	4	yes	—
95	C₈×D₈₉	C₂×D₄	1	2	yes	—
96	C₈×D₈₉	C₂×D₄	1	2	yes	—
97	C₈×D₈₉	C₂×Q₈	1	2	yes	—
98	C₈×D₈₉	C₂×Q₈	1	2	yes	—
99	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
100	#13083553	C₂×C₈	1	4	yes	—
101	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
102	#13083551	C₂×C₈	1	4	yes	—
103	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
104	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
105	#13083549	—	2	2	no	—
106	#13083549	—	2	2	no	—
107	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	M₁₆×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
115	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
116	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
117	C₁₆×C₈₉	—	2	2	no	—
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂²×D₈₉	—	2	4	no	—
121	C₂²×D₈₉	C₄⋊C₈	1	4	yes	—
122	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
123	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
126	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₄⋊C₈	1	4	yes	—
130	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
131	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
133	C₄×D₈₉	—	2	8	no	—
134	C₄×D₈₉	C₄⋊C₈	1	4	yes	—
135	C₄×D₈₉	C₄⋊C₈	1	4	yes	—
136	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
137	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
138	C₄×D₈₉	C₂×M₁₆	1	2	yes	—
139	C₄×D₈₉	C₂×M₁₆	1	2	yes	—
140	C₈×C₈₉	—	2	2	no	—
141	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
142	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
143	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
144	#10623494	—	2	2	no	—
145	#10623494	C₂×C₄⋊C₄	1	2	yes	—
146	#10623494	C₂×C₄⋊C₄	1	2	yes	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
149	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
150	C₂×D₈₉	—	2	2	no	—
151	C₂×D₈₉	—	2	8	no	—
152	C₂×D₈₉	C₂×C₄⋊C₈	1	2	yes	—
153	C₂×D₈₉	C₂×C₄⋊C₈	1	2	yes	—
154	C₂×D₈₉	#504	1	4	yes	—
155	C₄×C₈₉	—	2	4	no	—
156	C₄×C₈₉	C₂×C₄⋊C₈	1	2	yes	—
157	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
159	Dic₈₉	—	2	4	no	—
160	Dic₈₉	C₂×C₄⋊C₈	1	2	yes	—
161	Dic₈₉	C₂×C₄⋊C₈	1	2	yes	—
162	C₂×#2151	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	—	2	2	no	—
164	C₂×C₈₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₈₉	C₂×#504	1	2	yes	—
166	D₈₉	—	2	2	no	—
167	D₈₉	#2151	1	4	yes	direct product factor
168	C₂²×M₃₂	—	89	1	no	—
169	C₂³×C₁₆	—	89	1	no	—
170	C₂×#405	—	89	1	no	—
171	#2151	—	89	1	no	—
172	#2151	—	89	1	no	—
173	#2151	—	89	1	no	—
174	#2151	—	89	4	no	—
175	#2151	D₈₉	1	1	yes	characteristic, direct product factor
176	C₂×C₄⋊C₁₆	—	89	4	no	—
177	C₈₉	C₂×#2151	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	89	1	no	—
179	C₂³×C₈	—	89	1	no	—
180	C₂×#117	—	89	1	no	—
181	C₂×M₃₂	—	89	1	no	—
182	C₂×M₃₂	—	89	2	no	—
183	C₂×M₃₂	—	178	4	no	—
184	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
185	C₂²×C₁₆	—	89	1	no	—
186	C₂²×C₁₆	—	89	2	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	C₂²×C₁₆	—	89	2	no	—
189	C₂²×C₁₆	—	178	4	no	—
190	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
191	#405	—	89	1	no	—
192	#405	—	89	1	no	—
193	#405	—	89	1	no	—
194	#405	—	89	2	no	—
195	#405	—	89	2	no	—
196	#405	C₂×D₈₉	1	1	yes	characteristic
197	C₂×#97	—	89	2	no	—
198	C₂×C₄×C₈	—	89	2	no	—
199	C₄⋊C₁₆	—	89	4	no	—
200	C₄⋊C₁₆	—	89	4	no	—
201	C₄⋊C₁₆	—	89	4	no	—
202	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	complemented
203	C₂³×C₄	—	89	1	no	—
204	C₂×M₁₆	—	89	1	no	—
205	C₂×M₁₆	—	89	2	no	—
206	C₂×M₁₆	—	178	4	no	—
207	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	—	89	1	no	—
211	C₂²×C₈	—	89	4	no	—
212	C₂²×C₈	—	89	4	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
217	#117	—	89	1	no	—
218	#117	—	89	2	no	—
219	#117	—	89	2	no	—
220	#117	—	89	2	no	—
221	#117	C₄×D₈₉	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	89	2	no	—
223	C₂×C₂²⋊C₄	—	89	2	no	—
224	C₂×C₄²	—	89	2	no	—
225	M₃₂	—	2	2	no	—
226	M₃₂	—	178	4	no	—
227	M₃₂	—	178	2	no	—
228	C₂×C₁₆	—	2	2	no	—
229	C₂×C₁₆	—	89	2	no	—
230	C₂×C₁₆	—	89	2	no	—
231	C₂×C₁₆	—	178	4	no	—
232	C₂×C₁₆	—	178	4	no	—
233	C₂×C₁₆	—	178	4	no	—
234	C₂×C₁₆	—	178	2	no	—
235	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
236	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
237	#97	—	89	2	no	—
238	#97	—	89	2	no	—
239	#97	—	89	2	no	—
240	#97	C₄×D₈₉	1	2	yes	—
241	C₄×C₈	—	89	2	no	—
242	C₄×C₈	—	89	2	no	—
243	C₄×C₈	—	89	2	no	—
244	C₄×C₈	C₂²×D₈₉	1	2	yes	—
245	C₂⁴	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	1	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	89	4	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	4	no	—
253	C₂²×C₄	—	89	4	no	—
254	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
255	M₁₆	—	2	2	no	—
256	M₁₆	—	178	4	no	—
257	M₁₆	—	178	2	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	4	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	—	89	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	4	no	—
266	C₂×C₈	—	89	4	no	—
267	C₂×C₈	—	89	2	no	—
268	C₂×C₈	—	178	4	no	—
269	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
270	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
271	C₂×C₈	D₄×D₈₉	1	2	yes	—
272	C₂×C₈	Q₈×D₈₉	1	2	yes	—
273	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	89	4	no	—
275	C₄⋊C₄	—	89	2	no	—
276	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
277	C₂²⋊C₄	—	89	2	no	—
278	C₂²⋊C₄	—	89	4	no	—
279	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
280	C₄²	—	89	4	no	—
281	C₄²	—	89	2	no	—
282	C₄²	C₂×C₄×D₈₉	1	2	yes	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	178	2	no	—
286	C₁₆	—	178	2	no	—
287	C₂³	—	89	1	no	—
288	C₂³	—	89	4	no	—
289	C₂³	—	89	1	no	—
290	C₂³	—	178	4	no	—
291	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
292	C₂×C₄	—	89	4	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	4	no	—
296	C₂×C₄	—	89	4	no	—
297	C₂×C₄	—	89	2	no	—
298	C₂×C₄	—	89	2	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	89	4	no	—
302	C₂×C₄	—	178	8	no	—
303	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
304	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
305	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
306	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
307	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	89	2	no	—
310	C₈	—	89	2	no	—
311	C₈	—	178	2	no	—
312	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
313	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
314	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	89	2	no	—
317	C₂²	—	89	4	no	—
318	C₂²	—	89	2	no	—
319	C₂²	—	178	8	no	—
320	C₂²	—	178	2	no	—
321	C₂²	C₄⋊C₈×D₈₉	1	2	yes	—
322	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
323	C₄	—	2	4	no	—
324	C₄	—	89	2	no	—
325	C₄	—	89	2	no	—
326	C₄	—	178	4	no	—
327	C₄	C₄⋊C₈×D₈₉	1	2	yes	—
328	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
329	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	89	4	no	—
332	C₂	—	178	2	no	—
333	C₂	C₂×C₄⋊C₈×D₈₉	1	1	yes	characteristic
334	C₂	#504×D₈₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2151	C₂
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2151	D₈₉
hypocenter	C₈₉	C₂×#2151
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
4. #26022012	C₂	semidirect
5. #26022011	C₂	semidirect
6. #26022010	C₂	semidirect
7. #26022009	C₂	semidirect
8. C₄⋊C₁₆×D₈₉	C₂	semidirect
9. C₈₉×#2151	C₂	semidirect
28. C₄⋊#13083549	C₂²	semidirect
29. #19151416	C₂²	semidirect
30. #19151416	C₂²	semidirect
34. C₄⋊C₁₆×C₈₉	C₂²	semidirect
167. D₈₉	#2151	direct (N × H)
175. #2151	D₈₉	direct (N × H)
177. C₈₉	C₂×#2151	semidirect
202. C₄⋊C₁₆	C₂×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
C₂²×C₁₆×D₈₉	2
#405×D₈₉	2
#26022012	2
#26022011	2
#26022010	2
#26022009	2
C₄⋊C₁₆×D₈₉	2
C₈₉×#2151	2
C₂×#2151	89

Subgroups — #2151×D89 #36707408

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2151×D₈₉ #36707408