Subgroups — #2241×D₈₉ #36709714

Order 22784.

622 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	#500×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#494×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022481	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022480	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022478	C₂	1	4	yes	maximal, complemented
7	#358×D₈₉	C₂	1	4	yes	maximal
8	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₁₆×D₈₉	—	2	1	no	—
11	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
12	C₄⋊Q₈×D₈₉	—	2	1	no	—
13	C₄⋊Q₈×D₈₉	—	2	1	no	—
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	—	2	1	no	—
17	#19152002	C₂²	1	2	yes	—
18	#19152001	—	2	1	no	—
19	#19151978	C₂²	1	1	yes	characteristic
20	#19151977	—	2	1	no	—
21	#19151976	—	2	1	no	—
22	#19151976	C₂²	1	2	yes	—
23	#107×D₈₉	—	2	1	no	—
24	#107×D₈₉	C₂²	1	2	yes	—
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	#19151403	C₂²	1	4	yes	—
27	#19151402	C₂²	1	4	yes	—
28	#19151401	C₂²	1	4	yes	—
29	C₄×#13083549	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₈₉	C₂²	1	2	yes	—
31	#500×C₈₉	C₂²	1	1	yes	characteristic
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#358×C₈₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×Q₈×D₈₉	—	2	2	no	—
36	C₂×Q₈×D₈₉	—	4	1	no	—
37	C₂×Dic₃₅₆	—	2	1	no	—
38	C₂×Dic₃₅₆	C₂³	1	2	yes	—
39	C₂×#13083564	—	2	2	no	—
40	C₂×#13083564	—	2	2	no	—
41	Q₁₆×D₈₉	—	2	8	no	—
42	Q₁₆×D₈₉	—	4	4	no	—
43	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668012	—	2	1	no	—
47	#15668011	—	2	2	no	—
48	#15668011	—	4	1	no	—
49	#15668011	—	4	1	no	—
50	#15668011	—	4	1	no	—
51	#15668010	—	2	2	no	—
52	#15668010	—	4	1	no	—
53	C₄⋊C₄×D₈₉	—	2	2	no	—
54	C₄⋊C₄×D₈₉	—	4	1	no	—
55	C₄⋊C₄×D₈₉	—	4	1	no	—
56	C₄⋊C₄×D₈₉	—	4	1	no	—
57	C₄²×D₈₉	D₄	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₁₆×D₈₉	—	2	4	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	2	yes	—
62	#15667945	—	2	2	no	—
63	#15667945	—	2	2	no	—
64	C₄×#10623494	D₄	1	1	yes	characteristic
65	C₈×Dic₈₉	D₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₉	—	2	1	no	—
68	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	C₂³	1	2	yes	—
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₇₈	—	2	2	no	—
76	C₂×Dic₁₇₈	—	4	1	no	—
77	C₂×Dic₁₇₈	—	4	2	no	—
78	C₂×Dic₁₇₈	—	4	2	no	—
79	Q₈×D₈₉	—	4	4	no	—
80	Q₈×D₈₉	—	4	4	no	—
81	Q₈×D₈₉	—	8	4	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
85	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
86	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
87	Dic₃₅₆	—	2	4	no	—
88	Dic₃₅₆	—	4	2	no	—
89	#13083564	—	4	4	no	—
90	#13083564	—	4	4	no	—
91	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
92	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
93	C₈×D₈₉	SD₁₆	1	4	yes	—
94	C₈×D₈₉	#41	1	4	yes	—
95	C₄⋊Dic₈₉	—	2	2	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	C₄⋊Dic₈₉	—	4	1	no	—
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	#13083553	—	4	2	no	—
102	#13083553	—	4	2	no	—
103	C₄×Dic₈₉	D₈	1	1	yes	characteristic
104	C₄×Dic₈₉	D₈	1	1	yes	characteristic
105	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
106	#13083549	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	Q₁₆×C₈₉	—	2	4	no	—
110	Q₁₆×C₈₉	—	4	2	no	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
133	C₄×D₈₉	—	8	4	no	—
134	C₄×D₈₉	—	8	4	no	—
135	C₄×D₈₉	SD₃₂	1	4	yes	—
136	C₄×D₈₉	Q₃₂	1	4	yes	—
137	C₄×D₈₉	#124	1	2	yes	—
138	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
139	Dic₁₇₈	—	4	2	no	—
140	Dic₁₇₈	—	4	2	no	—
141	Dic₁₇₈	—	8	2	no	—
142	Dic₁₇₈	—	8	4	no	—
143	Dic₁₇₈	—	8	4	no	—
144	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
145	C₈×C₈₉	C₂×#41	1	2	yes	—
146	#10623494	#124	1	2	yes	—
147	#10623494	C₂×SD₁₆	1	2	yes	—
148	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
149	C₂×D₈₉	#486	1	2	yes	—
150	C₂×D₈₉	C₂×SD₃₂	1	2	yes	—
151	C₂×D₈₉	C₂×Q₃₂	1	2	yes	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
155	C₄×C₈₉	C₂×Q₃₂	1	2	yes	—
156	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
157	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
158	Dic₈₉	—	8	2	no	—
159	Dic₈₉	—	8	2	no	—
160	Dic₈₉	#486	1	2	yes	—
161	Dic₈₉	C₂×SD₃₂	1	2	yes	—
162	Dic₈₉	C₂×Q₃₂	1	2	yes	—
163	C₂×#2241	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
165	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
166	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
167	D₈₉	#2241	1	4	yes	direct product factor
168	C₂×#500	—	89	1	no	—
169	C₂×#494	—	89	1	no	—
170	#2241	—	89	1	no	—
171	#2241	—	89	1	no	—
172	#2241	—	89	4	no	—
173	#2241	—	89	1	no	—
174	#2241	D₈₉	1	1	yes	characteristic, direct product factor
175	C₂×#358	—	89	4	no	—
176	C₂×C₄×C₁₆	—	89	1	no	—
177	C₈₉	C₂×#2241	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	89	2	no	—
179	C₂²×Q₁₆	—	178	1	no	—
180	C₂×C₄⋊Q₈	—	178	1	no	—
181	C₂×C₄⋊Q₈	—	178	1	no	—
182	C₂²×C₁₆	—	89	2	no	—
183	#500	—	89	2	no	—
184	#500	—	89	1	no	—
185	#500	—	178	1	no	—
186	#500	—	178	1	no	—
187	#500	C₂×D₈₉	1	1	yes	characteristic
188	#494	—	89	2	no	—
189	#494	—	89	1	no	—
190	#494	—	178	1	no	—
191	#494	—	178	1	no	—
192	#494	C₂×D₈₉	1	1	yes	characteristic
193	C₂×#107	—	89	2	no	—
194	C₂×#107	—	178	1	no	—
195	C₂×C₄×C₈	—	89	1	no	—
196	#358	—	89	4	no	—
197	#358	—	89	4	no	—
198	#358	—	89	4	no	—
199	#358	C₂×D₈₉	1	4	yes	—
200	C₄×C₁₆	—	89	2	no	—
201	C₄×C₁₆	—	89	1	no	—
202	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₂²×Q₈	—	356	1	no	—
204	C₂²×Q₈	—	178	2	no	—
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	89	2	no	—
207	C₂×Q₁₆	—	356	4	no	—
208	C₂×Q₁₆	—	178	8	no	—
209	C₂×Q₁₆	—	178	2	no	—
210	C₂×Q₁₆	—	178	2	no	—
211	C₂×Q₁₆	—	178	1	no	—
212	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₄⋊Q₈	—	2	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	356	1	no	—
218	C₄⋊Q₈	—	356	1	no	—
219	C₄⋊Q₈	—	356	1	no	—
220	C₄⋊Q₈	—	356	1	no	—
221	C₄⋊Q₈	—	178	2	no	—
222	C₄⋊Q₈	—	178	2	no	—
223	C₄⋊Q₈	—	178	1	no	—
224	C₄⋊Q₈	—	178	1	no	—
225	C₂×C₄⋊C₄	—	356	1	no	—
226	C₂×C₄⋊C₄	—	356	1	no	—
227	C₂×C₄⋊C₄	—	356	1	no	—
228	C₂×C₄⋊C₄	—	178	2	no	—
229	C₂×C₄²	—	89	1	no	—
230	C₂×C₁₆	—	89	2	no	—
231	C₂×C₁₆	—	178	4	no	—
232	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	89	2	no	—
235	#107	—	178	2	no	—
236	#107	—	178	2	no	—
237	#107	—	178	1	no	—
238	#107	C₂²×D₈₉	1	2	yes	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	—	89	1	no	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	356	2	no	—
246	C₂×Q₈	—	356	2	no	—
247	C₂×Q₈	—	356	4	no	—
248	C₂×Q₈	—	356	4	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×Q₈	—	178	2	no	—
251	C₂×Q₈	—	712	4	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	356	2	no	—
256	C₂²×C₄	—	356	2	no	—
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	4	2	no	—
259	Q₁₆	—	356	4	no	—
260	Q₁₆	—	356	4	no	—
261	Q₁₆	—	356	2	no	—
262	Q₁₆	—	178	4	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	4	no	—
266	C₂×C₈	—	89	4	no	—
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	356	2	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	356	1	no	—
278	C₄⋊C₄	—	356	1	no	—
279	C₄⋊C₄	—	356	1	no	—
280	C₄⋊C₄	—	178	2	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	D₄×D₈₉	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	178	2	no	—
287	C₂³	—	89	1	no	—
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	2	no	—
291	Q₈	—	356	2	no	—
292	Q₈	—	356	2	no	—
293	Q₈	—	712	4	no	—
294	Q₈	—	712	4	no	—
295	Q₈	—	712	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	4	no	—
302	C₂×C₄	—	89	4	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	712	4	no	—
308	C₂×C₄	—	712	4	no	—
309	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
310	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
312	C₈	—	89	2	no	—
313	C₈	—	89	2	no	—
314	C₈	SD₁₆×D₈₉	1	2	yes	—
315	C₈	#41×D₈₉	1	2	yes	—
316	C₂²	—	89	2	no	—
317	C₂²	—	89	2	no	—
318	C₂²	—	89	2	no	—
319	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	—	89	2	no	—
323	C₄	—	89	2	no	—
324	C₄	—	89	2	no	—
325	C₄	—	712	2	no	—
326	C₄	—	712	2	no	—
327	C₄	SD₃₂×D₈₉	1	2	yes	—
328	C₄	Q₃₂×D₈₉	1	2	yes	—
329	C₄	#124×D₈₉	1	1	yes	characteristic
330	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
331	C₂	—	89	4	no	—
332	C₂	#486×D₈₉	1	1	yes	characteristic
333	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
334	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2241	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2241	D₈₉
hypocenter	C₈₉	C₂×#2241
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
3. #26022481	C₂	semidirect
4. #26022480	C₂	semidirect
5. #26022479	C₂	semidirect
6. #26022478	C₂	semidirect
9. C₈₉×#2241	C₂	semidirect
167. D₈₉	#2241	direct (N × H)
174. #2241	D₈₉	direct (N × H)
177. C₈₉	C₂×#2241	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₉	2
#494×D₈₉	2
#26022481	2
#26022480	2
#26022479	2
#26022478	2
#358×D₈₉	2
C₄×C₁₆×D₈₉	2
C₈₉×#2241	2
C₂×#2241	89

Subgroups — #2241×D89 #36709714

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2241×D₈₉ #36709714