Subgroups — #2242×D₈₉ #36709719

Order 22784.

610 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#499×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#495×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022486	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022485	C₂	1	2	yes	maximal, complemented
5	#26022484	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022483	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022482	C₂	1	2	yes	maximal, complemented
8	#358×D₈₉	C₂	1	2	yes	maximal, complemented
9	#357×D₈₉	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2242	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	1	yes	characteristic
13	C₂×SD₁₆×D₈₉	—	2	1	no	—
14	C₂×D₈×D₈₉	C₂²	1	1	yes	characteristic
15	#125×D₈₉	—	2	1	no	—
16	#124×D₈₉	—	2	1	no	—
17	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
18	#19152000	C₂²	1	1	yes	characteristic, complemented
19	#19151999	C₂²	1	2	yes	complemented
20	#19151998	—	2	1	no	—
21	#19151997	—	2	1	no	—
22	#19151983	C₂²	1	1	yes	characteristic
23	#19151982	C₂²	1	1	yes	characteristic
24	#19151981	C₂²	1	1	yes	characteristic
25	#19151980	—	2	1	no	—
26	#19151979	—	2	1	no	—
27	#107×D₈₉	C₂²	1	2	yes	—
28	#106×D₈₉	—	2	1	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151403	C₂²	1	2	yes	complemented
31	#19151402	C₂²	1	2	yes	complemented
32	#19151401	C₂²	1	2	yes	complemented
33	#19151400	C₂²	1	2	yes	—
34	#19151399	C₂²	1	2	yes	—
35	#19151398	C₂²	1	2	yes	—
36	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
38	#499×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	#495×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	complemented
41	#357×C₈₉	C₂²	1	2	yes	—
42	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₈₉	—	2	1	no	—
44	C₂×D₄×D₈₉	—	2	1	no	—
45	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×#13083564	—	2	1	no	—
47	Q₁₆×D₈₉	—	2	4	no	—
48	C₂×#13083563	—	2	1	no	—
49	C₂×#13083562	—	2	1	no	—
50	C₂×#13083561	—	2	1	no	—
51	SD₁₆×D₈₉	—	4	4	no	—
52	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
53	C₂×#13083559	—	2	1	no	—
54	D₈×D₈₉	—	2	4	no	—
55	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
56	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
57	#15668004	—	2	1	no	—
58	#15668003	—	2	2	no	—
59	#15668002	—	4	2	no	—
60	#15668001	—	2	1	no	—
61	#15668000	—	2	1	no	—
62	#15667999	—	2	1	no	—
63	#15667998	—	4	2	no	—
64	C₄⋊C₄×D₈₉	—	2	2	no	—
65	C₄⋊C₄×D₈₉	—	4	2	no	—
66	C₂²⋊C₄×D₈₉	—	4	2	no	—
67	C₄²×D₈₉	D₄	1	1	yes	characteristic
68	C₂×#13083549	C₂³	1	2	yes	—
69	C₁₆×D₈₉	—	2	4	no	—
70	#15667946	C₂³	1	2	yes	—
71	#15667945	—	2	2	no	—
72	#15667944	—	2	1	no	—
73	#15667943	—	2	2	no	—
74	C₄×#10623494	D₄	1	1	yes	characteristic
75	C₈×Dic₈₉	D₄	1	1	yes	characteristic
76	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
77	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₈₉	—	2	1	no	—
79	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
80	#125×C₈₉	—	2	1	no	—
81	#124×C₈₉	—	2	1	no	—
82	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
83	#107×C₈₉	C₂³	1	2	yes	—
84	#106×C₈₉	—	2	1	no	—
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₂³×D₈₉	—	4	1	no	—
87	C₂×Dic₁₇₈	—	2	1	no	—
88	C₂×Dic₁₇₈	—	4	1	no	—
89	Q₈×D₈₉	—	4	2	no	—
90	Q₈×D₈₉	—	4	2	no	—
91	C₂×D₃₅₆	—	2	1	no	—
92	C₂×C₂²⋊D₈₉	—	4	1	no	—
93	D₄×D₈₉	—	4	2	no	—
94	D₄×D₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	—	4	1	no	—
96	C₂×C₄×D₈₉	—	4	2	no	—
97	C₂×C₄×D₈₉	D₈	1	2	yes	—
98	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
99	Dic₃₅₆	—	2	2	no	—
100	#13083564	—	4	2	no	—
101	#13083563	—	4	2	no	—
102	#13083562	—	4	2	no	—
103	#13083561	—	4	2	no	—
104	D₇₁₂	—	2	2	no	—
105	#13083559	—	4	2	no	—
106	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₈×D₈₉	SD₁₆	1	4	yes	—
109	C₈×D₈₉	#41	1	4	yes	—
110	C₄⋊Dic₈₉	—	2	2	no	—
111	C₄⋊Dic₈₉	—	4	2	no	—
112	#13083553	—	4	2	no	—
113	#13083553	—	4	2	no	—
114	#13083553	—	4	2	no	—
115	C₂²⋊Dic₈₉	—	4	2	no	—
116	#13083551	—	4	2	no	—
117	#13083551	—	4	2	no	—
118	C₄×Dic₈₉	D₈	1	2	yes	—
119	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
120	#13083549	—	2	2	no	—
121	C₂×Q₈×C₈₉	—	2	1	no	—
122	C₂×D₄×C₈₉	—	2	1	no	—
123	Q₁₆×C₈₉	—	2	2	no	—
124	SD₁₆×C₈₉	—	4	2	no	—
125	D₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄⋊C₄×C₈₉	—	4	2	no	—
130	C₂²⋊C₄×C₈₉	—	4	2	no	—
131	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₁₆×C₈₉	—	2	2	no	—
133	C₂³×C₈₉	—	4	1	no	—
134	C₂²×D₈₉	—	4	1	no	—
135	C₂²×D₈₉	—	8	2	no	—
136	C₂²×D₈₉	—	8	4	no	—
137	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
138	Q₈×C₈₉	—	4	1	no	—
139	Q₈×C₈₉	—	4	1	no	—
140	D₄×C₈₉	—	4	1	no	—
141	D₄×C₈₉	—	4	1	no	—
142	C₂×C₄×C₈₉	—	4	1	no	—
143	C₂×C₄×C₈₉	—	4	2	no	—
144	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₈₉	—	8	2	no	—
147	C₂×Dic₈₉	—	4	1	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
150	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
151	D₃₅₆	—	4	1	no	—
152	D₃₅₆	—	4	1	no	—
153	C₄×D₈₉	—	2	4	no	—
154	C₄×D₈₉	—	8	2	no	—
155	C₄×D₈₉	—	8	4	no	—
156	C₄×D₈₉	#124	1	2	yes	—
157	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
158	Dic₁₇₈	—	4	1	no	—
159	Dic₁₇₈	—	4	1	no	—
160	Dic₁₇₈	—	8	2	no	—
161	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
162	C₈×C₈₉	C₂×#41	1	2	yes	—
163	#10623494	#124	1	2	yes	—
164	#10623494	C₂×SD₁₆	1	2	yes	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	2	no	—
167	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₈₉	—	8	1	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	16	4	no	—
171	C₂×D₈₉	#486	1	2	yes	—
172	C₂×D₈₉	#508	1	4	yes	—
173	C₄×C₈₉	—	2	2	no	—
174	C₄×C₈₉	—	8	1	no	—
175	C₄×C₈₉	—	8	2	no	—
176	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
177	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
178	Dic₈₉	—	2	2	no	—
179	Dic₈₉	—	8	1	no	—
180	Dic₈₉	—	8	2	no	—
181	Dic₈₉	#486	1	2	yes	—
182	C₂×#2242	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	16	1	no	—
184	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#508	1	2	yes	—
186	D₈₉	—	16	1	no	—
187	D₈₉	#2242	1	4	yes	direct product factor
188	C₂×#499	—	89	1	no	—
189	C₂×#495	—	89	1	no	—
190	#2242	—	89	1	no	—
191	#2242	—	89	1	no	—
192	#2242	—	89	2	no	—
193	#2242	—	89	2	no	—
194	#2242	—	89	1	no	—
195	#2242	D₈₉	1	1	yes	characteristic, direct product factor
196	C₂×#358	—	89	2	no	—
197	C₂×#357	—	89	2	no	—
198	C₂×C₄×C₁₆	—	89	1	no	—
199	C₈₉	C₂×#2242	1	1	yes	characteristic, complemented, Sylow
200	C₂²×Q₁₆	—	89	1	no	—
201	C₂²×SD₁₆	—	178	1	no	—
202	C₂²×D₈	—	89	1	no	—
203	C₂×#125	—	178	1	no	—
204	C₂×#124	—	178	1	no	—
205	C₂²×C₁₆	—	89	2	no	—
206	#499	—	89	2	no	—
207	#499	—	89	1	no	—
208	#499	—	178	1	no	—
209	#499	—	178	1	no	—
210	#499	C₂×D₈₉	1	1	yes	characteristic, complemented
211	#495	—	89	1	no	—
212	#495	—	89	1	no	—
213	#495	—	89	1	no	—
214	#495	—	178	1	no	—
215	#495	—	178	1	no	—
216	#495	C₂×D₈₉	1	1	yes	characteristic
217	C₂×#107	—	89	2	no	—
218	C₂×#106	—	178	1	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	#358	—	89	2	no	—
221	#358	—	89	2	no	—
222	#358	—	89	2	no	—
223	#358	C₂×D₈₉	1	2	yes	complemented
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	#357	C₂×D₈₉	1	2	yes	—
228	C₄×C₁₆	—	89	2	no	—
229	C₄×C₁₆	—	89	1	no	—
230	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
231	C₂²×Q₈	—	178	1	no	—
232	C₂²×D₄	—	178	1	no	—
233	C₂×Q₁₆	—	89	1	no	—
234	C₂×Q₁₆	—	178	4	no	—
235	C₂×Q₁₆	—	178	1	no	—
236	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×SD₁₆	—	356	4	no	—
239	C₂×SD₁₆	—	178	1	no	—
240	C₂×SD₁₆	—	178	1	no	—
241	C₂×SD₁₆	—	178	1	no	—
242	C₂×D₈	—	89	1	no	—
243	C₂×D₈	—	178	4	no	—
244	C₂×D₈	—	178	1	no	—
245	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂²×C₈	—	89	1	no	—
247	C₂²×C₈	—	89	1	no	—
248	#125	—	2	1	no	—
249	#125	—	356	2	no	—
250	#125	—	178	2	no	—
251	#125	—	178	1	no	—
252	#124	—	2	1	no	—
253	#124	—	356	2	no	—
254	#124	—	178	1	no	—
255	#124	—	178	1	no	—
256	#124	—	178	1	no	—
257	C₂×C₄⋊C₄	—	356	2	no	—
258	C₂×C₄⋊C₄	—	178	2	no	—
259	C₂×C₂²⋊C₄	—	356	2	no	—
260	C₂×C₄²	—	89	1	no	—
261	C₂×C₁₆	—	89	2	no	—
262	C₂×C₁₆	—	178	4	no	—
263	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
264	#107	—	89	2	no	—
265	#107	—	178	2	no	—
266	#107	C₂²×D₈₉	1	2	yes	—
267	#106	—	2	1	no	—
268	#106	—	178	2	no	—
269	#106	—	178	1	no	—
270	C₄×C₈	—	89	1	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₂⁴	—	356	1	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	356	1	no	—
277	C₂×Q₈	—	356	2	no	—
278	C₂×Q₈	—	356	2	no	—
279	C₂×Q₈	—	178	1	no	—
280	C₂×D₄	—	2	1	no	—
281	C₂×D₄	—	356	2	no	—
282	C₂×D₄	—	356	2	no	—
283	C₂×D₄	—	356	1	no	—
284	C₂×D₄	—	178	1	no	—
285	C₂²×C₄	—	89	1	no	—
286	C₂²×C₄	—	89	2	no	—
287	C₂²×C₄	—	356	1	no	—
288	C₂²×C₄	—	356	2	no	—
289	Q₁₆	—	2	2	no	—
290	Q₁₆	—	356	2	no	—
291	Q₁₆	—	178	2	no	—
292	SD₁₆	—	4	2	no	—
293	SD₁₆	—	356	2	no	—
294	SD₁₆	—	356	2	no	—
295	SD₁₆	—	356	2	no	—
296	D₈	—	2	2	no	—
297	D₈	—	356	2	no	—
298	D₈	—	178	2	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	89	1	no	—
301	C₂×C₈	—	89	4	no	—
302	C₂×C₈	—	89	4	no	—
303	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	—	356	2	no	—
308	C₄⋊C₄	—	356	2	no	—
309	C₄⋊C₄	—	356	2	no	—
310	C₄⋊C₄	—	356	2	no	—
311	C₄⋊C₄	—	178	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₂²⋊C₄	—	356	2	no	—
314	C₂²⋊C₄	—	356	2	no	—
315	C₂²⋊C₄	—	356	2	no	—
316	C₄²	—	89	2	no	—
317	C₄²	—	89	1	no	—
318	C₄²	D₄×D₈₉	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	178	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	89	1	no	—
323	C₂³	—	356	1	no	—
324	C₂³	—	712	2	no	—
325	C₂³	—	712	4	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	4	1	no	—
328	Q₈	—	356	1	no	—
329	Q₈	—	356	1	no	—
330	Q₈	—	712	2	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	356	1	no	—
334	D₄	—	356	1	no	—
335	D₄	—	712	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	89	1	no	—
339	C₂×C₄	—	89	2	no	—
340	C₂×C₄	—	89	2	no	—
341	C₂×C₄	—	89	2	no	—
342	C₂×C₄	—	356	1	no	—
343	C₂×C₄	—	356	2	no	—
344	C₂×C₄	—	178	4	no	—
345	C₂×C₄	—	712	4	no	—
346	C₂×C₄	—	712	2	no	—
347	C₂×C₄	D₈×D₈₉	1	2	yes	—
348	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
349	C₈	—	89	2	no	—
350	C₈	—	89	2	no	—
351	C₈	SD₁₆×D₈₉	1	2	yes	—
352	C₈	#41×D₈₉	1	2	yes	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	89	4	no	—
356	C₂²	—	89	2	no	—
357	C₂²	—	1424	4	no	—
358	C₂²	—	712	2	no	—
359	C₂²	—	712	1	no	—
360	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
361	C₄	—	2	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	8	1	no	—
364	C₄	—	89	2	no	—
365	C₄	—	178	2	no	—
366	C₄	—	712	2	no	—
367	C₄	—	712	1	no	—
368	C₄	#124×D₈₉	1	1	yes	characteristic
369	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
370	C₂	—	16	1	no	—
371	C₂	—	89	4	no	—
372	C₂	—	1424	1	no	—
373	C₂	#486×D₈₉	1	1	yes	characteristic
374	C₂	#508×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2242	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₈×D₈₉	C₂²
hypercenter	#2242	D₈₉
hypocenter	C₈₉	C₂×#2242
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2242

Complements

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

N	H ≅ G/N	type
1. #499×D₈₉	C₂	semidirect
3. #26022486	C₂	semidirect
4. #26022485	C₂	semidirect
5. #26022484	C₂	semidirect
6. #26022483	C₂	semidirect
7. #26022482	C₂	semidirect
8. #358×D₈₉	C₂	semidirect
10. C₄×C₁₆×D₈₉	C₂	semidirect
11. C₈₉×#2242	C₂	semidirect
18. #19152000	C₂²	semidirect
19. #19151999	C₂²	semidirect
30. #19151403	C₂²	semidirect
31. #19151402	C₂²	semidirect
32. #19151401	C₂²	semidirect
36. C₄×#13083549	C₂²	semidirect
37. C₁₆×Dic₈₉	C₂²	semidirect
38. #499×C₈₉	C₂²	semidirect
40. #358×C₈₉	C₂²	semidirect
42. C₄×C₁₆×C₈₉	C₂²	semidirect
187. D₈₉	#2242	direct (N × H)
195. #2242	D₈₉	direct (N × H)
199. C₈₉	C₂×#2242	semidirect
210. #499	C₂×D₈₉	semidirect
223. #358	C₂×D₈₉	semidirect
230. 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
#499×D₈₉	2
#495×D₈₉	2
#26022486	2
#26022485	2
#26022484	2
#26022483	2
#26022482	2
#358×D₈₉	2
#357×D₈₉	2
C₄×C₁₆×D₈₉	2
C₈₉×#2242	2
C₂×#2242	89

Subgroups — #2242×D89 #36709719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2242×D₈₉ #36709719