Subgroups — #36469331

Order 22784.

570 conjugacy classes of subgroups.

The subgroup lattice has 352 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	C₂×#108×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151633	C₂	1	1	yes	maximal, characteristic
4	#26018352	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018351	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018350	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1383	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₈₉	—	2	1	no	—
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂×#15667970	—	2	1	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667951	C₄	1	2	yes	complemented
13	M₃₂×D₈₉	—	2	2	no	—
14	M₃₂×D₈₉	—	2	2	no	—
15	C₂×C₁₆×D₈₉	C₄	1	2	yes	complemented
16	C₂×#15667948	C₂²	1	1	yes	characteristic
17	C₂×#15667947	—	2	1	no	—
18	#108×D₈₉	—	2	2	no	—
19	#108×D₈₉	C₄	1	4	yes	complemented
20	#19151633	—	2	2	no	—
21	#19151633	C₄	1	4	yes	—
22	C₂×#15667945	—	2	1	no	—
23	C₂×#15667944	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
28	C₂×#13083558	—	2	1	no	—
29	C₂×#13083557	—	4	1	no	—
30	M₁₆×D₈₉	—	2	4	no	—
31	M₁₆×D₈₉	—	4	2	no	—
32	C₂²×#10623494	D₄	1	1	yes	characteristic
33	C₂×C₈×D₈₉	—	4	1	no	—
34	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	D₄	1	2	yes	—
39	C₂×C₈×D₈₉	D₄	1	2	yes	—
40	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
41	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
42	#15667970	—	2	4	no	—
43	#15667970	—	4	2	no	—
44	C₂×C₄⋊Dic₈₉	—	2	1	no	—
45	C₂×#13083551	—	4	1	no	—
46	C₂×C₄×Dic₈₉	—	4	1	no	—
47	#15667952	—	2	1	no	—
48	#15667952	—	2	1	no	—
49	#15667951	—	2	4	no	—
50	C₂×#13083549	C₂×C₄	1	2	yes	—
51	C₁₆×D₈₉	—	2	4	no	—
52	#15667948	—	2	1	no	—
53	#15667948	C₂×C₄	1	2	yes	—
54	#15667947	—	2	4	no	—
55	#15667945	—	2	4	no	—
56	#15667944	—	2	1	no	—
57	#15667944	C₂×C₄	1	2	yes	—
58	C₂×M₁₆×C₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
61	M₃₂×C₈₉	—	2	1	no	—
62	M₃₂×C₈₉	—	2	1	no	—
63	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
64	#108×C₈₉	—	2	1	no	—
65	#108×C₈₉	C₂×C₄	1	2	yes	—
66	#106×C₈₉	—	2	1	no	—
67	#106×C₈₉	C₂×C₄	1	2	yes	—
68	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
69	C₂²×Dic₈₉	—	4	1	no	—
70	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
71	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
73	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×D₈₉	D₈	1	2	yes	—
76	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
77	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
78	C₂×C₄×D₈₉	Q₁₆	1	2	yes	—
79	#13083558	—	2	2	no	—
80	#13083558	—	4	1	no	—
81	#13083557	—	4	4	no	—
82	C₂×#10623494	—	4	1	no	—
83	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
84	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
85	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
86	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
87	C₈×D₈₉	—	4	4	no	—
88	C₈×D₈₉	C₄²	1	2	yes	—
89	C₈×D₈₉	C₄²	1	2	yes	—
90	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
91	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
92	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
93	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
94	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
95	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
96	C₄⋊Dic₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	#13083551	—	4	4	no	—
99	C₄×Dic₈₉	—	4	4	no	—
100	#13083549	—	2	2	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	M₁₆×C₈₉	—	2	2	no	—
104	M₁₆×C₈₉	—	4	1	no	—
105	C₂×C₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₁₆×C₈₉	—	2	2	no	—
115	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
116	C₂²×D₈₉	—	4	2	no	—
117	C₂²×D₈₉	#102	1	2	yes	—
118	C₂²×D₈₉	#103	1	2	yes	—
119	C₂²×D₈₉	#106	1	1	yes	characteristic
120	C₂²×D₈₉	#106	1	1	yes	characteristic
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	8	2	no	—
131	C₂×Dic₈₉	#102	1	2	yes	—
132	C₂×Dic₈₉	#103	1	2	yes	—
133	C₂×Dic₈₉	#106	1	1	yes	characteristic
134	C₂×Dic₈₉	#106	1	1	yes	characteristic
135	C₄×D₈₉	#95	1	2	yes	—
136	C₄×D₈₉	#95	1	2	yes	—
137	C₄×D₈₉	#102	1	2	yes	—
138	C₄×D₈₉	#102	1	2	yes	—
139	C₄×D₈₉	#103	1	2	yes	—
140	C₄×D₈₉	#103	1	2	yes	—
141	C₄×D₈₉	#107	1	2	yes	—
142	C₄×D₈₉	#107	1	2	yes	—
143	C₈×C₈₉	—	4	2	no	—
144	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
145	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
147	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
148	#10623494	—	4	2	no	—
149	#10623494	#95	1	2	yes	—
150	#10623494	#95	1	2	yes	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
154	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	4	2	no	—
158	C₂×D₈₉	—	4	2	no	—
159	C₂×D₈₉	#340	1	2	yes	—
160	C₂×D₈₉	#340	1	2	yes	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
164	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
165	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
166	Dic₈₉	—	8	2	no	—
167	Dic₈₉	#340	1	2	yes	—
168	Dic₈₉	#340	1	2	yes	—
169	#12563	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	—	2	1	no	—
172	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
173	C₂×C₈₉	#2155	1	2	yes	—
174	D₈₉	—	2	2	no	—
175	D₈₉	—	2	2	no	—
176	C₂²×M₃₂	—	89	1	no	—
177	C₂²×#108	—	89	1	no	—
178	C₂×#427	—	89	1	no	—
179	#1383	—	89	1	no	—
180	#1383	—	89	1	no	—
181	#1383	—	89	1	no	—
182	#1383	D₈₉	1	1	yes	characteristic, complemented
183	C₈₉	#12563	1	1	yes	characteristic, complemented, Sylow
184	C₂²×M₁₆	—	178	1	no	—
185	C₂³×C₈	—	89	1	no	—
186	C₂×#117	—	178	1	no	—
187	C₂×M₃₂	—	89	1	no	—
188	C₂×M₃₂	—	89	2	no	—
189	C₂×M₃₂	—	178	2	no	—
190	C₂×M₃₂	—	178	2	no	—
191	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
192	C₂²×C₁₆	—	89	2	no	—
193	C₂×#108	—	89	1	no	—
194	C₂×#108	—	89	4	no	—
195	C₂×#108	—	178	2	no	—
196	C₂×#108	—	178	1	no	—
197	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
198	#427	—	89	4	no	—
199	#427	—	178	2	no	—
200	C₂×#107	—	178	1	no	—
201	C₂×#106	—	89	1	no	—
202	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
203	C₂³×C₄	—	89	1	no	—
204	C₂×M₁₆	—	2	1	no	—
205	C₂×M₁₆	—	356	1	no	—
206	C₂×M₁₆	—	356	2	no	—
207	C₂×M₁₆	—	178	4	no	—
208	C₂×M₁₆	—	178	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	—	89	2	no	—
211	C₂²×C₈	—	89	2	no	—
212	C₂²×C₈	—	89	1	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	2	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	—	89	1	no	—
217	C₂²×C₈	—	89	1	no	—
218	C₂²×C₈	—	356	1	no	—
219	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
220	#117	—	356	2	no	—
221	#117	—	178	4	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄⋊C₄	—	178	1	no	—
224	C₂×C₂²⋊C₄	—	356	1	no	—
225	C₂×C₄²	—	356	1	no	—
226	M₃₂	—	2	1	no	—
227	M₃₂	—	2	1	no	—
228	M₃₂	—	178	4	no	—
229	M₃₂	—	178	1	no	—
230	M₃₂	—	178	1	no	—
231	C₂×C₁₆	—	89	2	no	—
232	C₂×C₁₆	—	178	4	no	—
233	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
234	#108	—	2	1	no	—
235	#108	—	89	2	no	—
236	#108	—	178	4	no	—
237	#108	—	178	1	no	—
238	#108	C₄×D₈₉	1	2	yes	—
239	#107	—	178	4	no	—
240	#106	—	2	1	no	—
241	#106	—	89	2	no	—
242	#106	—	178	1	no	—
243	#106	C₄×D₈₉	1	2	yes	—
244	C₂⁴	—	89	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	—	89	1	no	—
249	C₂²×C₄	—	89	1	no	—
250	C₂²×C₄	—	89	2	no	—
251	C₂²×C₄	—	89	2	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	356	1	no	—
256	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	356	4	no	—
260	M₁₆	—	356	1	no	—
261	M₁₆	—	178	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	—	89	1	no	—
267	C₂×C₈	—	89	2	no	—
268	C₂×C₈	—	89	2	no	—
269	C₂×C₈	—	89	2	no	—
270	C₂×C₈	—	89	2	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	—	356	4	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
278	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
281	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
282	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	356	1	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₂²⋊C₄	—	356	4	no	—
288	C₄²	—	356	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	178	2	no	—
291	C₂³	—	89	2	no	—
292	C₂³	—	89	2	no	—
293	C₂³	—	89	1	no	—
294	C₂³	—	89	1	no	—
295	C₂³	—	356	2	no	—
296	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	8	2	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	89	2	no	—
307	C₂×C₄	—	89	2	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	89	2	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	712	2	no	—
313	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
316	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
318	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	89	2	no	—
321	C₈	—	89	2	no	—
322	C₈	—	356	2	no	—
323	C₈	C₄²×D₈₉	1	1	yes	characteristic
324	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
325	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
326	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
327	C₂²	—	2	2	no	—
328	C₂²	—	89	2	no	—
329	C₂²	—	89	2	no	—
330	C₂²	—	356	2	no	—
331	C₂²	—	356	2	no	—
332	C₂²	—	178	2	no	—
333	C₂²	—	178	2	no	—
334	C₂²	#102×D₈₉	1	1	yes	characteristic
335	C₂²	#103×D₈₉	1	1	yes	characteristic
336	C₂²	#106×D₈₉	1	1	yes	characteristic
337	C₄	—	8	2	no	—
338	C₄	—	89	2	no	—
339	C₄	—	89	2	no	—
340	C₄	—	712	2	no	—
341	C₄	#95×D₈₉	1	1	yes	characteristic
342	C₄	#102×D₈₉	1	1	yes	characteristic
343	C₄	#103×D₈₉	1	1	yes	characteristic
344	C₄	#107×D₈₉	1	1	yes	characteristic
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	178	2	no	—
348	C₂	—	178	2	no	—
349	C₂	#340×D₈₉	1	1	yes	characteristic
350	C₂	#26022023	1	2	yes	—

	subgroup	G/H
center	C₂²	#106×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1383	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×D₈₉	SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1383	D₈₉
hypocenter	C₈₉	#12563
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1383

Complements

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

N	H ≅ G/N	type
4. #26018352	C₂	semidirect
5. #26018351	C₂	semidirect
6. #26018350	C₂	semidirect
7. C₈₉×#1383	C₂	semidirect
12. C₂×#15667951	C₄	semidirect
15. C₂×C₁₆×D₈₉	C₄	semidirect
19. #108×D₈₉	C₄	semidirect
182. #1383	D₈₉	semidirect
183. C₈₉	#12563	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
C₂×#108×D₈₉	2
C₂×#19151633	2
#26018352	2
#26018351	2
#26018350	2
C₈₉×#1383	2
#12563	89