Subgroups — #36469353

Order 22784.

570 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	C₂×M₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151635	C₂	1	1	yes	maximal, characteristic
4	#26018362	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018361	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018360	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1387	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	—	2	1	no	—
13	M₃₂×D₈₉	—	2	2	no	—
14	M₃₂×D₈₉	C₄	1	4	yes	complemented
15	C₂×C₁₆×D₈₉	—	2	1	no	—
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	#19151635	—	2	1	no	—
21	#19151635	—	2	1	no	—
22	#19151635	C₄	1	2	yes	—
23	#19151635	C₄	1	2	yes	—
24	C₂×#15667946	C₂²	1	1	yes	characteristic
25	C₂×#15667943	—	2	1	no	—
26	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
30	C₂×#13083558	—	2	1	no	—
31	C₂×#13083557	—	4	1	no	—
32	M₁₆×D₈₉	—	2	4	no	—
33	M₁₆×D₈₉	—	4	2	no	—
34	C₂²×#10623494	D₄	1	1	yes	characteristic
35	C₂×C₈×D₈₉	—	4	1	no	—
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	D₄	1	1	yes	characteristic
41	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
42	C₂×C₈×D₈₉	D₄	1	2	yes	—
43	C₂×C₈×D₈₉	Q₈	1	2	yes	—
44	#15667970	—	2	2	no	—
45	#15667970	—	2	2	no	—
46	#15667970	—	4	1	no	—
47	#15667970	—	4	1	no	—
48	C₂×C₄⋊Dic₈₉	—	2	1	no	—
49	C₂×#13083551	—	4	1	no	—
50	C₂×C₄×Dic₈₉	—	4	1	no	—
51	#15667952	—	2	1	no	—
52	#15667952	C₂×C₄	1	2	yes	—
53	#15667951	—	2	4	no	—
54	C₂×#13083549	—	2	1	no	—
55	C₁₆×D₈₉	—	2	4	no	—
56	#15667948	—	2	1	no	—
57	#15667948	C₂×C₄	1	2	yes	—
58	#15667947	—	2	4	no	—
59	#15667946	—	2	1	no	—
60	#15667946	C₂×C₄	1	2	yes	—
61	#15667943	—	2	2	no	—
62	#15667943	—	2	2	no	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
66	M₃₂×C₈₉	—	2	1	no	—
67	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
68	C₂×C₁₆×C₈₉	—	2	1	no	—
69	#108×C₈₉	—	2	1	no	—
70	#108×C₈₉	C₂×C₄	1	2	yes	—
71	#107×C₈₉	—	2	1	no	—
72	#107×C₈₉	C₂×C₄	1	2	yes	—
73	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₈₉	—	4	1	no	—
75	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
76	C₂×C₄×D₈₉	C₂²⋊C₄	1	2	yes	—
77	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
80	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
81	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
82	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
83	C₂×C₄×D₈₉	Q₁₆	1	2	yes	—
84	#13083558	—	2	2	no	—
85	#13083558	—	4	1	no	—
86	#13083557	—	4	4	no	—
87	C₂×#10623494	—	4	1	no	—
88	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
91	C₂×#10623494	C₄⋊C₄	1	2	yes	—
92	C₈×D₈₉	—	4	4	no	—
93	C₈×D₈₉	C₄²	1	2	yes	—
94	C₈×D₈₉	C₄²	1	2	yes	—
95	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
96	C₈×D₈₉	C₂²⋊C₄	1	2	yes	—
97	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
98	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
99	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
100	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
101	C₄⋊Dic₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	4	1	no	—
103	#13083551	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	C₄×Dic₈₉	—	4	2	no	—
106	C₄×Dic₈₉	—	4	2	no	—
107	#13083549	—	2	2	no	—
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	M₁₆×C₈₉	—	2	2	no	—
111	M₁₆×C₈₉	—	4	1	no	—
112	C₂×C₈×C₈₉	—	4	1	no	—
113	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₁₆×C₈₉	—	2	2	no	—
122	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂²×D₈₉	—	4	1	no	—
124	C₂²×D₈₉	—	4	1	no	—
125	C₂²×D₈₉	#102	1	1	yes	characteristic
126	C₂²×D₈₉	#102	1	1	yes	characteristic
127	C₂²×D₈₉	#103	1	2	yes	—
128	C₂²×D₈₉	#106	1	2	yes	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	8	2	no	—
139	C₂×Dic₈₉	#102	1	1	yes	characteristic
140	C₂×Dic₈₉	#102	1	1	yes	characteristic
141	C₂×Dic₈₉	#103	1	2	yes	—
142	C₂×Dic₈₉	#106	1	2	yes	—
143	C₄×D₈₉	#95	1	2	yes	—
144	C₄×D₈₉	#95	1	2	yes	—
145	C₄×D₈₉	#102	1	2	yes	—
146	C₄×D₈₉	#102	1	2	yes	—
147	C₄×D₈₉	#103	1	2	yes	—
148	C₄×D₈₉	#103	1	2	yes	—
149	C₄×D₈₉	#107	1	2	yes	—
150	C₄×D₈₉	#107	1	2	yes	—
151	C₈×C₈₉	—	4	2	no	—
152	C₈×C₈₉	C₂×C₄²	1	1	yes	characteristic
153	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
154	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
156	#10623494	—	4	2	no	—
157	#10623494	#95	1	2	yes	—
158	#10623494	#95	1	2	yes	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
162	C₂²×C₈₉	C₂×#103	1	1	yes	characteristic
163	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	—	2	1	no	—
166	C₂×D₈₉	—	2	1	no	—
167	C₂×D₈₉	—	2	1	no	—
168	C₂×D₈₉	—	4	2	no	—
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	#340	1	2	yes	—
171	C₂×D₈₉	#340	1	2	yes	—
172	C₄×C₈₉	—	8	2	no	—
173	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
174	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
175	C₄×C₈₉	C₂×#103	1	1	yes	characteristic
176	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	#340	1	2	yes	—
179	Dic₈₉	#340	1	2	yes	—
180	#12569	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	—	2	1	no	—
182	C₂×C₈₉	—	2	1	no	—
183	C₂×C₈₉	C₂×#340	1	1	yes	characteristic
184	C₂×C₈₉	#2148	1	1	yes	characteristic
185	C₂×C₈₉	#2148	1	1	yes	characteristic
186	D₈₉	—	2	2	no	—
187	D₈₉	—	2	2	no	—
188	C₂²×M₃₂	—	89	1	no	—
189	C₂²×#108	—	89	1	no	—
190	C₂×#427	—	89	1	no	—
191	#1387	—	89	1	no	—
192	#1387	—	89	1	no	—
193	#1387	—	89	1	no	—
194	#1387	D₈₉	1	1	yes	characteristic, complemented
195	C₈₉	#12569	1	1	yes	characteristic, complemented, Sylow
196	C₂²×M₁₆	—	178	1	no	—
197	C₂³×C₈	—	89	1	no	—
198	C₂×#117	—	178	1	no	—
199	C₂×M₃₂	—	89	1	no	—
200	C₂×M₃₂	—	89	4	no	—
201	C₂×M₃₂	—	178	1	no	—
202	C₂×M₃₂	—	178	2	no	—
203	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
204	C₂²×C₁₆	—	178	1	no	—
205	C₂×#108	—	89	1	no	—
206	C₂×#108	—	89	4	no	—
207	C₂×#108	—	178	2	no	—
208	C₂×#108	—	178	1	no	—
209	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
210	#427	—	89	2	no	—
211	#427	—	89	2	no	—
212	#427	—	178	1	no	—
213	#427	—	178	1	no	—
214	C₂×#107	—	89	1	no	—
215	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
216	C₂×#106	—	178	1	no	—
217	C₂³×C₄	—	89	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	356	1	no	—
220	C₂×M₁₆	—	356	2	no	—
221	C₂×M₁₆	—	178	4	no	—
222	C₂×M₁₆	—	178	1	no	—
223	C₂²×C₈	—	89	1	no	—
224	C₂²×C₈	—	89	1	no	—
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	—	89	1	no	—
229	C₂²×C₈	—	89	1	no	—
230	C₂²×C₈	—	89	2	no	—
231	C₂²×C₈	—	89	2	no	—
232	C₂²×C₈	—	356	1	no	—
233	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
234	#117	—	356	1	no	—
235	#117	—	356	1	no	—
236	#117	—	178	2	no	—
237	#117	—	178	2	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	178	1	no	—
240	C₂×C₂²⋊C₄	—	356	1	no	—
241	C₂×C₄²	—	356	1	no	—
242	M₃₂	—	2	1	no	—
243	M₃₂	—	89	2	no	—
244	M₃₂	—	178	4	no	—
245	M₃₂	—	178	1	no	—
246	M₃₂	C₄×D₈₉	1	2	yes	—
247	C₂×C₁₆	—	2	1	no	—
248	C₂×C₁₆	—	178	4	no	—
249	C₂×C₁₆	—	178	1	no	—
250	#108	—	2	1	no	—
251	#108	—	89	2	no	—
252	#108	—	178	4	no	—
253	#108	—	178	1	no	—
254	#108	C₄×D₈₉	1	2	yes	—
255	#107	—	2	1	no	—
256	#107	—	89	2	no	—
257	#107	—	178	1	no	—
258	#107	C₄×D₈₉	1	2	yes	—
259	#106	—	178	2	no	—
260	#106	—	178	2	no	—
261	C₂⁴	—	89	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	89	1	no	—
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	89	1	no	—
266	C₂²×C₄	—	89	2	no	—
267	C₂²×C₄	—	89	2	no	—
268	C₂²×C₄	—	89	1	no	—
269	C₂²×C₄	—	89	1	no	—
270	C₂²×C₄	—	89	2	no	—
271	C₂²×C₄	—	89	2	no	—
272	C₂²×C₄	—	356	1	no	—
273	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
274	M₁₆	—	2	2	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	356	4	no	—
277	M₁₆	—	356	1	no	—
278	M₁₆	—	178	2	no	—
279	C₂×C₈	—	4	1	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	—	89	2	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	—	89	2	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	89	2	no	—
291	C₂×C₈	—	89	2	no	—
292	C₂×C₈	—	356	4	no	—
293	C₂×C₈	—	356	1	no	—
294	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
296	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
299	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	356	2	no	—
305	C₂²⋊C₄	—	356	2	no	—
306	C₄²	—	356	2	no	—
307	C₄²	—	356	2	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	178	2	no	—
310	C₂³	—	89	2	no	—
311	C₂³	—	89	2	no	—
312	C₂³	—	89	1	no	—
313	C₂³	—	89	1	no	—
314	C₂³	—	356	1	no	—
315	C₂³	—	356	1	no	—
316	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	8	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	—	89	2	no	—
324	C₂×C₄	—	89	2	no	—
325	C₂×C₄	—	89	2	no	—
326	C₂×C₄	—	89	2	no	—
327	C₂×C₄	—	89	2	no	—
328	C₂×C₄	—	89	2	no	—
329	C₂×C₄	—	89	2	no	—
330	C₂×C₄	—	89	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	712	2	no	—
333	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
336	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
337	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
338	C₂×C₄	Q₁₆×D₈₉	1	1	yes	characteristic
339	C₈	—	4	2	no	—
340	C₈	—	89	2	no	—
341	C₈	—	89	2	no	—
342	C₈	—	356	2	no	—
343	C₈	C₄²×D₈₉	1	1	yes	characteristic
344	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
345	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
346	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
347	C₂²	—	2	1	no	—
348	C₂²	—	2	1	no	—
349	C₂²	—	89	2	no	—
350	C₂²	—	89	2	no	—
351	C₂²	—	356	2	no	—
352	C₂²	—	356	2	no	—
353	C₂²	—	178	1	no	—
354	C₂²	—	178	1	no	—
355	C₂²	—	178	1	no	—
356	C₂²	—	178	1	no	—
357	C₂²	#102×D₈₉	1	1	yes	characteristic
358	C₂²	#103×D₈₉	1	1	yes	characteristic
359	C₂²	#106×D₈₉	1	1	yes	characteristic
360	C₄	—	8	2	no	—
361	C₄	—	89	2	no	—
362	C₄	—	89	2	no	—
363	C₄	—	712	2	no	—
364	C₄	#95×D₈₉	1	1	yes	characteristic
365	C₄	#102×D₈₉	1	1	yes	characteristic
366	C₄	#103×D₈₉	1	1	yes	characteristic
367	C₄	#107×D₈₉	1	1	yes	characteristic
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	—	178	2	no	—
371	C₂	—	178	2	no	—
372	C₂	#340×D₈₉	1	1	yes	characteristic
373	C₂	#26021998	1	1	yes	characteristic
374	C₂	#26022000	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#102×D₈₉
commutator	C₈×C₈₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1387	C₂
socle	C₂²×C₈₉	C₂×#102
prime radical	C₂³×D₈₉	SD₁₆
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1387	D₈₉
hypocenter	C₈₉	#12569
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1387

Complements

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

N	H ≅ G/N	type
4. #26018362	C₂	semidirect
5. #26018361	C₂	semidirect
6. #26018360	C₂	semidirect
7. C₈₉×#1387	C₂	semidirect
14. M₃₂×D₈₉	C₄	semidirect
19. #108×D₈₉	C₄	semidirect
194. #1387	D₈₉	semidirect
195. C₈₉	#12569	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×#19151635	2
#26018362	2
#26018361	2
#26018360	2
C₈₉×#1387	2
#12569	89