Subgroups — #36709334

Order 22784.

534 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029905	C₂	1	1	yes	maximal, characteristic
2	#26028862	C₂	1	1	yes	maximal, characteristic
3	#26022473	C₂	1	2	yes	maximal, complemented
4	#26022472	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022471	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022470	C₂	1	2	yes	maximal, complemented
7	#26022469	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022023	C₂	1	2	yes	maximal, complemented
9	#362×D₈₉	C₂	1	2	yes	maximal, complemented
10	#26021871	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2239	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	1	yes	characteristic
13	#19152364	—	2	1	no	—
14	C₂×#15668020	C₂²	1	1	yes	characteristic
15	#19152171	—	2	1	no	—
16	M₃₂×D₈₉	C₂²	1	2	yes	—
17	#19151857	C₂²	1	1	yes	characteristic
18	#19151856	C₂²	1	1	yes	characteristic
19	#19151853	C₂²	1	1	yes	characteristic
20	#19151838	—	2	1	no	—
21	#19151837	—	2	1	no	—
22	#19151647	C₂²	1	1	yes	characteristic
23	#19151645	—	2	1	no	—
24	#19151645	C₂²	1	2	yes	—
25	C₂×#15667947	—	2	1	no	—
26	#108×D₈₉	—	2	1	no	—
27	#108×D₈₉	C₂²	1	2	yes	—
28	#19151633	C₂²	1	1	yes	characteristic
29	#19151605	—	2	1	no	—
30	#19151422	C₂²	1	2	yes	—
31	#19151421	C₂²	1	2	yes	—
32	#19151420	C₂²	1	2	yes	—
33	#19151415	C₂²	1	2	yes	—
34	#19151414	C₂²	1	2	yes	—
35	#19151413	C₂²	1	2	yes	—
36	#19151380	C₂²	1	1	yes	characteristic
37	#19151379	C₂²	1	2	yes	—
38	#470×C₈₉	C₂²	1	1	yes	characteristic
39	#430×C₈₉	C₂²	1	1	yes	characteristic
40	#365×C₈₉	C₂²	1	2	yes	—
41	#362×C₈₉	C₂²	1	2	yes	—
42	#349×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083574	D₄	1	1	yes	characteristic
44	C₂×Q₈×D₈₉	—	2	1	no	—
45	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
46	C₂×#13083564	—	2	1	no	—
47	Q₁₆×D₈₉	—	2	2	no	—
48	Q₁₆×D₈₉	—	2	2	no	—
49	#15668020	—	2	2	no	—
50	#15668020	—	2	2	no	—
51	#15668020	—	4	2	no	—
52	#15668018	—	4	2	no	—
53	C₂×#13083558	—	2	1	no	—
54	C₂×#13083557	D₄	1	1	yes	characteristic
55	C₂×#13083557	C₂³	1	1	yes	characteristic
56	M₁₆×D₈₉	—	2	2	no	—
57	M₁₆×D₈₉	—	2	2	no	—
58	C₂²×#10623494	D₄	1	1	yes	characteristic
59	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
60	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
61	#15668011	—	4	1	no	—
62	#15667998	—	4	1	no	—
63	#15667997	—	4	1	no	—
64	#15667994	—	4	1	no	—
65	#15667992	—	2	1	no	—
66	#15667991	—	2	1	no	—
67	#15667988	—	2	1	no	—
68	#15667970	—	2	1	no	—
69	#15667952	C₂³	1	2	yes	—
70	#15667951	—	2	2	no	—
71	C₁₆×D₈₉	—	2	2	no	—
72	#15667948	—	2	1	no	—
73	#15667948	C₂³	1	2	yes	—
74	#15667947	—	2	2	no	—
75	#15667947	—	2	2	no	—
76	#15667945	—	2	1	no	—
77	#15667944	C₂³	1	1	yes	characteristic
78	#15667937	—	2	1	no	—
79	#15667936	—	2	1	no	—
80	#15667935	—	2	1	no	—
81	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×M₁₆×C₈₉	—	2	1	no	—
83	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²⋊Q₈×C₈₉	—	2	1	no	—
85	M₃₂×C₈₉	C₂³	1	2	yes	—
86	#108×C₈₉	—	2	1	no	—
87	#108×C₈₉	C₂³	1	2	yes	—
88	#106×C₈₉	C₂³	1	1	yes	characteristic
89	#103×C₈₉	—	2	1	no	—
90	#13083574	—	2	2	no	—
91	#13083574	—	2	2	no	—
92	C₂×Dic₁₇₈	—	2	1	no	—
93	C₂×Dic₁₇₈	—	4	1	no	—
94	C₂×Dic₁₇₈	SD₁₆	1	1	yes	characteristic
95	Q₈×D₈₉	—	4	2	no	—
96	Q₈×D₈₉	—	4	2	no	—
97	C₂×D₃₅₆	SD₁₆	1	1	yes	characteristic
98	C₂×C₂²⋊D₈₉	SD₁₆	1	1	yes	characteristic
99	C₂×C₂²⋊D₈₉	SD₁₆	1	1	yes	characteristic
100	C₂×C₄×D₈₉	—	4	1	no	—
101	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
103	Dic₃₅₆	—	2	2	no	—
104	#13083564	—	4	1	no	—
105	#13083564	—	4	1	no	—
106	#13083558	—	2	2	no	—
107	#13083558	—	4	1	no	—
108	#13083557	—	2	2	no	—
109	#13083557	—	2	2	no	—
110	#13083557	—	4	2	no	—
111	#13083557	—	4	2	no	—
112	C₂×#10623494	—	2	1	no	—
113	C₂×#10623494	—	2	1	no	—
114	C₂×#10623494	—	4	1	no	—
115	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
116	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
117	C₈×D₈₉	—	2	1	no	—
118	C₈×D₈₉	—	2	1	no	—
119	C₈×D₈₉	—	4	2	no	—
120	C₈×D₈₉	—	4	2	no	—
121	C₈×D₈₉	#41	1	2	yes	—
122	C₈×D₈₉	#41	1	2	yes	—
123	C₄⋊Dic₈₉	—	2	1	no	—
124	C₄⋊Dic₈₉	—	4	1	no	—
125	#13083553	—	4	1	no	—
126	#13083553	—	4	1	no	—
127	C₂²⋊Dic₈₉	—	4	1	no	—
128	#13083551	—	4	1	no	—
129	#13083551	—	4	1	no	—
130	#13083551	—	4	1	no	—
131	C₄×Dic₈₉	—	4	1	no	—
132	#13083549	—	2	2	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	Q₁₆×C₈₉	—	2	2	no	—
136	M₁₆×C₈₉	—	2	2	no	—
137	M₁₆×C₈₉	—	2	2	no	—
138	M₁₆×C₈₉	—	4	1	no	—
139	C₂×C₈×C₈₉	—	4	1	no	—
140	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
142	C₄⋊C₄×C₈₉	—	2	1	no	—
143	C₄⋊C₄×C₈₉	—	4	1	no	—
144	C₂²⋊C₄×C₈₉	—	4	1	no	—
145	C₁₆×C₈₉	—	2	2	no	—
146	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
147	C₂²×D₈₉	C₂×SD₁₆	1	1	yes	characteristic
148	C₂²×D₈₉	C₂×SD₁₆	1	1	yes	characteristic
149	Q₈×C₈₉	—	4	1	no	—
150	Q₈×C₈₉	—	4	1	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₈₉	—	2	2	no	—
158	C₂²⋊D₈₉	—	2	2	no	—
159	C₂×Dic₈₉	—	4	1	no	—
160	C₂×Dic₈₉	—	4	1	no	—
161	C₂×Dic₈₉	C₂×SD₁₆	1	1	yes	characteristic
162	C₂×Dic₈₉	C₂×SD₁₆	1	1	yes	characteristic
163	D₃₅₆	—	2	2	no	—
164	C₄×D₈₉	—	2	1	no	—
165	C₄×D₈₉	—	2	1	no	—
166	C₄×D₈₉	—	8	2	no	—
167	C₄×D₈₉	#123	1	2	yes	—
168	C₄×D₈₉	AGL(1,ℤ/8)	1	2	yes	—
169	Dic₁₇₈	—	2	2	no	—
170	Dic₁₇₈	—	4	1	no	—
171	Dic₁₇₈	—	4	1	no	—
172	Dic₁₇₈	—	8	2	no	—
173	C₈×C₈₉	—	2	1	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	C₂×#41	1	2	yes	—
176	#10623494	—	2	1	no	—
177	#10623494	—	4	2	no	—
178	#10623494	#123	1	1	yes	characteristic
179	#10623494	#123	1	1	yes	characteristic
180	C₂²×C₈₉	—	2	1	no	—
181	C₂²×C₈₉	—	4	1	no	—
182	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
183	C₂×D₈₉	—	2	1	no	—
184	C₂×D₈₉	—	4	1	no	—
185	C₂×D₈₉	—	4	1	no	—
186	C₂×D₈₉	#481	1	2	yes	—
187	C₄×C₈₉	—	2	1	no	—
188	C₄×C₈₉	—	8	1	no	—
189	C₄×C₈₉	—	8	1	no	—
190	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
191	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	Dic₈₉	—	2	1	no	—
193	Dic₈₉	—	8	1	no	—
194	Dic₈₉	—	8	1	no	—
195	Dic₈₉	#481	1	2	yes	—
196	#33658	—	89	1	no	maximal, Sylow
197	C₂×C₈₉	—	2	1	no	—
198	C₂×C₈₉	—	4	1	no	—
199	C₂×C₈₉	C₂×#481	1	1	yes	characteristic
200	D₈₉	—	2	2	no	—
201	D₈₉	—	4	1	no	—
202	#3059	—	89	1	no	—
203	#2916	—	89	1	no	—
204	#2239	—	89	1	no	—
205	#2239	—	89	1	no	—
206	#2239	—	89	2	no	—
207	#2239	—	89	2	no	—
208	#2239	—	89	1	no	—
209	#2239	D₈₉	1	1	yes	characteristic, complemented
210	#2155	—	89	2	no	—
211	C₂×#362	—	89	2	no	—
212	#2115	—	89	1	no	—
213	C₈₉	#33658	1	1	yes	characteristic, complemented, Sylow
214	C₂²×Q₁₆	—	89	1	no	—
215	#568	—	178	1	no	—
216	C₂×#131	—	89	1	no	—
217	#536	—	178	1	no	—
218	C₂×M₃₂	—	89	2	no	—
219	#470	—	89	1	no	—
220	#470	—	89	1	no	—
221	#470	—	89	1	no	—
222	#470	C₂×D₈₉	1	1	yes	characteristic
223	#467	—	178	1	no	—
224	#467	—	178	1	no	—
225	#430	—	89	1	no	—
226	#430	—	89	2	no	—
227	#430	—	178	1	no	—
228	#430	C₂×D₈₉	1	1	yes	characteristic
229	C₂×#108	—	89	2	no	—
230	C₂×#108	—	178	1	no	—
231	C₂×#108	—	178	1	no	—
232	#427	—	89	1	no	—
233	#419	—	178	1	no	—
234	#365	—	89	2	no	—
235	#365	—	89	2	no	—
236	#365	—	89	2	no	—
237	#365	C₂×D₈₉	1	2	yes	—
238	#362	—	89	2	no	—
239	#362	—	89	2	no	—
240	#362	—	89	2	no	—
241	#362	C₂×D₈₉	1	2	yes	—
242	#349	—	89	1	no	—
243	#349	—	89	2	no	—
244	#349	C₂×D₈₉	1	1	yes	characteristic
245	C₂×#41	—	89	1	no	—
246	C₂²×Q₈	—	178	1	no	—
247	C₂×Q₁₆	—	89	1	no	—
248	C₂×Q₁₆	—	178	2	no	—
249	C₂×Q₁₆	—	178	2	no	—
250	C₂×Q₁₆	—	178	1	no	—
251	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
252	#131	—	356	2	no	—
253	#131	—	356	2	no	—
254	#131	—	178	2	no	—
255	#131	—	178	2	no	—
256	C₂×M₁₆	—	2	1	no	—
257	C₂×M₁₆	—	89	1	no	—
258	C₂×M₁₆	—	89	1	no	—
259	C₂×M₁₆	—	178	2	no	—
260	C₂×M₁₆	—	178	2	no	—
261	C₂×M₁₆	—	178	1	no	—
262	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₈	—	89	1	no	—
264	C₂²×C₈	—	89	1	no	—
265	C₂²×C₈	—	89	1	no	—
266	C₄⋊Q₈	—	356	1	no	—
267	#124	—	356	1	no	—
268	#123	—	356	1	no	—
269	#123	—	356	1	no	—
270	C₂²⋊Q₈	—	2	1	no	—
271	C₂²⋊Q₈	—	178	1	no	—
272	C₂²⋊Q₈	—	178	1	no	—
273	C₂²⋊Q₈	—	178	1	no	—
274	#117	—	178	1	no	—
275	M₃₂	—	89	2	no	—
276	M₃₂	—	178	2	no	—
277	M₃₂	C₂²×D₈₉	1	2	yes	—
278	C₂×C₁₆	—	178	2	no	—
279	#108	—	2	1	no	—
280	#108	—	89	2	no	—
281	#108	—	178	2	no	—
282	#108	—	178	2	no	—
283	#108	—	178	1	no	—
284	#108	C₂²×D₈₉	1	2	yes	—
285	#107	—	178	1	no	—
286	#106	—	89	1	no	—
287	#106	C₂²×D₈₉	1	1	yes	characteristic
288	#103	—	2	1	no	—
289	#103	—	178	1	no	—
290	#103	—	178	1	no	—
291	#103	—	178	1	no	—
292	#41	—	178	2	no	—
293	#41	—	178	2	no	—
294	C₂×Q₈	—	2	1	no	—
295	C₂×Q₈	—	89	1	no	—
296	C₂×Q₈	—	356	1	no	—
297	C₂×Q₈	—	356	2	no	—
298	C₂×Q₈	—	356	2	no	—
299	C₂×Q₈	—	178	1	no	—
300	C₂×D₄	—	89	1	no	—
301	C₂×D₄	—	89	1	no	—
302	C₂×D₄	—	89	1	no	—
303	C₂²×C₄	—	89	1	no	—
304	C₂²×C₄	—	89	1	no	—
305	C₂²×C₄	—	356	1	no	—
306	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	356	1	no	—
309	Q₁₆	—	356	1	no	—
310	Q₁₆	—	178	2	no	—
311	M₁₆	—	2	2	no	—
312	M₁₆	—	2	2	no	—
313	M₁₆	—	4	1	no	—
314	M₁₆	—	356	2	no	—
315	M₁₆	—	356	2	no	—
316	M₁₆	—	356	1	no	—
317	M₁₆	—	178	2	no	—
318	M₁₆	—	178	2	no	—
319	M₁₆	—	178	2	no	—
320	C₂×C₈	—	4	1	no	—
321	C₂×C₈	—	89	1	no	—
322	C₂×C₈	—	89	2	no	—
323	C₂×C₈	—	89	2	no	—
324	C₂×C₈	—	89	1	no	—
325	C₂×C₈	—	356	2	no	—
326	C₂×C₈	—	356	2	no	—
327	C₂×C₈	—	356	1	no	—
328	C₂×C₈	—	178	1	no	—
329	C₂×C₈	—	178	1	no	—
330	C₂×C₈	—	178	1	no	—
331	C₂×C₈	—	178	1	no	—
332	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	356	1	no	—
337	C₄⋊C₄	—	356	1	no	—
338	C₄⋊C₄	—	356	1	no	—
339	C₄⋊C₄	—	178	1	no	—
340	C₂²⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	356	1	no	—
342	C₂²⋊C₄	—	356	1	no	—
343	C₂²⋊C₄	—	356	1	no	—
344	C₂²⋊C₄	—	356	1	no	—
345	C₄²	—	356	1	no	—
346	C₁₆	—	2	2	no	—
347	C₁₆	—	178	2	no	—
348	C₂³	—	89	1	no	—
349	C₂³	—	89	1	no	—
350	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
351	Q₈	—	4	1	no	—
352	Q₈	—	4	1	no	—
353	Q₈	—	356	1	no	—
354	Q₈	—	356	1	no	—
355	Q₈	—	178	2	no	—
356	Q₈	—	712	2	no	—
357	D₄	—	178	2	no	—
358	D₄	—	178	2	no	—
359	D₄	—	178	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	89	1	no	—
365	C₂×C₄	—	89	2	no	—
366	C₂×C₄	—	89	2	no	—
367	C₂×C₄	—	89	1	no	—
368	C₂×C₄	—	356	1	no	—
369	C₂×C₄	—	356	1	no	—
370	C₂×C₄	—	178	1	no	—
371	C₂×C₄	—	178	1	no	—
372	C₂×C₄	—	712	2	no	—
373	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
374	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
375	C₈	—	2	1	no	—
376	C₈	—	4	2	no	—
377	C₈	—	89	1	no	—
378	C₈	—	89	1	no	—
379	C₈	—	356	2	no	—
380	C₈	—	178	1	no	—
381	C₈	#41×D₈₉	1	2	yes	—
382	C₂²	—	2	1	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	89	2	no	—
385	C₂²	—	356	1	no	—
386	C₂²	—	356	1	no	—
387	C₂²	—	178	1	no	—
388	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	8	1	no	—
391	C₄	—	8	1	no	—
392	C₄	—	89	2	no	—
393	C₄	—	178	1	no	—
394	C₄	—	712	1	no	—
395	C₄	—	712	1	no	—
396	C₄	#123×D₈₉	1	1	yes	characteristic
397	C₄	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
398	C₂	—	2	1	no	—
399	C₂	—	4	1	no	—
400	C₂	—	356	1	no	—
401	C₂	—	178	2	no	—
402	C₂	#481×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#481×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2239	C₂
socle	C₂×C₈₉	C₂×#481
prime radical	C₂×#13083574	D₄
hypercenter	#2239	D₈₉
hypocenter	C₈₉	#33658
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #2239

Complements

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

N	H ≅ G/N	type
3. #26022473	C₂	semidirect
4. #26022472	C₂	semidirect
5. #26022471	C₂	semidirect
6. #26022470	C₂	semidirect
7. #26022469	C₂	semidirect
8. #26022023	C₂	semidirect
9. #362×D₈₉	C₂	semidirect
11. C₈₉×#2239	C₂	semidirect
209. #2239	D₈₉	semidirect
213. C₈₉	#33658	semidirect

Maximal subgroups

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

subgroup	index
#26029905	2
#26028862	2
#26022473	2
#26022472	2
#26022471	2
#26022470	2
#26022469	2
#26022023	2
#362×D₈₉	2
#26021871	2
C₈₉×#2239	2
#33658	89