Subgroups — #36707838

Order 22784.

652 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033385	C₂	1	1	yes	maximal, characteristic
2	#443×D₈₉	C₂	1	2	yes	maximal, complemented
3	#26022139	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022136	C₂	1	2	yes	maximal, complemented
5	#26022133	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022132	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022130	C₂	1	2	yes	maximal, complemented
8	#26022045	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021978	C₂	1	2	yes	maximal, complemented
10	#26021845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2179	C₂	1	1	yes	maximal, characteristic, complemented
12	#135×D₈₉	C₂²	1	2	yes	—
13	#131×D₈₉	—	2	2	no	—
14	#19152040	C₄	1	2	yes	complemented
15	#19152037	C₄	1	2	yes	complemented
16	#19152035	C₄	1	2	yes	complemented
17	#19152032	C₄	1	2	yes	complemented
18	#19152030	C₂²	1	1	yes	characteristic
19	#19152028	C₂²	1	2	yes	—
20	C₂×#15667951	C₂²	1	1	yes	characteristic
21	C₂×#15667951	C₂²	1	1	yes	characteristic
22	#19151716	C₂²	1	2	yes	complemented
23	#19151715	C₂²	1	2	yes	complemented
24	#19151714	C₂²	1	2	yes	complemented
25	#19151713	—	2	2	no	—
26	#19151712	—	2	2	no	—
27	#108×D₈₉	C₂²	1	2	yes	—
28	#104×D₈₉	—	2	2	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151428	C₂²	1	1	yes	characteristic, complemented
31	#19151427	C₂²	1	2	yes	complemented
32	#19151406	C₂²	1	2	yes	complemented
33	#19151405	C₂²	1	2	yes	complemented
34	#19151404	C₂²	1	2	yes	complemented
35	#19151374	C₂²	1	1	yes	characteristic, complemented
36	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
37	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
38	#508×C₈₉	C₂²	1	1	yes	characteristic
39	#443×C₈₉	C₂²	1	2	yes	complemented
40	#368×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	#359×C₈₉	C₂²	1	2	yes	complemented
42	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#41×D₈₉	—	2	2	no	—
44	#15668035	C₂×C₄	1	2	yes	—
45	#15668034	C₂³	1	2	yes	—
46	#15668033	—	2	2	no	—
47	#15668032	—	2	2	no	—
48	#15668031	C₂×C₄	1	2	yes	—
49	Q₁₆×D₈₉	C₂×C₄	1	2	yes	—
50	SD₁₆×D₈₉	—	2	2	no	—
51	D₈×D₈₉	C₂×C₄	1	2	yes	—
52	#15668020	—	2	2	no	—
53	#15668020	—	4	2	no	—
54	#15668019	—	2	2	no	—
55	#15668018	—	2	2	no	—
56	#15668018	—	4	2	no	—
57	M₁₆×D₈₉	—	2	2	no	—
58	M₁₆×D₈₉	—	4	2	no	—
59	C₂×C₈×D₈₉	—	4	2	no	—
60	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
61	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
62	C₄²×D₈₉	D₄	1	1	yes	characteristic
63	Dic₇₁₂	C₂×C₄	1	1	yes	characteristic, complemented
64	#15667958	C₂×C₄	1	2	yes	complemented
65	#15667957	C₂×C₄	1	2	yes	complemented
66	#15667956	C₂×C₄	1	2	yes	complemented
67	#15667955	C₂×C₄	1	2	yes	complemented
68	D₁₄₂₄	C₂×C₄	1	1	yes	characteristic, complemented
69	#15667953	C₂×C₄	1	2	yes	complemented
70	#15667951	—	2	2	no	—
71	#15667951	C₂×C₄	1	2	yes	—
72	#15667951	C₂×C₄	1	2	yes	—
73	C₂×#13083549	C₂³	1	1	yes	characteristic
74	C₂×#13083549	C₂³	1	1	yes	characteristic
75	#15667948	C₂³	1	2	yes	—
76	#15667947	—	2	2	no	—
77	#15667940	—	2	2	no	—
78	#15667939	—	2	2	no	—
79	#15667938	—	2	2	no	—
80	C₄×#10623494	D₄	1	1	yes	characteristic
81	C₈×Dic₈₉	D₄	1	1	yes	characteristic
82	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
83	#135×C₈₉	C₂³	1	2	yes	—
84	#131×C₈₉	—	2	2	no	—
85	Q₃₂×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
86	SD₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
87	D₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
88	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
90	#108×C₈₉	C₂³	1	2	yes	—
91	#104×C₈₉	—	2	2	no	—
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	#13083575	—	2	2	no	—
94	#13083574	—	2	2	no	—
95	#13083574	—	4	2	no	—
96	#13083573	—	2	2	no	—
97	#13083573	—	4	2	no	—
98	Q₈×D₈₉	—	2	2	no	—
99	D₄×D₈₉	—	2	2	no	—
100	D₄×D₈₉	—	4	2	no	—
101	C₂×C₄×D₈₉	—	2	1	no	—
102	C₂×C₄×D₈₉	—	4	2	no	—
103	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
104	Dic₃₅₆	C₂²×C₄	1	2	yes	—
105	#13083564	—	2	2	no	—
106	#13083563	—	2	2	no	—
107	#13083562	—	2	2	no	—
108	#13083561	—	2	2	no	—
109	D₇₁₂	C₂²×C₄	1	2	yes	—
110	#13083559	—	2	2	no	—
111	#13083558	—	2	2	no	—
112	#13083558	—	4	2	no	—
113	#13083557	—	4	2	no	—
114	#13083557	—	4	2	no	—
115	#13083557	—	4	2	no	—
116	C₂×#10623494	—	4	2	no	—
117	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
118	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
119	C₈×D₈₉	—	4	2	no	—
120	C₈×D₈₉	—	4	2	no	—
121	C₈×D₈₉	—	4	2	no	—
122	C₈×D₈₉	D₈	1	2	yes	—
123	C₈×D₈₉	D₈	1	2	yes	—
124	C₈×D₈₉	C₂²×C₄	1	2	yes	—
125	C₈×D₈₉	#41	1	2	yes	—
126	C₄×Dic₈₉	—	2	1	no	—
127	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	#13083549	—	2	1	no	—
129	#13083549	C₂²×C₄	1	2	yes	—
130	#41×C₈₉	—	2	2	no	—
131	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
132	SD₁₆×C₈₉	—	2	2	no	—
133	D₈×C₈₉	C₂²×C₄	1	2	yes	—
134	M₁₆×C₈₉	—	2	2	no	—
135	M₁₆×C₈₉	—	4	2	no	—
136	C₂×C₈×C₈₉	—	4	2	no	—
137	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₁₆×C₈₉	—	2	1	no	—
141	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
142	C₁₆×C₈₉	C₂²×C₄	1	1	yes	characteristic
143	C₂²×D₈₉	—	4	2	no	—
144	C₂²×D₈₉	#135	1	1	yes	characteristic
145	Q₈×C₈₉	—	2	2	no	—
146	D₄×C₈₉	—	2	2	no	—
147	D₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂²⋊D₈₉	—	4	2	no	—
152	C₂²⋊D₈₉	—	4	2	no	—
153	C₂²⋊D₈₉	—	4	2	no	—
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	#135	1	1	yes	characteristic
157	D₃₅₆	—	2	2	no	—
158	D₃₅₆	—	4	2	no	—
159	C₄×D₈₉	—	4	2	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₄×D₈₉	—	4	2	no	—
163	C₄×D₈₉	C₄×D₄	1	2	yes	—
164	C₄×D₈₉	C₂×D₈	1	2	yes	—
165	Dic₁₇₈	—	2	2	no	—
166	Dic₁₇₈	—	4	2	no	—
167	C₈×C₈₉	—	4	2	no	—
168	C₈×C₈₉	C₂×D₈	1	2	yes	—
169	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
170	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
171	#10623494	—	4	2	no	—
172	#10623494	C₄×D₄	1	2	yes	—
173	#10623494	C₂×D₈	1	1	yes	characteristic
174	#10623494	C₂×D₈	1	1	yes	characteristic
175	C₂²×C₈₉	—	4	2	no	—
176	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
177	C₂×D₈₉	—	4	1	no	—
178	C₂×D₈₉	—	4	2	no	—
179	C₂×D₈₉	—	8	4	no	—
180	C₂×D₈₉	C₄×D₈	1	2	yes	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	—	4	2	no	—
183	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
184	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
185	Dic₈₉	—	2	2	no	—
186	Dic₈₉	—	4	2	no	—
187	Dic₈₉	C₄×D₈	1	2	yes	—
188	#33463	—	89	1	no	maximal, Sylow
189	C₂×C₈₉	—	2	1	no	—
190	C₂×C₈₉	—	8	2	no	—
191	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
192	D₈₉	—	2	2	no	—
193	D₈₉	—	8	2	no	—
194	#3414	—	89	1	no	—
195	C₂×#443	—	89	2	no	—
196	#2179	—	89	2	no	—
197	#2179	—	89	1	no	—
198	#2179	D₈₉	1	1	yes	characteristic, complemented
199	#2178	—	89	1	no	—
200	#2178	—	89	1	no	—
201	#2178	—	89	2	no	—
202	#2161	—	89	1	no	—
203	#2143	—	89	2	no	—
204	C₄×M₃₂	—	89	1	no	—
205	C₈₉	#33463	1	1	yes	characteristic, complemented, Sylow
206	C₂×#135	—	89	2	no	—
207	C₂×#131	—	178	2	no	—
208	#510	—	89	2	no	—
209	#510	—	89	2	no	—
210	#509	—	89	2	no	—
211	#509	—	89	2	no	—
212	#508	—	89	2	no	—
213	#508	—	89	1	no	—
214	#508	C₂×D₈₉	1	1	yes	characteristic
215	C₂×M₃₂	—	89	1	no	—
216	C₂×M₃₂	—	89	1	no	—
217	#443	—	89	2	no	—
218	#443	—	89	2	no	—
219	#443	—	89	2	no	—
220	#443	—	178	2	no	—
221	#443	—	178	2	no	—
222	#443	C₂×D₈₉	1	2	yes	complemented
223	C₂×#108	—	89	2	no	—
224	C₂×#104	—	178	2	no	—
225	C₂×C₄×C₈	—	89	1	no	—
226	#368	—	89	1	no	—
227	#368	—	89	2	no	—
228	#368	C₂×D₈₉	1	1	yes	characteristic, complemented
229	#359	—	89	2	no	—
230	#359	—	89	2	no	—
231	#359	—	89	2	no	—
232	#359	C₂×D₈₉	1	2	yes	complemented
233	#346	—	89	1	no	—
234	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
235	C₄×C₁₆	—	89	1	no	—
236	C₄×C₁₆	—	89	1	no	—
237	C₂×#41	—	178	2	no	—
238	#135	—	89	2	no	—
239	#135	—	89	2	no	—
240	#135	—	89	2	no	—
241	#135	—	178	2	no	—
242	#135	—	178	2	no	—
243	#135	C₂²×D₈₉	1	2	yes	—
244	C₂×Q₁₆	—	89	2	no	—
245	C₂×SD₁₆	—	178	2	no	—
246	C₂×D₈	—	89	2	no	—
247	#131	—	2	2	no	—
248	#131	—	356	2	no	—
249	#131	—	356	2	no	—
250	#131	—	178	2	no	—
251	#131	—	178	2	no	—
252	#131	—	178	2	no	—
253	C₂×M₁₆	—	356	2	no	—
254	C₂×M₁₆	—	178	2	no	—
255	C₂²×C₈	—	89	1	no	—
256	C₂²×C₈	—	89	1	no	—
257	C₂²×C₈	—	356	2	no	—
258	C₂×C₄²	—	89	1	no	—
259	Q₃₂	—	89	2	no	—
260	Q₃₂	—	89	1	no	—
261	Q₃₂	C₄×D₈₉	1	1	yes	characteristic, complemented
262	SD₃₂	—	89	2	no	—
263	SD₃₂	—	89	2	no	—
264	SD₃₂	—	89	2	no	—
265	SD₃₂	C₄×D₈₉	1	2	yes	complemented
266	D₁₆	—	89	2	no	—
267	D₁₆	—	89	1	no	—
268	D₁₆	C₄×D₈₉	1	1	yes	characteristic, complemented
269	M₃₂	—	89	2	no	—
270	M₃₂	—	89	2	no	—
271	M₃₂	—	178	2	no	—
272	C₂×C₁₆	—	89	1	no	—
273	C₂×C₁₆	—	89	1	no	—
274	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
275	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
276	#108	—	89	2	no	—
277	#108	—	178	2	no	—
278	#108	C₂²×D₈₉	1	2	yes	—
279	#104	—	2	2	no	—
280	#104	—	178	2	no	—
281	#104	—	178	2	no	—
282	#104	—	178	2	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	—	89	1	no	—
285	C₄×C₈	—	89	1	no	—
286	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
287	#41	—	2	2	no	—
288	#41	—	356	2	no	—
289	#41	—	356	2	no	—
290	#41	—	178	2	no	—
291	#41	—	178	2	no	—
292	#41	—	178	2	no	—
293	C₂×Q₈	—	178	2	no	—
294	C₂×D₄	—	356	2	no	—
295	C₂×D₄	—	178	2	no	—
296	C₂²×C₄	—	89	1	no	—
297	C₂²×C₄	—	356	2	no	—
298	C₂²×C₄	—	178	1	no	—
299	Q₁₆	—	89	2	no	—
300	Q₁₆	—	178	2	no	—
301	Q₁₆	C₂×C₄×D₈₉	1	2	yes	—
302	SD₁₆	—	2	2	no	—
303	SD₁₆	—	178	2	no	—
304	SD₁₆	—	178	2	no	—
305	SD₁₆	—	178	2	no	—
306	D₈	—	89	2	no	—
307	D₈	—	178	2	no	—
308	D₈	C₂×C₄×D₈₉	1	2	yes	—
309	M₁₆	—	2	2	no	—
310	M₁₆	—	4	2	no	—
311	M₁₆	—	356	2	no	—
312	M₁₆	—	356	2	no	—
313	M₁₆	—	356	2	no	—
314	M₁₆	—	356	2	no	—
315	M₁₆	—	178	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	—	89	1	no	—
319	C₂×C₈	—	89	2	no	—
320	C₂×C₈	—	89	2	no	—
321	C₂×C₈	—	89	2	no	—
322	C₂×C₈	—	89	2	no	—
323	C₂×C₈	—	356	2	no	—
324	C₂×C₈	—	356	2	no	—
325	C₂×C₈	—	356	2	no	—
326	C₂×C₈	—	356	2	no	—
327	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
328	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
329	C₄²	—	89	1	no	—
330	C₄²	—	178	1	no	—
331	C₄²	D₄×D₈₉	1	1	yes	characteristic
332	C₁₆	—	2	1	no	—
333	C₁₆	—	89	2	no	—
334	C₁₆	—	178	1	no	—
335	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
336	C₁₆	C₂×C₄×D₈₉	1	1	yes	characteristic
337	C₂³	—	89	1	no	—
338	C₂³	—	356	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	356	2	no	—
341	Q₈	—	178	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	4	2	no	—
344	D₄	—	356	2	no	—
345	D₄	—	356	2	no	—
346	D₄	—	356	2	no	—
347	D₄	—	356	2	no	—
348	D₄	—	178	2	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	89	1	no	—
352	C₂×C₄	—	89	2	no	—
353	C₂×C₄	—	89	2	no	—
354	C₂×C₄	—	356	2	no	—
355	C₂×C₄	—	356	2	no	—
356	C₂×C₄	—	356	2	no	—
357	C₂×C₄	—	356	2	no	—
358	C₂×C₄	—	356	2	no	—
359	C₂×C₄	—	178	1	no	—
360	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
361	C₈	—	4	2	no	—
362	C₈	—	89	2	no	—
363	C₈	—	89	1	no	—
364	C₈	—	89	1	no	—
365	C₈	—	356	2	no	—
366	C₈	D₈×D₈₉	1	2	yes	—
367	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
368	C₈	#41×D₈₉	1	1	yes	characteristic
369	C₂²	—	4	2	no	—
370	C₂²	—	89	2	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	356	2	no	—
373	C₂²	—	712	4	no	—
374	C₂²	#135×D₈₉	1	1	yes	characteristic
375	C₄	—	4	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	89	2	no	—
378	C₄	—	356	2	no	—
379	C₄	—	178	2	no	—
380	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
381	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	8	2	no	—
384	C₂	—	178	2	no	—
385	C₂	—	712	2	no	—
386	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2179	C₂
socle	C₂×C₈₉	C₂×C₄×D₈
prime radical	#26033385	C₂
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2179	D₈₉
hypocenter	C₈₉	#33463
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

Complements

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

N	H ≅ G/N	type
2. #443×D₈₉	C₂	semidirect
3. #26022139	C₂	semidirect
4. #26022136	C₂	semidirect
5. #26022133	C₂	semidirect
6. #26022132	C₂	semidirect
7. #26022130	C₂	semidirect
8. #26022045	C₂	semidirect
9. #26021978	C₂	semidirect
10. #26021845	C₂	semidirect
11. C₈₉×#2179	C₂	semidirect
14. #19152040	C₄	semidirect
15. #19152037	C₄	semidirect
16. #19152035	C₄	semidirect
17. #19152032	C₄	semidirect
22. #19151716	C₂²	semidirect
23. #19151715	C₂²	semidirect
24. #19151714	C₂²	semidirect
30. #19151428	C₂²	semidirect
31. #19151427	C₂²	semidirect
32. #19151406	C₂²	semidirect
33. #19151405	C₂²	semidirect
34. #19151404	C₂²	semidirect
35. #19151374	C₂²	semidirect
36. C₁₆×Dic₈₉	C₂²	semidirect
37. C₁₆×Dic₈₉	C₂²	semidirect
39. #443×C₈₉	C₂²	semidirect
40. #368×C₈₉	C₂²	semidirect
41. #359×C₈₉	C₂²	semidirect
42. #346×C₈₉	C₂²	semidirect
63. Dic₇₁₂	C₂×C₄	semidirect
64. #15667958	C₂×C₄	semidirect
65. #15667957	C₂×C₄	semidirect
66. #15667956	C₂×C₄	semidirect
67. #15667955	C₂×C₄	semidirect
68. D₁₄₂₄	C₂×C₄	semidirect
69. #15667953	C₂×C₄	semidirect
85. Q₃₂×C₈₉	C₂×C₄	semidirect
86. SD₃₂×C₈₉	C₂×C₄	semidirect
87. D₁₆×C₈₉	C₂×C₄	semidirect
198. #2179	D₈₉	semidirect
205. C₈₉	#33463	semidirect
222. #443	C₂×D₈₉	semidirect
228. #368	C₂×D₈₉	semidirect
232. #359	C₂×D₈₉	semidirect
234. #346	C₂×D₈₉	semidirect
261. Q₃₂	C₄×D₈₉	semidirect
265. SD₃₂	C₄×D₈₉	semidirect
268. D₁₆	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26033385	2
#443×D₈₉	2
#26022139	2
#26022136	2
#26022133	2
#26022132	2
#26022130	2
#26022045	2
#26021978	2
#26021845	2
C₈₉×#2179	2
#33463	89