Subgroups — #36709517

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030839	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15668035	C₂	1	1	yes	maximal, characteristic
3	#26022462	C₂	1	2	yes	maximal, complemented
4	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022298	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022195	C₂	1	2	yes	maximal
8	#26022120	C₂	1	2	yes	maximal, complemented
9	#26022115	C₂	1	2	yes	maximal, complemented
10	#26022054	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668035	C₂²	1	1	yes	characteristic
13	#19152153	—	2	1	no	—
14	C₄×#13083575	—	2	1	no	—
15	C₂×#15667958	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151950	—	2	1	no	—
19	#19151840	—	2	1	no	—
20	#19151840	C₂²	1	2	yes	—
21	Q₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
22	#19151692	—	2	1	no	—
23	#19151692	C₂²	1	2	yes	—
24	C₄×#13083561	—	2	1	no	—
25	#19151683	—	2	1	no	—
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	#19151680	C₂²	1	1	yes	characteristic
28	#19151635	C₂²	1	1	yes	characteristic
29	#19151593	—	2	1	no	—
30	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
31	#19151425	C₂²	1	2	yes	—
32	C₄⋊#13083549	C₂²	1	1	yes	characteristic
33	#19151403	C₂²	1	2	yes	—
34	#19151401	C₂²	1	2	yes	—
35	#19151398	C₂²	1	2	yes	—
36	#19151377	C₂²	1	2	yes	—
37	#19151373	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#366×C₈₉	C₂²	1	2	yes	—
41	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#358×C₈₉	C₂²	1	2	yes	—
43	C₂×#13083575	—	2	1	no	—
44	#15668035	—	2	2	no	—
45	#15668035	—	2	2	no	—
46	C₂×Dic₃₅₆	—	2	1	no	—
47	C₂×Dic₃₅₆	C₂³	1	2	yes	—
48	C₂×#13083561	—	2	1	no	—
49	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
50	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
52	#15668012	—	2	1	no	—
53	#15668003	—	4	1	no	—
54	#15667992	—	2	2	no	—
55	#15667988	—	4	1	no	—
56	Q₈×Dic₈₉	—	2	1	no	—
57	Q₈×Dic₈₉	—	4	1	no	—
58	#15667977	—	2	2	no	—
59	C₄×D₃₅₆	—	2	1	no	—
60	C₄×D₃₅₆	—	4	1	no	—
61	#15667974	—	2	1	no	—
62	#15667974	—	4	1	no	—
63	#15667970	—	2	1	no	—
64	#15667970	—	4	1	no	—
65	#15667970	—	4	1	no	—
66	C₄²×D₈₉	—	4	1	no	—
67	C₄²×D₈₉	D₄	1	1	yes	characteristic
68	#15667958	—	2	4	no	—
69	#15667957	—	2	4	no	—
70	C₂×#13083549	C₂³	1	2	yes	—
71	#15667946	C₂³	1	1	yes	characteristic
72	#15667943	—	2	1	no	—
73	#15667936	—	2	1	no	—
74	#15667935	—	2	2	no	—
75	#15667935	—	2	2	no	—
76	#15667934	—	2	1	no	—
77	#15667932	—	2	1	no	—
78	C₄×#10623494	D₄	1	1	yes	characteristic
79	C₈×Dic₈₉	D₄	1	1	yes	characteristic
80	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
81	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₈₉	—	2	1	no	—
83	C₄×Q₈×C₈₉	—	2	1	no	—
84	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
85	#107×C₈₉	—	2	1	no	—
86	#107×C₈₉	C₂³	1	1	yes	characteristic
87	#107×C₈₉	C₂³	1	2	yes	—
88	#103×C₈₉	—	2	1	no	—
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	#13083575	—	2	4	no	—
91	#13083575	—	4	2	no	—
92	C₂×Dic₁₇₈	—	2	2	no	—
93	C₂×D₃₅₆	—	2	1	no	—
94	C₂×D₃₅₆	—	4	1	no	—
95	C₂×C₄×D₈₉	—	4	1	no	—
96	C₂×C₄×D₈₉	—	4	1	no	—
97	C₂×C₄×D₈₉	—	4	1	no	—
98	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
99	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
100	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
101	Dic₃₅₆	—	2	2	no	—
102	Dic₃₅₆	—	2	2	no	—
103	Dic₃₅₆	—	4	2	no	—
104	#13083561	—	2	4	no	—
105	D₇₁₂	—	2	1	no	—
106	D₇₁₂	#41	1	2	yes	—
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
109	C₈×D₈₉	—	2	1	no	—
110	C₈×D₈₉	—	2	1	no	—
111	C₈×D₈₉	—	4	1	no	—
112	C₄⋊Dic₈₉	—	2	1	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	—	4	1	no	—
115	C₄⋊Dic₈₉	—	4	1	no	—
116	C₄⋊Dic₈₉	—	4	2	no	—
117	#13083553	—	4	2	no	—
118	#13083551	—	4	1	no	—
119	#13083551	—	4	2	no	—
120	#13083551	—	4	1	no	—
121	#13083551	—	4	1	no	—
122	C₄×Dic₈₉	—	4	1	no	—
123	C₄×Dic₈₉	—	4	2	no	—
124	C₄×Dic₈₉	—	4	1	no	—
125	C₄×Dic₈₉	D₈	1	1	yes	characteristic
126	C₄×Dic₈₉	D₈	1	1	yes	characteristic
127	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	#13083549	—	2	2	no	—
129	C₂×Q₈×C₈₉	—	2	1	no	—
130	C₂×Q₈×C₈₉	—	4	1	no	—
131	Q₁₆×C₈₉	—	2	1	no	—
132	Q₁₆×C₈₉	#41	1	2	yes	—
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
135	C₄⋊C₄×C₈₉	—	2	1	no	—
136	C₄⋊C₄×C₈₉	—	2	2	no	—
137	C₄⋊C₄×C₈₉	—	4	1	no	—
138	C₄⋊C₄×C₈₉	—	4	1	no	—
139	C₄²×C₈₉	—	4	1	no	—
140	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₁₆×C₈₉	—	2	2	no	—
142	C₂²×D₈₉	—	4	1	no	—
143	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
144	Q₈×C₈₉	—	2	2	no	—
145	Q₈×C₈₉	—	4	1	no	—
146	Q₈×C₈₉	—	8	2	no	—
147	C₂×C₄×C₈₉	—	4	1	no	—
148	C₂×C₄×C₈₉	—	4	1	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	C₂×Dic₈₉	—	4	1	no	—
154	C₂×Dic₈₉	—	4	2	no	—
155	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
156	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
157	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
158	D₃₅₆	—	2	2	no	—
159	D₃₅₆	—	4	4	no	—
160	D₃₅₆	—	4	1	no	—
161	C₄×D₈₉	—	2	1	no	—
162	C₄×D₈₉	—	2	1	no	—
163	C₄×D₈₉	—	2	2	no	—
164	C₄×D₈₉	—	4	1	no	—
165	C₄×D₈₉	—	4	4	no	—
166	C₄×D₈₉	—	8	2	no	—
167	C₄×D₈₉	—	8	2	no	—
168	Dic₁₇₈	—	4	2	no	—
169	Dic₁₇₈	—	4	2	no	—
170	C₈×C₈₉	—	2	1	no	—
171	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
172	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
173	#10623494	—	2	1	no	—
174	#10623494	—	2	1	no	—
175	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	2	1	no	—
178	C₂×D₈₉	—	2	1	no	—
179	C₂×D₈₉	—	4	2	no	—
180	C₂×D₈₉	—	8	1	no	—
181	C₂×D₈₉	—	8	1	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	4	2	no	—
184	C₄×C₈₉	—	8	1	no	—
185	C₄×C₈₉	—	8	2	no	—
186	C₄×C₈₉	#510	1	2	yes	—
187	C₄×C₈₉	#525	1	1	yes	characteristic
188	C₄×C₈₉	#575	1	1	yes	characteristic
189	Dic₈₉	—	2	1	no	—
190	Dic₈₉	—	2	1	no	—
191	Dic₈₉	—	2	1	no	—
192	Dic₈₉	—	8	2	no	—
193	Dic₈₉	—	8	1	no	—
194	#33676	—	89	1	no	maximal, Sylow
195	C₂×C₈₉	#3100	1	1	yes	characteristic
196	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
197	C₂×C₈₉	#3414	1	1	yes	characteristic
198	D₈₉	—	4	1	no	—
199	D₈₉	—	8	2	no	—
200	#3170	—	89	1	no	—
201	C₄×#135	—	89	1	no	—
202	#2237	—	89	2	no	—
203	#2227	D₈₉	1	1	yes	characteristic, complemented
204	#2226	—	89	1	no	—
205	#2209	—	89	1	no	—
206	#2208	—	89	1	no	—
207	#2189	—	89	2	no	—
208	#2176	—	89	2	no	—
209	#2175	—	89	2	no	—
210	#2163	—	89	1	no	—
211	C₈₉	#33676	1	1	yes	characteristic, complemented, Sylow
212	C₂×#135	—	89	1	no	—
213	#533	—	178	1	no	—
214	C₄×#41	—	178	1	no	—
215	C₂×Q₃₂	—	89	2	no	—
216	C₂×SD₃₂	—	89	2	no	—
217	#500	C₂×D₈₉	1	1	yes	characteristic
218	#494	—	89	1	no	—
219	#487	—	178	1	no	—
220	#467	—	89	2	no	—
221	#467	—	178	1	no	—
222	C₄×Q₁₆	—	89	1	no	—
223	C₄×Q₁₆	—	89	2	no	—
224	C₄×Q₁₆	—	178	1	no	—
225	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
226	C₄×SD₁₆	—	178	1	no	—
227	C₄×SD₁₆	—	178	1	no	—
228	C₄×D₈	—	89	1	no	—
229	C₄×D₈	—	89	1	no	—
230	#427	—	89	1	no	—
231	#416	—	178	1	no	—
232	C₂×C₄×C₈	—	89	1	no	—
233	#367	—	89	2	no	—
234	#366	C₂×D₈₉	1	2	yes	—
235	C₄⋊C₁₆	—	89	1	no	—
236	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
237	#358	—	89	2	no	—
238	#358	—	89	2	no	—
239	#358	C₂×D₈₉	1	2	yes	—
240	#357	—	89	2	no	—
241	C₂²⋊C₁₆	—	89	2	no	—
242	#346	—	89	2	no	—
243	C₂×#41	—	178	1	no	—
244	#135	—	178	2	no	—
245	#135	—	178	2	no	—
246	C₂×Q₁₆	—	89	2	no	—
247	C₂×Q₁₆	—	178	1	no	—
248	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
249	C₂×SD₁₆	—	178	1	no	—
250	C₂×D₈	—	89	1	no	—
251	C₂²×C₈	—	89	1	no	—
252	C₂²×C₈	—	89	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	178	1	no	—
255	#125	—	356	1	no	—
256	C₂²⋊Q₈	—	356	1	no	—
257	C₂²⋊Q₈	—	178	2	no	—
258	C₄×Q₈	—	2	1	no	—
259	C₄×Q₈	—	356	1	no	—
260	C₄×Q₈	—	178	2	no	—
261	C₄×Q₈	—	178	1	no	—
262	C₄×D₄	—	356	1	no	—
263	C₄×D₄	—	356	1	no	—
264	C₄×D₄	—	178	1	no	—
265	C₄×D₄	—	178	1	no	—
266	#117	—	356	1	no	—
267	#117	—	356	1	no	—
268	#117	—	178	1	no	—
269	C₂×C₄²	—	89	1	no	—
270	C₂×C₄²	—	356	1	no	—
271	Q₃₂	—	178	4	no	—
272	SD₃₂	—	178	4	no	—
273	C₂×C₁₆	—	89	2	no	—
274	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
275	#107	—	2	1	no	—
276	#107	—	89	1	no	—
277	#107	C₂²×D₈₉	1	2	yes	—
278	#107	C₂²×D₈₉	1	1	yes	characteristic
279	#106	—	178	1	no	—
280	#103	—	2	1	no	—
281	#103	—	178	2	no	—
282	#103	—	178	2	no	—
283	#103	—	178	1	no	—
284	#102	—	178	1	no	—
285	#102	—	178	1	no	—
286	C₄×C₈	—	89	1	no	—
287	C₄×C₈	—	89	1	no	—
288	C₄×C₈	—	89	1	no	—
289	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	#41	—	356	2	no	—
291	#41	—	178	4	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	4	1	no	—
294	C₂×Q₈	—	178	2	no	—
295	C₂×D₄	—	356	1	no	—
296	C₂×D₄	—	178	1	no	—
297	C₂²×C₄	—	89	1	no	—
298	C₂²×C₄	—	89	1	no	—
299	C₂²×C₄	—	89	1	no	—
300	C₂²×C₄	—	356	1	no	—
301	C₂²×C₄	—	356	1	no	—
302	C₂²×C₄	—	356	1	no	—
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	356	2	no	—
305	Q₁₆	—	178	2	no	—
306	Q₁₆	—	178	2	no	—
307	Q₁₆	#13083575	1	2	yes	—
308	SD₁₆	—	178	4	no	—
309	D₈	—	89	2	no	—
310	D₈	—	178	1	no	—
311	C₂×C₈	—	89	1	no	—
312	C₂×C₈	—	89	1	no	—
313	C₂×C₈	—	356	1	no	—
314	C₂×C₈	—	178	1	no	—
315	C₂×C₈	—	178	1	no	—
316	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
317	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	4	1	no	—
321	C₄⋊C₄	—	4	1	no	—
322	C₄⋊C₄	—	356	1	no	—
323	C₄⋊C₄	—	356	2	no	—
324	C₄⋊C₄	—	356	2	no	—
325	C₄⋊C₄	—	356	1	no	—
326	C₄⋊C₄	—	356	1	no	—
327	C₄⋊C₄	—	178	1	no	—
328	C₂²⋊C₄	—	356	1	no	—
329	C₂²⋊C₄	—	356	2	no	—
330	C₂²⋊C₄	—	356	1	no	—
331	C₂²⋊C₄	—	356	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	—	89	1	no	—
334	C₄²	—	89	1	no	—
335	C₄²	—	89	1	no	—
336	C₄²	—	356	1	no	—
337	C₄²	—	356	1	no	—
338	C₄²	—	356	2	no	—
339	C₄²	D₄×D₈₉	1	1	yes	characteristic
340	C₁₆	—	2	2	no	—
341	C₁₆	—	178	2	no	—
342	C₂³	—	89	1	no	—
343	C₂³	—	356	1	no	—
344	Q₈	—	2	2	no	—
345	Q₈	—	4	1	no	—
346	Q₈	—	8	2	no	—
347	Q₈	—	356	2	no	—
348	Q₈	—	356	2	no	—
349	D₄	—	356	4	no	—
350	D₄	—	356	1	no	—
351	D₄	—	178	2	no	—
352	C₂×C₄	—	4	1	no	—
353	C₂×C₄	—	4	1	no	—
354	C₂×C₄	—	4	2	no	—
355	C₂×C₄	—	89	1	no	—
356	C₂×C₄	—	89	1	no	—
357	C₂×C₄	—	89	1	no	—
358	C₂×C₄	—	356	1	no	—
359	C₂×C₄	—	356	4	no	—
360	C₂×C₄	—	356	1	no	—
361	C₂×C₄	—	356	2	no	—
362	C₂×C₄	—	178	1	no	—
363	C₂×C₄	—	178	2	no	—
364	C₂×C₄	—	178	1	no	—
365	C₂×C₄	—	712	2	no	—
366	C₂×C₄	—	712	2	no	—
367	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
368	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
369	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	178	1	no	—
372	C₈	—	178	1	no	—
373	C₈	#41×D₈₉	1	1	yes	characteristic
374	C₈	C₂×#13083575	1	1	yes	characteristic
375	C₂²	—	356	2	no	—
376	C₂²	—	178	1	no	—
377	C₂²	—	178	1	no	—
378	C₂²	—	178	1	no	—
379	C₂²	—	712	1	no	—
380	C₂²	—	712	1	no	—
381	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
382	C₄	—	2	1	no	—
383	C₄	—	4	2	no	—
384	C₄	—	8	1	no	—
385	C₄	—	8	2	no	—
386	C₄	—	178	1	no	—
387	C₄	—	178	1	no	—
388	C₄	—	178	1	no	—
389	C₄	—	712	2	no	—
390	C₄	—	712	1	no	—
391	C₄	#19152040	1	2	yes	—
392	C₄	#19152111	1	1	yes	characteristic
393	C₄	#19152397	1	1	yes	characteristic
394	C₂	—	356	1	no	—
395	C₂	—	712	2	no	—
396	C₂	#26030254	1	1	yes	characteristic
397	C₂	C₂×#19152040	1	1	yes	characteristic
398	C₂	#26033384	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2227	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15668035	C₂²
hypercenter	#2227	D₈₉
hypocenter	C₈₉	#33676
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
1. #26030839	C₂	semidirect
3. #26022462	C₂	semidirect
4. #26022396	C₂	semidirect
5. #26022302	C₂	semidirect
6. #26022298	C₂	semidirect
8. #26022120	C₂	semidirect
9. #26022115	C₂	semidirect
10. #26022054	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
203. #2227	D₈₉	semidirect
211. C₈₉	#33676	semidirect

Maximal subgroups

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

subgroup	index
#26030839	2
C₄×#15668035	2
#26022462	2
#26022396	2
#26022302	2
#26022298	2
#26022195	2
#26022120	2
#26022115	2
#26022054	2
C₈₉×#2227	2
#33676	89