Subgroups — #36538280

Order 22784.

602 conjugacy classes of subgroups.

The subgroup lattice has 420 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
2	#26030058	C₂	1	2	yes	maximal
3	C₈×Q₈×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#26019713	C₂	1	2	yes	maximal, complemented
5	#26019712	C₂	1	2	yes	maximal, complemented
6	#26019653	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019623	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019622	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019009	C₂	1	1	yes	maximal, characteristic
10	#26018939	C₂	1	2	yes	maximal
11	C₈₉×#1676	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	C₄×Q₈×D₈₉	C₂²	1	1	yes	characteristic
14	#19151978	C₂²	1	1	yes	characteristic
15	#19151950	C₂²	1	2	yes	—
16	#19151904	—	2	2	no	—
17	#19151903	—	2	2	no	—
18	#19151890	C₂²	1	2	yes	—
19	#19151884	C₂²	1	2	yes	—
20	#19151883	C₂²	1	2	yes	—
21	#19151840	—	2	1	no	—
22	#19151840	—	2	1	no	—
23	Q₈×#10623494	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₇₈	—	2	1	no	—
25	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	#19151722	—	2	1	no	—
27	#19151722	C₂²	1	1	yes	characteristic
28	#19151692	—	2	1	no	—
29	#19151692	—	2	1	no	—
30	#19151639	—	2	2	no	—
31	#19151620	C₂²	1	2	yes	—
32	C₄⋊C₈×D₈₉	—	2	1	no	—
33	C₄⋊C₈×D₈₉	C₂²	1	1	yes	characteristic
34	#19151605	—	2	2	no	—
35	C₄×C₈×D₈₉	—	2	1	no	—
36	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
37	#19151351	C₂²	1	1	yes	characteristic
38	#19151350	C₂²	1	2	yes	—
39	#19151347	C₂²	1	2	yes	—
40	#19151345	C₂²	1	2	yes	—
41	#19151343	C₂²	1	2	yes	—
42	#500×C₈₉	C₂²	1	1	yes	characteristic
43	#477×C₈₉	C₂²	1	2	yes	—
44	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
45	#335×C₈₉	C₂²	1	1	yes	characteristic
46	#333×C₈₉	C₂²	1	2	yes	—
47	C₂×Q₈×D₈₉	D₄	1	1	yes	characteristic
48	C₂×Dic₃₅₆	—	2	1	no	—
49	C₂×Dic₃₅₆	—	2	1	no	—
50	C₂×#13083557	—	2	2	no	—
51	C₂×C₈×D₈₉	—	2	1	no	—
52	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
53	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
54	#15668012	C₂³	1	2	yes	—
55	#15668003	—	2	2	no	—
56	#15667992	—	2	2	no	—
57	#15667988	—	2	2	no	—
58	Q₈×Dic₈₉	—	2	2	no	—
59	Q₈×Dic₈₉	D₄	1	1	yes	characteristic
60	C₄×Dic₁₇₈	—	2	1	no	—
61	C₄×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	#15667977	—	2	1	no	—
63	#15667977	—	2	2	no	—
64	#15667977	D₄	1	1	yes	characteristic
65	#15667970	—	2	2	no	—
66	C₄⋊C₄×D₈₉	—	2	1	no	—
67	C₄⋊C₄×D₈₉	D₄	1	1	yes	characteristic
68	C₄²×D₈₉	—	2	1	no	—
69	C₄²×D₈₉	C₂³	1	1	yes	characteristic
70	#15667945	—	2	2	no	—
71	#15667944	—	2	2	no	—
72	#15667943	—	2	2	no	—
73	C₄⋊#10623494	—	2	1	no	—
74	C₄⋊#10623494	C₂³	1	1	yes	characteristic
75	C₄⋊#10623494	C₂³	1	2	yes	—
76	#15667941	—	2	1	no	—
77	#15667941	—	2	1	no	—
78	#15667941	—	2	2	no	—
79	#15667941	—	2	1	no	—
80	#15667937	—	2	2	no	—
81	#15667936	—	2	2	no	—
82	#15667935	—	2	2	no	—
83	#15667935	—	2	2	no	—
84	#15667935	—	2	2	no	—
85	C₄×#10623494	—	2	1	no	—
86	C₄×#10623494	C₂³	1	1	yes	characteristic
87	C₈×Dic₈₉	—	2	1	no	—
88	C₈×Dic₈₉	—	2	1	no	—
89	C₈×Dic₈₉	D₄	1	1	yes	characteristic
90	C₈×Dic₈₉	D₄	1	1	yes	characteristic
91	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
92	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
93	#107×C₈₉	—	2	1	no	—
94	#107×C₈₉	—	2	1	no	—
95	#107×C₈₉	—	2	2	no	—
96	C₄⋊C₈×C₈₉	—	2	1	no	—
97	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
99	#103×C₈₉	—	2	2	no	—
100	C₄×C₈×C₈₉	—	2	1	no	—
101	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₇₈	—	2	1	no	—
103	C₂×Dic₁₇₈	—	2	2	no	—
104	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₈₉	—	2	2	no	—
106	Q₈×D₈₉	Q₁₆	1	4	yes	complemented
107	C₂×C₄×D₈₉	—	2	1	no	—
108	C₂×C₄×D₈₉	—	2	1	no	—
109	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
112	Dic₃₅₆	—	4	2	no	—
113	Dic₃₅₆	—	4	2	no	—
114	#13083557	—	4	4	no	—
115	C₂×#10623494	—	2	1	no	—
116	C₂×#10623494	—	2	2	no	—
117	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
118	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
119	C₈×D₈₉	—	2	2	no	—
120	C₈×D₈₉	—	4	1	no	—
121	C₈×D₈₉	—	4	2	no	—
122	C₄⋊Dic₈₉	—	2	1	no	—
123	C₄⋊Dic₈₉	—	2	2	no	—
124	C₄⋊Dic₈₉	—	2	2	no	—
125	C₄⋊Dic₈₉	—	4	2	no	—
126	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	#13083553	—	2	1	no	—
128	#13083553	—	2	1	no	—
129	#13083553	—	2	1	no	—
130	#13083553	—	4	2	no	—
131	#13083551	—	4	2	no	—
132	C₄×Dic₈₉	—	2	1	no	—
133	C₄×Dic₈₉	—	2	1	no	—
134	C₄×Dic₈₉	—	2	1	no	—
135	C₄×Dic₈₉	—	4	2	no	—
136	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄×Dic₈₉	#41	1	1	yes	characteristic
139	C₂×Q₈×C₈₉	—	2	2	no	—
140	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₈₉	—	2	1	no	—
142	C₂×C₈×C₈₉	—	2	2	no	—
143	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₈₉	—	2	1	no	—
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	2	2	no	—
148	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₈₉	—	2	1	no	—
150	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
151	C₂²×D₈₉	#121	1	1	yes	characteristic
152	Q₈×C₈₉	—	2	1	no	—
153	Q₈×C₈₉	—	8	2	no	—
154	Q₈×C₈₉	C₂×Q₁₆	1	2	yes	—
155	C₂×C₄×C₈₉	—	2	1	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	4	2	no	—
158	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂×Dic₈₉	—	2	1	no	—
162	C₂×Dic₈₉	—	2	1	no	—
163	C₂×Dic₈₉	—	4	2	no	—
164	C₂×Dic₈₉	#121	1	1	yes	characteristic
165	C₂×Dic₈₉	#121	1	1	yes	characteristic
166	C₂×Dic₈₉	#121	1	1	yes	characteristic
167	C₄×D₈₉	—	2	4	no	—
168	C₄×D₈₉	—	2	1	no	—
169	C₄×D₈₉	—	2	1	no	—
170	C₄×D₈₉	—	4	1	no	—
171	C₄×D₈₉	—	4	2	no	—
172	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
173	C₄×D₈₉	#135	1	2	yes	—
174	Dic₁₇₈	—	2	1	no	—
175	Dic₁₇₈	—	2	4	no	—
176	Dic₁₇₈	—	4	2	no	—
177	Dic₁₇₈	—	4	2	no	—
178	Dic₁₇₈	C₂×Q₁₆	1	2	yes	—
179	C₈×C₈₉	—	2	1	no	—
180	C₈×C₈₉	—	4	2	no	—
181	C₈×C₈₉	—	4	1	no	—
182	C₈×C₈₉	#137	1	2	yes	—
183	#10623494	—	2	1	no	—
184	#10623494	—	2	1	no	—
185	#10623494	—	4	2	no	—
186	#10623494	—	4	1	no	—
187	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
188	C₂×D₈₉	—	2	1	no	—
189	C₂×D₈₉	—	2	1	no	—
190	C₂×D₈₉	#467	1	2	yes	—
191	C₄×C₈₉	—	2	2	no	—
192	C₄×C₈₉	—	2	1	no	—
193	C₄×C₈₉	—	4	1	no	—
194	C₄×C₈₉	—	8	2	no	—
195	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
196	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
197	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
198	C₄×C₈₉	#575	1	1	yes	characteristic
199	Dic₈₉	—	2	1	no	—
200	Dic₈₉	—	2	2	no	—
201	Dic₈₉	—	2	1	no	—
202	Dic₈₉	—	4	1	no	—
203	Dic₈₉	—	8	2	no	—
204	Dic₈₉	#467	1	2	yes	—
205	#19262	—	89	1	no	maximal, Sylow
206	C₂×C₈₉	#3037	1	1	yes	characteristic
207	C₂×C₈₉	C₂×#467	1	1	yes	characteristic
208	C₂×C₈₉	#3067	1	1	yes	characteristic
209	D₈₉	—	2	2	no	—
210	#3170	—	89	1	no	—
211	#3078	—	89	2	no	—
212	C₂×C₈×Q₈	—	89	1	no	—
213	#1684	—	89	2	no	—
214	#1684	—	89	2	no	—
215	#1676	—	89	1	no	—
216	#1676	D₈₉	1	1	yes	characteristic, complemented
217	#1670	—	89	1	no	—
218	#1670	—	89	1	no	—
219	#1567	—	89	1	no	—
220	#1550	—	89	2	no	—
221	C₈₉	#19262	1	1	yes	characteristic, complemented, Sylow
222	#533	—	89	2	no	—
223	C₂×C₄×Q₈	—	89	1	no	—
224	#500	C₂×D₈₉	1	1	yes	characteristic
225	#494	—	89	1	no	—
226	#487	—	89	2	no	—
227	#479	—	178	2	no	—
228	#479	—	178	2	no	—
229	#477	—	89	2	no	—
230	#477	C₂×D₈₉	1	2	yes	—
231	#475	—	89	2	no	—
232	#475	—	89	2	no	—
233	#467	—	178	1	no	—
234	#467	—	178	1	no	—
235	C₈×Q₈	—	89	1	no	—
236	C₈×Q₈	—	89	1	no	—
237	C₈×Q₈	—	89	1	no	—
238	C₈×Q₈	—	178	1	no	—
239	C₈×Q₈	—	178	1	no	—
240	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
241	C₄×Q₁₆	—	178	1	no	—
242	C₄×Q₁₆	—	178	1	no	—
243	#428	—	178	2	no	—
244	#423	—	89	2	no	—
245	C₂×C₄⋊C₈	—	89	1	no	—
246	C₂×C₄⋊C₈	—	178	1	no	—
247	#419	—	178	2	no	—
248	C₂×C₄×C₈	—	89	1	no	—
249	C₂×C₄×C₈	—	178	1	no	—
250	#335	—	89	2	no	—
251	#335	—	89	1	no	—
252	#335	C₂×D₈₉	1	1	yes	characteristic
253	#333	—	89	2	no	—
254	#333	C₂×D₈₉	1	2	yes	—
255	#332	—	89	2	no	—
256	#332	—	89	2	no	—
257	C₂²×Q₈	—	89	1	no	—
258	C₂×Q₁₆	—	178	1	no	—
259	C₂×Q₁₆	—	178	1	no	—
260	C₂×M₁₆	—	178	2	no	—
261	C₂²×C₈	—	89	1	no	—
262	C₂²×C₈	—	89	1	no	—
263	C₂²×C₈	—	178	1	no	—
264	C₄⋊Q₈	—	89	2	no	—
265	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
266	#125	—	178	2	no	—
267	C₂²⋊Q₈	—	178	2	no	—
268	C₂²⋊Q₈	—	178	2	no	—
269	C₄×Q₈	—	89	1	no	—
270	C₄×Q₈	—	89	1	no	—
271	C₄×Q₈	—	89	1	no	—
272	C₄×Q₈	—	178	1	no	—
273	C₄×Q₈	—	178	2	no	—
274	C₄×Q₈	—	178	2	no	—
275	C₄×Q₈	—	178	1	no	—
276	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
277	#117	—	178	2	no	—
278	C₂×C₄⋊C₄	—	89	1	no	—
279	C₂×C₄⋊C₄	—	178	1	no	—
280	C₂×C₄²	—	89	1	no	—
281	C₂×C₄²	—	178	1	no	—
282	#107	—	2	1	no	—
283	#107	—	2	1	no	—
284	#107	—	2	2	no	—
285	#107	—	178	2	no	—
286	#106	—	178	2	no	—
287	#106	—	178	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	89	1	no	—
290	C₄⋊C₈	—	89	2	no	—
291	C₄⋊C₈	—	178	1	no	—
292	C₄⋊C₈	—	178	1	no	—
293	C₄⋊C₈	—	178	2	no	—
294	C₄⋊C₈	—	178	1	no	—
295	C₄⋊C₈	—	178	1	no	—
296	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
298	#103	—	2	2	no	—
299	#103	—	178	2	no	—
300	#103	—	178	2	no	—
301	#103	—	178	2	no	—
302	#103	—	178	2	no	—
303	#103	—	178	2	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	—	89	1	no	—
306	C₄×C₈	—	89	1	no	—
307	C₄×C₈	—	89	1	no	—
308	C₄×C₈	—	178	1	no	—
309	C₄×C₈	—	178	1	no	—
310	C₄×C₈	—	178	1	no	—
311	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	89	4	no	—
314	C₂×Q₈	—	89	1	no	—
315	C₂×Q₈	—	178	1	no	—
316	C₂×Q₈	—	178	2	no	—
317	C₂×Q₈	—	178	2	no	—
318	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
319	C₂²×C₄	—	89	1	no	—
320	C₂²×C₄	—	89	1	no	—
321	C₂²×C₄	—	89	1	no	—
322	C₂²×C₄	—	178	1	no	—
323	C₂²×C₄	—	178	1	no	—
324	Q₁₆	—	356	2	no	—
325	Q₁₆	—	356	2	no	—
326	M₁₆	—	356	4	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	89	1	no	—
330	C₂×C₈	—	89	1	no	—
331	C₂×C₈	—	356	1	no	—
332	C₂×C₈	—	356	2	no	—
333	C₂×C₈	—	178	1	no	—
334	C₂×C₈	—	178	2	no	—
335	C₂×C₈	—	178	2	no	—
336	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
337	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	89	1	no	—
342	C₄⋊C₄	—	356	2	no	—
343	C₄⋊C₄	—	356	2	no	—
344	C₄⋊C₄	—	178	1	no	—
345	C₄⋊C₄	—	178	1	no	—
346	C₄⋊C₄	—	178	1	no	—
347	C₄⋊C₄	—	178	2	no	—
348	C₄⋊C₄	—	178	2	no	—
349	C₄⋊C₄	—	178	1	no	—
350	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
351	C₂²⋊C₄	—	356	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	89	1	no	—
354	C₄²	—	89	1	no	—
355	C₄²	—	89	1	no	—
356	C₄²	—	356	2	no	—
357	C₄²	—	178	1	no	—
358	C₄²	—	178	1	no	—
359	C₄²	—	178	1	no	—
360	C₄²	C₂³×D₈₉	1	1	yes	characteristic
361	C₂³	—	89	1	no	—
362	Q₈	—	2	1	no	—
363	Q₈	—	8	2	no	—
364	Q₈	—	89	2	no	—
365	Q₈	—	356	2	no	—
366	Q₈	—	356	2	no	—
367	Q₈	—	178	4	no	—
368	Q₈	—	178	1	no	—
369	Q₈	Q₁₆×D₈₉	1	2	yes	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	89	1	no	—
374	C₂×C₄	—	89	2	no	—
375	C₂×C₄	—	89	2	no	—
376	C₂×C₄	—	89	1	no	—
377	C₂×C₄	—	89	1	no	—
378	C₂×C₄	—	356	1	no	—
379	C₂×C₄	—	356	2	no	—
380	C₂×C₄	—	356	2	no	—
381	C₂×C₄	—	178	4	no	—
382	C₂×C₄	—	178	1	no	—
383	C₂×C₄	—	178	1	no	—
384	C₂×C₄	—	178	1	no	—
385	C₂×C₄	—	178	1	no	—
386	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
388	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
389	C₈	—	2	1	no	—
390	C₈	—	4	1	no	—
391	C₈	—	4	2	no	—
392	C₈	—	356	1	no	—
393	C₈	—	356	2	no	—
394	C₈	—	178	1	no	—
395	C₈	—	178	1	no	—
396	C₈	#15668044	1	2	yes	—
397	C₂²	—	89	2	no	—
398	C₂²	—	178	1	no	—
399	C₂²	—	178	1	no	—
400	C₂²	#121×D₈₉	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	2	no	—
403	C₄	—	4	1	no	—
404	C₄	—	8	2	no	—
405	C₄	—	89	2	no	—
406	C₄	—	356	1	no	—
407	C₄	—	178	1	no	—
408	C₄	—	178	1	no	—
409	C₄	—	178	2	no	—
410	C₄	—	712	2	no	—
411	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
412	C₄	#135×D₈₉	1	1	yes	characteristic
413	C₄	C₂×#15668044	1	1	yes	characteristic
414	C₄	#19152397	1	1	yes	characteristic
415	C₂	—	178	2	no	—
416	C₂	#26029725	1	1	yes	characteristic
417	C₂	#467×D₈₉	1	1	yes	characteristic
418	C₂	#26029970	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1676	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂²×D₈₉	#121
hypercenter	#1676	D₈₉
hypocenter	C₈₉	#19262
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1676

Complements

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

N	H ≅ G/N	type
4. #26019713	C₂	semidirect
5. #26019712	C₂	semidirect
6. #26019653	C₂	semidirect
7. #26019623	C₂	semidirect
8. #26019622	C₂	semidirect
11. C₈₉×#1676	C₂	semidirect
106. Q₈×D₈₉	Q₁₆	semidirect
216. #1676	D₈₉	semidirect
221. C₈₉	#19262	semidirect

Maximal subgroups

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

subgroup	index
#26030839	2
#26030058	2
C₈×Q₈×D₈₉	2
#26019713	2
#26019712	2
#26019653	2
#26019623	2
#26019622	2
#26019009	2
#26018939	2
C₈₉×#1676	2
#19262	89