Subgroups — #36537530

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031382	C₂	1	1	yes	maximal, characteristic
2	#26031179	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151996	C₂	1	1	yes	maximal, characteristic, complemented
4	#26028965	C₂	1	1	yes	maximal, characteristic
5	#26019752	C₂	1	4	yes	maximal, complemented
6	#26019628	C₂	1	4	yes	maximal, complemented
7	#26018936	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018932	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	—	2	1	no	—
11	#19152148	C₂²	1	1	yes	characteristic
12	C₂×#15668012	C₂²	1	1	yes	characteristic
13	C₂×#15668012	C₂²	1	1	yes	characteristic
14	#19151996	—	2	1	no	—
15	#19151996	—	2	1	no	—
16	#19151996	—	2	1	no	—
17	#19151996	C₂²	1	2	yes	—
18	#19151996	C₂²	1	2	yes	—
19	#19151971	—	2	1	no	—
20	#19151949	—	2	1	no	—
21	#19151945	—	2	1	no	—
22	#19151917	—	2	2	no	—
23	#19151889	C₂²	1	4	yes	—
24	#19151804	C₂²	1	4	yes	—
25	#19151739	—	2	2	no	—
26	#19151677	—	2	1	no	—
27	#19151677	C₂²	1	2	yes	—
28	C₈×D₃₅₆	—	2	1	no	—
29	C₈×D₃₅₆	C₂²	1	2	yes	—
30	C₂×#15667944	—	2	1	no	—
31	C₂×#15667944	—	2	1	no	—
32	#19151622	C₂²	1	1	yes	characteristic
33	#19151564	—	2	1	no	—
34	#19151346	C₂²	1	4	yes	—
35	#19151345	C₂²	1	4	yes	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#431×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	#335×C₈₉	C₂²	1	2	yes	—
40	#334×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083574	—	2	1	no	—
42	C₂²×Dic₁₇₈	—	2	1	no	—
43	C₂²×Dic₁₇₈	—	2	1	no	—
44	C₂×#13083563	—	2	4	no	—
45	#15668018	—	4	4	no	—
46	C₂×#13083558	—	2	1	no	—
47	C₂×#13083557	—	2	2	no	—
48	C₂×C₈×D₈₉	—	2	2	no	—
49	#15668012	—	2	1	no	—
50	#15668012	—	2	1	no	—
51	#15668012	—	2	1	no	—
52	#15668012	—	2	1	no	—
53	#15668012	—	2	1	no	—
54	#15668012	—	2	1	no	—
55	#15668012	D₄	1	1	yes	characteristic
56	#15668012	C₂³	1	2	yes	—
57	#15668012	C₂³	1	2	yes	—
58	#15668009	D₄	1	1	yes	characteristic
59	#15668000	—	2	1	no	—
60	#15667986	—	2	1	no	—
61	#15667986	D₄	1	2	yes	—
62	C₄×D₃₅₆	—	2	1	no	—
63	C₄×D₃₅₆	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₈₉	—	2	1	no	—
65	C₂×C₄⋊Dic₈₉	—	2	1	no	—
66	C₂×C₄⋊Dic₈₉	—	2	1	no	—
67	C₂×C₄⋊Dic₈₉	—	2	1	no	—
68	#15667944	—	2	4	no	—
69	#15667944	—	2	2	no	—
70	#15667944	—	2	2	no	—
71	#15667944	—	2	2	no	—
72	#15667944	—	2	2	no	—
73	C₄⋊#10623494	—	2	1	no	—
74	C₄⋊#10623494	C₂³	1	2	yes	—
75	#15667937	—	2	4	no	—
76	#15667934	—	2	4	no	—
77	#15667924	—	2	2	no	—
78	#15667924	—	2	2	no	—
79	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
83	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
84	C₂²⋊C₈×C₈₉	—	2	1	no	—
85	C₂²⋊C₈×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	—	2	1	no	—
87	C₄×C₈×C₈₉	—	2	1	no	—
88	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
89	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	#13083574	—	4	2	no	—
93	#13083574	—	4	2	no	—
94	C₂×Dic₁₇₈	—	2	1	no	—
95	C₂×Dic₁₇₈	—	2	2	no	—
96	C₂×Dic₁₇₈	—	2	2	no	—
97	C₂×Dic₁₇₈	—	4	1	no	—
98	C₂×Dic₁₇₈	—	4	1	no	—
99	C₂×Dic₁₇₈	—	8	2	no	—
100	C₂×Dic₁₇₈	—	8	2	no	—
101	C₂×D₃₅₆	—	2	1	no	—
102	C₂×D₃₅₆	—	2	1	no	—
103	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
104	C₂×C₂²⋊D₈₉	—	2	2	no	—
105	C₂²×Dic₈₉	—	4	1	no	—
106	C₂²×Dic₈₉	—	4	1	no	—
107	C₂×C₄×D₈₉	—	2	2	no	—
108	#13083563	—	8	4	no	—
109	#13083558	—	4	2	no	—
110	#13083557	—	4	4	no	—
111	C₂×#10623494	—	2	2	no	—
112	C₈×D₈₉	—	4	4	no	—
113	C₄⋊Dic₈₉	—	2	1	no	—
114	C₄⋊Dic₈₉	—	2	2	no	—
115	C₄⋊Dic₈₉	—	2	2	no	—
116	C₄⋊Dic₈₉	—	2	2	no	—
117	C₄⋊Dic₈₉	—	2	2	no	—
118	C₄⋊Dic₈₉	—	4	1	no	—
119	C₄⋊Dic₈₉	—	4	1	no	—
120	C₄⋊Dic₈₉	—	4	1	no	—
121	C₄⋊Dic₈₉	—	4	1	no	—
122	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
123	#13083551	—	2	2	no	—
124	#13083551	—	2	2	no	—
125	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	2	no	—
130	C₂×C₈×C₈₉	—	2	2	no	—
131	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
132	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₈₉	#41	1	1	yes	characteristic
139	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
140	C₂³×C₈₉	#121	1	1	yes	characteristic
141	C₂²×D₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
146	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
148	C₂×C₄×C₈₉	#137	1	2	yes	—
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₈₉	—	4	4	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	4	2	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	C₂×Dic₈₉	—	8	2	no	—
157	C₂×Dic₈₉	—	8	2	no	—
158	D₃₅₆	—	2	4	no	—
159	D₃₅₆	—	4	2	no	—
160	D₃₅₆	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₄×D₈₉	—	4	2	no	—
163	Dic₁₇₈	—	4	2	no	—
164	Dic₁₇₈	—	4	4	no	—
165	Dic₁₇₈	—	8	2	no	—
166	Dic₁₇₈	—	8	2	no	—
167	Dic₁₇₈	—	8	2	no	—
168	C₈×C₈₉	—	2	1	no	—
169	C₈×C₈₉	—	2	1	no	—
170	C₈×C₈₉	—	4	2	no	—
171	C₈×C₈₉	—	4	2	no	—
172	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
173	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
174	#10623494	—	4	2	no	—
175	C₂²×C₈₉	—	2	1	no	—
176	C₂²×C₈₉	—	2	1	no	—
177	C₂²×C₈₉	#461	1	2	yes	—
178	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
179	C₂×D₈₉	—	4	2	no	—
180	C₂×D₈₉	—	4	2	no	—
181	C₂×D₈₉	—	4	2	no	—
182	C₄×C₈₉	—	2	1	no	—
183	C₄×C₈₉	—	2	1	no	—
184	C₄×C₈₉	#461	1	2	yes	—
185	C₄×C₈₉	#471	1	2	yes	—
186	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
187	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
188	C₄×C₈₉	#577	1	2	yes	—
189	Dic₈₉	—	4	2	no	—
190	Dic₈₉	—	8	2	no	—
191	Dic₈₉	—	8	2	no	—
192	#19191	—	89	1	no	maximal, Sylow
193	C₂×C₈₉	—	2	2	no	—
194	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
195	C₂×C₈₉	#3057	1	1	yes	characteristic
196	C₂×C₈₉	C₂×#471	1	1	yes	characteristic
197	D₈₉	—	8	2	no	—
198	#3227	—	89	1	no	—
199	#3206	—	89	1	no	—
200	C₂×#498	—	89	1	no	—
201	#2931	—	89	1	no	—
202	#1691	—	89	4	no	—
203	#1671	—	89	4	no	—
204	#1564	D₈₉	1	1	yes	characteristic, complemented
205	#1549	—	89	1	no	—
206	#1548	—	89	1	no	—
207	C₈₉	#19191	1	1	yes	characteristic, complemented, Sylow
208	C₂×#131	—	178	1	no	—
209	#532	—	89	1	no	—
210	C₂×C₄⋊Q₈	—	89	1	no	—
211	C₂×C₄⋊Q₈	—	89	1	no	—
212	#498	—	89	2	no	—
213	#498	—	89	2	no	—
214	#498	—	178	1	no	—
215	#498	—	178	1	no	—
216	#498	—	178	1	no	—
217	#492	—	178	1	no	—
218	#487	—	178	1	no	—
219	#486	—	178	1	no	—
220	#481	—	178	2	no	—
221	#476	—	89	4	no	—
222	#461	—	89	4	no	—
223	#448	—	178	2	no	—
224	#436	—	89	2	no	—
225	#436	—	178	1	no	—
226	C₈×D₄	—	89	2	no	—
227	C₈×D₄	—	178	1	no	—
228	#431	C₂×D₈₉	1	1	yes	characteristic
229	#431	C₂×D₈₉	1	1	yes	characteristic
230	C₂×#106	—	178	1	no	—
231	C₂×#106	—	178	1	no	—
232	#423	—	89	1	no	—
233	#408	—	178	1	no	—
234	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
235	#335	C₂×D₈₉	1	2	yes	—
236	#334	C₂×D₈₉	1	2	yes	—
237	#333	—	89	4	no	—
238	#332	—	89	4	no	—
239	C₂×#41	—	178	1	no	—
240	C₂²×Q₈	—	178	1	no	—
241	C₂²×Q₈	—	178	1	no	—
242	C₂×SD₁₆	—	178	4	no	—
243	#131	—	356	4	no	—
244	C₂×M₁₆	—	178	1	no	—
245	C₂×M₁₆	—	178	2	no	—
246	C₂²×C₈	—	178	2	no	—
247	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
248	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
249	C₄⋊Q₈	—	89	1	no	—
250	C₄⋊Q₈	—	89	2	no	—
251	C₄⋊Q₈	—	89	2	no	—
252	C₄⋊Q₈	—	178	1	no	—
253	C₄⋊Q₈	—	178	1	no	—
254	C₄⋊Q₈	—	178	1	no	—
255	C₄⋊Q₈	—	178	1	no	—
256	C₄⋊Q₈	—	178	1	no	—
257	C₄⋊Q₈	—	178	1	no	—
258	#127	—	89	1	no	—
259	#124	—	178	1	no	—
260	#121	—	89	2	no	—
261	#121	—	178	1	no	—
262	C₄×D₄	—	89	2	no	—
263	C₄×D₄	—	178	1	no	—
264	C₂×C₄⋊C₄	—	178	1	no	—
265	C₂×C₄⋊C₄	—	178	1	no	—
266	C₂×C₄⋊C₄	—	178	1	no	—
267	C₂×C₄⋊C₄	—	178	1	no	—
268	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
269	#106	—	178	4	no	—
270	#106	—	178	2	no	—
271	#106	—	178	2	no	—
272	#106	—	178	2	no	—
273	#106	—	178	2	no	—
274	C₄⋊C₈	—	89	2	no	—
275	C₄⋊C₈	—	178	1	no	—
276	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
277	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
278	#103	—	178	4	no	—
279	#102	—	178	4	no	—
280	C₂²⋊C₈	—	2	1	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	178	2	no	—
283	C₂²⋊C₈	—	178	2	no	—
284	C₄×C₈	—	2	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	D₃₅₆	1	1	yes	characteristic
287	C₄×C₈	D₃₅₆	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
290	#41	—	356	2	no	—
291	#41	—	356	2	no	—
292	C₂×Q₈	—	356	1	no	—
293	C₂×Q₈	—	356	1	no	—
294	C₂×Q₈	—	178	2	no	—
295	C₂×Q₈	—	178	2	no	—
296	C₂×Q₈	—	178	1	no	—
297	C₂×Q₈	—	712	2	no	—
298	C₂×Q₈	—	712	2	no	—
299	C₂×D₄	—	89	2	no	—
300	C₂×D₄	—	178	2	no	—
301	C₂×D₄	—	178	1	no	—
302	C₂×D₄	—	178	1	no	—
303	C₂²×C₄	—	356	1	no	—
304	C₂²×C₄	—	356	1	no	—
305	C₂²×C₄	—	178	2	no	—
306	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
308	C₂²×C₄	#13083575	1	1	yes	characteristic
309	SD₁₆	—	712	4	no	—
310	M₁₆	—	356	2	no	—
311	M₁₆	—	356	4	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	356	4	no	—
316	C₂×C₈	—	178	2	no	—
317	C₂×C₈	#13083563	1	4	yes	complemented
318	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
319	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
320	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
321	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
322	C₄⋊C₄	—	89	2	no	—
323	C₄⋊C₄	—	356	1	no	—
324	C₄⋊C₄	—	356	1	no	—
325	C₄⋊C₄	—	356	1	no	—
326	C₄⋊C₄	—	356	1	no	—
327	C₄⋊C₄	—	178	2	no	—
328	C₄⋊C₄	—	178	2	no	—
329	C₄⋊C₄	—	178	2	no	—
330	C₄⋊C₄	—	178	2	no	—
331	C₄⋊C₄	—	178	1	no	—
332	C₂²⋊C₄	—	178	2	no	—
333	C₂²⋊C₄	—	178	2	no	—
334	C₄²	D₄×D₈₉	1	1	yes	characteristic
335	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
336	C₄²	#13083575	1	1	yes	characteristic
337	C₄²	C₂³×D₈₉	1	1	yes	characteristic
338	C₂³	—	178	2	no	—
339	C₂³	#15667987	1	1	yes	characteristic
340	Q₈	—	356	2	no	—
341	Q₈	—	356	4	no	—
342	Q₈	—	712	2	no	—
343	Q₈	—	712	2	no	—
344	Q₈	—	712	2	no	—
345	D₄	—	356	4	no	—
346	D₄	—	356	2	no	—
347	D₄	—	356	2	no	—
348	D₄	—	178	4	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	356	2	no	—
353	C₂×C₄	—	356	2	no	—
354	C₂×C₄	—	356	2	no	—
355	C₂×C₄	—	178	2	no	—
356	C₂×C₄	—	712	2	no	—
357	C₂×C₄	—	712	2	no	—
358	C₂×C₄	#15667987	1	1	yes	characteristic
359	C₂×C₄	#15667987	1	1	yes	characteristic
360	C₂×C₄	#15667987	1	1	yes	characteristic
361	C₂×C₄	C₂×#13083563	1	2	yes	—
362	C₂×C₄	#15668045	1	2	yes	—
363	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
365	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
366	C₈	—	2	1	no	—
367	C₈	—	2	1	no	—
368	C₈	—	4	2	no	—
369	C₈	—	4	2	no	—
370	C₈	—	356	2	no	—
371	C₈	C₂×#13083563	1	2	yes	—
372	C₈	C₂×#13083563	1	2	yes	—
373	C₂²	—	2	1	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	356	2	no	—
376	C₂²	—	356	2	no	—
377	C₂²	—	356	2	no	—
378	C₂²	#19151804	1	2	yes	—
379	C₂²	C₂×#15667987	1	1	yes	characteristic
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	356	2	no	—
383	C₄	—	712	2	no	—
384	C₄	—	712	2	no	—
385	C₄	#19151804	1	2	yes	—
386	C₄	#19151862	1	2	yes	—
387	C₄	C₂²×#13083563	1	1	yes	characteristic
388	C₄	C₂×#15668045	1	1	yes	characteristic
389	C₄	#19152405	1	2	yes	—
390	C₂	—	2	2	no	—
391	C₂	—	712	2	no	—
392	C₂	C₂×#19151804	1	1	yes	characteristic
393	C₂	#26029892	1	1	yes	characteristic
394	C₂	C₂×#19151862	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667987
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#19152148	C₂²
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#19191
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. C₂×#19151996	C₂	semidirect
5. #26019752	C₂	semidirect
6. #26019628	C₂	semidirect
7. #26018936	C₂	semidirect
8. #26018932	C₂	semidirect
9. C₈₉×#1564	C₂	semidirect
131. C₂×C₈×C₈₉	SD₁₆	semidirect
204. #1564	D₈₉	semidirect
207. C₈₉	#19191	semidirect
317. C₂×C₈	#13083563	semidirect

Maximal subgroups

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

subgroup	index
#26031382	2
#26031179	2
C₂×#19151996	2
#26028965	2
#26019752	2
#26019628	2
#26018936	2
#26018932	2
C₈₉×#1564	2
#19191	89