Subgroups — #36711682

Order 22784.

620 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032676	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022446	C₂	1	2	yes	maximal, complemented
5	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022354	C₂	1	2	yes	maximal, complemented
7	#26022287	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022162	C₂	1	2	yes	maximal, complemented
9	#26022085	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021966	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₈×D₈₉	C₂²	1	2	yes	—
13	#19152313	—	2	1	no	—
14	D₄×D₃₅₆	—	2	1	no	—
15	C₂×#15667953	C₂²	1	2	yes	complemented
16	C₂×#15667951	C₂²	1	2	yes	complemented
17	#19152003	C₂²	1	1	yes	characteristic, complemented
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151920	—	2	2	no	—
20	#19151888	—	2	1	no	—
21	#19151835	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	complemented
23	#19151793	—	2	1	no	—
24	#19151733	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	#19151635	C₂²	1	2	yes	complemented
27	#19151425	C₂²	1	2	yes	complemented
28	C₄⋊#13083549	C₂²	1	1	yes	characteristic, complemented
29	#19151400	C₂²	1	2	yes	complemented
30	#19151400	C₂²	1	2	yes	complemented
31	#19151399	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	complemented
33	#19151377	C₂²	1	2	yes	complemented
34	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#366×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#357×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	C₂²×D₃₅₆	—	4	1	no	—
42	C₂×D₄×D₈₉	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	C₂×#13083559	—	2	2	no	—
46	D₈×D₈₉	—	2	4	no	—
47	D₈×D₈₉	—	4	2	no	—
48	C₂×C₈×D₈₉	C₂³	1	2	yes	—
49	#15668012	—	2	1	no	—
50	#15668009	—	2	1	no	—
51	#15667998	—	4	2	no	—
52	#15667987	—	2	2	no	—
53	#15667986	—	4	1	no	—
54	#15667986	—	4	1	no	—
55	#15667983	—	2	2	no	—
56	#15667982	—	4	2	no	—
57	C₄×D₃₅₆	D₄	1	1	yes	characteristic
58	#15667971	—	4	1	no	—
59	#15667970	—	2	2	no	—
60	#15667953	—	4	2	no	—
61	#15667951	—	4	2	no	—
62	C₂×#13083549	C₂³	1	2	yes	—
63	#15667946	—	2	1	no	—
64	#15667946	C₂³	1	2	yes	—
65	#15667943	—	2	2	no	—
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
67	#15667933	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	D₁₆×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#102×C₈₉	—	2	1	no	—
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×D₈₉	—	4	2	no	—
83	C₂×Dic₁₇₈	—	4	1	no	—
84	C₂×D₃₅₆	—	2	2	no	—
85	C₂×D₃₅₆	—	4	2	no	—
86	C₂×D₃₅₆	—	4	1	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×D₃₅₆	—	8	4	no	—
89	C₂×D₃₅₆	D₈	1	1	yes	characteristic
90	C₂×C₂²⋊D₈₉	—	4	2	no	—
91	C₂×C₂²⋊D₈₉	—	4	2	no	—
92	D₄×D₈₉	—	4	4	no	—
93	D₄×D₈₉	—	8	2	no	—
94	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
95	D₇₁₂	—	2	2	no	—
96	D₇₁₂	—	2	2	no	—
97	D₇₁₂	—	8	1	no	—
98	#13083559	—	4	4	no	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₈×D₈₉	—	2	2	no	—
101	C₈×D₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
105	C₂²⋊Dic₈₉	—	4	2	no	—
106	#13083551	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	#13083551	D₈	1	2	yes	—
109	C₄×Dic₈₉	—	4	2	no	—
110	#13083549	—	2	2	no	—
111	C₂×D₄×C₈₉	—	2	1	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	4	1	no	—
114	C₂²×C₄×C₈₉	—	4	1	no	—
115	D₈×C₈₉	—	2	1	no	—
116	D₈×C₈₉	—	2	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	D₈×C₈₉	#41	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₂³×C₈₉	—	4	1	no	—
129	C₂³×C₈₉	—	4	1	no	—
130	C₂²×D₈₉	—	4	2	no	—
131	C₂²×D₈₉	—	8	4	no	—
132	C₂²×D₈₉	—	8	4	no	—
133	C₂²×D₈₉	—	16	2	no	—
134	C₂²×D₈₉	C₂×D₈	1	2	yes	—
135	D₄×C₈₉	—	2	2	no	—
136	D₄×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	4	1	no	—
138	D₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	—	4	1	no	—
141	C₂×C₄×C₈₉	—	8	2	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂²⋊D₈₉	—	8	4	no	—
147	C₂²⋊D₈₉	—	16	2	no	—
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
150	D₃₅₆	—	2	2	no	—
151	D₃₅₆	—	4	2	no	—
152	D₃₅₆	—	4	2	no	—
153	D₃₅₆	—	8	4	no	—
154	D₃₅₆	—	8	2	no	—
155	D₃₅₆	—	16	1	no	—
156	C₄×D₈₉	—	2	2	no	—
157	C₄×D₈₉	—	2	2	no	—
158	Dic₁₇₈	—	16	1	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
162	#10623494	—	2	2	no	—
163	C₂²×C₈₉	—	4	2	no	—
164	C₂²×C₈₉	—	8	1	no	—
165	C₂²×C₈₉	—	8	1	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	1	no	—
168	C₂²×C₈₉	—	8	1	no	—
169	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
170	C₂×D₈₉	—	2	2	no	—
171	C₂×D₈₉	—	2	2	no	—
172	C₂×D₈₉	—	2	2	no	—
173	C₂×D₈₉	—	8	2	no	—
174	C₂×D₈₉	—	8	2	no	—
175	C₂×D₈₉	—	8	2	no	—
176	C₂×D₈₉	—	16	8	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	8	1	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	#509	1	2	yes	—
181	C₄×C₈₉	#546	1	1	yes	characteristic
182	C₄×C₈₉	#577	1	1	yes	characteristic
183	Dic₈₉	—	2	2	no	—
184	Dic₈₉	—	8	2	no	—
185	#33892	—	89	1	no	maximal, Sylow
186	C₂×C₈₉	—	8	2	no	—
187	C₂×C₈₉	—	16	1	no	—
188	C₂×C₈₉	#3294	1	1	yes	characteristic
189	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
190	C₂×C₈₉	#3415	1	1	yes	characteristic
191	D₈₉	—	4	2	no	—
192	D₈₉	—	16	2	no	—
193	#3341	—	89	1	no	—
194	D₄×D₈	—	89	1	no	—
195	#2240	—	89	1	no	—
196	#2234	—	89	2	no	—
197	#2226	—	89	1	no	—
198	#2218	—	89	2	no	—
199	#2206	—	89	1	no	—
200	#2184	—	89	2	no	—
201	C₄×D₁₆	D₈₉	1	1	yes	characteristic, complemented
202	#2169	—	89	1	no	—
203	#2140	—	89	2	no	—
204	C₈₉	#33892	1	1	yes	characteristic, complemented, Sylow
205	C₂²×D₈	—	89	2	no	—
206	#555	—	178	1	no	—
207	D₄²	—	178	1	no	—
208	C₂×D₁₆	—	89	2	no	—
209	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₂×M₃₂	—	89	2	no	—
211	#500	—	89	1	no	—
212	#493	—	89	1	no	—
213	#482	—	178	2	no	—
214	#476	—	178	1	no	—
215	#466	—	89	2	no	—
216	#466	—	89	2	no	—
217	#459	—	178	1	no	—
218	#447	—	178	2	no	—
219	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
220	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic, complemented
221	C₈×D₄	—	89	1	no	—
222	#427	—	89	2	no	—
223	#367	—	89	2	no	—
224	#366	C₂×D₈₉	1	1	yes	characteristic, complemented
225	C₄⋊C₁₆	—	89	1	no	—
226	#357	—	89	2	no	—
227	#357	—	89	2	no	—
228	#357	—	89	2	no	—
229	#357	—	89	2	no	—
230	#357	C₂×D₈₉	1	1	yes	characteristic
231	#357	C₂×D₈₉	1	1	yes	characteristic, complemented
232	C₂²⋊C₁₆	—	89	2	no	—
233	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
234	C₂²×D₄	—	356	1	no	—
235	C₂²×D₄	—	178	2	no	—
236	C₂×D₈	—	89	2	no	—
237	C₂×D₈	—	356	2	no	—
238	C₂×D₈	—	178	2	no	—
239	C₂×D₈	—	178	4	no	—
240	C₂×D₈	—	178	1	no	—
241	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂²×C₈	—	89	2	no	—
244	C₄⋊Q₈	—	178	1	no	—
245	#127	—	178	1	no	—
246	#124	—	356	2	no	—
247	#121	—	356	1	no	—
248	#121	—	356	1	no	—
249	#121	—	178	2	no	—
250	#121	—	178	2	no	—
251	#120	—	356	2	no	—
252	C₄×D₄	—	2	1	no	—
253	C₄×D₄	—	2	1	no	—
254	C₄×D₄	—	89	1	no	—
255	#117	—	356	1	no	—
256	#117	—	178	2	no	—
257	D₁₆	—	2	2	no	—
258	D₁₆	—	356	2	no	—
259	M₃₂	—	356	2	no	—
260	C₂×C₁₆	—	89	2	no	—
261	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
262	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
263	#107	—	89	2	no	—
264	#107	—	178	1	no	—
265	#107	C₂²×D₈₉	1	1	yes	characteristic
266	#107	C₂²×D₈₉	1	1	yes	characteristic
267	#106	—	178	2	no	—
268	C₄⋊C₈	—	89	1	no	—
269	#102	—	2	1	no	—
270	#102	—	2	1	no	—
271	#102	—	178	2	no	—
272	#102	—	178	2	no	—
273	C₂²⋊C₈	—	89	2	no	—
274	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₂⁴	—	356	2	no	—
276	C₂×Q₈	—	356	1	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	—	2	1	no	—
279	C₂×D₄	—	89	1	no	—
280	C₂×D₄	—	356	2	no	—
281	C₂×D₄	—	356	2	no	—
282	C₂×D₄	—	356	2	no	—
283	C₂×D₄	—	356	4	no	—
284	C₂×D₄	—	356	1	no	—
285	C₂×D₄	—	356	1	no	—
286	C₂×D₄	—	178	2	no	—
287	C₂×D₄	—	712	4	no	—
288	C₂×D₄	—	712	2	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	4	1	no	—
291	C₂²×C₄	—	89	2	no	—
292	D₈	—	2	1	no	—
293	D₈	—	2	1	no	—
294	D₈	—	2	1	no	—
295	D₈	—	356	4	no	—
296	D₈	—	178	2	no	—
297	D₈	—	178	2	no	—
298	D₈	—	712	1	no	—
299	D₈	#13083574	1	2	yes	—
300	C₂×C₈	—	89	2	no	—
301	C₂×C₈	—	178	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
305	C₄⋊C₄	—	2	1	no	—
306	C₄⋊C₄	—	2	1	no	—
307	C₄⋊C₄	—	89	1	no	—
308	C₄⋊C₄	—	356	1	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₂²⋊C₄	—	4	1	no	—
311	C₂²⋊C₄	—	4	1	no	—
312	C₂²⋊C₄	—	89	2	no	—
313	C₂²⋊C₄	—	356	2	no	—
314	C₂²⋊C₄	—	356	2	no	—
315	C₂²⋊C₄	—	356	2	no	—
316	C₄²	—	356	2	no	—
317	C₄²	D₄×D₈₉	1	1	yes	characteristic
318	C₁₆	—	2	1	no	—
319	C₁₆	—	2	1	no	—
320	C₁₆	—	178	2	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	89	2	no	—
324	C₂³	—	356	2	no	—
325	C₂³	—	1424	2	no	—
326	C₂³	—	712	4	no	—
327	C₂³	—	712	4	no	—
328	Q₈	—	1424	1	no	—
329	D₄	—	2	2	no	—
330	D₄	—	4	1	no	—
331	D₄	—	4	1	no	—
332	D₄	—	4	1	no	—
333	D₄	—	356	2	no	—
334	D₄	—	356	2	no	—
335	D₄	—	1424	1	no	—
336	D₄	—	1424	2	no	—
337	D₄	—	178	2	no	—
338	D₄	—	712	4	no	—
339	D₄	—	712	4	no	—
340	D₄	—	712	2	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	8	2	no	—
344	C₂×C₄	—	8	2	no	—
345	C₂×C₄	—	89	2	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	178	2	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
350	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
351	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	178	2	no	—
354	C₈	C₂×#13083574	1	1	yes	characteristic
355	C₈	#15668058	1	1	yes	characteristic
356	C₂²	—	4	2	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	8	1	no	—
360	C₂²	—	8	1	no	—
361	C₂²	—	8	1	no	—
362	C₂²	—	1424	8	no	—
363	C₂²	—	178	2	no	—
364	C₂²	—	178	2	no	—
365	C₂²	—	178	2	no	—
366	C₂²	—	712	2	no	—
367	C₂²	—	712	2	no	—
368	C₂²	—	712	2	no	—
369	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	8	1	no	—
372	C₄	—	8	1	no	—
373	C₄	—	178	2	no	—
374	C₄	—	712	2	no	—
375	C₄	#19152032	1	2	yes	—
376	C₄	#19152244	1	1	yes	characteristic
377	C₄	#19152405	1	1	yes	characteristic
378	C₂	—	8	2	no	—
379	C₂	—	16	1	no	—
380	C₂	—	356	2	no	—
381	C₂	—	1424	2	no	—
382	C₂	#26032067	1	1	yes	characteristic
383	C₂	C₂×#19152032	1	1	yes	characteristic
384	C₂	#26033390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×D₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×D₁₆	D₈₉
hypocenter	C₈₉	#33892
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #26032676	C₂	semidirect
2. D₈×D₃₅₆	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022446	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022354	C₂	semidirect
7. #26022287	C₂	semidirect
8. #26022162	C₂	semidirect
9. #26022085	C₂	semidirect
10. #26021966	C₂	semidirect
11. C₄×D₁₆×C₈₉	C₂	semidirect
15. C₂×#15667953	C₂²	semidirect
16. C₂×#15667951	C₂²	semidirect
17. #19152003	C₂²	semidirect
22. #19151831	C₂²	semidirect
26. #19151635	C₂²	semidirect
27. #19151425	C₂²	semidirect
28. C₄⋊#13083549	C₂²	semidirect
29. #19151400	C₂²	semidirect
30. #19151400	C₂²	semidirect
32. #19151398	C₂²	semidirect
33. #19151377	C₂²	semidirect
36. C₄×D₈×C₈₉	C₂²	semidirect
37. #366×C₈₉	C₂²	semidirect
39. #357×C₈₉	C₂²	semidirect
40. C₄×C₁₆×C₈₉	C₂²	semidirect
201. C₄×D₁₆	D₈₉	semidirect
204. C₈₉	#33892	semidirect
220. C₄×D₈	C₂×D₈₉	semidirect
224. #366	C₂×D₈₉	semidirect
231. #357	C₂×D₈₉	semidirect
233. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26032676	2
D₈×D₃₅₆	2
#26022476	2
#26022446	2
#26022399	2
#26022354	2
#26022287	2
#26022162	2
#26022085	2
#26021966	2
C₄×D₁₆×C₈₉	2
#33892	89