Subgroups — #36712143

Order 22784.

532 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033226	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033117	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022492	C₂	1	2	yes	maximal, complemented
4	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022412	C₂	1	2	yes	maximal, complemented
6	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022287	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022285	C₂	1	2	yes	maximal, complemented
9	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022125	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152305	—	2	1	no	—
13	#19152269	—	2	1	no	—
14	C₂×#15667953	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151999	C₂²	1	2	yes	—
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151973	C₂²	1	2	yes	—
19	#19151896	—	2	2	no	—
20	#19151888	—	2	1	no	—
21	#19151793	—	2	1	no	—
22	#19151792	—	2	2	no	—
23	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
24	D₈×Dic₈₉	C₂²	1	2	yes	—
25	#19151680	C₂²	1	2	yes	—
26	#19151425	C₂²	1	2	yes	—
27	C₄⋊#13083549	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	2	yes	—
29	#19151400	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151399	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#19151373	C₂²	1	2	yes	—
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
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#357×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	4	1	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	C₂³	1	2	yes	—
44	C₂×#13083559	—	2	2	no	—
45	#15668012	—	2	1	no	—
46	#15668009	—	2	1	no	—
47	#15668008	—	2	2	no	—
48	#15668003	—	2	2	no	—
49	#15667991	—	4	1	no	—
50	#15667989	—	4	2	no	—
51	#15667986	—	4	1	no	—
52	#15667985	—	4	2	no	—
53	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
54	D₄×Dic₈₉	—	2	2	no	—
55	#15667974	—	2	2	no	—
56	C₂×C₄⋊Dic₈₉	—	4	1	no	—
57	#15667953	—	2	4	no	—
58	C₂×#13083549	C₂³	1	2	yes	—
59	#15667946	—	2	1	no	—
60	#15667946	C₂³	1	2	yes	—
61	#15667943	—	2	2	no	—
62	C₄⋊#10623494	D₄	1	1	yes	characteristic
63	#15667941	D₄	1	2	yes	—
64	#15667933	—	2	2	no	—
65	#15667932	—	2	2	no	—
66	C₈×Dic₈₉	C₂³	1	2	yes	—
67	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₈₉	—	2	1	no	—
70	C₄×D₄×C₈₉	—	2	1	no	—
71	D₁₆×C₈₉	—	2	2	no	—
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#102×C₈₉	—	2	1	no	—
77	#102×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083574	—	8	4	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×C₂²⋊D₈₉	—	4	2	no	—
86	C₂×C₂²⋊D₈₉	—	4	2	no	—
87	C₂²×Dic₈₉	—	4	2	no	—
88	C₂×C₄×D₈₉	—	4	2	no	—
89	D₇₁₂	—	2	2	no	—
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	—	8	1	no	—
92	#13083559	—	8	2	no	—
93	C₂×#10623494	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
98	#13083553	—	4	2	no	—
99	#13083553	D₈	1	2	yes	—
100	C₂²⋊Dic₈₉	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	#13083549	—	2	2	no	—
104	C₂×D₄×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	—	4	1	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	D₈×C₈₉	—	2	1	no	—
109	D₈×C₈₉	—	2	1	no	—
110	D₈×C₈₉	—	2	1	no	—
111	D₈×C₈₉	#41	1	2	yes	complemented
112	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₈₉	—	2	1	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₂²⋊C₄×C₈₉	—	4	1	no	—
117	C₂²⋊C₄×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	1	no	—
120	C₁₆×C₈₉	—	2	1	no	—
121	C₂³×C₈₉	—	4	1	no	—
122	C₂³×C₈₉	—	4	1	no	—
123	C₂²×D₈₉	—	4	2	no	—
124	D₄×C₈₉	—	2	2	no	—
125	D₄×C₈₉	—	4	1	no	—
126	D₄×C₈₉	—	4	1	no	—
127	D₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	—	8	2	no	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₈₉	—	8	4	no	—
136	C₂²⋊D₈₉	—	16	2	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	16	2	no	—
139	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
140	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
141	D₃₅₆	—	4	2	no	—
142	D₃₅₆	—	4	2	no	—
143	D₃₅₆	—	8	2	no	—
144	D₃₅₆	—	16	1	no	—
145	C₄×D₈₉	—	8	4	no	—
146	Dic₁₇₈	—	2	2	no	—
147	Dic₁₇₈	—	16	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
150	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
151	#10623494	—	2	2	no	—
152	C₂²×C₈₉	—	4	2	no	—
153	C₂²×C₈₉	—	8	1	no	—
154	C₂²×C₈₉	—	8	1	no	—
155	C₂²×C₈₉	—	8	1	no	—
156	C₂²×C₈₉	—	8	1	no	—
157	C₂²×C₈₉	—	8	1	no	—
158	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₈₉	—	8	2	no	—
160	C₂×D₈₉	—	8	2	no	—
161	C₂×D₈₉	—	8	2	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	8	1	no	—
164	C₄×C₈₉	—	8	1	no	—
165	C₄×C₈₉	#509	1	2	yes	—
166	C₄×C₈₉	#548	1	1	yes	characteristic
167	C₄×C₈₉	#577	1	1	yes	characteristic
168	Dic₈₉	—	2	2	no	—
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	8	2	no	—
171	#33933	—	89	1	no	maximal, Sylow
172	C₂×C₈₉	—	8	2	no	—
173	C₂×C₈₉	—	16	1	no	—
174	C₂×C₈₉	#3358	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
176	C₂×C₈₉	#3415	1	1	yes	characteristic
177	D₈₉	—	16	2	no	—
178	#3391	—	89	1	no	—
179	#3380	—	89	1	no	—
180	#2244	—	89	2	no	—
181	#2240	—	89	1	no	—
182	#2228	—	89	2	no	—
183	#2226	—	89	1	no	—
184	#2206	—	89	2	no	—
185	#2206	—	89	1	no	—
186	#2183	—	89	1	no	—
187	#2177	—	89	2	no	—
188	C₄×D₁₆	D₈₉	1	1	yes	characteristic, complemented
189	C₈₉	#33933	1	1	yes	characteristic, complemented, Sylow
190	#554	—	178	1	no	—
191	#550	—	178	1	no	—
192	C₂×D₁₆	—	89	2	no	—
193	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
194	#500	—	89	1	no	—
195	#499	—	89	2	no	—
196	#493	—	89	2	no	—
197	#493	—	89	1	no	—
198	#478	—	178	2	no	—
199	#476	—	178	1	no	—
200	#459	—	178	2	no	—
201	#459	—	178	1	no	—
202	C₈×Q₈	—	89	1	no	—
203	C₄×D₈	—	89	2	no	—
204	C₄×D₈	—	89	2	no	—
205	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
206	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
207	#367	—	89	2	no	—
208	#366	C₂×D₈₉	1	1	yes	characteristic
209	C₄⋊C₁₆	—	89	2	no	—
210	C₄⋊C₁₆	—	89	1	no	—
211	#357	—	89	2	no	—
212	#357	—	89	2	no	—
213	#357	—	89	2	no	—
214	#357	—	89	2	no	—
215	#357	C₂×D₈₉	1	1	yes	characteristic
216	#357	C₂×D₈₉	1	1	yes	characteristic
217	#346	—	89	2	no	—
218	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
219	C₂×#41	—	356	1	no	—
220	C₂×D₈	—	89	2	no	—
221	C₂×D₈	—	178	2	no	—
222	C₂×D₈	—	178	1	no	—
223	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
224	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₄⋊Q₈	—	178	1	no	—
226	#127	—	178	2	no	—
227	#127	—	178	1	no	—
228	#125	—	178	2	no	—
229	C₂²⋊Q₈	—	356	2	no	—
230	C₂²⋊Q₈	—	356	1	no	—
231	#121	—	356	2	no	—
232	#121	—	356	1	no	—
233	C₄×Q₈	—	89	1	no	—
234	C₄×D₄	—	2	1	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	178	2	no	—
237	C₄×D₄	—	178	2	no	—
238	C₂×C₄⋊C₄	—	356	1	no	—
239	D₁₆	—	2	2	no	—
240	D₁₆	—	178	4	no	—
241	C₂×C₁₆	—	89	2	no	—
242	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
244	#107	—	89	2	no	—
245	#107	—	178	1	no	—
246	#107	C₂²×D₈₉	1	1	yes	characteristic
247	#107	C₂²×D₈₉	1	1	yes	characteristic
248	#106	—	178	2	no	—
249	C₄⋊C₈	—	89	2	no	—
250	C₄⋊C₈	—	89	1	no	—
251	#102	—	2	1	no	—
252	#102	—	2	1	no	—
253	#102	—	178	2	no	—
254	#102	—	178	2	no	—
255	C₄×C₈	—	89	2	no	—
256	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	#41	—	712	4	no	—
258	C₂×Q₈	—	89	1	no	—
259	C₂×Q₈	—	356	1	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	356	2	no	—
263	C₂×D₄	—	356	2	no	—
264	C₂×D₄	—	356	1	no	—
265	C₂×D₄	—	356	1	no	—
266	C₂×D₄	—	178	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	356	2	no	—
270	C₂²×C₄	—	356	2	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	1	no	—
273	D₈	—	2	1	no	—
274	D₈	—	178	2	no	—
275	D₈	—	178	2	no	—
276	D₈	—	712	1	no	—
277	D₈	—	712	2	no	—
278	D₈	#13083574	1	2	yes	complemented
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
281	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	89	1	no	—
285	C₄⋊C₄	—	89	2	no	—
286	C₄⋊C₄	—	356	2	no	—
287	C₄⋊C₄	—	356	2	no	—
288	C₄⋊C₄	—	356	1	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	1	no	—
292	C₂²⋊C₄	—	356	2	no	—
293	C₂²⋊C₄	—	356	2	no	—
294	C₄²	—	89	2	no	—
295	C₄²	D₄×D₈₉	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	1	no	—
298	C₁₆	—	178	2	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	1	no	—
301	C₂³	—	356	2	no	—
302	Q₈	—	1424	1	no	—
303	Q₈	—	178	2	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	1	no	—
307	D₄	—	4	1	no	—
308	D₄	—	356	2	no	—
309	D₄	—	356	2	no	—
310	D₄	—	1424	1	no	—
311	D₄	—	1424	2	no	—
312	D₄	—	712	4	no	—
313	D₄	—	712	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	8	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	1424	2	no	—
322	C₂×C₄	—	712	4	no	—
323	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
324	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	178	2	no	—
328	C₈	C₂×#13083574	1	1	yes	characteristic
329	C₈	#15668059	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	712	2	no	—
337	C₂²	—	712	2	no	—
338	C₂²	—	712	2	no	—
339	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	178	2	no	—
344	C₄	—	178	2	no	—
345	C₄	—	712	2	no	—
346	C₄	#19152032	1	2	yes	—
347	C₄	#19152257	1	1	yes	characteristic
348	C₄	#19152405	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	1	no	—
351	C₂	—	1424	2	no	—
352	C₂	#26032885	1	1	yes	characteristic
353	C₂	C₂×#19152032	1	1	yes	characteristic
354	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	#33933
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. #26033226	C₂	semidirect
2. #26033117	C₂	semidirect
3. #26022492	C₂	semidirect
4. #26022476	C₂	semidirect
5. #26022412	C₂	semidirect
6. #26022399	C₂	semidirect
7. #26022287	C₂	semidirect
8. #26022285	C₂	semidirect
9. #26022160	C₂	semidirect
10. #26022125	C₂	semidirect
11. C₄×D₁₆×C₈₉	C₂	semidirect
111. D₈×C₈₉	#41	semidirect
188. C₄×D₁₆	D₈₉	semidirect
189. C₈₉	#33933	semidirect
278. D₈	#13083574	semidirect

Maximal subgroups

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

subgroup	index
#26033226	2
#26033117	2
#26022492	2
#26022476	2
#26022412	2
#26022399	2
#26022287	2
#26022285	2
#26022160	2
#26022125	2
C₄×D₁₆×C₈₉	2
#33933	89