Subgroups — #36711859

Order 22784.

492 conjugacy classes of subgroups.

The subgroup lattice has 344 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	#26032674	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022445	C₂	1	2	yes	maximal, complemented
5	#26022442	C₂	1	2	yes	maximal, complemented
6	#26022400	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022337	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022071	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022035	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2226	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152314	—	2	1	no	—
13	#19152313	—	2	1	no	—
14	C₂×#15667953	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151925	—	2	2	no	—
18	#19151920	—	2	2	no	—
19	#19151888	—	2	1	no	—
20	#19151885	—	2	1	no	—
21	#19151835	C₂²	1	2	yes	—
22	#19151831	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	#19151635	C₂²	1	2	yes	—
26	#19151635	C₂²	1	2	yes	—
27	#19151426	C₂²	1	2	yes	—
28	#19151423	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	#19151423	C₂²	1	1	yes	characteristic
31	#19151400	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151377	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#357×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15668012	—	2	1	no	—
44	#15668001	—	4	2	no	—
45	#15667998	—	4	2	no	—
46	#15667987	—	2	2	no	—
47	#15667987	—	4	1	no	—
48	#15667986	—	4	1	no	—
49	#15667983	—	2	2	no	—
50	C₄×D₃₅₆	—	2	1	no	—
51	C₄×D₃₅₆	D₄	1	1	yes	characteristic
52	#15667971	—	4	1	no	—
53	#15667970	—	2	2	no	—
54	#15667970	—	2	2	no	—
55	#15667970	—	4	1	no	—
56	#15667953	—	2	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₁₆×D₈₉	—	2	4	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	2	yes	—
63	#15667943	—	2	2	no	—
64	#15667943	—	2	2	no	—
65	C₄⋊#10623494	D₄	1	1	yes	characteristic
66	#15667934	—	2	1	no	—
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	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄×D₄×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	C₂³	1	2	yes	—
77	#102×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×D₃₅₆	—	4	2	no	—
82	C₂×D₃₅₆	D₈	1	1	yes	characteristic
83	C₂×C₂²⋊D₈₉	—	4	2	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	—	2	1	no	—
88	C₂×#10623494	C₂×D₄	1	2	yes	—
89	C₈×D₈₉	—	2	2	no	—
90	C₈×D₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
95	C₂²⋊Dic₈₉	—	4	2	no	—
96	#13083551	—	4	2	no	—
97	#13083551	—	4	2	no	—
98	#13083551	—	4	1	no	—
99	#13083551	—	4	2	no	—
100	#13083551	D₈	1	2	yes	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	#13083549	—	2	1	no	—
104	#13083549	#41	1	2	yes	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	D₈×C₈₉	—	2	1	no	—
109	D₈×C₈₉	—	2	1	no	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	1	no	—
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₂²⋊C₄×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₂³×C₈₉	—	4	1	no	—
119	C₂²×D₈₉	—	4	1	no	—
120	C₂²×D₈₉	C₂×D₈	1	2	yes	—
121	Q₈×C₈₉	—	16	1	no	—
122	D₄×C₈₉	—	4	1	no	—
123	D₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	—	8	2	no	—
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	16	2	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	D₃₅₆	—	2	2	no	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	16	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	8	2	no	—
141	Dic₁₇₈	—	16	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
144	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
145	#10623494	—	2	2	no	—
146	C₂²×C₈₉	—	8	1	no	—
147	C₂²×C₈₉	—	8	1	no	—
148	C₂²×C₈₉	—	8	1	no	—
149	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₈₉	—	2	2	no	—
151	C₂×D₈₉	—	2	2	no	—
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	8	1	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	#508	1	2	yes	—
160	C₄×C₈₉	#547	1	1	yes	characteristic
161	C₄×C₈₉	#578	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	8	1	no	—
164	Dic₈₉	—	8	2	no	—
165	#33908	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	16	1	no	—
167	C₂×C₈₉	#3334	1	1	yes	characteristic
168	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
169	C₂×C₈₉	#3415	1	1	yes	characteristic
170	D₈₉	—	4	2	no	—
171	D₈₉	—	16	1	no	—
172	#3341	—	89	1	no	—
173	#3341	—	89	1	no	—
174	#2252	—	89	1	no	—
175	#2234	—	89	2	no	—
176	#2234	—	89	2	no	—
177	#2226	—	89	1	no	—
178	#2226	D₈₉	1	1	yes	characteristic, complemented
179	#2215	—	89	2	no	—
180	C₄×D₁₆	—	89	1	no	—
181	D₄×C₁₆	—	89	1	no	—
182	#2158	—	89	2	no	—
183	C₈₉	#33908	1	1	yes	characteristic, complemented, Sylow
184	#555	—	178	1	no	—
185	#555	—	178	1	no	—
186	C₂×D₁₆	—	89	1	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	#500	—	89	1	no	—
189	#500	C₂×D₈₉	1	1	yes	characteristic
190	#482	—	178	2	no	—
191	#482	—	178	2	no	—
192	#476	—	178	1	no	—
193	#476	—	178	1	no	—
194	#466	—	89	2	no	—
195	#466	—	89	2	no	—
196	C₄×D₈	—	89	1	no	—
197	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
198	C₈×D₄	—	89	1	no	—
199	#427	—	89	2	no	—
200	#427	—	89	2	no	—
201	#367	—	89	2	no	—
202	#367	C₂×D₈₉	1	2	yes	—
203	#366	—	89	1	no	—
204	#366	—	89	2	no	—
205	#366	—	89	2	no	—
206	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	#357	—	89	1	no	—
210	#357	C₂×D₈₉	1	2	yes	—
211	C₂²⋊C₁₆	—	89	2	no	—
212	C₄×C₁₆	—	89	1	no	—
213	C₂×D₈	—	89	1	no	—
214	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	—	89	2	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	178	1	no	—
218	#124	—	356	2	no	—
219	#124	—	356	2	no	—
220	#121	—	356	1	no	—
221	#121	—	356	1	no	—
222	#121	—	178	2	no	—
223	#121	—	178	2	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	89	1	no	—
226	C₄×D₄	—	178	1	no	—
227	#117	—	356	1	no	—
228	#117	—	356	1	no	—
229	#117	—	178	2	no	—
230	#117	—	178	2	no	—
231	D₁₆	—	178	2	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	89	1	no	—
234	C₂×C₁₆	—	178	4	no	—
235	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
236	#107	—	2	1	no	—
237	#107	—	89	2	no	—
238	#107	—	89	1	no	—
239	#107	—	178	1	no	—
240	#107	C₂²×D₈₉	1	1	yes	characteristic
241	#107	C₂²×D₈₉	1	2	yes	—
242	#106	—	178	2	no	—
243	#106	—	178	2	no	—
244	C₄⋊C₈	—	89	1	no	—
245	#102	—	2	1	no	—
246	#102	—	178	2	no	—
247	#102	—	178	2	no	—
248	#102	—	178	1	no	—
249	C₂²⋊C₈	—	89	2	no	—
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	89	1	no	—
255	C₂×D₄	—	356	2	no	—
256	C₂×D₄	—	356	2	no	—
257	C₂×D₄	—	178	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	356	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	178	1	no	—
264	D₈	—	178	1	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	—	178	2	no	—
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	2	no	—
277	C₂²⋊C₄	—	4	1	no	—
278	C₂²⋊C₄	—	89	2	no	—
279	C₂²⋊C₄	—	356	2	no	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₂²⋊C₄	—	356	2	no	—
282	C₂²⋊C₄	—	356	2	no	—
283	C₂²⋊C₄	—	356	1	no	—
284	C₄²	—	356	2	no	—
285	C₄²	—	356	2	no	—
286	C₄²	D₄×D₈₉	1	1	yes	characteristic
287	C₁₆	—	2	2	no	—
288	C₁₆	—	89	2	no	—
289	C₁₆	—	178	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	89	2	no	—
292	C₂³	—	356	1	no	—
293	Q₈	—	16	1	no	—
294	Q₈	—	1424	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	356	1	no	—
298	D₄	—	356	1	no	—
299	D₄	—	1424	2	no	—
300	D₄	—	1424	2	no	—
301	D₄	—	178	2	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	8	2	no	—
305	C₂×C₄	—	89	2	no	—
306	C₂×C₄	—	356	1	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	712	2	no	—
311	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
312	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	178	2	no	—
316	C₈	#41×D₈₉	1	1	yes	characteristic
317	C₈	#15668060	1	1	yes	characteristic
318	C₂²	—	8	1	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	178	2	no	—
322	C₂²	—	178	2	no	—
323	C₂²	—	178	2	no	—
324	C₂²	—	712	1	no	—
325	C₂²	—	712	1	no	—
326	C₂²	—	712	1	no	—
327	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	178	2	no	—
332	C₄	—	712	1	no	—
333	C₄	—	712	2	no	—
334	C₄	#19152029	1	2	yes	—
335	C₄	#19152250	1	1	yes	characteristic
336	C₄	#19152414	1	1	yes	characteristic
337	C₂	—	16	1	no	—
338	C₂	—	356	2	no	—
339	C₂	—	1424	1	no	—
340	C₂	#26032598	1	1	yes	characteristic
341	C₂	C₂×#19152029	1	1	yes	characteristic
342	C₂	#26033392	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×#2226	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2226	D₈₉
hypocenter	C₈₉	#33908
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2226

Complements

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

N	H ≅ G/N	type
1. #26032676	C₂	semidirect
2. #26032674	C₂	semidirect
3. #26022525	C₂	semidirect
4. #26022445	C₂	semidirect
5. #26022442	C₂	semidirect
6. #26022400	C₂	semidirect
7. #26022337	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. #26022071	C₂	semidirect
10. #26022035	C₂	semidirect
11. C₈₉×#2226	C₂	semidirect
178. #2226	D₈₉	semidirect
183. C₈₉	#33908	semidirect

Maximal subgroups

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

subgroup	index
#26032676	2
#26032674	2
#26022525	2
#26022445	2
#26022442	2
#26022400	2
#26022337	2
C₄×#15667953	2
#26022071	2
#26022035	2
C₈₉×#2226	2
#33908	89