Subgroups — #36709951

Order 22784.

514 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	#26030765	C₂	1	1	yes	maximal, characteristic
2	#500×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022539	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022538	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022535	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022533	C₂	1	2	yes	maximal, complemented
7	#26022526	C₂	1	2	yes	maximal, complemented
8	#26022039	C₂	1	2	yes	maximal
9	#26022037	C₂	1	2	yes	maximal
10	#26021845	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2255	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152154	—	2	1	no	—
13	C₄⋊Q₈×D₈₉	—	2	1	no	—
14	C₂×#15667951	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	—	2	1	no	—
17	#19152002	C₂²	1	1	yes	characteristic
18	#19152002	C₂²	1	2	yes	—
19	#19152002	C₂²	1	1	yes	characteristic
20	#19152001	—	2	1	no	—
21	#19152000	C₂²	1	1	yes	characteristic
22	#19151997	—	2	1	no	—
23	#19151995	—	2	1	no	—
24	#19151635	C₂²	1	1	yes	characteristic
25	#19151635	C₂²	1	1	yes	characteristic
26	#19151633	—	2	1	no	—
27	#107×D₈₉	—	2	1	no	—
28	#107×D₈₉	C₂²	1	2	yes	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	#19151426	C₂²	1	2	yes	—
31	#19151425	C₂²	1	2	yes	—
32	#19151425	C₂²	1	2	yes	—
33	#19151424	C₂²	1	2	yes	—
34	#19151423	C₂²	1	2	yes	—
35	#19151423	C₂²	1	2	yes	—
36	#19151374	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₈₉	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#499×C₈₉	C₂²	1	1	yes	characteristic
40	#367×C₈₉	C₂²	1	2	yes	—
41	#366×C₈₉	C₂²	1	2	yes	—
42	#346×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₈₉	—	4	1	no	—
44	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
45	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
46	#15668012	—	2	1	no	—
47	#15668011	—	2	2	no	—
48	#15668011	—	2	1	no	—
49	#15668011	—	2	1	no	—
50	#15668011	—	4	1	no	—
51	#15668010	—	4	1	no	—
52	#15668004	—	2	1	no	—
53	#15667990	—	4	1	no	—
54	#15667990	—	4	1	no	—
55	#15667977	—	4	1	no	—
56	#15667977	—	4	1	no	—
57	#15667970	—	2	1	no	—
58	#15667970	—	2	1	no	—
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	C₄⋊C₄×D₈₉	—	4	1	no	—
61	C₄²×D₈₉	D₄	1	1	yes	characteristic
62	#15667951	Q₈	1	8	yes	complemented
63	C₂×#13083549	C₂³	1	2	yes	—
64	#15667946	—	2	1	no	—
65	#15667946	C₂³	1	2	yes	—
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667946	C₂³	1	1	yes	characteristic
68	#15667945	—	2	2	no	—
69	#15667945	—	2	2	no	—
70	#15667945	—	2	1	no	—
71	#15667945	—	2	1	no	—
72	#15667944	—	2	1	no	—
73	#15667943	—	2	1	no	—
74	#15667943	—	2	1	no	—
75	C₄×#10623494	D₄	1	1	yes	characteristic
76	C₈×Dic₈₉	D₄	1	1	yes	characteristic
77	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₈₉	—	2	1	no	—
79	#125×C₈₉	—	2	1	no	—
80	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
81	#107×C₈₉	—	2	1	no	—
82	#107×C₈₉	C₂³	1	2	yes	—
83	#107×C₈₉	C₂³	1	1	yes	characteristic
84	#107×C₈₉	C₂³	1	1	yes	characteristic
85	#106×C₈₉	—	2	1	no	—
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₇₈	—	4	1	no	—
88	C₂×Dic₁₇₈	—	4	1	no	—
89	C₂×Dic₁₇₈	—	4	1	no	—
90	C₂×Dic₁₇₈	—	4	2	no	—
91	Q₈×D₈₉	—	8	4	no	—
92	C₂×C₄×D₈₉	—	4	2	no	—
93	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
94	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
95	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
97	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
98	C₈×D₈₉	Q₁₆	1	4	yes	—
99	C₈×D₈₉	C₂×Q₈	1	4	yes	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	2	1	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	#13083553	—	4	1	no	—
105	#13083553	—	4	1	no	—
106	#13083553	—	4	1	no	—
107	#13083553	—	4	1	no	—
108	#13083553	—	4	2	no	—
109	#13083553	—	4	2	no	—
110	#13083551	—	4	1	no	—
111	#13083551	—	4	1	no	—
112	C₄×Dic₈₉	—	4	1	no	—
113	C₄×Dic₈₉	—	4	1	no	—
114	C₄×Dic₈₉	D₈	1	1	yes	characteristic
115	C₄×Dic₈₉	D₈	1	1	yes	characteristic
116	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	#13083549	C₂×Q₈	1	4	yes	complemented
118	C₂×Q₈×C₈₉	—	4	1	no	—
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	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	4	1	no	—
125	C₄⋊C₄×C₈₉	—	4	1	no	—
126	C₄⋊C₄×C₈₉	—	4	1	no	—
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₈₉	C₂×Q₈	1	4	yes	complemented
129	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	2	no	—
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	—	4	1	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
143	C₄×D₈₉	—	2	2	no	—
144	C₄×D₈₉	—	2	2	no	—
145	C₄×D₈₉	—	8	4	no	—
146	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
147	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
148	Dic₁₇₈	—	8	2	no	—
149	Dic₁₇₈	—	8	4	no	—
150	Dic₁₇₈	—	8	2	no	—
151	Dic₁₇₈	—	8	2	no	—
152	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
153	C₈×C₈₉	C₂²×Q₈	1	2	yes	—
154	#10623494	C₄⋊Q₈	1	2	yes	—
155	#10623494	C₂×Q₁₆	1	2	yes	—
156	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	2	1	no	—
159	C₂×D₈₉	#500	1	2	yes	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	8	2	no	—
163	C₄×C₈₉	—	8	1	no	—
164	C₄×C₈₉	—	8	1	no	—
165	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
166	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
167	Dic₈₉	—	8	2	no	—
168	Dic₈₉	—	8	1	no	—
169	Dic₈₉	—	8	1	no	—
170	Dic₈₉	#500	1	2	yes	—
171	Dic₈₉	#509	1	2	yes	—
172	Dic₈₉	#510	1	2	yes	—
173	#33731	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
177	D₈₉	—	2	2	no	—
178	#3160	—	89	1	no	—
179	C₂×#500	—	89	1	no	—
180	#2255	—	89	1	no	—
181	#2255	—	89	1	no	—
182	#2255	—	89	1	no	—
183	#2255	D₈₉	1	1	yes	characteristic, complemented
184	#2254	—	89	2	no	—
185	#2252	—	89	2	no	—
186	#2159	—	89	2	no	—
187	#2159	—	89	2	no	—
188	C₄×M₃₂	—	89	1	no	—
189	C₈₉	#33731	1	1	yes	characteristic, complemented, Sylow
190	#533	—	178	1	no	—
191	C₂×C₄⋊Q₈	—	178	1	no	—
192	C₂×M₃₂	—	89	2	no	—
193	#500	—	89	2	no	—
194	#500	—	89	1	no	—
195	#500	—	89	1	no	—
196	#500	—	89	1	no	—
197	#500	—	178	1	no	—
198	#500	—	178	1	no	—
199	#500	C₂×D₈₉	1	1	yes	characteristic
200	#499	—	89	1	no	—
201	#499	—	178	1	no	—
202	#499	C₂×D₈₉	1	1	yes	characteristic
203	#498	—	178	1	no	—
204	#427	—	89	1	no	—
205	#427	—	89	1	no	—
206	#427	—	178	1	no	—
207	C₂×#107	—	89	2	no	—
208	C₂×#107	—	178	1	no	—
209	C₂×C₄×C₈	—	89	1	no	—
210	#367	—	89	2	no	—
211	#367	—	89	2	no	—
212	#367	—	89	2	no	—
213	#367	C₂×D₈₉	1	2	yes	—
214	#366	—	89	2	no	—
215	#366	—	89	2	no	—
216	#366	—	89	2	no	—
217	#366	C₂×D₈₉	1	2	yes	—
218	#346	—	89	1	no	—
219	#346	C₂×D₈₉	1	1	yes	characteristic
220	C₄×C₁₆	—	89	2	no	—
221	C₂²×Q₈	—	356	1	no	—
222	C₂²×C₈	—	89	1	no	—
223	C₂²×C₈	—	89	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	356	1	no	—
226	C₄⋊Q₈	—	356	1	no	—
227	C₄⋊Q₈	—	178	2	no	—
228	C₄⋊Q₈	—	178	1	no	—
229	C₄⋊Q₈	—	178	1	no	—
230	C₄⋊Q₈	—	178	1	no	—
231	#125	—	2	1	no	—
232	#125	—	178	1	no	—
233	C₂²⋊Q₈	—	356	1	no	—
234	C₂²⋊Q₈	—	356	1	no	—
235	C₄×Q₈	—	356	1	no	—
236	C₄×Q₈	—	356	1	no	—
237	#117	—	178	1	no	—
238	#117	—	178	1	no	—
239	C₂×C₄⋊C₄	—	356	1	no	—
240	C₂×C₄⋊C₄	—	178	2	no	—
241	C₂×C₄²	—	89	1	no	—
242	M₃₂	—	89	8	no	—
243	C₂×C₁₆	—	89	2	no	—
244	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
245	#107	—	2	1	no	—
246	#107	—	89	1	no	—
247	#107	—	89	2	no	—
248	#107	—	89	1	no	—
249	#107	—	178	2	no	—
250	#107	—	178	2	no	—
251	#107	—	178	1	no	—
252	#107	—	178	1	no	—
253	#107	—	178	1	no	—
254	#107	C₂²×D₈₉	1	1	yes	characteristic
255	#107	C₂²×D₈₉	1	2	yes	—
256	#107	C₂²×D₈₉	1	1	yes	characteristic
257	#106	—	2	1	no	—
258	#106	—	178	1	no	—
259	#106	—	178	1	no	—
260	#106	—	178	1	no	—
261	C₄×C₈	—	89	1	no	—
262	C₄×C₈	—	89	1	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×Q₈	—	356	1	no	—
268	C₂×Q₈	—	356	2	no	—
269	C₂×Q₈	—	356	1	no	—
270	C₂×Q₈	—	712	4	no	—
271	C₂²×C₄	—	89	1	no	—
272	C₂²×C₄	—	89	1	no	—
273	C₂²×C₄	—	89	1	no	—
274	C₂²×C₄	—	356	2	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	89	4	no	—
278	C₂×C₈	—	89	4	no	—
279	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	356	2	no	—
288	C₄⋊C₄	—	356	2	no	—
289	C₄⋊C₄	—	356	1	no	—
290	C₄⋊C₄	—	356	1	no	—
291	C₄⋊C₄	—	356	1	no	—
292	C₄⋊C₄	—	356	1	no	—
293	C₄⋊C₄	—	356	1	no	—
294	C₄⋊C₄	—	178	1	no	—
295	C₄⋊C₄	—	178	1	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₂²⋊C₄	—	356	1	no	—
298	C₂²⋊C₄	—	356	1	no	—
299	C₄²	—	89	1	no	—
300	C₄²	—	89	1	no	—
301	C₄²	—	89	1	no	—
302	C₄²	—	356	1	no	—
303	C₄²	—	356	1	no	—
304	C₄²	D₄×D₈₉	1	1	yes	characteristic
305	C₁₆	—	89	4	no	—
306	C₁₆	Q₈×D₈₉	1	4	yes	complemented
307	C₂³	—	89	1	no	—
308	Q₈	—	8	2	no	—
309	Q₈	—	712	2	no	—
310	Q₈	—	712	2	no	—
311	Q₈	—	712	2	no	—
312	Q₈	—	712	4	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	89	2	no	—
321	C₂×C₄	—	356	1	no	—
322	C₂×C₄	—	356	1	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	712	4	no	—
327	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
328	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
330	C₈	—	89	2	no	—
331	C₈	—	89	2	no	—
332	C₈	Q₁₆×D₈₉	1	2	yes	—
333	C₈	C₂×Q₈×D₈₉	1	2	yes	—
334	C₂²	—	89	2	no	—
335	C₂²	—	178	1	no	—
336	C₂²	—	178	1	no	—
337	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	89	2	no	—
344	C₄	—	89	2	no	—
345	C₄	—	89	2	no	—
346	C₄	—	712	2	no	—
347	C₄	—	712	1	no	—
348	C₄	—	712	1	no	—
349	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
350	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
351	C₂	—	178	2	no	—
352	C₂	#500×D₈₉	1	1	yes	characteristic
353	C₂	#509×D₈₉	1	1	yes	characteristic
354	C₂	#510×D₈₉	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×#2255	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2255	D₈₉
hypocenter	C₈₉	#33731
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2255

Complements

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

N	H ≅ G/N	type
3. #26022539	C₂	semidirect
4. #26022538	C₂	semidirect
5. #26022535	C₂	semidirect
6. #26022533	C₂	semidirect
7. #26022526	C₂	semidirect
11. C₈₉×#2255	C₂	semidirect
62. #15667951	Q₈	semidirect
117. #13083549	C₂×Q₈	semidirect
128. C₁₆×C₈₉	C₂×Q₈	semidirect
183. #2255	D₈₉	semidirect
189. C₈₉	#33731	semidirect
306. C₁₆	Q₈×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030765	2
#500×D₈₉	2
#26022539	2
#26022538	2
#26022535	2
#26022533	2
#26022526	2
#26022039	2
#26022037	2
#26021845	2
C₈₉×#2255	2
#33731	89