Subgroups — #36712107

Order 22784.

528 conjugacy classes of subgroups.

The subgroup lattice has 348 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
2	Q₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022474	C₂	1	2	yes	maximal, complemented
5	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022398	C₂	1	2	yes	maximal, complemented
7	#26022292	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022290	C₂	1	2	yes	maximal, complemented
9	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022101	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₇₈	—	2	1	no	—
13	#19152269	—	2	1	no	—
14	C₂×#15667955	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19151975	C₂²	1	1	yes	characteristic
18	#19151973	C₂²	1	2	yes	—
19	#19151893	—	2	1	no	—
20	#19151891	—	2	2	no	—
21	#19151793	—	2	1	no	—
22	#19151792	—	2	2	no	—
23	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
24	Q₁₆×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	C₁₆×Dic₈₉	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×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	#15668012	—	4	1	no	—
47	#15668011	—	2	2	no	—
48	#15668011	—	4	2	no	—
49	#15668009	—	2	1	no	—
50	#15668008	—	2	2	no	—
51	#15667986	—	4	1	no	—
52	#15667985	—	4	2	no	—
53	Q₈×Dic₈₉	—	2	2	no	—
54	C₄×Dic₁₇₈	—	4	1	no	—
55	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
56	#15667974	—	2	2	no	—
57	#15667955	—	2	4	no	—
58	C₂×#13083549	C₂³	1	2	yes	—
59	#15667946	—	2	1	no	—
60	#15667946	C₂³	1	2	yes	—
61	#15667945	—	2	2	no	—
62	C₄⋊#10623494	D₄	1	1	yes	characteristic
63	#15667941	D₄	1	2	yes	—
64	#15667936	—	2	2	no	—
65	#15667932	—	2	2	no	—
66	C₈×Dic₈₉	C₂³	1	2	yes	—
67	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×D₄×C₈₉	—	2	1	no	—
71	SD₃₂×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	#103×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	—	4	2	no	—
82	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×D₃₅₆	—	4	1	no	—
86	C₂×C₂²⋊D₈₉	—	4	2	no	—
87	C₂×C₂²⋊D₈₉	—	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	#13083551	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	#13083549	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	—	4	1	no	—
107	Q₁₆×C₈₉	—	2	1	no	—
108	Q₁₆×C₈₉	#41	1	2	yes	complemented
109	D₈×C₈₉	—	2	1	no	—
110	D₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₄⋊C₄×C₈₉	—	2	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₂²⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×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	C₂²×D₈₉	—	4	2	no	—
123	Q₈×C₈₉	—	2	2	no	—
124	Q₈×C₈₉	—	4	1	no	—
125	D₄×C₈₉	—	4	1	no	—
126	D₄×C₈₉	—	4	1	no	—
127	C₂×C₄×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	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂²⋊D₈₉	—	8	4	no	—
135	C₂²⋊D₈₉	—	16	2	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
138	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
139	D₃₅₆	—	4	2	no	—
140	D₃₅₆	—	4	2	no	—
141	D₃₅₆	—	8	2	no	—
142	D₃₅₆	—	16	1	no	—
143	C₄×D₈₉	—	8	4	no	—
144	Dic₁₇₈	—	16	2	no	—
145	Dic₁₇₈	—	16	1	no	—
146	Dic₁₇₈	SD₃₂	1	4	yes	complemented
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
150	#10623494	—	2	2	no	—
151	C₂²×C₈₉	—	8	1	no	—
152	C₂²×C₈₉	—	8	1	no	—
153	C₂²×C₈₉	—	8	1	no	—
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	8	2	no	—
157	C₂×D₈₉	—	8	2	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	4	2	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	—	8	1	no	—
162	C₄×C₈₉	C₂×SD₃₂	1	2	yes	—
163	C₄×C₈₉	#548	1	1	yes	characteristic
164	C₄×C₈₉	#577	1	1	yes	characteristic
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	8	2	no	—
167	Dic₈₉	C₂×SD₃₂	1	4	yes	—
168	#33929	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	16	1	no	—
170	C₂×C₈₉	#3358	1	1	yes	characteristic
171	C₂×C₈₉	C₂²×SD₃₂	1	1	yes	characteristic
172	C₂×C₈₉	#3415	1	1	yes	characteristic
173	D₈₉	—	16	2	no	—
174	#3391	—	89	1	no	—
175	Q₈×Q₁₆	—	89	1	no	—
176	#2240	—	89	2	no	—
177	#2240	—	89	1	no	—
178	#2226	—	89	1	no	—
179	#2226	—	89	2	no	—
180	#2207	—	89	2	no	—
181	#2207	—	89	1	no	—
182	Q₈×C₁₆	—	89	1	no	—
183	C₄×SD₃₂	—	89	2	no	—
184	C₄×SD₃₂	D₈₉	1	1	yes	characteristic, complemented
185	C₈₉	#33929	1	1	yes	characteristic, complemented, Sylow
186	Q₈²	—	178	1	no	—
187	#550	—	178	1	no	—
188	C₂×SD₃₂	—	89	2	no	—
189	C₂×SD₃₂	C₂×D₈₉	1	1	yes	characteristic
190	#500	—	89	2	no	—
191	#500	—	89	1	no	—
192	#493	—	89	2	no	—
193	#493	—	89	1	no	—
194	#477	—	178	2	no	—
195	#477	—	178	1	no	—
196	#459	—	178	2	no	—
197	#459	—	178	1	no	—
198	C₈×Q₈	—	89	1	no	—
199	C₄×Q₁₆	—	89	2	no	—
200	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₄×D₈	—	89	2	no	—
202	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
203	#367	—	89	2	no	—
204	#367	C₂×D₈₉	1	1	yes	characteristic
205	C₄⋊C₁₆	—	89	2	no	—
206	C₄⋊C₁₆	—	89	1	no	—
207	#358	C₂×D₈₉	1	1	yes	characteristic
208	#357	—	89	2	no	—
209	#357	—	89	2	no	—
210	#357	—	89	2	no	—
211	#357	—	89	2	no	—
212	#357	C₂×D₈₉	1	1	yes	characteristic
213	C₄×C₁₆	—	89	2	no	—
214	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×#41	—	356	1	no	—
216	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
217	C₂×D₈	—	89	2	no	—
218	C₂×D₈	—	178	2	no	—
219	C₂×D₈	—	178	1	no	—
220	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊Q₈	—	356	1	no	—
222	C₄⋊Q₈	—	356	2	no	—
223	C₄⋊Q₈	—	178	2	no	—
224	C₄⋊Q₈	—	178	1	no	—
225	#127	—	178	2	no	—
226	#127	—	178	1	no	—
227	#121	—	356	2	no	—
228	#121	—	356	1	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	89	1	no	—
231	C₄×Q₈	—	356	1	no	—
232	C₄×Q₈	—	178	2	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	—	178	2	no	—
235	SD₃₂	—	2	2	no	—
236	SD₃₂	—	178	4	no	—
237	C₂×C₁₆	—	89	2	no	—
238	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
239	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
240	#107	—	89	2	no	—
241	#107	—	178	2	no	—
242	#107	—	178	1	no	—
243	#107	C₂²×D₈₉	1	1	yes	characteristic
244	#107	C₂²×D₈₉	1	1	yes	characteristic
245	C₄⋊C₈	—	89	2	no	—
246	C₄⋊C₈	—	89	1	no	—
247	#103	—	2	1	no	—
248	#103	—	178	2	no	—
249	#102	—	2	1	no	—
250	#102	—	178	2	no	—
251	C₄×C₈	—	89	2	no	—
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	#41	—	712	4	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	89	1	no	—
256	C₂×Q₈	—	356	2	no	—
257	C₂×Q₈	—	356	1	no	—
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	356	2	no	—
260	C₂×D₄	—	356	2	no	—
261	C₂×D₄	—	356	1	no	—
262	C₂×D₄	—	356	1	no	—
263	C₂×D₄	—	178	2	no	—
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	—	356	2	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	#13083574	1	2	yes	complemented
268	D₈	—	2	1	no	—
269	D₈	—	2	1	no	—
270	D₈	—	178	2	no	—
271	D₈	—	178	2	no	—
272	D₈	—	712	1	no	—
273	D₈	—	712	2	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	89	2	no	—
281	C₄⋊C₄	—	89	1	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	2	no	—
284	C₄⋊C₄	—	356	1	no	—
285	C₄⋊C₄	—	178	2	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	356	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	—	89	2	no	—
290	C₄²	—	356	2	no	—
291	C₄²	D₄×D₈₉	1	1	yes	characteristic
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₁₆	—	178	2	no	—
295	C₂³	—	4	1	no	—
296	C₂³	—	356	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	89	4	no	—
300	Q₈	—	1424	1	no	—
301	Q₈	—	1424	2	no	—
302	D₄	—	4	1	no	—
303	D₄	—	4	1	no	—
304	D₄	—	356	2	no	—
305	D₄	—	356	2	no	—
306	D₄	—	1424	1	no	—
307	D₄	—	1424	2	no	—
308	D₄	—	712	4	no	—
309	D₄	—	712	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	89	2	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	712	4	no	—
318	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	178	2	no	—
323	C₈	C₂×#13083574	1	1	yes	characteristic
324	C₈	#15668059	1	1	yes	characteristic
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	—	8	1	no	—
328	C₂²	—	712	2	no	—
329	C₂²	—	712	2	no	—
330	C₂²	—	712	2	no	—
331	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	89	4	no	—
337	C₄	—	178	2	no	—
338	C₄	—	712	2	no	—
339	C₄	SD₃₂×D₈₉	1	2	yes	—
340	C₄	#19152257	1	1	yes	characteristic
341	C₄	#19152405	1	1	yes	characteristic
342	C₂	—	16	1	no	—
343	C₂	—	1424	2	no	—
344	C₂	#26032885	1	1	yes	characteristic
345	C₂	C₂×SD₃₂×D₈₉	1	1	yes	characteristic
346	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×SD₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26033226	C₂
hypercenter	C₄×SD₃₂	D₈₉
hypocenter	C₈₉	#33929
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
2. Q₁₆×Dic₁₇₈	C₂	semidirect
3. #26022476	C₂	semidirect
4. #26022474	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022398	C₂	semidirect
7. #26022292	C₂	semidirect
8. #26022290	C₂	semidirect
9. C₁₆×Dic₁₇₈	C₂	semidirect
10. #26022101	C₂	semidirect
11. C₄×SD₃₂×C₈₉	C₂	semidirect
108. Q₁₆×C₈₉	#41	semidirect
146. Dic₁₇₈	SD₃₂	semidirect
184. C₄×SD₃₂	D₈₉	semidirect
185. C₈₉	#33929	semidirect
267. Q₁₆	#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
Q₁₆×Dic₁₇₈	2
#26022476	2
#26022474	2
#26022399	2
#26022398	2
#26022292	2
#26022290	2
C₁₆×Dic₁₇₈	2
#26022101	2
C₄×SD₃₂×C₈₉	2
#33929	89