Subgroups — #36712167

Order 22784.

518 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033227	C₂	1	1	yes	maximal, characteristic
2	#26022495	C₂	1	2	yes	maximal, complemented
3	#26022477	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022406	C₂	1	2	yes	maximal, complemented
5	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022292	C₂	1	2	yes	maximal, complemented
7	#26022292	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022114	C₂	1	2	yes	maximal, complemented
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3383	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152270	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	C₂×#15667955	C₂²	1	2	yes	—
15	#19151975	C₂²	1	1	yes	characteristic
16	#19151975	C₂²	1	2	yes	—
17	#19151794	—	2	1	no	—
18	#19151794	—	2	2	no	—
19	C₄×D₇₁₂	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
21	#19151425	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	C₄⋊#13083549	C₂²	1	2	yes	—
24	C₄⋊#13083549	C₂²	1	1	yes	characteristic
25	#19151402	C₂²	1	1	yes	characteristic
26	#19151398	C₂²	1	2	yes	—
27	#19151398	C₂²	1	2	yes	—
28	#19151398	C₂²	1	2	yes	—
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#557×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#499×C₈₉	C₂²	1	2	yes	—
36	#479×C₈₉	—	2	2	no	—
37	#475×C₈₉	—	2	1	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083575	—	4	1	no	—
42	C₂×D₇₁₂	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
46	#15668009	—	2	2	no	—
47	#15668009	—	2	1	no	—
48	#15667987	—	4	1	no	—
49	#15667987	—	4	2	no	—
50	C₄×D₃₅₆	—	2	2	no	—
51	C₄×D₃₅₆	—	2	1	no	—
52	#15667955	—	2	2	no	—
53	#15667955	—	2	4	no	—
54	C₂×#13083549	C₂³	1	1	yes	characteristic
55	C₂×#13083549	C₂³	1	1	yes	characteristic
56	C₂×#13083549	C₂³	1	2	yes	—
57	#15667946	C₂³	1	1	yes	characteristic
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
61	C₄⋊Q₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	—	4	1	no	—
63	#125×C₈₉	—	2	2	no	—
64	#125×C₈₉	—	4	2	no	—
65	C₄×Q₈×C₈₉	—	2	1	no	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	C₄×Q₈×C₈₉	—	4	1	no	—
68	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
69	#107×C₈₉	—	2	1	no	—
70	#107×C₈₉	C₂³	1	2	yes	—
71	#107×C₈₉	C₂³	1	1	yes	characteristic
72	#106×C₈₉	—	2	2	no	—
73	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
75	#103×C₈₉	—	2	2	no	—
76	#103×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	#13083575	—	8	4	no	—
80	C₂×D₃₅₆	—	2	2	no	—
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×D₃₅₆	—	4	2	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	4	2	no	—
85	C₂×D₃₅₆	—	4	1	no	—
86	C₂×C₄×D₈₉	—	4	2	no	—
87	C₂×C₄×D₈₉	—	4	1	no	—
88	D₇₁₂	—	2	2	no	—
89	D₇₁₂	—	2	1	no	—
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	—	2	1	no	—
92	D₇₁₂	—	8	2	no	—
93	D₇₁₂	—	8	1	no	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	#13083551	—	4	2	no	—
96	#13083551	—	4	1	no	—
97	#13083549	—	2	1	no	—
98	#13083549	—	2	2	no	—
99	#13083549	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
103	Q₁₆×C₈₉	—	2	1	no	—
104	Q₁₆×C₈₉	#41	1	2	yes	complemented
105	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	—	4	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄⋊C₄×C₈₉	—	4	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
116	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
117	C₄²×C₈₉	—	4	2	no	—
118	C₄²×C₈₉	—	4	1	no	—
119	C₄²×C₈₉	C₂×D₄	1	2	yes	—
120	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×D₈₉	—	4	2	no	—
122	C₂²×D₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	2	2	no	—
124	Q₈×C₈₉	—	2	2	no	—
125	Q₈×C₈₉	—	4	1	no	—
126	Q₈×C₈₉	—	8	2	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	—	4	1	no	—
136	D₃₅₆	—	4	2	no	—
137	D₃₅₆	—	4	1	no	—
138	D₃₅₆	—	4	1	no	—
139	D₃₅₆	—	4	2	no	—
140	D₃₅₆	—	8	4	no	—
141	D₃₅₆	—	8	2	no	—
142	D₃₅₆	—	16	2	no	—
143	D₃₅₆	—	16	1	no	—
144	C₄×D₈₉	—	8	4	no	—
145	C₄×D₈₉	—	8	2	no	—
146	C₈×C₈₉	—	2	2	no	—
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	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	8	1	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	8	1	no	—
157	C₄×C₈₉	—	2	2	no	—
158	C₄×C₈₉	—	2	2	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	4	2	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	—	8	1	no	—
163	C₄×C₈₉	#509	1	2	yes	—
164	C₄×C₈₉	#548	1	1	yes	characteristic
165	C₄×C₈₉	#577	1	1	yes	characteristic
166	Dic₈₉	—	8	1	no	—
167	#33935	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	#3358	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
170	C₂×C₈₉	#3416	1	1	yes	characteristic
171	D₈₉	—	16	2	no	—
172	D₈₉	—	16	1	no	—
173	#3391	—	89	1	no	—
174	#3383	D₈₉	1	1	yes	characteristic, complemented
175	#2244	—	89	2	no	—
176	#2240	—	89	1	no	—
177	#2228	—	89	2	no	—
178	#2224	—	89	1	no	—
179	#2207	—	89	2	no	—
180	#2207	—	89	1	no	—
181	#2183	—	89	1	no	—
182	#2175	—	89	2	no	—
183	C₄×SD₃₂	—	89	1	no	—
184	C₈₉	#33935	1	1	yes	characteristic, complemented, Sylow
185	#557	—	2	1	no	—
186	#550	—	178	1	no	—
187	C₂×SD₃₂	—	89	1	no	—
188	C₂×SD₃₂	—	89	2	no	—
189	#500	C₂×D₈₉	1	1	yes	characteristic
190	#499	C₂×D₈₉	1	2	yes	—
191	#493	—	89	1	no	—
192	#493	—	89	2	no	—
193	#479	—	2	2	no	—
194	#475	—	2	1	no	—
195	#459	—	178	1	no	—
196	#459	—	178	2	no	—
197	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
198	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
199	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
200	C₄×D₈	—	89	1	no	—
201	C₄×D₈	—	89	2	no	—
202	#367	—	89	1	no	—
203	#366	—	89	2	no	—
204	C₄⋊C₁₆	—	89	1	no	—
205	C₄⋊C₁₆	—	89	2	no	—
206	#358	—	89	1	no	—
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	#357	—	89	2	no	—
210	#357	—	89	2	no	—
211	#357	—	89	2	no	—
212	#346	—	89	2	no	—
213	C₄×C₁₆	—	89	1	no	—
214	C₂×#41	—	356	1	no	—
215	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×D₈	—	89	1	no	—
217	C₂×D₈	—	89	2	no	—
218	C₂×D₈	—	178	2	no	—
219	C₂×D₈	—	178	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	4	1	no	—
222	#127	—	178	2	no	—
223	#127	—	178	1	no	—
224	#125	—	2	2	no	—
225	#125	—	4	2	no	—
226	#121	—	356	1	no	—
227	#121	—	356	2	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	4	1	no	—
231	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₄×D₄	—	178	2	no	—
233	C₄×D₄	—	178	1	no	—
234	SD₃₂	—	178	2	no	—
235	SD₃₂	—	178	4	no	—
236	C₂×C₁₆	—	89	1	no	—
237	C₂×C₁₆	—	89	1	no	—
238	C₂×C₁₆	—	89	2	no	—
239	#107	—	2	1	no	—
240	#107	—	89	1	no	—
241	#107	C₂²×D₈₉	1	2	yes	—
242	#107	C₂²×D₈₉	1	1	yes	characteristic
243	#106	—	2	2	no	—
244	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
246	#103	—	2	2	no	—
247	#103	—	2	1	no	—
248	#102	—	178	2	no	—
249	#102	—	178	1	no	—
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₄×C₈	C₂²×D₈₉	1	2	yes	—
252	#41	—	712	4	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	#13083559	1	1	yes	characteristic
256	C₂×D₄	—	356	2	no	—
257	C₂×D₄	—	356	1	no	—
258	C₂×D₄	—	356	2	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	178	2	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	—	356	1	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	#13083575	1	2	yes	complemented
266	D₈	—	178	1	no	—
267	D₈	—	178	2	no	—
268	D₈	—	178	1	no	—
269	D₈	—	178	2	no	—
270	D₈	—	712	2	no	—
271	D₈	—	712	1	no	—
272	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
273	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	178	1	no	—
283	C₄⋊C₄	#13083559	1	1	yes	characteristic
284	C₄⋊C₄	#13083559	1	2	yes	—
285	C₂²⋊C₄	—	356	2	no	—
286	C₂²⋊C₄	—	356	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
290	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
291	C₁₆	—	178	1	no	—
292	C₁₆	—	178	2	no	—
293	C₁₆	—	178	1	no	—
294	C₂³	—	356	2	no	—
295	C₂³	—	356	1	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	2	2	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	2	no	—
300	D₄	—	356	1	no	—
301	D₄	—	356	2	no	—
302	D₄	—	356	1	no	—
303	D₄	—	356	2	no	—
304	D₄	—	1424	2	no	—
305	D₄	—	1424	1	no	—
306	D₄	—	712	4	no	—
307	D₄	—	712	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	356	1	no	—
312	C₂×C₄	—	712	4	no	—
313	C₂×C₄	—	712	2	no	—
314	C₂×C₄	C₂×#13083559	1	2	yes	—
315	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
316	C₂×C₄	C₂×#13083559	1	2	yes	—
317	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
318	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	1	no	—
321	C₈	C₂×#13083575	1	1	yes	characteristic
322	C₈	#15668060	1	1	yes	characteristic
323	C₂²	—	712	2	no	—
324	C₂²	—	712	1	no	—
325	C₂²	—	712	2	no	—
326	C₂²	—	712	1	no	—
327	C₂²	—	712	2	no	—
328	C₂²	—	712	1	no	—
329	C₂²	C₂²×#13083559	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	4	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	712	1	no	—
337	C₄	#19152033	1	2	yes	—
338	C₄	#19152259	1	1	yes	characteristic
339	C₄	#19152406	1	1	yes	characteristic
340	C₂	—	1424	2	no	—
341	C₂	—	1424	1	no	—
342	C₂	#26032887	1	1	yes	characteristic
343	C₂	C₂×#19152033	1	1	yes	characteristic
344	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3383	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26033227	C₂
hypercenter	#3383	D₈₉
hypocenter	C₈₉	#33935
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #3383

Complements

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

N	H ≅ G/N	type
2. #26022495	C₂	semidirect
3. #26022477	C₂	semidirect
4. #26022406	C₂	semidirect
5. #26022386	C₂	semidirect
6. #26022292	C₂	semidirect
7. #26022292	C₂	semidirect
8. #26022161	C₂	semidirect
9. #26022114	C₂	semidirect
10. C₄×#15667955	C₂	semidirect
11. C₈₉×#3383	C₂	semidirect
104. Q₁₆×C₈₉	#41	semidirect
174. #3383	D₈₉	semidirect
184. C₈₉	#33935	semidirect
265. Q₁₆	#13083575	semidirect

Maximal subgroups

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

subgroup	index
#26033227	2
#26022495	2
#26022477	2
#26022406	2
#26022386	2
#26022292	2
#26022161	2
#26022114	2
C₄×#15667955	2
C₈₉×#3383	2
#33935	89