Subgroups — #36712132

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033262	C₂	1	1	yes	maximal, characteristic
2	#26033141	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022485	C₂	1	2	yes	maximal, complemented
4	#26022476	C₂	1	1	yes	maximal, characteristic
5	#26022412	C₂	1	2	yes	maximal, complemented
6	#26022394	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022316	C₂	1	2	yes	maximal, complemented
8	#26022292	C₂	1	1	yes	maximal, characteristic
9	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022101	C₂	1	2	yes	maximal, complemented
11	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	#19152327	—	2	1	no	—
14	C₂×#15667955	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151999	C₂²	1	2	yes	—
17	#19151983	C₂²	1	2	yes	—
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151907	—	2	2	no	—
20	#19151883	—	2	1	no	—
21	#19151816	—	2	2	no	—
22	#19151798	—	2	1	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	#19151402	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151400	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	C₁₆×Dic₈₉	C₂²	1	2	yes	—
34	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	1	yes	characteristic
39	#358×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083561	—	2	2	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	C₂³	1	2	yes	—
44	#15668012	—	2	1	no	—
45	#15668012	—	4	1	no	—
46	#15668009	—	2	1	no	—
47	#15668003	—	2	2	no	—
48	#15668003	—	4	2	no	—
49	#15668001	—	2	2	no	—
50	#15668000	—	4	1	no	—
51	#15667997	—	4	2	no	—
52	Q₈×Dic₈₉	—	2	2	no	—
53	C₄×Dic₁₇₈	—	4	1	no	—
54	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
55	C₄×D₃₅₆	—	4	1	no	—
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	#15667943	—	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×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	C₄×Q₈×C₈₉	—	2	1	no	—
71	Q₃₂×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	#103×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
81	C₂×D₃₅₆	—	2	2	no	—
82	C₂×D₃₅₆	—	4	1	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	#13083561	—	8	2	no	—
86	D₇₁₂	—	2	2	no	—
87	D₇₁₂	—	2	2	no	—
88	D₇₁₂	—	8	1	no	—
89	C₂×#10623494	C₂×D₄	1	2	yes	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	4	1	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	—	4	2	no	—
94	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
95	#13083553	—	4	2	no	—
96	#13083553	D₈	1	2	yes	—
97	#13083551	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
102	#13083549	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	2	1	no	—
104	C₂×Q₈×C₈₉	—	2	1	no	—
105	Q₁₆×C₈₉	—	2	1	no	—
106	Q₁₆×C₈₉	—	2	1	no	—
107	Q₁₆×C₈₉	—	2	1	no	—
108	Q₁₆×C₈₉	#41	1	2	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	2	1	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄²×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	2	2	no	—
122	Q₈×C₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	4	1	no	—
124	Q₈×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	—	4	2	no	—
137	D₃₅₆	—	8	2	no	—
138	D₃₅₆	—	16	1	no	—
139	C₄×D₈₉	—	16	2	no	—
140	Dic₁₇₈	—	2	2	no	—
141	Dic₁₇₈	—	8	2	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	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	—	8	2	no	—
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	2	no	—
152	C₄×C₈₉	—	8	1	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	#508	1	2	yes	—
156	C₄×C₈₉	#548	1	1	yes	characteristic
157	C₄×C₈₉	#577	1	1	yes	characteristic
158	Dic₈₉	—	2	2	no	—
159	Dic₈₉	—	2	2	no	—
160	Dic₈₉	—	8	2	no	—
161	#33932	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	#3358	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
164	C₂×C₈₉	#3416	1	1	yes	characteristic
165	D₈₉	—	16	2	no	—
166	#3396	—	89	1	no	—
167	#3383	—	89	1	no	—
168	#2242	—	89	2	no	—
169	#2240	—	89	1	no	—
170	#2228	—	89	2	no	—
171	#2225	—	89	1	no	—
172	#2211	—	89	2	no	—
173	#2207	—	89	1	no	—
174	Q₈×C₁₆	—	89	1	no	—
175	C₄×Q₃₂	D₈₉	1	1	yes	characteristic, complemented
176	C₄×SD₃₂	—	89	2	no	—
177	C₈₉	#33932	1	1	yes	characteristic, complemented, Sylow
178	#559	—	178	1	no	—
179	#557	—	178	1	no	—
180	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
181	C₂×SD₃₂	—	89	2	no	—
182	#500	—	89	1	no	—
183	#499	—	89	2	no	—
184	#495	—	89	2	no	—
185	#493	—	89	1	no	—
186	#479	—	178	2	no	—
187	#475	—	178	1	no	—
188	#463	—	178	2	no	—
189	#460	—	178	1	no	—
190	C₈×Q₈	—	89	1	no	—
191	C₄×Q₁₆	—	89	2	no	—
192	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
193	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
194	C₄×D₈	—	89	2	no	—
195	#367	—	89	2	no	—
196	#366	C₂×D₈₉	1	1	yes	characteristic
197	C₄⋊C₁₆	—	89	2	no	—
198	C₄⋊C₁₆	—	89	1	no	—
199	#358	—	89	2	no	—
200	#358	C₂×D₈₉	1	1	yes	characteristic
201	#358	C₂×D₈₉	1	1	yes	characteristic
202	#357	—	89	2	no	—
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	C₄×C₁₆	—	89	2	no	—
206	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
207	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
208	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
209	C₂×SD₁₆	—	178	2	no	—
210	C₂×D₈	—	89	2	no	—
211	C₂×D₈	—	178	1	no	—
212	C₄⋊Q₈	—	356	1	no	—
213	C₄⋊Q₈	—	178	1	no	—
214	#127	—	178	1	no	—
215	#125	—	356	2	no	—
216	#125	—	178	2	no	—
217	#124	—	356	1	no	—
218	#124	—	178	2	no	—
219	#123	—	356	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	—	89	1	no	—
223	C₄×Q₈	—	356	1	no	—
224	C₄×Q₈	—	178	2	no	—
225	C₄×D₄	—	356	1	no	—
226	C₄×D₄	—	178	2	no	—
227	Q₃₂	—	2	2	no	—
228	SD₃₂	—	178	4	no	—
229	C₂×C₁₆	—	89	2	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	#107	—	89	2	no	—
233	#107	—	178	1	no	—
234	#107	C₂²×D₈₉	1	1	yes	characteristic
235	#107	C₂²×D₈₉	1	1	yes	characteristic
236	#106	—	178	2	no	—
237	C₄⋊C₈	—	89	2	no	—
238	C₄⋊C₈	—	89	1	no	—
239	#103	—	2	1	no	—
240	#103	—	2	1	no	—
241	#103	—	178	2	no	—
242	#102	—	178	2	no	—
243	C₄×C₈	—	89	2	no	—
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	89	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×D₄	—	356	1	no	—
250	C₂×D₄	—	356	1	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂²×C₄	—	356	2	no	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	#13083574	1	2	yes	—
257	SD₁₆	—	712	2	no	—
258	D₈	—	178	2	no	—
259	D₈	—	178	2	no	—
260	D₈	—	712	1	no	—
261	C₂×C₈	—	89	2	no	—
262	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	2	no	—
270	C₄⋊C₄	—	356	2	no	—
271	C₄⋊C₄	—	356	2	no	—
272	C₄⋊C₄	—	356	2	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	178	2	no	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	89	2	no	—
281	C₄²	—	356	2	no	—
282	C₄²	D₄×D₈₉	1	1	yes	characteristic
283	C₁₆	—	2	1	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	178	2	no	—
286	C₂³	—	356	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	178	2	no	—
292	Q₈	—	712	2	no	—
293	D₄	—	356	2	no	—
294	D₄	—	356	2	no	—
295	D₄	—	1424	1	no	—
296	D₄	—	712	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	89	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	1424	2	no	—
305	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
306	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	178	2	no	—
310	C₈	C₂×#13083574	1	1	yes	characteristic
311	C₈	#15668059	1	1	yes	characteristic
312	C₂²	—	712	2	no	—
313	C₂²	—	712	2	no	—
314	C₂²	—	712	2	no	—
315	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	178	2	no	—
322	C₄	—	178	2	no	—
323	C₄	—	712	2	no	—
324	C₄	#19152031	1	2	yes	—
325	C₄	#19152257	1	1	yes	characteristic
326	C₄	#19152405	1	1	yes	characteristic
327	C₂	—	1424	2	no	—
328	C₂	#26032885	1	1	yes	characteristic
329	C₂	C₂×#19152031	1	1	yes	characteristic
330	C₂	#26033395	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×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#19151975	C₂²
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#33932
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #26033141	C₂	semidirect
3. #26022485	C₂	semidirect
5. #26022412	C₂	semidirect
6. #26022394	C₂	semidirect
7. #26022316	C₂	semidirect
9. C₁₆×Dic₁₇₈	C₂	semidirect
10. #26022101	C₂	semidirect
11. C₄×Q₃₂×C₈₉	C₂	semidirect
175. C₄×Q₃₂	D₈₉	semidirect
177. C₈₉	#33932	semidirect

Maximal subgroups

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

subgroup	index
#26033262	2
#26033141	2
#26022485	2
#26022476	2
#26022412	2
#26022394	2
#26022316	2
#26022292	2
C₁₆×Dic₁₇₈	2
#26022101	2
C₄×Q₃₂×C₈₉	2
#33932	89