Subgroups — #36712368

Order 22784.

450 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033141	C₂	1	1	yes	maximal, characteristic
2	Q₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
3	#26022536	C₂	1	2	yes	maximal
4	#26022534	C₂	1	1	yes	maximal, characteristic
5	#26022414	C₂	1	2	yes	maximal
6	#26022404	C₂	1	1	yes	maximal, characteristic
7	#26022394	C₂	1	1	yes	maximal, characteristic
8	#26022392	C₂	1	2	yes	maximal
9	#26022160	C₂	1	1	yes	maximal, characteristic
10	#26022123	C₂	1	2	yes	maximal
11	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
12	Q₈×Dic₁₇₈	—	2	1	no	—
13	#19152327	—	2	1	no	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19151999	C₂²	1	2	yes	—
18	#19151907	—	2	2	no	—
19	#19151893	—	2	1	no	—
20	#19151891	—	2	2	no	—
21	#19151883	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
24	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
25	#19151426	C₂²	1	2	yes	—
26	#19151426	C₂²	1	2	yes	—
27	#19151425	C₂²	1	2	yes	—
28	#19151423	C₂²	1	2	yes	—
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151402	C₂²	1	2	yes	—
32	#19151402	C₂²	1	2	yes	—
33	#19151373	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	#15668012	—	2	1	no	—
42	#15668012	—	2	1	no	—
43	#15668012	—	4	1	no	—
44	#15668012	—	4	1	no	—
45	#15668011	—	2	2	no	—
46	#15668011	—	4	2	no	—
47	#15668003	—	2	2	no	—
48	#15668003	—	4	2	no	—
49	Q₈×Dic₈₉	—	2	2	no	—
50	Q₈×Dic₈₉	—	2	2	no	—
51	C₄×Dic₁₇₈	—	4	1	no	—
52	C₄×Dic₁₇₈	—	4	1	no	—
53	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
54	C₂×#13083549	C₂³	1	2	yes	—
55	#15667946	—	2	1	no	—
56	#15667946	—	2	1	no	—
57	#15667946	C₂³	1	2	yes	—
58	#15667946	C₂³	1	2	yes	—
59	#15667945	—	2	2	no	—
60	#15667943	—	2	2	no	—
61	C₄⋊#10623494	D₄	1	1	yes	characteristic
62	#15667941	D₄	1	2	yes	—
63	#15667936	—	2	2	no	—
64	#15667936	—	2	2	no	—
65	C₈×Dic₈₉	C₂³	1	2	yes	—
66	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₄×Q₈×C₈₉	—	2	1	no	—
69	C₄×Q₈×C₈₉	—	2	1	no	—
70	Q₃₂×C₈₉	Q₈	1	4	yes	complemented
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	C₂³	1	1	yes	characteristic
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#103×C₈₉	—	2	1	no	—
76	#103×C₈₉	—	2	1	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	4	1	no	—
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
82	C₂×#10623494	C₂×D₄	1	2	yes	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	—	4	1	no	—
86	C₄⋊Dic₈₉	—	4	1	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
88	C₄⋊Dic₈₉	—	4	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
91	#13083553	—	4	2	no	—
92	#13083553	—	4	2	no	—
93	#13083553	D₈	1	2	yes	—
94	C₄×Dic₈₉	—	4	2	no	—
95	C₄×Dic₈₉	—	4	2	no	—
96	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
97	#13083549	—	2	2	no	—
98	C₂×Q₈×C₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	Q₁₆×C₈₉	—	2	1	no	—
101	Q₁₆×C₈₉	—	2	1	no	—
102	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
103	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₄²×C₈₉	—	4	1	no	—
111	C₄²×C₈₉	—	4	1	no	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₈₉	—	2	1	no	—
114	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
115	Q₈×C₈₉	—	2	2	no	—
116	Q₈×C₈₉	—	2	2	no	—
117	Q₈×C₈₉	—	4	1	no	—
118	Q₈×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂×Dic₈₉	—	4	2	no	—
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
129	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
130	Dic₁₇₈	—	2	2	no	—
131	Dic₁₇₈	—	8	2	no	—
132	Dic₁₇₈	—	16	2	no	—
133	Dic₁₇₈	—	16	1	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
136	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
137	#10623494	—	2	2	no	—
138	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
139	C₄×C₈₉	—	2	1	no	—
140	C₄×C₈₉	—	4	2	no	—
141	C₄×C₈₉	—	4	2	no	—
142	C₄×C₈₉	—	8	1	no	—
143	C₄×C₈₉	—	8	1	no	—
144	C₄×C₈₉	#510	1	2	yes	—
145	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
146	C₄×C₈₉	#578	1	1	yes	characteristic
147	Dic₈₉	—	2	2	no	—
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	—	8	2	no	—
151	#33954	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
153	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
154	C₂×C₈₉	#3416	1	1	yes	characteristic
155	#3383	—	89	1	no	—
156	Q₈×Q₁₆	—	89	1	no	—
157	#2255	—	89	2	no	—
158	#2254	—	89	1	no	—
159	#2229	—	89	2	no	—
160	#2227	—	89	1	no	—
161	#2225	—	89	1	no	—
162	#2225	—	89	2	no	—
163	#2183	—	89	1	no	—
164	#2176	—	89	2	no	—
165	C₄×Q₃₂	D₈₉	1	1	yes	characteristic
166	C₈₉	#33954	1	1	yes	characteristic, complemented, Sylow
167	Q₈²	—	178	1	no	—
168	#557	—	178	1	no	—
169	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
170	#500	—	89	2	no	—
171	#500	—	89	1	no	—
172	#500	—	89	1	no	—
173	#499	—	89	2	no	—
174	#479	—	178	2	no	—
175	#477	—	178	2	no	—
176	#477	—	178	1	no	—
177	#475	—	178	1	no	—
178	C₈×Q₈	—	89	1	no	—
179	C₄×Q₁₆	—	89	2	no	—
180	C₄×Q₁₆	—	89	2	no	—
181	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
182	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
183	#367	—	89	2	no	—
184	#367	—	89	2	no	—
185	#367	—	89	2	no	—
186	#366	—	89	2	no	—
187	#366	C₂×D₈₉	1	1	yes	characteristic
188	C₄⋊C₁₆	—	89	2	no	—
189	C₄⋊C₁₆	—	89	1	no	—
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#358	C₂×D₈₉	1	1	yes	characteristic
193	#358	C₂×D₈₉	1	1	yes	characteristic
194	#346	—	89	2	no	—
195	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
197	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	356	1	no	—
199	C₄⋊Q₈	—	356	1	no	—
200	C₄⋊Q₈	—	356	2	no	—
201	C₄⋊Q₈	—	178	2	no	—
202	C₄⋊Q₈	—	178	1	no	—
203	C₄⋊Q₈	—	178	1	no	—
204	#125	—	356	2	no	—
205	#125	—	178	2	no	—
206	C₄×Q₈	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	89	1	no	—
209	C₄×Q₈	—	356	1	no	—
210	C₄×Q₈	—	356	1	no	—
211	C₄×Q₈	—	178	2	no	—
212	C₄×Q₈	—	178	2	no	—
213	Q₃₂	Dic₁₇₈	1	4	yes	complemented
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	89	2	no	—
218	#107	—	89	2	no	—
219	#107	—	178	2	no	—
220	#107	—	178	1	no	—
221	#107	—	178	1	no	—
222	#107	C₂²×D₈₉	1	1	yes	characteristic
223	#107	C₂²×D₈₉	1	1	yes	characteristic
224	#106	—	178	2	no	—
225	C₄⋊C₈	—	89	2	no	—
226	C₄⋊C₈	—	89	1	no	—
227	#103	—	2	1	no	—
228	#103	—	2	1	no	—
229	#103	—	178	2	no	—
230	#103	—	178	2	no	—
231	C₄×C₈	—	89	2	no	—
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	89	1	no	—
236	C₂×Q₈	—	356	2	no	—
237	C₂×Q₈	—	356	1	no	—
238	C₂×Q₈	—	356	1	no	—
239	Q₁₆	—	2	1	no	—
240	Q₁₆	—	2	1	no	—
241	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
242	Q₁₆	C₂×Dic₁₇₈	1	2	yes	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
245	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	4	1	no	—
249	C₄⋊C₄	—	4	1	no	—
250	C₄⋊C₄	—	89	1	no	—
251	C₄⋊C₄	—	89	2	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	356	1	no	—
258	C₄⋊C₄	—	356	1	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	—	178	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	—	89	2	no	—
264	C₄²	—	356	2	no	—
265	C₄²	—	356	2	no	—
266	C₄²	D₄×D₈₉	1	1	yes	characteristic
267	C₁₆	—	2	1	no	—
268	C₁₆	—	178	2	no	—
269	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
270	Q₈	—	2	2	no	—
271	Q₈	—	2	2	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	1424	1	no	—
275	Q₈	—	1424	2	no	—
276	Q₈	—	178	2	no	—
277	Q₈	—	712	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	89	2	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
287	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	178	2	no	—
291	C₈	C₂²×Dic₁₇₈	1	1	yes	characteristic
292	C₈	#15668059	1	1	yes	characteristic
293	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	1	no	—
298	C₄	—	8	1	no	—
299	C₄	—	178	2	no	—
300	C₄	—	178	2	no	—
301	C₄	—	712	2	no	—
302	C₄	—	712	2	no	—
303	C₄	#19152040	1	2	yes	—
304	C₄	D₄×Dic₁₇₈	1	1	yes	characteristic
305	C₄	#19152412	1	1	yes	characteristic
306	C₂	D₈×Dic₁₇₈	1	1	yes	characteristic
307	C₂	C₂×#19152040	1	1	yes	characteristic
308	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, prime radical	C₂²×C₈₉	C₂²×D₈
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#33954
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
70. Q₃₂×C₈₉	Q₈	semidirect
166. C₈₉	#33954	semidirect
213. Q₃₂	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26033141	2
Q₁₆×Dic₁₇₈	2
#26022536	2
#26022534	2
#26022414	2
#26022404	2
#26022394	2
#26022392	2
#26022160	2
#26022123	2
C₄×Q₃₂×C₈₉	2
#33954	89