Subgroups — #36712347

Order 22784.

464 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033121	C₂	1	1	yes	maximal, characteristic
3	#26022536	C₂	1	2	yes	maximal, complemented
4	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022414	C₂	1	2	yes	maximal, complemented
6	#26022400	C₂	1	1	yes	maximal, characteristic
7	#26022388	C₂	1	2	yes	maximal
8	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022119	C₂	1	2	yes	maximal
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₈×Dic₁₇₈	—	2	1	no	—
13	#19152309	—	2	1	no	—
14	C₂×#15667955	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	#19151901	—	2	2	no	—
19	#19151893	—	2	1	no	—
20	#19151891	—	2	2	no	—
21	#19151885	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151680	C₂²	1	2	yes	—
26	#19151426	C₂²	1	2	yes	—
27	#19151425	C₂²	1	1	yes	characteristic
28	#19151423	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	#19151416	C₂²	1	2	yes	—
31	#19151402	C₂²	1	1	yes	characteristic
32	#19151400	C₂²	1	2	yes	—
33	#19151398	C₂²	1	1	yes	characteristic
34	#19151373	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668012	—	4	1	no	—
44	#15668011	—	2	2	no	—
45	#15668011	—	4	2	no	—
46	#15668003	—	2	2	no	—
47	#15667992	—	4	1	no	—
48	#15667988	—	4	2	no	—
49	Q₈×Dic₈₉	—	2	2	no	—
50	C₄×Dic₁₇₈	—	4	1	no	—
51	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
52	C₄×D₃₅₆	—	2	1	no	—
53	#15667974	—	2	2	no	—
54	C₄⋊C₄×D₈₉	—	4	1	no	—
55	#15667955	Q₈	1	4	yes	complemented
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	#15667946	—	2	1	no	—
59	#15667946	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667945	—	2	2	no	—
62	#15667943	—	2	2	no	—
63	C₄⋊#10623494	D₄	1	1	yes	characteristic
64	#15667941	D₄	1	2	yes	—
65	#15667936	—	2	2	no	—
66	#15667934	—	2	1	no	—
67	#15667932	—	2	2	no	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
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	2	no	—
80	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	D₇₁₂	—	2	1	no	—
85	D₇₁₂	C₂×Q₈	1	2	yes	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	—	4	2	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
93	#13083553	—	4	2	no	—
94	#13083553	—	4	2	no	—
95	#13083553	D₈	1	2	yes	—
96	#13083551	—	4	1	no	—
97	#13083551	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
100	#13083549	—	2	1	no	—
101	#13083549	C₂×Q₈	1	2	yes	—
102	C₂×Q₈×C₈₉	—	2	1	no	—
103	C₂×Q₈×C₈₉	—	4	1	no	—
104	Q₁₆×C₈₉	—	2	1	no	—
105	Q₁₆×C₈₉	C₂×Q₈	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	2	no	—
109	C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	—	4	1	no	—
112	C₄²×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₈₉	—	2	2	no	—
115	C₂²×D₈₉	—	4	1	no	—
116	Q₈×C₈₉	—	2	2	no	—
117	Q₈×C₈₉	—	4	1	no	—
118	Q₈×C₈₉	—	16	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	—	4	1	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	4	1	no	—
127	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
128	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
129	D₃₅₆	—	2	2	no	—
130	D₃₅₆	—	4	1	no	—
131	C₄×D₈₉	—	8	2	no	—
132	C₄×D₈₉	—	16	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	16	2	no	—
135	Dic₁₇₈	—	16	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₈₉	—	4	2	no	—
142	C₂×D₈₉	—	8	1	no	—
143	C₂×D₈₉	—	8	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	4	2	no	—
146	C₄×C₈₉	—	8	1	no	—
147	C₄×C₈₉	—	8	2	no	—
148	C₄×C₈₉	#510	1	2	yes	—
149	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
150	C₄×C₈₉	#578	1	1	yes	characteristic
151	Dic₈₉	—	2	2	no	—
152	Dic₈₉	—	2	2	no	—
153	Dic₈₉	—	8	1	no	—
154	Dic₈₉	—	8	2	no	—
155	#33952	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
158	C₂×C₈₉	#3415	1	1	yes	characteristic
159	D₈₉	—	8	2	no	—
160	Q₈×Q₁₆	—	89	1	no	—
161	#3380	—	89	1	no	—
162	#2255	—	89	2	no	—
163	#2252	—	89	1	no	—
164	#2229	—	89	2	no	—
165	#2227	D₈₉	1	1	yes	characteristic, complemented
166	#2226	—	89	1	no	—
167	#2224	—	89	2	no	—
168	#2183	—	89	1	no	—
169	#2175	—	89	2	no	—
170	C₄×SD₃₂	—	89	1	no	—
171	C₈₉	#33952	1	1	yes	characteristic, complemented, Sylow
172	Q₈²	—	178	1	no	—
173	#554	—	178	1	no	—
174	C₂×SD₃₂	—	89	1	no	—
175	#500	—	89	2	no	—
176	#500	—	89	1	no	—
177	#500	C₂×D₈₉	1	1	yes	characteristic
178	#499	—	89	2	no	—
179	#478	—	178	2	no	—
180	#477	—	178	2	no	—
181	#477	—	178	1	no	—
182	#476	—	178	1	no	—
183	C₈×Q₈	—	89	1	no	—
184	C₄×Q₁₆	—	89	2	no	—
185	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
186	C₄×D₈	—	89	2	no	—
187	C₄×D₈	—	89	1	no	—
188	#367	—	89	2	no	—
189	#367	—	89	1	no	—
190	#366	—	89	2	no	—
191	#366	—	89	2	no	—
192	#366	C₂×D₈₉	1	2	yes	—
193	C₄⋊C₁₆	—	89	2	no	—
194	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
195	#358	—	89	1	no	—
196	#358	C₂×D₈₉	1	2	yes	—
197	#357	—	89	2	no	—
198	#357	—	89	1	no	—
199	#346	—	89	2	no	—
200	C₄×C₁₆	—	89	1	no	—
201	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
202	C₂×D₈	—	89	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	356	1	no	—
205	C₄⋊Q₈	—	356	2	no	—
206	C₄⋊Q₈	—	178	2	no	—
207	C₄⋊Q₈	—	178	1	no	—
208	#125	—	178	2	no	—
209	C₂²⋊Q₈	—	356	1	no	—
210	C₂²⋊Q₈	—	356	2	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	89	1	no	—
213	C₄×Q₈	—	356	1	no	—
214	C₄×Q₈	—	178	2	no	—
215	C₄×D₄	—	178	2	no	—
216	C₄×D₄	—	178	1	no	—
217	C₂×C₄⋊C₄	—	356	1	no	—
218	SD₃₂	—	89	4	no	—
219	C₂×C₁₆	—	89	1	no	—
220	C₂×C₁₆	—	89	1	no	—
221	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
222	#107	—	2	1	no	—
223	#107	—	89	2	no	—
224	#107	—	89	1	no	—
225	#107	—	178	2	no	—
226	#107	—	178	1	no	—
227	#107	C₂²×D₈₉	1	1	yes	characteristic
228	#107	C₂²×D₈₉	1	2	yes	—
229	#106	—	178	2	no	—
230	C₄⋊C₈	—	89	2	no	—
231	C₄⋊C₈	—	89	1	no	—
232	#103	—	2	1	no	—
233	#103	—	178	2	no	—
234	#102	—	178	2	no	—
235	#102	—	178	1	no	—
236	C₄×C₈	—	89	2	no	—
237	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	—	356	2	no	—
242	C₂×Q₈	—	356	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂²×C₄	—	356	2	no	—
245	C₂²×C₄	—	356	1	no	—
246	Q₁₆	—	2	1	no	—
247	Q₁₆	Q₈×D₈₉	1	2	yes	—
248	D₈	—	89	2	no	—
249	D₈	—	178	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	2	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₄⋊C₄	—	356	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	356	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	2	no	—
266	C₂²⋊C₄	—	356	2	no	—
267	C₂²⋊C₄	—	356	1	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	89	2	no	—
270	C₄²	—	356	2	no	—
271	C₄²	D₄×D₈₉	1	1	yes	characteristic
272	C₁₆	—	2	2	no	—
273	C₁₆	—	89	2	no	—
274	C₁₆	—	178	1	no	—
275	C₂³	—	356	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	4	1	no	—
278	Q₈	—	16	1	no	—
279	Q₈	—	1424	1	no	—
280	Q₈	—	1424	2	no	—
281	Q₈	—	178	2	no	—
282	D₄	—	356	1	no	—
283	D₄	—	178	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	89	2	no	—
289	C₂×C₄	—	356	2	no	—
290	C₂×C₄	—	356	1	no	—
291	C₂×C₄	—	1424	2	no	—
292	C₂×C₄	—	712	2	no	—
293	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
294	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	178	2	no	—
298	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
299	C₈	#15668059	1	1	yes	characteristic
300	C₂²	—	356	2	no	—
301	C₂²	—	712	1	no	—
302	C₂²	—	712	1	no	—
303	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	1	no	—
308	C₄	—	178	2	no	—
309	C₄	—	178	2	no	—
310	C₄	—	712	1	no	—
311	C₄	—	712	2	no	—
312	C₄	#19152040	1	2	yes	—
313	C₄	#19152261	1	1	yes	characteristic
314	C₄	#19152414	1	1	yes	characteristic
315	C₂	—	712	2	no	—
316	C₂	#26033102	1	1	yes	characteristic
317	C₂	C₂×#19152040	1	1	yes	characteristic
318	C₂	#26033392	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×#2227	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2227	D₈₉
hypocenter	C₈₉	#33952
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
1. Q₁₆×Dic₁₇₈	C₂	semidirect
3. #26022536	C₂	semidirect
4. #26022525	C₂	semidirect
5. #26022414	C₂	semidirect
8. #26022157	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
55. #15667955	Q₈	semidirect
165. #2227	D₈₉	semidirect
171. C₈₉	#33952	semidirect

Maximal subgroups

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

subgroup	index
Q₁₆×Dic₁₇₈	2
#26033121	2
#26022536	2
#26022525	2
#26022414	2
#26022400	2
#26022388	2
#26022157	2
#26022119	2
C₄×#15667955	2
C₈₉×#2227	2
#33952	89