Subgroups — #36542131

Order 22784.

706 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030839	C₂	1	1	yes	maximal, characteristic
2	#26030837	C₂	1	2	yes	maximal
3	#26030836	C₂	1	1	yes	maximal, characteristic
4	#26030761	C₂	1	2	yes	maximal
5	#26019869	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019868	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019860	C₂	1	2	yes	maximal, complemented
8	#26019849	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019847	C₂	1	2	yes	maximal, complemented
10	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1707	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152153	C₂²	1	2	yes	—
13	#19152153	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152001	C₂²	1	2	yes	—
16	#19152001	C₂²	1	2	yes	—
17	#19152001	C₂²	1	2	yes	—
18	#19151998	—	2	4	no	—
19	#19151996	C₂²	1	2	yes	—
20	#19151978	C₂²	1	1	yes	characteristic
21	#19151977	C₂²	1	2	yes	—
22	#19151950	C₂²	1	2	yes	—
23	#19151949	C₂²	1	2	yes	—
24	#19151948	C₂²	1	2	yes	—
25	#19151840	—	2	2	no	—
26	#19151838	—	2	4	no	—
27	#19151836	—	2	2	no	—
28	Q₁₆×Dic₈₉	—	2	2	no	—
29	#19151692	—	2	2	no	—
30	#19151690	—	2	4	no	—
31	#19151633	—	2	4	no	—
32	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
35	C₈×#10623494	C₂²	1	1	yes	characteristic
36	C₈×#10623494	C₂²	1	1	yes	characteristic
37	C₈×#10623494	C₂²	1	1	yes	characteristic
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#498×C₈₉	C₂²	1	2	yes	—
40	#494×C₈₉	C₂²	1	1	yes	characteristic
41	#487×C₈₉	C₂²	1	2	yes	—
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₃₅₆	—	2	2	no	—
44	C₂×#13083564	—	2	4	no	—
45	C₂×C₈×D₈₉	D₄	1	2	yes	—
46	C₂×C₈×D₈₉	D₄	1	2	yes	—
47	C₂×C₈×D₈₉	D₄	1	2	yes	—
48	#15668012	C₂³	1	2	yes	—
49	#15668012	C₂³	1	2	yes	—
50	#15668003	—	2	2	no	—
51	#15668003	—	2	2	no	—
52	#15667992	—	2	2	no	—
53	#15667992	—	2	2	no	—
54	#15667988	—	2	2	no	—
55	#15667988	—	2	2	no	—
56	Q₈×Dic₈₉	—	2	2	no	—
57	Q₈×Dic₈₉	—	2	2	no	—
58	#15667977	—	2	2	no	—
59	#15667977	—	2	2	no	—
60	#15667970	—	2	2	no	—
61	#15667970	—	2	2	no	—
62	C₄²×D₈₉	C₂³	1	1	yes	characteristic
63	#15667946	—	2	2	no	—
64	#15667945	—	2	4	no	—
65	#15667945	—	2	4	no	—
66	#15667945	—	2	4	no	—
67	#15667944	—	2	4	no	—
68	#15667937	—	2	4	no	—
69	#15667936	—	2	4	no	—
70	#15667935	—	2	4	no	—
71	C₄×#10623494	C₂³	1	1	yes	characteristic
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₈×Dic₈₉	D₄	1	2	yes	—
75	C₈×Dic₈₉	D₄	1	2	yes	—
76	C₈×Dic₈₉	D₄	1	2	yes	—
77	C₂×Q₁₆×C₈₉	—	2	2	no	—
78	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
79	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
80	#107×C₈₉	—	2	2	no	—
81	#106×C₈₉	—	2	4	no	—
82	#103×C₈₉	—	2	4	no	—
83	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×Dic₁₇₈	—	2	2	no	—
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
91	Dic₃₅₆	—	4	4	no	—
92	#13083564	—	4	8	no	—
93	C₂×#10623494	C₂×D₄	1	2	yes	—
94	C₂×#10623494	C₂×D₄	1	2	yes	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₈×D₈₉	—	2	4	no	—
97	C₈×D₈₉	—	2	4	no	—
98	C₈×D₈₉	Q₁₆	1	8	yes	complemented
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	2	no	—
101	C₄⋊Dic₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	4	2	no	—
104	C₄⋊Dic₈₉	—	4	2	no	—
105	#13083553	—	4	2	no	—
106	#13083553	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	C₄×Dic₈₉	—	4	2	no	—
110	C₄×Dic₈₉	—	4	2	no	—
111	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₈₉	—	2	2	no	—
115	C₂×Q₈×C₈₉	—	2	2	no	—
116	Q₁₆×C₈₉	—	4	4	no	—
117	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
118	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
119	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²×D₈₉	#127	1	1	yes	characteristic
126	Q₈×C₈₉	—	4	4	no	—
127	Q₈×C₈₉	—	4	4	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	#127	1	1	yes	characteristic
136	C₂×Dic₈₉	#127	1	1	yes	characteristic
137	C₂×Dic₈₉	#127	1	1	yes	characteristic
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
141	Dic₁₇₈	—	4	4	no	—
142	Dic₁₇₈	—	4	4	no	—
143	C₈×C₈₉	—	2	2	no	—
144	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
145	C₈×C₈₉	#135	1	4	yes	—
146	#10623494	—	2	2	no	—
147	#10623494	C₂×Q₁₆	1	4	yes	—
148	#10623494	#135	1	4	yes	—
149	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	#494	1	2	yes	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
156	C₄×C₈₉	C₂×#135	1	2	yes	—
157	C₄×C₈₉	C₂×#135	1	2	yes	—
158	Dic₈₉	—	2	1	no	—
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	—	8	2	no	—
162	Dic₈₉	#494	1	2	yes	—
163	#19604	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
165	C₂×C₈₉	#3147	1	1	yes	characteristic
166	C₂×C₈₉	#3147	1	1	yes	characteristic
167	D₈₉	—	2	2	no	—
168	#3170	—	89	1	no	—
169	#3170	—	89	1	no	—
170	#3170	—	89	2	no	—
171	#3160	—	89	2	no	—
172	#1713	—	89	1	no	—
173	#1713	—	89	1	no	—
174	#1710	—	89	2	no	—
175	#1707	—	89	2	no	—
176	#1707	—	89	1	no	—
177	#1707	D₈₉	1	1	yes	characteristic, complemented
178	C₂×C₈²	—	89	1	no	—
179	C₈₉	#19604	1	1	yes	characteristic, complemented, Sylow
180	#533	—	89	2	no	—
181	#533	—	89	2	no	—
182	#500	—	89	2	no	—
183	#500	—	89	2	no	—
184	#500	—	89	2	no	—
185	#500	—	89	1	no	—
186	#500	C₂×D₈₉	1	1	yes	characteristic
187	#499	—	178	4	no	—
188	#498	—	89	2	no	—
189	#498	C₂×D₈₉	1	2	yes	—
190	#494	—	89	2	no	—
191	#494	—	89	1	no	—
192	#494	C₂×D₈₉	1	1	yes	characteristic
193	#487	—	89	2	no	—
194	#487	—	89	2	no	—
195	#487	—	89	2	no	—
196	#487	C₂×D₈₉	1	2	yes	—
197	#467	—	178	4	no	—
198	#467	—	178	2	no	—
199	#467	—	178	2	no	—
200	C₄×Q₁₆	—	178	4	no	—
201	C₄×Q₁₆	—	178	2	no	—
202	C₄×Q₁₆	—	178	2	no	—
203	#427	—	178	4	no	—
204	C₂×C₄×C₈	—	89	1	no	—
205	C₂×C₄×C₈	—	89	1	no	—
206	C₂×C₄×C₈	—	89	1	no	—
207	C₈²	—	89	1	no	—
208	C₈²	—	89	1	no	—
209	C₈²	—	89	1	no	—
210	C₈²	C₂×D₈₉	1	1	yes	characteristic
211	C₂×Q₁₆	—	2	2	no	—
212	C₂×Q₁₆	—	178	4	no	—
213	C₂×Q₁₆	—	178	2	no	—
214	C₂²×C₈	—	89	2	no	—
215	C₂²×C₈	—	89	2	no	—
216	C₂²×C₈	—	89	2	no	—
217	C₄⋊Q₈	—	89	2	no	—
218	C₄⋊Q₈	—	89	2	no	—
219	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
220	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
221	#125	—	178	2	no	—
222	#125	—	178	2	no	—
223	C₂²⋊Q₈	—	178	2	no	—
224	C₂²⋊Q₈	—	178	2	no	—
225	C₂²⋊Q₈	—	178	2	no	—
226	C₂²⋊Q₈	—	178	2	no	—
227	C₄×Q₈	—	178	2	no	—
228	C₄×Q₈	—	178	2	no	—
229	C₄×Q₈	—	178	2	no	—
230	C₄×Q₈	—	178	2	no	—
231	#117	—	178	2	no	—
232	#117	—	178	2	no	—
233	C₂×C₄²	—	89	1	no	—
234	#107	—	2	2	no	—
235	#107	—	178	4	no	—
236	#107	—	178	4	no	—
237	#107	—	178	4	no	—
238	#107	—	178	2	no	—
239	#106	—	2	4	no	—
240	#106	—	178	4	no	—
241	#103	—	2	4	no	—
242	#103	—	178	4	no	—
243	#103	—	178	4	no	—
244	#103	—	178	4	no	—
245	C₄×C₈	—	89	2	no	—
246	C₄×C₈	—	89	2	no	—
247	C₄×C₈	—	89	2	no	—
248	C₄×C₈	—	89	1	no	—
249	C₄×C₈	—	89	1	no	—
250	C₄×C₈	—	89	1	no	—
251	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
253	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	178	2	no	—
257	C₂×Q₈	—	178	2	no	—
258	C₂²×C₄	—	89	1	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	89	1	no	—
261	Q₁₆	—	4	4	no	—
262	Q₁₆	—	356	8	no	—
263	Q₁₆	—	356	4	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	89	8	no	—
267	C₂×C₈	—	89	2	no	—
268	C₂×C₈	—	178	4	no	—
269	C₂×C₈	—	178	4	no	—
270	C₂×C₈	D₄×D₈₉	1	2	yes	—
271	C₂×C₈	D₄×D₈₉	1	2	yes	—
272	C₂×C₈	D₄×D₈₉	1	2	yes	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	356	2	no	—
278	C₄⋊C₄	—	356	2	no	—
279	C₄⋊C₄	—	356	2	no	—
280	C₄⋊C₄	—	356	2	no	—
281	C₄⋊C₄	—	178	2	no	—
282	C₄⋊C₄	—	178	2	no	—
283	C₄⋊C₄	—	178	2	no	—
284	C₄⋊C₄	—	178	2	no	—
285	C₂²⋊C₄	—	356	2	no	—
286	C₂²⋊C₄	—	356	2	no	—
287	C₄²	—	89	1	no	—
288	C₄²	—	89	1	no	—
289	C₄²	—	89	1	no	—
290	C₄²	—	356	2	no	—
291	C₄²	—	356	2	no	—
292	C₄²	C₂³×D₈₉	1	1	yes	characteristic
293	C₂³	—	89	1	no	—
294	Q₈	—	4	4	no	—
295	Q₈	—	4	4	no	—
296	Q₈	—	356	4	no	—
297	Q₈	—	356	4	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	4	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	178	2	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
311	C₈	—	2	2	no	—
312	C₈	—	89	4	no	—
313	C₈	—	89	4	no	—
314	C₈	—	178	2	no	—
315	C₈	Q₁₆×D₈₉	1	4	yes	—
316	C₈	#15668033	1	4	yes	—
317	C₂²	—	89	2	no	—
318	C₂²	—	178	1	no	—
319	C₂²	—	178	1	no	—
320	C₂²	#127×D₈₉	1	1	yes	characteristic
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	89	2	no	—
324	C₄	—	178	1	no	—
325	C₄	—	178	1	no	—
326	C₄	—	712	2	no	—
327	C₄	—	712	2	no	—
328	C₄	C₂×Q₁₆×D₈₉	1	2	yes	—
329	C₄	#135×D₈₉	1	2	yes	—
330	C₄	C₂×#15668033	1	2	yes	—
331	C₂	—	178	2	no	—
332	C₂	#494×D₈₉	1	1	yes	characteristic
333	C₂	#26030675	1	1	yes	characteristic
334	C₂	#26030677	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1707	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂²×D₈₉	#127
hypercenter	#1707	D₈₉
hypocenter	C₈₉	#19604
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1707

Complements

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

N	H ≅ G/N	type
5. #26019869	C₂	semidirect
6. #26019868	C₂	semidirect
7. #26019860	C₂	semidirect
8. #26019849	C₂	semidirect
9. #26019847	C₂	semidirect
11. C₈₉×#1707	C₂	semidirect
98. C₈×D₈₉	Q₁₆	semidirect
177. #1707	D₈₉	semidirect
179. C₈₉	#19604	semidirect

Maximal subgroups

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

subgroup	index
#26030839	2
#26030837	2
#26030836	2
#26030761	2
#26019869	2
#26019868	2
#26019860	2
#26019849	2
#26019847	2
C₈²×D₈₉	2
C₈₉×#1707	2
#19604	89