Subgroups — #36709801

Order 22784.

622 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	#500×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030681	C₂	1	1	yes	maximal, characteristic
3	#26022490	C₂	1	4	yes	maximal, complemented
4	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022477	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021963	C₂	1	4	yes	maximal, complemented
8	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668035	—	2	1	no	—
11	C₂×#15668035	C₂²	1	2	yes	—
12	#19152149	—	2	1	no	—
13	C₄⋊Q₈×D₈₉	—	2	1	no	—
14	C₂×#15667951	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152002	—	2	1	no	—
17	#19152002	C₂²	1	2	yes	—
18	#19152001	—	2	1	no	—
19	#19151983	—	2	1	no	—
20	#19151983	C₂²	1	2	yes	—
21	#19151975	C₂²	1	1	yes	characteristic
22	#19151972	—	2	1	no	—
23	#107×D₈₉	—	2	1	no	—
24	#107×D₈₉	C₂²	1	2	yes	—
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	#19151402	C₂²	1	4	yes	—
27	#19151400	C₂²	1	4	yes	—
28	#19151398	C₂²	1	4	yes	—
29	#19151373	C₂²	1	2	yes	—
30	C₄×#13083549	C₂²	1	1	yes	characteristic
31	#500×C₈₉	C₂²	1	1	yes	characteristic
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#358×C₈₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#13083575	—	2	2	no	—
36	C₂×Q₈×D₈₉	—	4	1	no	—
37	#15668035	—	2	8	no	—
38	#15668035	—	4	4	no	—
39	C₂×#13083561	—	2	2	no	—
40	C₂×#13083561	—	2	2	no	—
41	C₂×D₇₁₂	—	2	1	no	—
42	C₂×D₇₁₂	C₂³	1	2	yes	—
43	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15668012	—	2	1	no	—
46	#15668011	—	2	2	no	—
47	#15668011	—	4	1	no	—
48	#15668010	—	4	1	no	—
49	#15668009	—	2	1	no	—
50	#15668001	—	2	2	no	—
51	#15667987	—	4	1	no	—
52	#15667987	—	4	1	no	—
53	#15667974	—	4	1	no	—
54	#15667974	—	4	1	no	—
55	C₄⋊C₄×D₈₉	—	2	2	no	—
56	C₄⋊C₄×D₈₉	—	4	1	no	—
57	C₄²×D₈₉	D₄	1	1	yes	characteristic
58	#15667951	—	2	4	no	—
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	2	yes	—
62	#15667945	—	2	2	no	—
63	#15667945	—	2	2	no	—
64	C₄×#10623494	D₄	1	1	yes	characteristic
65	C₈×Dic₈₉	D₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₉	—	2	1	no	—
68	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	C₂³	1	2	yes	—
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#13083575	—	4	4	no	—
76	#13083575	—	4	4	no	—
77	C₂×Dic₁₇₈	—	4	1	no	—
78	C₂×Dic₁₇₈	—	4	2	no	—
79	Q₈×D₈₉	—	8	4	no	—
80	C₂×D₃₅₆	—	2	2	no	—
81	C₂×D₃₅₆	—	4	2	no	—
82	C₂×D₃₅₆	—	4	1	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×C₄×D₈₉	—	4	2	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
87	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
88	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
89	#13083561	—	4	4	no	—
90	#13083561	—	4	4	no	—
91	D₇₁₂	—	2	4	no	—
92	D₇₁₂	—	4	2	no	—
93	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
94	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
95	C₈×D₈₉	SD₁₆	1	4	yes	—
96	C₈×D₈₉	#41	1	4	yes	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	4	1	no	—
99	#13083553	—	4	2	no	—
100	#13083553	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	#13083551	—	4	2	no	—
103	C₄×Dic₈₉	D₈	1	1	yes	characteristic
104	C₄×Dic₈₉	D₈	1	1	yes	characteristic
105	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
106	#13083549	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	Q₁₆×C₈₉	—	2	4	no	—
110	Q₁₆×C₈₉	—	4	2	no	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂²×D₈₉	—	4	2	no	—
120	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	Q₈×C₈₉	—	8	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	4	2	no	—
135	D₃₅₆	—	8	4	no	—
136	D₃₅₆	—	8	2	no	—
137	D₃₅₆	—	8	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	8	4	no	—
141	C₄×D₈₉	—	8	4	no	—
142	C₄×D₈₉	#124	1	2	yes	—
143	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
144	Dic₁₇₈	—	8	4	no	—
145	Dic₁₇₈	—	8	2	no	—
146	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
147	C₈×C₈₉	C₂×#41	1	2	yes	—
148	#10623494	#124	1	2	yes	—
149	#10623494	C₂×SD₁₆	1	2	yes	—
150	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	#486	1	2	yes	—
157	C₄×C₈₉	—	8	2	no	—
158	C₄×C₈₉	—	8	2	no	—
159	C₄×C₈₉	#509	1	2	yes	—
160	C₄×C₈₉	#510	1	2	yes	—
161	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
162	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	#486	1	2	yes	—
167	#33707	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
171	D₈₉	—	2	2	no	—
172	D₈₉	—	16	2	no	—
173	C₂×#500	—	89	1	no	—
174	#3147	—	89	1	no	—
175	#2243	—	89	4	no	—
176	#2241	—	89	1	no	—
177	#2241	D₈₉	1	1	yes	characteristic, complemented
178	#2240	—	89	1	no	—
179	#2240	—	89	1	no	—
180	#2139	—	89	4	no	—
181	C₄×M₃₂	—	89	1	no	—
182	C₈₉	#33707	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	89	2	no	—
184	C₂×#135	—	178	1	no	—
185	#532	—	178	1	no	—
186	C₂×C₄⋊Q₈	—	178	1	no	—
187	C₂×M₃₂	—	89	2	no	—
188	#500	—	89	2	no	—
189	#500	—	89	1	no	—
190	#500	—	178	1	no	—
191	#500	—	178	1	no	—
192	#500	C₂×D₈₉	1	1	yes	characteristic
193	#495	—	89	2	no	—
194	#495	—	178	1	no	—
195	#494	C₂×D₈₉	1	1	yes	characteristic
196	#493	—	89	1	no	—
197	#492	—	178	1	no	—
198	C₂×#107	—	89	2	no	—
199	C₂×#107	—	178	1	no	—
200	C₂×C₄×C₈	—	89	1	no	—
201	#358	—	89	4	no	—
202	#358	C₂×D₈₉	1	4	yes	—
203	#357	—	89	4	no	—
204	#357	—	89	4	no	—
205	#346	—	89	2	no	—
206	C₄×C₁₆	—	89	1	no	—
207	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₂×#41	—	178	2	no	—
209	C₂²×Q₈	—	356	1	no	—
210	#135	—	356	4	no	—
211	#135	—	178	8	no	—
212	C₂×Q₁₆	—	2	1	no	—
213	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
214	C₂×SD₁₆	—	178	2	no	—
215	C₂×SD₁₆	—	178	2	no	—
216	C₂×D₈	—	89	2	no	—
217	C₂×D₈	—	178	1	no	—
218	C₂²×C₈	—	89	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	2	1	no	—
222	C₄⋊Q₈	—	356	1	no	—
223	C₄⋊Q₈	—	356	1	no	—
224	C₄⋊Q₈	—	178	2	no	—
225	C₄⋊Q₈	—	178	1	no	—
226	#127	—	178	1	no	—
227	#124	—	178	2	no	—
228	#121	—	356	1	no	—
229	#121	—	356	1	no	—
230	C₄×D₄	—	356	1	no	—
231	C₄×D₄	—	356	1	no	—
232	C₂×C₄⋊C₄	—	356	1	no	—
233	C₂×C₄⋊C₄	—	178	2	no	—
234	C₂×C₄²	—	89	1	no	—
235	M₃₂	—	178	4	no	—
236	C₂×C₁₆	—	89	2	no	—
237	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
238	#107	—	2	1	no	—
239	#107	—	89	2	no	—
240	#107	—	178	2	no	—
241	#107	—	178	2	no	—
242	#107	—	178	1	no	—
243	#107	C₂²×D₈₉	1	2	yes	—
244	C₄×C₈	—	89	1	no	—
245	C₄×C₈	—	89	1	no	—
246	C₄×C₈	—	89	1	no	—
247	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	#41	—	356	4	no	—
249	#41	—	356	4	no	—
250	C₂×Q₈	—	2	2	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	356	2	no	—
253	C₂×Q₈	—	356	1	no	—
254	C₂×Q₈	—	712	4	no	—
255	C₂×D₄	—	356	2	no	—
256	C₂×D₄	—	356	1	no	—
257	C₂×D₄	—	356	1	no	—
258	C₂×D₄	—	178	2	no	—
259	C₂²×C₄	—	89	1	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	89	1	no	—
262	C₂²×C₄	—	356	2	no	—
263	C₂²×C₄	—	356	2	no	—
264	Q₁₆	—	2	4	no	—
265	Q₁₆	—	4	2	no	—
266	SD₁₆	—	356	4	no	—
267	SD₁₆	—	356	4	no	—
268	D₈	—	356	2	no	—
269	D₈	—	178	4	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	89	4	no	—
273	C₂×C₈	—	89	4	no	—
274	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	356	2	no	—
281	C₄⋊C₄	—	356	2	no	—
282	C₄⋊C₄	—	356	1	no	—
283	C₄⋊C₄	—	178	2	no	—
284	C₂²⋊C₄	—	356	2	no	—
285	C₂²⋊C₄	—	356	2	no	—
286	C₄²	—	89	1	no	—
287	C₄²	—	89	1	no	—
288	C₄²	—	89	1	no	—
289	C₄²	D₄×D₈₉	1	1	yes	characteristic
290	C₁₆	—	2	2	no	—
291	C₁₆	—	178	2	no	—
292	C₂³	—	89	1	no	—
293	C₂³	—	356	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	Q₈	—	8	2	no	—
297	Q₈	—	712	4	no	—
298	Q₈	—	712	2	no	—
299	D₄	—	356	2	no	—
300	D₄	—	356	2	no	—
301	D₄	—	712	4	no	—
302	D₄	—	712	2	no	—
303	D₄	—	712	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	89	1	no	—
307	C₂×C₄	—	89	2	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	356	2	no	—
312	C₂×C₄	—	178	2	no	—
313	C₂×C₄	—	178	2	no	—
314	C₂×C₄	—	712	4	no	—
315	C₂×C₄	—	712	4	no	—
316	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
317	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
319	C₈	—	89	2	no	—
320	C₈	—	89	2	no	—
321	C₈	SD₁₆×D₈₉	1	2	yes	—
322	C₈	#41×D₈₉	1	2	yes	—
323	C₂²	—	89	2	no	—
324	C₂²	—	178	1	no	—
325	C₂²	—	178	1	no	—
326	C₂²	—	712	2	no	—
327	C₂²	—	712	2	no	—
328	C₂²	—	712	2	no	—
329	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	89	2	no	—
333	C₄	—	178	1	no	—
334	C₄	—	178	1	no	—
335	C₄	—	712	2	no	—
336	C₄	#19152035	1	2	yes	—
337	C₄	#19152040	1	2	yes	—
338	C₄	#124×D₈₉	1	1	yes	characteristic
339	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
340	C₂	—	1424	2	no	—
341	C₂	—	178	2	no	—
342	C₂	#486×D₈₉	1	1	yes	characteristic
343	C₂	C₂×#19152035	1	1	yes	characteristic
344	C₂	C₂×#19152040	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×#2241	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26030681	C₂
hypercenter	#2241	D₈₉
hypocenter	C₈₉	#33707
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #500×D₈₉	C₂	semidirect
3. #26022490	C₂	semidirect
4. #26022479	C₂	semidirect
5. #26022477	C₂	semidirect
6. #26022476	C₂	semidirect
7. #26021963	C₂	semidirect
8. C₄×#15667951	C₂	semidirect
9. C₈₉×#2241	C₂	semidirect
177. #2241	D₈₉	semidirect
182. C₈₉	#33707	semidirect

Maximal subgroups

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

subgroup	index
#500×D₈₉	2
#26030681	2
#26022490	2
#26022479	2
#26022477	2
#26022476	2
#26021963	2
C₄×#15667951	2
C₈₉×#2241	2
#33707	89