Subgroups — #36712251

Order 22784.

486 conjugacy classes of subgroups.

The subgroup lattice has 334 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	#26033135	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022490	C₂	1	2	yes	maximal, complemented
4	#26022477	C₂	1	1	yes	maximal, characteristic
5	#26022388	C₂	1	2	yes	maximal, complemented
6	#26022317	C₂	1	2	yes	maximal, complemented
7	#26022293	C₂	1	1	yes	maximal, characteristic
8	#26022157	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022115	C₂	1	2	yes	maximal, complemented
10	C₄×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152333	—	2	1	no	—
13	#19152328	—	2	1	no	—
14	C₂×#15667958	C₂²	1	1	yes	characteristic
15	C₂×#15667957	C₂²	1	2	yes	—
16	#19152002	C₂²	1	2	yes	—
17	#19151983	C₂²	1	2	yes	—
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151892	—	2	2	no	—
20	#19151890	—	2	1	no	—
21	#19151816	—	2	2	no	—
22	#19151798	—	2	1	no	—
23	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
24	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
25	#19151692	C₂²	1	2	yes	—
26	#19151680	C₂²	1	2	yes	—
27	#19151423	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	2	yes	—
29	#19151402	C₂²	1	1	yes	characteristic
30	#19151401	C₂²	1	1	yes	characteristic
31	#19151400	C₂²	1	2	yes	—
32	#19151398	C₂²	1	2	yes	—
33	#19151398	C₂²	1	2	yes	—
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×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×#13083561	—	2	2	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	#15668011	—	2	2	no	—
46	#15668011	—	4	1	no	—
47	#15668010	—	4	2	no	—
48	#15668009	—	2	1	no	—
49	#15668001	—	2	2	no	—
50	#15668000	—	4	1	no	—
51	#15667997	—	4	2	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
54	#15667977	—	2	2	no	—
55	#15667977	—	4	1	no	—
56	C₄×D₃₅₆	—	4	1	no	—
57	#15667974	—	2	2	no	—
58	#15667958	—	2	2	no	—
59	#15667957	—	2	4	no	—
60	C₂×#13083549	C₂³	1	1	yes	characteristic
61	C₂×#13083549	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667945	—	2	2	no	—
64	C₄⋊#10623494	D₄	1	1	yes	characteristic
65	#15667941	D₄	1	2	yes	—
66	#15667937	—	2	1	no	—
67	#15667935	—	2	2	no	—
68	#15667932	—	2	2	no	—
69	C₈×Dic₈₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#103×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
82	C₂×D₃₅₆	—	2	2	no	—
83	C₂×D₃₅₆	—	4	1	no	—
84	C₂×D₃₅₆	—	4	1	no	—
85	C₂×C₄×D₈₉	—	4	2	no	—
86	Dic₃₅₆	—	2	1	no	—
87	Dic₃₅₆	#41	1	2	yes	—
88	#13083561	—	8	2	no	—
89	D₇₁₂	—	2	2	no	—
90	D₇₁₂	—	2	2	no	—
91	D₇₁₂	—	8	1	no	—
92	C₂×#10623494	C₂×D₄	1	2	yes	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
95	#13083553	—	4	1	no	—
96	#13083553	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	#13083553	D₈	1	2	yes	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	#13083551	—	4	2	no	—
102	C₄×Dic₈₉	—	4	1	no	—
103	C₄×Dic₈₉	—	4	2	no	—
104	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
105	#13083549	—	2	1	no	—
106	#13083549	—	2	1	no	—
107	C₂×Q₈×C₈₉	—	2	1	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	Q₁₆×C₈₉	—	2	1	no	—
110	Q₁₆×C₈₉	—	2	1	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	1	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	—	4	1	no	—
118	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₈₉	—	2	2	no	—
120	C₂²×D₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	4	1	no	—
122	Q₈×C₈₉	—	4	1	no	—
123	Q₈×C₈₉	—	16	1	no	—
124	C₂×C₄×C₈₉	—	4	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	D₃₅₆	—	4	2	no	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	—	8	2	no	—
137	D₃₅₆	—	16	1	no	—
138	C₄×D₈₉	—	16	2	no	—
139	Dic₁₇₈	—	2	2	no	—
140	Dic₁₇₈	—	2	2	no	—
141	Dic₁₇₈	—	4	1	no	—
142	Dic₁₇₈	—	16	2	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	#10623494	—	2	2	no	—
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	8	1	no	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	—	8	1	no	—
155	C₄×C₈₉	#510	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	—	4	2	no	—
161	Dic₈₉	—	8	1	no	—
162	#33942	—	89	1	no	maximal, Sylow
163	C₂×C₈₉	#3358	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
165	C₂×C₈₉	#3415	1	1	yes	characteristic
166	D₈₉	—	16	2	no	—
167	#3396	—	89	1	no	—
168	Q₈×Q₁₆	—	89	1	no	—
169	#2243	—	89	2	no	—
170	#2240	—	89	1	no	—
171	#2227	D₈₉	1	1	yes	characteristic, complemented
172	#2224	—	89	2	no	—
173	#2211	—	89	2	no	—
174	#2207	—	89	1	no	—
175	#2183	—	89	1	no	—
176	#2175	—	89	2	no	—
177	C₄×Q₃₂	—	89	1	no	—
178	C₈₉	#33942	1	1	yes	characteristic, complemented, Sylow
179	#559	—	178	1	no	—
180	Q₈²	—	178	1	no	—
181	C₂×Q₃₂	—	89	1	no	—
182	C₂×SD₃₂	—	89	2	no	—
183	#500	—	89	2	no	—
184	#500	C₂×D₈₉	1	1	yes	characteristic
185	#495	—	89	2	no	—
186	#493	—	89	1	no	—
187	#477	—	178	2	no	—
188	#477	—	178	1	no	—
189	#463	—	178	2	no	—
190	#460	—	178	1	no	—
191	C₈×Q₈	—	89	1	no	—
192	C₄×Q₁₆	—	89	2	no	—
193	C₄×Q₁₆	—	89	1	no	—
194	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₄×D₈	—	89	2	no	—
196	#366	—	89	1	no	—
197	#366	C₂×D₈₉	1	2	yes	—
198	C₄⋊C₁₆	—	89	2	no	—
199	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
200	#358	—	89	1	no	—
201	#358	—	89	1	no	—
202	#358	C₂×D₈₉	1	2	yes	—
203	#357	—	89	2	no	—
204	#357	—	89	2	no	—
205	#357	—	89	2	no	—
206	#346	—	89	2	no	—
207	C₄×C₁₆	—	89	1	no	—
208	C₂×Q₁₆	—	89	1	no	—
209	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
210	C₂×SD₁₆	—	178	2	no	—
211	C₂×D₈	—	89	2	no	—
212	C₂×D₈	—	178	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	—	356	1	no	—
215	C₄⋊Q₈	—	356	2	no	—
216	C₄⋊Q₈	—	178	2	no	—
217	#127	—	178	1	no	—
218	#124	—	356	1	no	—
219	#124	—	178	2	no	—
220	#123	—	356	2	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×Q₈	—	178	1	no	—
226	C₄×D₄	—	356	1	no	—
227	C₄×D₄	—	178	2	no	—
228	Q₃₂	—	178	2	no	—
229	SD₃₂	—	178	4	no	—
230	C₂×C₁₆	—	89	1	no	—
231	C₂×C₁₆	—	89	1	no	—
232	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
233	#107	—	2	1	no	—
234	#107	—	89	1	no	—
235	#107	—	178	2	no	—
236	#107	C₂²×D₈₉	1	2	yes	—
237	#107	C₂²×D₈₉	1	1	yes	characteristic
238	C₄⋊C₈	—	89	2	no	—
239	C₄⋊C₈	—	89	1	no	—
240	#103	—	2	1	no	—
241	#103	—	178	2	no	—
242	#103	—	178	1	no	—
243	#102	—	178	2	no	—
244	C₄×C₈	—	89	2	no	—
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	89	1	no	—
249	C₂×Q₈	—	356	2	no	—
250	C₂×Q₈	—	178	1	no	—
251	C₂×D₄	—	356	1	no	—
252	C₂×D₄	—	356	1	no	—
253	C₂×D₄	—	178	2	no	—
254	C₂²×C₄	—	356	2	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	—	89	2	no	—
258	Q₁₆	—	178	1	no	—
259	SD₁₆	—	712	2	no	—
260	D₈	—	178	2	no	—
261	D₈	—	178	2	no	—
262	D₈	—	712	1	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	2	no	—
272	C₄⋊C₄	—	356	2	no	—
273	C₄⋊C₄	—	356	2	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	356	2	no	—
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	89	2	no	—
281	C₄²	—	356	2	no	—
282	C₄²	—	356	1	no	—
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₁₆	—	2	2	no	—
285	C₁₆	—	178	1	no	—
286	C₁₆	—	178	1	no	—
287	C₂³	—	356	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	1	no	—
291	Q₈	—	356	1	no	—
292	Q₈	—	1424	2	no	—
293	Q₈	—	178	2	no	—
294	Q₈	—	178	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	356	2	no	—
297	D₄	—	1424	1	no	—
298	D₄	—	712	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	356	1	no	—
305	C₂×C₄	—	356	1	no	—
306	C₂×C₄	—	1424	2	no	—
307	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
308	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	178	2	no	—
312	C₈	#41×D₈₉	1	1	yes	characteristic
313	C₈	#15668060	1	1	yes	characteristic
314	C₂²	—	712	2	no	—
315	C₂²	—	712	2	no	—
316	C₂²	—	712	2	no	—
317	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	356	2	no	—
323	C₄	—	178	2	no	—
324	C₄	—	178	2	no	—
325	C₄	—	712	1	no	—
326	C₄	#19152040	1	2	yes	—
327	C₄	#19152258	1	1	yes	characteristic
328	C₄	#19152407	1	1	yes	characteristic
329	C₂	—	1424	2	no	—
330	C₂	#26032886	1	1	yes	characteristic
331	C₂	C₂×#19152040	1	1	yes	characteristic
332	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	#19151975	C₂²
hypercenter	#2227	D₈₉
hypocenter	C₈₉	#33942
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
2. #26033135	C₂	semidirect
3. #26022490	C₂	semidirect
5. #26022388	C₂	semidirect
6. #26022317	C₂	semidirect
8. #26022157	C₂	semidirect
9. #26022115	C₂	semidirect
10. C₄×#15667958	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
171. #2227	D₈₉	semidirect
178. C₈₉	#33942	semidirect

Maximal subgroups

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

subgroup	index
#26033262	2
#26033135	2
#26022490	2
#26022477	2
#26022388	2
#26022317	2
#26022293	2
#26022157	2
#26022115	2
C₄×#15667958	2
C₈₉×#2227	2
#33942	89