Subgroups — #36542113

Order 22784.

750 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030838	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030831	C₂	1	1	yes	maximal, characteristic
3	#26030811	C₂	1	2	yes	maximal, complemented
4	#26030575	C₂	1	2	yes	maximal, complemented
5	#26019864	C₂	1	2	yes	maximal, complemented
6	#26019862	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019858	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019853	C₂	1	2	yes	maximal, complemented
9	#26019849	C₂	1	1	yes	maximal, characteristic, complemented
10	#26019844	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152155	C₂²	1	2	yes	—
13	#19152148	C₂²	1	2	yes	—
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151996	C₂²	1	2	yes	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151971	C₂²	1	2	yes	—
20	#19151953	—	2	4	no	—
21	#19151949	C₂²	1	2	yes	—
22	#19151949	C₂²	1	2	yes	—
23	#19151949	C₂²	1	2	yes	—
24	#19151949	C₂²	1	2	yes	—
25	#19151945	C₂²	1	2	yes	—
26	#19151945	C₂²	1	2	yes	—
27	#19151945	C₂²	1	2	yes	—
28	#19151945	C₂²	1	2	yes	—
29	#19151839	—	2	2	no	—
30	#19151834	—	2	2	no	—
31	#19151827	—	2	4	no	—
32	C₄×Dic₃₅₆	—	2	2	no	—
33	C₄×#13083563	—	2	4	no	—
34	C₄×D₇₁₂	—	2	2	no	—
35	#19151595	—	2	4	no	—
36	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈²×C₈₉	C₂²	1	1	yes	characteristic
40	C₈²×C₈₉	C₂²	1	1	yes	characteristic
41	C₈²×C₈₉	C₂²	1	1	yes	characteristic
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083574	—	2	2	no	—
44	C₂×Dic₃₅₆	—	2	2	no	—
45	C₂×#13083563	—	2	4	no	—
46	C₂×D₇₁₂	—	2	2	no	—
47	#15668012	C₂³	1	2	yes	—
48	#15668012	C₂³	1	2	yes	—
49	#15668012	C₂³	1	2	yes	—
50	#15668009	C₂³	1	2	yes	—
51	#15668004	—	2	2	no	—
52	#15668000	—	2	2	no	—
53	#15667991	—	2	2	no	—
54	#15667991	—	2	2	no	—
55	#15667986	—	2	2	no	—
56	#15667986	—	2	2	no	—
57	C₄×Dic₁₇₈	—	2	2	no	—
58	C₄×Dic₁₇₈	—	2	2	no	—
59	C₄×D₃₅₆	—	2	2	no	—
60	C₄×D₃₅₆	—	2	2	no	—
61	#15667971	—	2	2	no	—
62	#15667946	—	2	2	no	—
63	#15667946	—	2	2	no	—
64	#15667944	—	2	4	no	—
65	#15667937	—	2	4	no	—
66	#15667937	—	2	4	no	—
67	#15667937	—	2	4	no	—
68	#15667937	—	2	4	no	—
69	#15667934	—	2	4	no	—
70	#15667934	—	2	4	no	—
71	#15667934	—	2	4	no	—
72	#15667934	—	2	4	no	—
73	C₂²×C₈×C₈₉	D₄	1	2	yes	—
74	C₂²×C₈×C₈₉	D₄	1	2	yes	—
75	C₂²×C₈×C₈₉	D₄	1	2	yes	—
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	D₄	1	2	yes	—
79	C₄×C₈×C₈₉	D₄	1	2	yes	—
80	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
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	#13083574	—	4	8	no	—
87	C₂×Dic₁₇₈	—	2	2	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	—	2	2	no	—
90	C₂×D₃₅₆	—	2	2	no	—
91	C₂×D₃₅₆	—	2	2	no	—
92	C₂×D₃₅₆	—	2	2	no	—
93	C₂×C₂²⋊D₈₉	—	4	2	no	—
94	C₂×C₄×D₈₉	—	4	2	no	—
95	Dic₃₅₆	—	4	4	no	—
96	#13083563	—	4	8	no	—
97	D₇₁₂	—	4	4	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
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	—	2	2	no	—
104	#13083553	—	4	2	no	—
105	#13083553	—	4	2	no	—
106	C₂²⋊Dic₈₉	—	4	2	no	—
107	#13083551	—	4	2	no	—
108	#13083551	—	4	2	no	—
109	C₄×Dic₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	—	2	4	no	—
114	C₂×C₈×C₈₉	—	2	4	no	—
115	C₂×C₈×C₈₉	—	2	4	no	—
116	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
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	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₈₉	#127	1	1	yes	characteristic
127	C₂²×D₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	2	2	no	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#135	1	4	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₈₉	—	8	4	no	—
138	C₂×Dic₈₉	—	4	2	no	—
139	C₂×Dic₈₉	—	4	2	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	D₃₅₆	—	4	4	no	—
142	D₃₅₆	—	4	4	no	—
143	D₃₅₆	—	4	4	no	—
144	C₄×D₈₉	—	8	4	no	—
145	Dic₁₇₈	—	4	4	no	—
146	Dic₁₇₈	—	4	4	no	—
147	Dic₁₇₈	—	4	4	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	—	2	2	no	—
150	C₈×C₈₉	—	2	2	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	#135	1	4	yes	—
153	C₈×C₈₉	#135	1	4	yes	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	#495	1	2	yes	—
157	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
158	C₂×D₈₉	—	8	2	no	—
159	C₂×D₈₉	—	8	2	no	—
160	C₂×D₈₉	—	8	2	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	#495	1	2	yes	—
164	C₄×C₈₉	C₂×#135	1	2	yes	—
165	C₄×C₈₉	C₂×#135	1	2	yes	—
166	C₄×C₈₉	C₂×#135	1	2	yes	—
167	Dic₈₉	—	8	2	no	—
168	Dic₈₉	—	8	2	no	—
169	Dic₈₉	—	8	2	no	—
170	#19600	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	—	2	2	no	—
172	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
173	C₂×C₈₉	#3147	1	1	yes	characteristic
174	C₂×C₈₉	#3147	1	1	yes	characteristic
175	D₈₉	—	16	2	no	—
176	#3170	—	89	1	no	—
177	#3169	—	89	1	no	—
178	#3167	—	89	2	no	—
179	#3131	—	89	2	no	—
180	#1711	—	89	2	no	—
181	#1710	—	89	1	no	—
182	#1709	—	89	1	no	—
183	#1708	—	89	2	no	—
184	#1707	—	89	1	no	—
185	#1706	—	89	1	no	—
186	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
187	C₈₉	#19600	1	1	yes	characteristic, complemented, Sylow
188	#533	—	89	2	no	—
189	#532	—	89	2	no	—
190	#500	—	89	1	no	—
191	#500	—	89	1	no	—
192	#498	—	89	2	no	—
193	#494	—	89	1	no	—
194	#493	—	89	1	no	—
195	#492	—	89	2	no	—
196	#488	—	178	4	no	—
197	#487	—	89	2	no	—
198	#487	—	89	2	no	—
199	#487	—	89	2	no	—
200	#487	—	89	2	no	—
201	#486	—	89	2	no	—
202	#486	—	89	2	no	—
203	#486	—	89	2	no	—
204	#486	—	89	2	no	—
205	#467	—	178	2	no	—
206	#466	—	178	2	no	—
207	#465	—	178	4	no	—
208	C₄×Q₁₆	—	178	2	no	—
209	C₄×SD₁₆	—	178	4	no	—
210	C₄×D₈	—	178	2	no	—
211	#416	—	178	4	no	—
212	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
213	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
214	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
215	C₈²	C₂×D₈₉	1	1	yes	characteristic
216	C₈²	C₂×D₈₉	1	1	yes	characteristic
217	C₈²	C₂×D₈₉	1	1	yes	characteristic
218	C₈²	C₂×D₈₉	1	1	yes	characteristic
219	C₂×#41	—	178	2	no	—
220	C₂×Q₁₆	—	178	2	no	—
221	C₂×SD₁₆	—	178	4	no	—
222	C₂×D₈	—	178	2	no	—
223	C₂²×C₈	D₃₅₆	1	2	yes	—
224	C₂²×C₈	D₃₅₆	1	2	yes	—
225	C₂²×C₈	D₃₅₆	1	2	yes	—
226	C₄⋊Q₈	—	89	2	no	—
227	C₄⋊Q₈	—	89	2	no	—
228	C₄⋊Q₈	—	89	2	no	—
229	#127	—	89	2	no	—
230	#125	—	178	2	no	—
231	#124	—	178	2	no	—
232	C₂²⋊Q₈	—	178	2	no	—
233	C₂²⋊Q₈	—	178	2	no	—
234	#121	—	178	2	no	—
235	#121	—	178	2	no	—
236	C₄×Q₈	—	178	2	no	—
237	C₄×Q₈	—	178	2	no	—
238	C₄×D₄	—	178	2	no	—
239	C₄×D₄	—	178	2	no	—
240	#117	—	178	2	no	—
241	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
242	#107	—	178	2	no	—
243	#107	—	178	2	no	—
244	#106	—	178	4	no	—
245	#103	—	178	4	no	—
246	#103	—	178	4	no	—
247	#103	—	178	4	no	—
248	#103	—	178	4	no	—
249	#102	—	178	4	no	—
250	#102	—	178	4	no	—
251	#102	—	178	4	no	—
252	#102	—	178	4	no	—
253	C₄×C₈	D₃₅₆	1	2	yes	—
254	C₄×C₈	D₃₅₆	1	2	yes	—
255	C₄×C₈	D₃₅₆	1	2	yes	—
256	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
260	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
262	#41	—	356	8	no	—
263	C₂×Q₈	—	178	2	no	—
264	C₂×Q₈	—	178	2	no	—
265	C₂×Q₈	—	178	2	no	—
266	C₂×D₄	—	356	2	no	—
267	C₂×D₄	—	178	2	no	—
268	C₂×D₄	—	178	2	no	—
269	C₂×D₄	—	178	2	no	—
270	C₂²×C₄	—	356	2	no	—
271	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
274	Q₁₆	—	356	4	no	—
275	SD₁₆	—	356	8	no	—
276	D₈	—	356	4	no	—
277	C₂×C₈	—	2	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	2	4	no	—
280	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
281	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
282	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
283	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
284	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
285	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
286	C₄⋊C₄	—	356	2	no	—
287	C₄⋊C₄	—	356	2	no	—
288	C₄⋊C₄	—	178	2	no	—
289	C₄⋊C₄	—	178	2	no	—
290	C₄⋊C₄	—	178	2	no	—
291	C₄⋊C₄	—	178	2	no	—
292	C₄⋊C₄	—	178	2	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₂²⋊C₄	—	356	2	no	—
295	C₂²⋊C₄	—	356	2	no	—
296	C₂²⋊C₄	—	356	2	no	—
297	C₄²	—	356	2	no	—
298	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
299	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
300	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
301	C₄²	C₂³×D₈₉	1	1	yes	characteristic
302	C₂³	—	356	2	no	—
303	C₂³	#15668009	1	1	yes	characteristic
304	Q₈	—	356	4	no	—
305	Q₈	—	356	4	no	—
306	Q₈	—	356	4	no	—
307	D₄	—	356	4	no	—
308	D₄	—	356	4	no	—
309	D₄	—	356	4	no	—
310	D₄	—	712	4	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	712	4	no	—
317	C₂×C₄	#15668009	1	1	yes	characteristic
318	C₂×C₄	#15668009	1	1	yes	characteristic
319	C₂×C₄	#15668009	1	1	yes	characteristic
320	C₂×C₄	#15668034	1	4	yes	—
321	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	2	2	no	—
327	C₈	—	2	2	no	—
328	C₈	#15668034	1	4	yes	—
329	C₈	#15668034	1	4	yes	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	712	2	no	—
333	C₂²	—	712	2	no	—
334	C₂²	—	712	2	no	—
335	C₂²	#19151982	1	2	yes	—
336	C₂²	C₂×#15668009	1	1	yes	characteristic
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	712	2	no	—
340	C₄	—	712	2	no	—
341	C₄	—	712	2	no	—
342	C₄	#19151982	1	2	yes	—
343	C₄	C₂×#15668034	1	2	yes	—
344	C₄	C₂×#15668034	1	2	yes	—
345	C₄	C₂×#15668034	1	2	yes	—
346	C₂	—	2	2	no	—
347	C₂	—	1424	2	no	—
348	C₂	C₂×#19151982	1	1	yes	characteristic
349	C₂	#26030680	1	1	yes	characteristic
350	C₂	#26030680	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	#26030831	C₂
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#19600
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #26030838	C₂	semidirect
3. #26030811	C₂	semidirect
4. #26030575	C₂	semidirect
5. #26019864	C₂	semidirect
6. #26019862	C₂	semidirect
7. #26019858	C₂	semidirect
8. #26019853	C₂	semidirect
9. #26019849	C₂	semidirect
10. #26019844	C₂	semidirect
11. C₂×C₈²×C₈₉	C₂	semidirect
186. C₂×C₈²	D₈₉	semidirect
187. C₈₉	#19600	semidirect

Maximal subgroups

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

subgroup	index
#26030838	2
#26030831	2
#26030811	2
#26030575	2
#26019864	2
#26019862	2
#26019858	2
#26019853	2
#26019849	2
#26019844	2
C₂×C₈²×C₈₉	2
#19600	89