Subgroups — #36542089

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030810	C₂	1	2	yes	maximal, complemented
2	#26030809	C₂	1	2	yes	maximal, complemented
3	#26030763	C₂	1	1	yes	maximal, characteristic, complemented
4	#26030679	C₂	1	1	yes	maximal, characteristic
5	#26019852	C₂	1	2	yes	maximal, complemented
6	#26019850	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019849	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019841	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019838	C₂	1	2	yes	maximal, complemented
10	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	—	2	2	no	—
13	#19152153	C₂²	1	2	yes	—
14	#19152146	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151999	—	2	2	no	—
17	#19151996	C₂²	1	2	yes	—
18	#19151994	C₂²	1	2	yes	—
19	#19151994	C₂²	1	2	yes	—
20	#19151981	—	2	2	no	—
21	#19151978	C₂²	1	1	yes	characteristic
22	#19151970	C₂²	1	2	yes	—
23	#19151970	C₂²	1	2	yes	—
24	#19151949	C₂²	1	2	yes	—
25	#19151948	C₂²	1	2	yes	—
26	#19151944	C₂²	1	2	yes	—
27	#19151826	—	2	4	no	—
28	#19151824	—	2	4	no	—
29	#19151689	—	2	4	no	—
30	SD₁₆×Dic₈₉	—	2	4	no	—
31	#19151635	—	2	2	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	#493×C₈₉	C₂²	1	1	yes	characteristic
40	#492×C₈₉	C₂²	1	2	yes	—
41	#486×C₈₉	C₂²	1	2	yes	—
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083573	—	2	2	no	—
44	#15668031	—	4	8	no	—
45	C₂×Dic₃₅₆	—	2	2	no	—
46	C₂×#13083562	—	2	4	no	—
47	C₂×#13083562	—	2	4	no	—
48	C₂×C₈×D₈₉	D₄	1	2	yes	—
49	C₂×C₈×D₈₉	D₄	1	2	yes	—
50	C₂×C₈×D₈₉	D₄	1	2	yes	—
51	#15668012	C₂³	1	2	yes	—
52	#15668012	C₂³	1	2	yes	—
53	#15668003	—	2	2	no	—
54	#15667999	—	2	2	no	—
55	#15667992	—	2	2	no	—
56	#15667988	—	2	2	no	—
57	#15667983	—	2	2	no	—
58	#15667983	—	2	2	no	—
59	Q₈×Dic₈₉	—	2	2	no	—
60	#15667977	—	2	2	no	—
61	D₄×Dic₈₉	—	2	2	no	—
62	D₄×Dic₈₉	—	2	2	no	—
63	#15667970	—	2	2	no	—
64	C₄²×D₈₉	C₂³	1	1	yes	characteristic
65	#15667946	—	2	2	no	—
66	#15667944	—	2	4	no	—
67	#15667943	—	2	4	no	—
68	#15667943	—	2	4	no	—
69	#15667937	—	2	4	no	—
70	#15667936	—	2	4	no	—
71	#15667933	—	2	4	no	—
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₈×Dic₈₉	D₄	1	2	yes	—
76	C₈×Dic₈₉	D₄	1	2	yes	—
77	C₈×Dic₈₉	D₄	1	2	yes	—
78	C₂×SD₁₆×C₈₉	—	2	4	no	—
79	C₂×D₈×C₈₉	—	2	2	no	—
80	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
81	#127×C₈₉	C₂³	1	2	yes	—
82	#107×C₈₉	—	2	2	no	—
83	#102×C₈₉	—	2	4	no	—
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	#13083573	—	4	8	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	—	2	2	no	—
90	C₂×C₂²⋊D₈₉	—	4	2	no	—
91	C₂²×Dic₈₉	—	4	2	no	—
92	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
95	Dic₃₅₆	—	4	4	no	—
96	#13083562	—	4	8	no	—
97	#13083562	—	4	8	no	—
98	C₂×#10623494	C₂×D₄	1	2	yes	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₂×#10623494	C₂×D₄	1	2	yes	—
101	C₈×D₈₉	—	2	4	no	—
102	C₈×D₈₉	—	2	4	no	—
103	C₈×D₈₉	SD₁₆	1	8	yes	complemented
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	2	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	4	2	no	—
109	#13083553	—	4	2	no	—
110	C₂²⋊Dic₈₉	—	4	2	no	—
111	C₂²⋊Dic₈₉	—	4	2	no	—
112	#13083551	—	4	2	no	—
113	C₄×Dic₈₉	—	4	2	no	—
114	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₈₉	—	2	2	no	—
118	C₂×D₄×C₈₉	—	2	2	no	—
119	C₂×D₄×C₈₉	—	2	2	no	—
120	C₂×D₄×C₈₉	—	2	2	no	—
121	SD₁₆×C₈₉	—	4	8	no	—
122	D₈×C₈₉	—	4	4	no	—
123	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	—	4	2	no	—
130	C₂²×D₈₉	#127	1	1	yes	characteristic
131	Q₈×C₈₉	—	4	4	no	—
132	D₄×C₈₉	—	4	4	no	—
133	D₄×C₈₉	—	4	4	no	—
134	D₄×C₈₉	—	4	4	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₈₉	—	8	4	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	8	4	no	—
143	C₂×Dic₈₉	#127	1	1	yes	characteristic
144	C₂×Dic₈₉	#127	1	1	yes	characteristic
145	C₂×Dic₈₉	#127	1	1	yes	characteristic
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	C₂×SD₁₆	1	4	yes	—
149	Dic₁₇₈	—	4	4	no	—
150	Dic₁₇₈	—	4	4	no	—
151	C₈×C₈₉	—	2	2	no	—
152	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
153	C₈×C₈₉	#135	1	4	yes	complemented
154	#10623494	—	2	2	no	—
155	#10623494	C₂×SD₁₆	1	4	yes	—
156	#10623494	#135	1	4	yes	complemented
157	C₂²×C₈₉	—	8	2	no	—
158	C₂²×C₈₉	—	8	2	no	—
159	C₂²×C₈₉	—	8	2	no	—
160	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	#492	1	2	yes	—
164	C₄×C₈₉	—	8	2	no	—
165	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
166	C₄×C₈₉	C₂×#135	1	2	yes	—
167	C₄×C₈₉	C₂×#135	1	2	yes	—
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	8	2	no	—
171	Dic₈₉	—	8	2	no	—
172	Dic₈₉	#492	1	2	yes	—
173	#19596	—	89	1	no	maximal, Sylow
174	C₂×C₈₉	—	16	2	no	—
175	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
176	C₂×C₈₉	#3147	1	1	yes	characteristic
177	C₂×C₈₉	#3147	1	1	yes	characteristic
178	D₈₉	—	2	2	no	—
179	#3167	—	89	2	no	—
180	#3167	—	89	2	no	—
181	#3160	—	89	1	no	—
182	#3147	—	89	1	no	—
183	#1708	—	89	2	no	—
184	#1707	—	89	1	no	—
185	#1707	—	89	1	no	—
186	#1706	—	89	1	no	—
187	#1706	D₈₉	1	1	yes	characteristic, complemented
188	#1705	—	89	2	no	—
189	C₂×C₈²	—	89	1	no	—
190	C₈₉	#19596	1	1	yes	characteristic, complemented, Sylow
191	C₂×#135	—	178	2	no	—
192	#533	—	89	2	no	—
193	#532	—	89	2	no	—
194	#500	—	89	1	no	—
195	#500	C₂×D₈₉	1	1	yes	characteristic
196	#499	—	178	2	no	—
197	#498	—	89	2	no	—
198	#498	—	89	2	no	—
199	#498	—	89	2	no	—
200	#495	—	178	2	no	—
201	#494	—	89	1	no	—
202	#493	C₂×D₈₉	1	1	yes	characteristic
203	#492	—	89	2	no	—
204	#492	—	89	2	no	—
205	#492	C₂×D₈₉	1	2	yes	—
206	#487	—	89	2	no	—
207	#487	—	89	2	no	—
208	#486	—	89	2	no	—
209	#486	C₂×D₈₉	1	2	yes	—
210	#465	—	178	4	no	—
211	#465	—	178	4	no	—
212	C₄×SD₁₆	—	178	4	no	—
213	C₄×SD₁₆	—	178	4	no	—
214	#427	—	178	2	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	C₂×C₄×C₈	—	89	1	no	—
217	C₂×C₄×C₈	—	89	1	no	—
218	C₈²	—	89	1	no	—
219	C₈²	—	89	1	no	—
220	C₈²	—	89	1	no	—
221	C₈²	C₂×D₈₉	1	1	yes	characteristic
222	C₂×#41	—	178	2	no	—
223	#135	—	356	8	no	—
224	C₂×Q₁₆	—	178	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	178	4	no	—
227	C₂×SD₁₆	—	178	4	no	—
228	C₂×D₈	—	2	2	no	—
229	C₂²×C₈	—	89	2	no	—
230	C₂²×C₈	—	89	2	no	—
231	C₂²×C₈	—	89	2	no	—
232	C₄⋊Q₈	—	89	2	no	—
233	C₄⋊Q₈	—	89	2	no	—
234	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
235	#127	C₂²×D₈₉	1	2	yes	—
236	#125	—	178	2	no	—
237	#124	—	178	2	no	—
238	C₂²⋊Q₈	—	178	2	no	—
239	C₂²⋊Q₈	—	178	2	no	—
240	#121	—	178	2	no	—
241	#121	—	178	2	no	—
242	C₄×Q₈	—	178	2	no	—
243	C₄×Q₈	—	178	2	no	—
244	C₄×D₄	—	178	2	no	—
245	C₄×D₄	—	178	2	no	—
246	#117	—	178	2	no	—
247	C₂×C₄²	—	89	1	no	—
248	#107	—	2	2	no	—
249	#107	—	178	2	no	—
250	#106	—	178	4	no	—
251	#106	—	178	4	no	—
252	#106	—	178	4	no	—
253	#103	—	178	4	no	—
254	#103	—	178	4	no	—
255	#102	—	2	4	no	—
256	#102	—	178	4	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	2	no	—
259	C₄×C₈	—	89	2	no	—
260	C₄×C₈	—	89	1	no	—
261	C₄×C₈	—	89	1	no	—
262	C₄×C₈	—	89	1	no	—
263	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	#41	—	356	8	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	178	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	356	2	no	—
274	C₂²×C₄	—	89	1	no	—
275	C₂²×C₄	—	89	1	no	—
276	C₂²×C₄	—	89	1	no	—
277	C₂²×C₄	—	356	2	no	—
278	Q₁₆	—	356	4	no	—
279	SD₁₆	—	4	8	no	—
280	SD₁₆	—	356	8	no	—
281	SD₁₆	—	356	8	no	—
282	D₈	—	4	4	no	—
283	C₂×C₈	—	89	2	no	—
284	C₂×C₈	—	89	2	no	—
285	C₂×C₈	—	89	8	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	178	4	no	—
288	C₂×C₈	—	178	4	no	—
289	C₂×C₈	D₄×D₈₉	1	2	yes	—
290	C₂×C₈	D₄×D₈₉	1	2	yes	—
291	C₂×C₈	D₄×D₈₉	1	2	yes	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	356	2	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₄⋊C₄	—	178	2	no	—
299	C₄⋊C₄	—	178	2	no	—
300	C₂²⋊C₄	—	356	2	no	—
301	C₂²⋊C₄	—	356	2	no	—
302	C₂²⋊C₄	—	356	2	no	—
303	C₄²	—	89	1	no	—
304	C₄²	—	89	1	no	—
305	C₄²	—	89	1	no	—
306	C₄²	—	356	2	no	—
307	C₄²	C₂³×D₈₉	1	1	yes	characteristic
308	C₂³	—	4	2	no	—
309	C₂³	—	89	1	no	—
310	Q₈	—	4	4	no	—
311	Q₈	—	356	4	no	—
312	Q₈	—	356	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	4	no	—
315	D₄	—	4	4	no	—
316	D₄	—	712	4	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	4	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	712	4	no	—
327	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	89	4	no	—
332	C₈	—	89	4	no	—
333	C₈	—	178	2	no	—
334	C₈	SD₁₆×D₈₉	1	4	yes	—
335	C₈	#15668031	1	4	yes	complemented
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	89	2	no	—
340	C₂²	—	178	1	no	—
341	C₂²	—	178	1	no	—
342	C₂²	#127×D₈₉	1	1	yes	characteristic
343	C₄	—	8	2	no	—
344	C₄	—	89	2	no	—
345	C₄	—	178	1	no	—
346	C₄	—	178	1	no	—
347	C₄	—	712	2	no	—
348	C₄	—	712	2	no	—
349	C₄	C₂×SD₁₆×D₈₉	1	2	yes	—
350	C₄	#135×D₈₉	1	2	yes	—
351	C₄	C₂×#15668031	1	2	yes	—
352	C₂	—	16	2	no	—
353	C₂	—	178	2	no	—
354	C₂	#492×D₈₉	1	1	yes	characteristic
355	C₂	#26030677	1	1	yes	characteristic
356	C₂	#26030679	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1706	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	#26030679	C₂
hypercenter	#1706	D₈₉
hypocenter	C₈₉	#19596
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #26030810	C₂	semidirect
2. #26030809	C₂	semidirect
3. #26030763	C₂	semidirect
5. #26019852	C₂	semidirect
6. #26019850	C₂	semidirect
7. #26019849	C₂	semidirect
8. #26019841	C₂	semidirect
9. #26019838	C₂	semidirect
10. C₈²×D₈₉	C₂	semidirect
11. C₈₉×#1706	C₂	semidirect
103. C₈×D₈₉	SD₁₆	semidirect
153. C₈×C₈₉	#135	semidirect
156. #10623494	#135	semidirect
187. #1706	D₈₉	semidirect
190. C₈₉	#19596	semidirect
335. C₈	#15668031	semidirect

Maximal subgroups

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

subgroup	index
#26030810	2
#26030809	2
#26030763	2
#26030679	2
#26019852	2
#26019850	2
#26019849	2
#26019841	2
#26019838	2
C₈²×D₈₉	2
C₈₉×#1706	2
#19596	89