Subgroups — #36709286

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029908	C₂	1	1	yes	maximal, characteristic
2	#26028849	C₂	1	1	yes	maximal, characteristic
3	#26022463	C₂	1	2	yes	maximal, complemented
4	#26022462	C₂	1	2	yes	maximal, complemented
5	#26022456	C₂	1	2	yes	maximal, complemented
6	#26022451	C₂	1	2	yes	maximal, complemented
7	#26022041	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021865	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2158	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
13	C₂×#15668018	C₂²	1	1	yes	characteristic
14	#19152174	—	2	1	no	—
15	#19152078	—	2	1	no	—
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	#19151857	—	2	2	no	—
18	#19151840	C₂²	1	2	yes	—
19	#19151840	C₂²	1	2	yes	—
20	#19151641	—	2	1	no	—
21	#19151637	—	2	2	no	—
22	#19151635	C₂²	1	2	yes	—
23	C₂×#15667946	C₂²	1	1	yes	characteristic
24	#107×D₈₉	C₂²	1	2	yes	—
25	#19151606	—	2	1	no	—
26	#19151425	C₂²	1	2	yes	—
27	#19151423	C₂²	1	2	yes	—
28	#19151403	C₂²	1	2	yes	—
29	#19151403	C₂²	1	2	yes	—
30	#19151401	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151377	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	#427×C₈₉	C₂²	1	1	yes	characteristic
36	#427×C₈₉	C₂²	1	1	yes	characteristic
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#367×C₈₉	C₂²	1	1	yes	characteristic
39	#366×C₈₉	C₂²	1	1	yes	characteristic
40	#366×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	D₄	1	1	yes	characteristic
42	C₂²×Dic₁₇₈	—	2	1	no	—
43	C₂×Dic₃₅₆	—	2	1	no	—
44	C₂×Dic₃₅₆	—	2	2	no	—
45	C₂×Dic₃₅₆	—	2	2	no	—
46	C₂×Dic₃₅₆	C₂³	1	2	yes	—
47	#15668018	—	2	2	no	—
48	#15668018	—	2	2	no	—
49	C₂×#13083558	D₄	1	1	yes	characteristic
50	C₂×#13083557	D₄	1	2	yes	—
51	C₂×C₈×D₈₉	C₂³	1	2	yes	—
52	#15668012	—	4	1	no	—
53	#15668000	—	4	1	no	—
54	#15667996	—	4	2	no	—
55	#15667992	—	2	2	no	—
56	#15667992	—	2	2	no	—
57	C₄×Dic₁₇₈	—	4	1	no	—
58	C₄×D₃₅₆	—	4	1	no	—
59	#15667972	—	4	2	no	—
60	#15667970	—	2	2	no	—
61	C₂×C₄⋊Dic₈₉	—	2	1	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	#15667952	—	2	2	no	—
64	C₂×#13083549	C₂³	1	2	yes	—
65	#15667946	—	2	1	no	—
66	#15667946	C₂³	1	2	yes	—
67	#15667945	—	2	2	no	—
68	#15667943	—	2	2	no	—
69	#15667935	—	2	2	no	—
70	#15667935	—	2	2	no	—
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	#117×C₈₉	—	2	1	no	—
73	#117×C₈₉	—	2	1	no	—
74	C₂×C₁₆×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#106×C₈₉	—	2	1	no	—
82	#106×C₈₉	—	2	1	no	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	C₂×Dic₁₇₈	—	2	2	no	—
86	C₂×Dic₁₇₈	—	4	2	no	—
87	C₂×Dic₁₇₈	—	4	2	no	—
88	C₂×Dic₁₇₈	—	4	1	no	—
89	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
90	C₂×D₃₅₆	D₈	1	1	yes	characteristic
91	C₂×C₂²⋊D₈₉	D₈	1	2	yes	—
92	C₂²×Dic₈₉	—	4	1	no	—
93	C₂²×Dic₈₉	—	4	1	no	—
94	C₂×C₄×D₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
96	Dic₃₅₆	—	2	2	no	—
97	Dic₃₅₆	—	2	2	no	—
98	Dic₃₅₆	—	4	2	no	—
99	#13083558	—	2	2	no	—
100	#13083557	—	2	4	no	—
101	C₂×#10623494	C₂×D₄	1	2	yes	—
102	C₈×D₈₉	—	2	2	no	—
103	C₈×D₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	4	1	no	—
106	C₄⋊Dic₈₉	—	4	2	no	—
107	C₄⋊Dic₈₉	—	4	2	no	—
108	#13083553	—	4	2	no	—
109	#13083551	—	4	2	no	—
110	#13083551	—	4	2	no	—
111	#13083551	—	4	2	no	—
112	C₄×Dic₈₉	—	4	2	no	—
113	#13083549	—	2	2	no	—
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	#41	1	2	yes	—
117	C₂×C₈×C₈₉	#41	1	2	yes	—
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₂²⋊C₄×C₈₉	—	4	1	no	—
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	—	4	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₁₆×C₈₉	—	2	1	no	—
129	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂²×D₈₉	C₂×D₈	1	2	yes	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	#123	1	2	yes	—
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	#137	1	2	yes	—
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₈₉	—	2	4	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	8	2	no	—
143	C₂×Dic₈₉	—	16	1	no	—
144	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
145	D₃₅₆	—	2	2	no	—
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	16	2	no	—
149	Dic₁₇₈	—	2	2	no	—
150	Dic₁₇₈	—	4	2	no	—
151	Dic₁₇₈	—	4	2	no	—
152	Dic₁₇₈	—	8	2	no	—
153	C₈×C₈₉	#123	1	2	yes	—
154	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
155	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
156	#10623494	—	2	2	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	—	2	1	no	—
159	C₂²×C₈₉	#480	1	2	yes	—
160	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
161	C₂×D₈₉	—	2	2	no	—
162	C₂×D₈₉	—	2	2	no	—
163	C₂×D₈₉	—	2	2	no	—
164	C₄×C₈₉	—	8	1	no	—
165	C₄×C₈₉	—	8	1	no	—
166	C₄×C₈₉	—	8	1	no	—
167	C₄×C₈₉	—	8	1	no	—
168	C₄×C₈₉	#480	1	2	yes	—
169	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
170	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
171	Dic₈₉	—	2	2	no	—
172	Dic₈₉	—	8	2	no	—
173	Dic₈₉	—	8	2	no	—
174	#33651	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	2	2	no	—
176	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
177	C₂×C₈₉	#3416	1	1	yes	characteristic
178	C₂×C₈₉	#3417	1	1	yes	characteristic
179	D₈₉	—	4	2	no	—
180	#3059	—	89	1	no	—
181	#2913	—	89	1	no	—
182	#2237	—	89	2	no	—
183	#2237	—	89	2	no	—
184	#2236	—	89	2	no	—
185	#2235	—	89	2	no	—
186	#2159	—	89	1	no	—
187	#2158	D₈₉	1	1	yes	characteristic, complemented
188	#2141	—	89	1	no	—
189	C₂×#358	—	89	1	no	—
190	#2113	—	89	1	no	—
191	C₈₉	#33651	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₁₆	—	89	1	no	—
193	C₂×#131	—	89	1	no	—
194	#536	—	178	1	no	—
195	#520	—	178	1	no	—
196	C₂×M₃₂	—	89	1	no	—
197	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
198	#470	—	178	2	no	—
199	#467	—	89	2	no	—
200	#467	—	89	2	no	—
201	#428	—	178	2	no	—
202	#428	—	178	1	no	—
203	#427	—	89	2	no	—
204	#427	C₂×D₈₉	1	1	yes	characteristic
205	#427	C₂×D₈₉	1	1	yes	characteristic
206	C₂×#107	—	89	1	no	—
207	C₂×#107	—	89	2	no	—
208	#419	—	178	1	no	—
209	#367	—	89	2	no	—
210	#367	C₂×D₈₉	1	1	yes	characteristic
211	#367	C₂×D₈₉	1	1	yes	characteristic
212	#366	—	89	2	no	—
213	#366	C₂×D₈₉	1	1	yes	characteristic
214	#366	C₂×D₈₉	1	1	yes	characteristic
215	#358	—	89	2	no	—
216	#358	—	89	2	no	—
217	#358	—	89	2	no	—
218	#358	—	89	2	no	—
219	C₂²⋊C₁₆	—	89	2	no	—
220	C₂²⋊C₁₆	—	89	2	no	—
221	C₂×#41	—	89	1	no	—
222	C₂²×Q₈	—	178	1	no	—
223	C₂×Q₁₆	—	89	2	no	—
224	C₂×Q₁₆	—	178	1	no	—
225	C₂×Q₁₆	—	178	2	no	—
226	C₂×Q₁₆	—	178	2	no	—
227	#131	—	178	2	no	—
228	#131	—	178	2	no	—
229	C₂×M₁₆	—	89	1	no	—
230	C₂×M₁₆	—	89	2	no	—
231	C₂²×C₈	—	89	2	no	—
232	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₄⋊Q₈	—	356	1	no	—
234	#124	—	356	1	no	—
235	#123	—	356	2	no	—
236	C₂²⋊Q₈	—	178	2	no	—
237	C₂²⋊Q₈	—	178	2	no	—
238	C₄×Q₈	—	356	1	no	—
239	C₄×D₄	—	356	1	no	—
240	C₄×D₄	—	356	2	no	—
241	#117	—	2	1	no	—
242	#117	—	2	1	no	—
243	#117	—	178	2	no	—
244	C₂×C₄⋊C₄	—	178	2	no	—
245	C₂×C₄⋊C₄	—	178	1	no	—
246	M₃₂	—	178	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	89	2	no	—
249	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
251	#107	—	89	2	no	—
252	#107	—	178	2	no	—
253	#107	—	178	1	no	—
254	#107	C₂²×D₈₉	1	1	yes	characteristic
255	#107	C₂²×D₈₉	1	1	yes	characteristic
256	#107	C₂²×D₈₉	1	1	yes	characteristic
257	#107	C₂²×D₈₉	1	1	yes	characteristic
258	#106	—	2	1	no	—
259	#106	—	2	1	no	—
260	#106	—	178	2	no	—
261	#103	—	178	2	no	—
262	#103	—	178	2	no	—
263	#41	—	178	2	no	—
264	#41	—	178	2	no	—
265	C₂×Q₈	—	89	1	no	—
266	C₂×Q₈	—	356	1	no	—
267	C₂×Q₈	—	356	2	no	—
268	C₂×Q₈	—	356	2	no	—
269	C₂×Q₈	—	178	2	no	—
270	C₂×D₄	—	89	2	no	—
271	C₂×D₄	—	89	1	no	—
272	C₂²×C₄	—	89	2	no	—
273	C₂²×C₄	—	356	2	no	—
274	C₂²×C₄	—	356	1	no	—
275	C₂²×C₄	—	356	1	no	—
276	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
277	Q₁₆	—	356	2	no	—
278	Q₁₆	—	178	2	no	—
279	Q₁₆	—	178	2	no	—
280	M₁₆	—	178	2	no	—
281	M₁₆	—	178	4	no	—
282	C₂×C₈	—	89	2	no	—
283	C₂×C₈	—	178	2	no	—
284	C₂×C₈	—	178	2	no	—
285	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
286	C₂×C₈	#13083574	1	2	yes	—
287	C₂×C₈	#13083575	1	2	yes	—
288	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	356	2	no	—
294	C₄⋊C₄	—	356	2	no	—
295	C₄⋊C₄	—	356	2	no	—
296	C₄⋊C₄	—	356	1	no	—
297	C₄⋊C₄	—	178	2	no	—
298	C₂²⋊C₄	—	4	1	no	—
299	C₂²⋊C₄	—	4	1	no	—
300	C₂²⋊C₄	—	356	2	no	—
301	C₂²⋊C₄	—	356	2	no	—
302	C₂²⋊C₄	—	356	2	no	—
303	C₄²	—	4	1	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	356	2	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	178	2	no	—
309	C₂³	—	89	2	no	—
310	C₂³	D₈×D₈₉	1	1	yes	characteristic
311	Q₈	—	356	2	no	—
312	Q₈	—	356	2	no	—
313	Q₈	—	178	2	no	—
314	Q₈	—	712	2	no	—
315	D₄	—	178	4	no	—
316	D₄	—	178	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	89	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	356	2	no	—
324	C₂×C₄	—	1424	2	no	—
325	C₂×C₄	—	1424	1	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	712	2	no	—
329	C₂×C₄	#15667997	1	2	yes	—
330	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
331	C₂×C₄	#15668044	1	2	yes	—
332	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
333	C₈	—	178	2	no	—
334	C₈	#15667997	1	2	yes	—
335	C₈	C₂×#13083574	1	1	yes	characteristic
336	C₈	C₂×#13083575	1	1	yes	characteristic
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	178	2	no	—
340	C₂²	—	178	2	no	—
341	C₂²	—	178	2	no	—
342	C₂²	#19151913	1	2	yes	—
343	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
344	C₄	—	8	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	178	2	no	—
349	C₄	—	712	2	no	—
350	C₄	—	712	2	no	—
351	C₄	#19151913	1	2	yes	—
352	C₄	C₂×#15667997	1	1	yes	characteristic
353	C₄	C₂×#15668044	1	1	yes	characteristic
354	C₂	—	2	2	no	—
355	C₂	—	356	2	no	—
356	C₂	C₂×#19151913	1	1	yes	characteristic
357	C₂	#26033395	1	1	yes	characteristic
358	C₂	#26033402	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×#2158	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#13083574	D₄
hypercenter	#2158	D₈₉
hypocenter	C₈₉	#33651
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2158

Complements

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

N	H ≅ G/N	type
3. #26022463	C₂	semidirect
4. #26022462	C₂	semidirect
5. #26022456	C₂	semidirect
6. #26022451	C₂	semidirect
7. #26022041	C₂	semidirect
8. #26021971	C₂	semidirect
9. C₂×#19151403	C₂	semidirect
11. C₈₉×#2158	C₂	semidirect
187. #2158	D₈₉	semidirect
191. C₈₉	#33651	semidirect

Maximal subgroups

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

subgroup	index
#26029908	2
#26028849	2
#26022463	2
#26022462	2
#26022456	2
#26022451	2
#26022041	2
#26021971	2
C₂×#19151403	2
#26021865	2
C₈₉×#2158	2
#33651	89