Subgroups — #36707541

Order 22784.

590 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028674	C₂	1	1	yes	maximal, characteristic
2	#26022060	C₂	1	2	yes	maximal, complemented
3	#26022057	C₂	1	2	yes	maximal, complemented
4	#26022053	C₂	1	2	yes	maximal, complemented
5	#26022049	C₂	1	2	yes	maximal, complemented
6	#26022007	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151416	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021867	C₂	1	1	yes	maximal, characteristic
9	#26021865	C₂	1	1	yes	maximal, characteristic
10	#26021849	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668020	C₄	1	1	yes	characteristic
13	C₂×#15668018	C₂²	1	1	yes	characteristic
14	#19152066	C₄	1	1	yes	characteristic
15	C₂×#15667952	C₂²	1	1	yes	characteristic
16	C₂²×#13083549	C₂²	1	1	yes	characteristic
17	#19151555	C₄	1	2	yes	—
18	#19151554	C₄	1	1	yes	characteristic
19	#19151552	C₂²	1	2	yes	—
20	#19151550	C₄	1	2	yes	—
21	M₁₆×Dic₈₉	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₈×D₈₉	C₂²	1	2	yes	—
24	#19151416	C₂²	1	1	yes	characteristic
25	#19151416	C₂²	1	1	yes	characteristic
26	#19151416	C₂²	1	1	yes	characteristic
27	#19151416	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	2	yes	—
29	#19151416	C₂²	1	2	yes	—
30	#19151377	C₂²	1	2	yes	—
31	#19151377	C₂²	1	2	yes	—
32	#19151377	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	#19151374	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	2	yes	—
36	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	#346×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
42	#15668020	—	2	4	no	—
43	#15668018	—	2	2	no	—
44	#15668018	—	2	2	no	—
45	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083557	C₂×C₄	1	2	yes	—
47	C₂×#13083557	C₂×C₄	1	2	yes	—
48	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₈×D₈₉	C₂³	1	2	yes	—
51	#15667977	C₈	1	1	yes	characteristic
52	#15667977	C₈	1	1	yes	characteristic
53	#15667974	C₈	1	1	yes	characteristic
54	#15667974	C₈	1	1	yes	characteristic
55	#15667972	C₈	1	2	yes	—
56	#15667972	C₈	1	2	yes	—
57	#15667969	C₂×C₄	1	2	yes	—
58	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₈₉	C₂×C₄	1	2	yes	—
60	#15667952	—	2	2	no	—
61	C₂×#13083549	—	2	2	no	—
62	C₂×#13083549	C₂³	1	1	yes	characteristic
63	C₂×#13083549	C₂³	1	2	yes	—
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	#15667923	C₂×C₄	1	2	yes	—
66	#15667922	C₂×C₄	1	2	yes	—
67	C₄×#10623494	C₂×C₄	1	2	yes	—
68	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
69	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
72	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
76	M₃₂×C₈₉	—	2	2	no	—
77	C₂×C₁₆×C₈₉	—	2	2	no	—
78	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
81	#97×C₈₉	C₂×C₄	1	2	yes	—
82	C₄×C₈×C₈₉	C₂³	1	2	yes	—
83	#13083574	—	2	2	no	—
84	#13083574	—	2	2	no	—
85	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
86	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
88	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
89	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
90	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
91	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
92	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
93	#13083558	—	2	2	no	—
94	#13083557	—	2	4	no	—
95	#13083557	—	2	4	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
98	C₂×#10623494	C₂²×C₄	1	2	yes	—
99	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
100	C₈×D₈₉	—	2	4	no	—
101	C₈×D₈₉	—	2	2	no	—
102	C₈×D₈₉	—	2	2	no	—
103	#13083553	C₂×C₈	1	2	yes	—
104	#13083553	C₂×C₈	1	2	yes	—
105	#13083551	C₂×C₈	1	2	yes	—
106	#13083551	C₂×C₈	1	2	yes	—
107	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
108	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
109	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
111	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
113	#13083549	—	2	1	no	—
114	#13083549	—	2	1	no	—
115	#13083549	—	2	2	no	—
116	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₈₉	M₁₆	1	4	yes	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₈₉	#41	1	2	yes	—
121	C₂×C₈×C₈₉	#41	1	2	yes	—
122	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
124	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
126	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₁₆×C₈₉	—	2	1	no	—
129	C₁₆×C₈₉	—	2	2	no	—
130	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
131	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₈₉	#117	1	2	yes	—
133	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
135	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
137	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₈₉	—	2	4	no	—
139	C₂×Dic₈₉	—	2	2	no	—
140	C₂×Dic₈₉	—	4	1	no	—
141	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
143	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
144	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
145	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
146	D₃₅₆	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	2	2	no	—
149	C₄×D₈₉	—	2	4	no	—
150	C₄×D₈₉	—	4	2	no	—
151	Dic₁₇₈	—	2	2	no	—
152	C₈×C₈₉	#117	1	2	yes	—
153	C₈×C₈₉	C₂×M₁₆	1	4	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	#10623494	—	2	1	no	—
158	#10623494	—	2	1	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	#431	1	2	yes	—
162	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
163	C₂×D₈₉	—	2	2	no	—
164	C₂×D₈₉	—	2	2	no	—
165	C₂×D₈₉	—	2	2	no	—
166	C₄×C₈₉	—	2	2	no	—
167	C₄×C₈₉	—	2	2	no	—
168	C₄×C₈₉	#431	1	2	yes	—
169	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
170	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	2	no	—
175	Dic₈₉	#504	1	2	yes	—
176	#33423	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	2	2	no	—
178	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#504	1	1	yes	characteristic
180	C₂×C₈₉	#3407	1	1	yes	characteristic
181	D₈₉	—	4	2	no	—
182	C₄×#131	—	89	1	no	—
183	#2164	—	89	2	no	—
184	#2164	—	89	2	no	—
185	#2163	—	89	2	no	—
186	#2162	—	89	2	no	—
187	#2150	—	89	1	no	—
188	C₂×C₄⋊C₁₆	—	89	1	no	—
189	#2113	—	89	1	no	—
190	#2113	—	89	1	no	—
191	#2108	—	89	1	no	—
192	#2108	D₈₉	1	1	yes	characteristic, complemented
193	C₈₉	#33423	1	1	yes	characteristic, complemented, Sylow
194	C₂×#131	—	89	1	no	—
195	C₂×#131	—	89	1	no	—
196	C₄×#41	—	89	1	no	—
197	C₂×M₃₂	—	89	1	no	—
198	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
199	C₂²×C₁₆	—	89	1	no	—
200	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
201	#405	—	89	1	no	—
202	#405	—	89	2	no	—
203	#405	—	89	2	no	—
204	#405	C₂×D₈₉	1	1	yes	characteristic
205	C₄×M₁₆	—	89	1	no	—
206	C₄×M₁₆	—	89	2	no	—
207	C₂×C₄×C₈	—	89	1	no	—
208	C₂×C₄×C₈	—	89	2	no	—
209	C₄⋊C₁₆	—	89	1	no	—
210	C₄⋊C₁₆	—	89	1	no	—
211	C₄⋊C₁₆	—	89	1	no	—
212	C₄⋊C₁₆	—	89	1	no	—
213	C₄⋊C₁₆	—	89	2	no	—
214	C₄⋊C₁₆	—	89	2	no	—
215	C₂²⋊C₁₆	—	89	2	no	—
216	C₂²⋊C₁₆	—	89	2	no	—
217	C₂²⋊C₁₆	—	89	2	no	—
218	C₂²⋊C₁₆	—	89	2	no	—
219	#346	—	89	2	no	—
220	#346	C₂×D₈₉	1	2	yes	—
221	C₄×C₁₆	—	89	2	no	—
222	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
223	C₂×#41	—	89	1	no	—
224	#131	—	178	4	no	—
225	#131	—	178	2	no	—
226	#131	—	178	2	no	—
227	C₂×M₁₆	—	89	1	no	—
228	C₂×M₁₆	—	89	2	no	—
229	C₂×M₁₆	—	89	2	no	—
230	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₈	—	89	1	no	—
232	C₂²×C₈	—	89	2	no	—
233	C₂²×C₈	—	89	2	no	—
234	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
235	C₄×Q₈	—	89	1	no	—
236	C₄×Q₈	—	89	1	no	—
237	C₄×D₄	—	89	1	no	—
238	C₄×D₄	—	89	1	no	—
239	C₄×D₄	—	89	2	no	—
240	C₄×D₄	—	89	2	no	—
241	#117	—	89	2	no	—
242	#117	C₄×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	—	89	1	no	—
244	C₂×C₄²	—	89	2	no	—
245	M₃₂	—	2	2	no	—
246	M₃₂	—	178	2	no	—
247	C₂×C₁₆	—	2	2	no	—
248	C₂×C₁₆	—	89	1	no	—
249	C₂×C₁₆	—	89	2	no	—
250	C₂×C₁₆	—	89	1	no	—
251	C₂×C₁₆	—	178	2	no	—
252	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
254	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
255	#97	—	89	2	no	—
256	#97	—	89	2	no	—
257	#97	C₄×D₈₉	1	2	yes	—
258	C₄×C₈	—	89	1	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	—	89	2	no	—
264	C₄×C₈	C₂²×D₈₉	1	2	yes	—
265	#41	—	178	2	no	—
266	#41	—	178	2	no	—
267	C₂×Q₈	—	89	1	no	—
268	C₂×D₄	—	89	2	no	—
269	C₂×D₄	—	89	1	no	—
270	C₂²×C₄	—	89	1	no	—
271	C₂²×C₄	—	89	1	no	—
272	C₂²×C₄	—	89	2	no	—
273	C₂²×C₄	—	89	2	no	—
274	C₂²×C₄	—	89	2	no	—
275	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
276	M₁₆	—	178	2	no	—
277	M₁₆	—	178	4	no	—
278	M₁₆	—	178	4	no	—
279	M₁₆	#13083557	1	4	yes	—
280	C₂×C₈	—	89	1	no	—
281	C₂×C₈	—	89	2	no	—
282	C₂×C₈	—	89	1	no	—
283	C₂×C₈	—	178	2	no	—
284	C₂×C₈	—	178	4	no	—
285	C₂×C₈	—	178	2	no	—
286	C₂×C₈	—	178	2	no	—
287	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
289	C₂×C₈	#13083574	1	2	yes	—
290	C₂×C₈	#13083574	1	2	yes	—
291	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
292	C₄⋊C₄	—	89	2	no	—
293	C₄⋊C₄	—	89	2	no	—
294	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
295	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
296	C₂²⋊C₄	—	89	2	no	—
297	C₂²⋊C₄	—	89	2	no	—
298	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
299	C₄²	—	89	1	no	—
300	C₄²	—	89	2	no	—
301	C₄²	—	89	1	no	—
302	C₄²	—	89	2	no	—
303	C₄²	—	89	1	no	—
304	C₄²	—	89	1	no	—
305	C₄²	C₂×C₄×D₈₉	1	2	yes	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	1	no	—
308	C₁₆	—	2	2	no	—
309	C₁₆	—	178	1	no	—
310	C₁₆	—	178	1	no	—
311	C₁₆	—	178	2	no	—
312	C₂³	—	89	2	no	—
313	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
314	Q₈	—	178	2	no	—
315	D₄	—	178	4	no	—
316	D₄	—	178	2	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	89	1	no	—
319	C₂×C₄	—	89	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	89	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	356	1	no	—
324	C₂×C₄	—	178	4	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	#15667969	1	2	yes	—
329	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
330	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
331	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂×#13083557	1	2	yes	—
333	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
334	C₈	—	178	2	no	—
335	C₈	—	178	1	no	—
336	C₈	—	178	1	no	—
337	C₈	#15667969	1	2	yes	—
338	C₈	C₂×#13083557	1	4	yes	—
339	C₈	C₂×#13083574	1	1	yes	characteristic
340	C₈	C₂×#13083574	1	1	yes	characteristic
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	178	2	no	—
344	C₂²	—	178	2	no	—
345	C₂²	—	178	2	no	—
346	C₂²	#19151649	1	2	yes	—
347	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	89	2	no	—
351	C₄	—	178	2	no	—
352	C₄	—	178	1	no	—
353	C₄	—	178	1	no	—
354	C₄	—	178	1	no	—
355	C₄	#19151649	1	2	yes	—
356	C₄	C₂×#15667969	1	1	yes	characteristic
357	C₄	C₂²×#13083557	1	1	yes	characteristic
358	C₂	—	2	2	no	—
359	C₂	—	356	2	no	—
360	C₂	C₂×#19151649	1	1	yes	characteristic
361	C₂	#504×D₈₉	1	1	yes	characteristic
362	C₂	#26033353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2108	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2108	D₈₉
hypocenter	C₈₉	#33423
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2108

Complements

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

N	H ≅ G/N	type
2. #26022060	C₂	semidirect
3. #26022057	C₂	semidirect
4. #26022053	C₂	semidirect
5. #26022049	C₂	semidirect
6. #26022007	C₂	semidirect
7. C₂×#19151416	C₂	semidirect
10. #26021849	C₂	semidirect
11. C₈₉×#2108	C₂	semidirect
192. #2108	D₈₉	semidirect
193. C₈₉	#33423	semidirect

Maximal subgroups

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

subgroup	index
#26028674	2
#26022060	2
#26022057	2
#26022053	2
#26022049	2
#26022007	2
C₂×#19151416	2
#26021867	2
#26021865	2
#26021849	2
C₈₉×#2108	2
#33423	89