Subgroups — #36707525

Order 22784.

582 conjugacy classes of subgroups.

The subgroup lattice has 386 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	1	yes	maximal, characteristic, complemented
3	#26022059	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022057	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022054	C₂	1	2	yes	maximal, complemented
6	#26022052	C₂	1	2	yes	maximal, complemented
7	#26022009	C₂	1	2	yes	maximal, complemented
8	#26021864	C₂	1	2	yes	maximal
9	C₄×#15667951	C₂	1	1	yes	maximal, characteristic, complemented
10	#346×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2164	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×#15667951	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19151555	C₄	1	1	yes	characteristic
18	#19151555	C₄	1	1	yes	characteristic
19	#19151554	C₄	1	1	yes	characteristic
20	#19151552	C₂²	1	1	yes	characteristic
21	#19151552	C₂²	1	1	yes	characteristic
22	#19151550	C₄	1	1	yes	characteristic
23	#19151550	C₄	1	1	yes	characteristic
24	M₁₆×Dic₈₉	C₄	1	1	yes	characteristic
25	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
26	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
27	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
28	C₄⋊#13083549	C₂²	1	2	yes	—
29	#19151416	C₂²	1	2	yes	—
30	#19151416	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	#19151377	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	#19151374	C₂²	1	1	yes	characteristic
35	#19151373	C₂²	1	2	yes	—
36	#19151373	C₂²	1	2	yes	—
37	C₄×#13083549	C₂²	1	1	yes	characteristic
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
41	#346×C₈₉	C₂²	1	1	yes	characteristic
42	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
44	#15668020	—	2	4	no	—
45	#15668018	—	2	4	no	—
46	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
48	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
49	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
50	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
51	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
56	#15667977	C₈	1	2	yes	—
57	#15667974	C₈	1	2	yes	—
58	#15667972	C₈	1	2	yes	—
59	#15667972	C₈	1	2	yes	—
60	#15667969	C₂×C₄	1	1	yes	characteristic
61	#15667969	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
64	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
65	#15667951	—	2	4	no	—
66	C₂×#13083549	C₂³	1	2	yes	—
67	C₂×#13083549	C₂³	1	2	yes	—
68	C₁₆×D₈₉	—	2	4	no	—
69	#15667923	C₂×C₄	1	1	yes	characteristic
70	#15667923	C₂×C₄	1	1	yes	characteristic
71	#15667922	C₂×C₄	1	1	yes	characteristic
72	#15667922	C₂×C₄	1	1	yes	characteristic
73	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
74	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
77	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
78	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
79	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
80	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
81	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
82	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
83	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
85	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
86	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
87	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	#13083574	—	2	2	no	—
91	#13083574	—	2	2	no	—
92	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
93	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
94	C₂×C₂²⋊D₈₉	C₂×C₈	1	1	yes	characteristic
95	C₂×C₂²⋊D₈₉	C₂×C₈	1	1	yes	characteristic
96	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
97	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
98	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
99	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
101	#13083558	—	2	2	no	—
102	#13083557	—	2	2	no	—
103	#13083557	—	2	2	no	—
104	#13083557	—	2	2	no	—
105	#13083557	—	2	2	no	—
106	C₂×#10623494	—	2	2	no	—
107	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
108	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
109	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
110	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
111	C₈×D₈₉	—	2	2	no	—
112	C₈×D₈₉	—	2	2	no	—
113	C₈×D₈₉	M₁₆	1	4	yes	—
114	C₈×D₈₉	#41	1	4	yes	—
115	#13083553	C₂×C₈	1	2	yes	—
116	#13083553	C₂×C₈	1	2	yes	—
117	#13083551	C₂×C₈	1	2	yes	—
118	#13083551	C₂×C₈	1	2	yes	—
119	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
120	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
121	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
122	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
124	#13083549	—	2	2	no	—
125	#13083549	—	2	2	no	—
126	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	M₁₆×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
134	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
135	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
136	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
137	C₁₆×C₈₉	—	2	2	no	—
138	C₁₆×C₈₉	—	2	2	no	—
139	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
140	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
141	C₂²×D₈₉	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	—	2	1	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
145	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
146	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
148	C₂²⋊D₈₉	—	2	2	no	—
149	C₂²⋊D₈₉	—	2	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
152	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
153	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
154	C₂×Dic₈₉	C₂²×C₈	1	1	yes	characteristic
155	D₃₅₆	—	2	2	no	—
156	C₄×D₈₉	—	2	1	no	—
157	C₄×D₈₉	—	2	1	no	—
158	C₄×D₈₉	—	4	2	no	—
159	C₄×D₈₉	—	4	2	no	—
160	C₄×D₈₉	#117	1	2	yes	—
161	C₄×D₈₉	C₂×M₁₆	1	2	yes	—
162	Dic₁₇₈	—	2	2	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	C₂×M₁₆	1	2	yes	—
166	C₈×C₈₉	C₂×#41	1	2	yes	—
167	#10623494	—	2	1	no	—
168	#10623494	—	2	1	no	—
169	#10623494	#117	1	2	yes	—
170	#10623494	C₂×M₁₆	1	2	yes	—
171	C₂²×C₈₉	—	2	2	no	—
172	C₂²×C₈₉	—	2	1	no	—
173	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
174	C₂×D₈₉	—	2	2	no	—
175	C₂×D₈₉	—	2	2	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	#431	1	2	yes	—
178	C₄×C₈₉	—	2	2	no	—
179	C₄×C₈₉	—	2	2	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
182	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
183	Dic₈₉	—	2	2	no	—
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	#431	1	2	yes	—
187	#33421	—	89	1	no	maximal, Sylow
188	C₂×C₈₉	—	4	1	no	—
189	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
190	C₂×C₈₉	#3407	1	2	yes	—
191	D₈₉	—	2	2	no	—
192	D₈₉	—	4	1	no	—
193	C₄×#131	—	89	1	no	—
194	#2164	—	89	1	no	—
195	#2164	—	89	1	no	—
196	#2164	—	89	1	no	—
197	#2164	D₈₉	1	1	yes	characteristic, complemented
198	#2163	—	89	2	no	—
199	#2163	—	89	2	no	—
200	#2151	—	89	2	no	—
201	#2113	—	89	2	no	—
202	C₄×M₃₂	—	89	1	no	—
203	C₂×#346	—	89	1	no	—
204	C₈₉	#33421	1	1	yes	characteristic, complemented, Sylow
205	C₂×#131	—	89	1	no	—
206	C₂×#131	—	89	1	no	—
207	C₄×#41	—	89	1	no	—
208	C₂×M₃₂	—	89	2	no	—
209	C₂²×C₁₆	—	89	2	no	—
210	#405	—	89	1	no	—
211	#405	—	89	1	no	—
212	#405	—	89	1	no	—
213	#405	—	89	1	no	—
214	#405	—	89	1	no	—
215	#405	C₂×D₈₉	1	1	yes	characteristic
216	C₄×M₁₆	—	89	1	no	—
217	C₄×M₁₆	—	89	1	no	—
218	C₄×M₁₆	—	89	1	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	C₂×C₄×C₈	—	89	1	no	—
222	C₄⋊C₁₆	—	89	2	no	—
223	C₄⋊C₁₆	—	89	2	no	—
224	C₄⋊C₁₆	—	89	2	no	—
225	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
226	C₂²⋊C₁₆	—	89	2	no	—
227	C₂²⋊C₁₆	—	89	2	no	—
228	C₂²⋊C₁₆	—	89	2	no	—
229	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
230	#346	—	89	2	no	—
231	#346	—	89	2	no	—
232	#346	—	89	1	no	—
233	#346	C₂×D₈₉	1	1	yes	characteristic
234	C₄×C₁₆	—	89	1	no	—
235	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
236	C₂×#41	—	89	1	no	—
237	#131	—	178	4	no	—
238	#131	—	178	4	no	—
239	C₂×M₁₆	—	89	1	no	—
240	C₂×M₁₆	—	89	1	no	—
241	C₂×M₁₆	—	89	1	no	—
242	C₂×M₁₆	—	89	1	no	—
243	C₂×M₁₆	—	89	1	no	—
244	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
245	C₂²×C₈	—	89	1	no	—
246	C₂²×C₈	—	89	1	no	—
247	C₂²×C₈	—	89	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	89	1	no	—
250	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₄×Q₈	—	89	2	no	—
252	C₄×D₄	—	89	2	no	—
253	C₄×D₄	—	89	2	no	—
254	C₄×D₄	—	89	2	no	—
255	#117	—	89	1	no	—
256	#117	—	89	1	no	—
257	#117	C₄×D₈₉	1	1	yes	characteristic
258	C₂×C₄²	—	89	1	no	—
259	C₂×C₄²	—	89	1	no	—
260	C₂×C₄²	—	89	1	no	—
261	M₃₂	—	178	4	no	—
262	C₂×C₁₆	—	89	2	no	—
263	C₂×C₁₆	—	89	2	no	—
264	C₂×C₁₆	—	178	4	no	—
265	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
266	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
267	#97	—	89	1	no	—
268	#97	—	89	1	no	—
269	#97	—	89	1	no	—
270	#97	—	89	1	no	—
271	#97	C₄×D₈₉	1	1	yes	characteristic
272	#97	C₄×D₈₉	1	1	yes	characteristic
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	1	no	—
276	C₄×C₈	—	89	1	no	—
277	C₄×C₈	—	89	1	no	—
278	C₄×C₈	—	89	1	no	—
279	C₄×C₈	—	89	1	no	—
280	C₄×C₈	—	89	1	no	—
281	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
283	#41	—	178	2	no	—
284	#41	—	178	2	no	—
285	C₂×Q₈	—	89	1	no	—
286	C₂×D₄	—	89	1	no	—
287	C₂×D₄	—	89	1	no	—
288	C₂×D₄	—	89	1	no	—
289	C₂²×C₄	—	89	1	no	—
290	C₂²×C₄	—	89	2	no	—
291	C₂²×C₄	—	89	2	no	—
292	C₂²×C₄	—	89	2	no	—
293	C₂²×C₄	—	89	1	no	—
294	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
295	M₁₆	—	2	2	no	—
296	M₁₆	—	178	2	no	—
297	M₁₆	—	178	2	no	—
298	M₁₆	—	178	2	no	—
299	M₁₆	—	178	2	no	—
300	M₁₆	—	178	2	no	—
301	C₂×C₈	—	2	2	no	—
302	C₂×C₈	—	89	1	no	—
303	C₂×C₈	—	89	1	no	—
304	C₂×C₈	—	89	4	no	—
305	C₂×C₈	—	89	4	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	178	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	2	no	—
311	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
314	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
315	C₄⋊C₄	—	89	2	no	—
316	C₄⋊C₄	—	89	2	no	—
317	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
318	C₂²⋊C₄	—	89	2	no	—
319	C₂²⋊C₄	—	89	2	no	—
320	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
321	C₄²	—	89	2	no	—
322	C₄²	—	89	2	no	—
323	C₄²	—	89	2	no	—
324	C₄²	—	89	1	no	—
325	C₄²	—	89	1	no	—
326	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
327	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
328	C₁₆	—	2	2	no	—
329	C₁₆	—	2	2	no	—
330	C₁₆	—	178	2	no	—
331	C₁₆	—	178	2	no	—
332	C₂³	—	89	1	no	—
333	C₂³	—	89	1	no	—
334	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
335	Q₈	—	178	2	no	—
336	D₄	—	178	2	no	—
337	D₄	—	178	2	no	—
338	D₄	—	178	2	no	—
339	C₂×C₄	—	2	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	89	1	no	—
342	C₂×C₄	—	89	2	no	—
343	C₂×C₄	—	89	2	no	—
344	C₂×C₄	—	89	2	no	—
345	C₂×C₄	—	89	2	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	178	1	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
354	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
355	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	89	2	no	—
359	C₈	—	89	2	no	—
360	C₈	—	178	1	no	—
361	C₈	—	178	1	no	—
362	C₈	M₁₆×D₈₉	1	2	yes	—
363	C₈	#41×D₈₉	1	2	yes	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	89	2	no	—
367	C₂²	—	178	1	no	—
368	C₂²	—	178	2	no	—
369	C₂²	—	178	2	no	—
370	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	89	2	no	—
375	C₄	—	178	1	no	—
376	C₄	—	178	2	no	—
377	C₄	—	178	2	no	—
378	C₄	#117×D₈₉	1	1	yes	characteristic
379	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
380	C₂	—	4	1	no	—
381	C₂	—	356	1	no	—
382	C₂	—	178	2	no	—
383	C₂	#431×D₈₉	1	1	yes	characteristic
384	C₂	#26033353	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2164	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2164	D₈₉
hypocenter	C₈₉	#33421
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #26022060	C₂	semidirect
3. #26022059	C₂	semidirect
4. #26022057	C₂	semidirect
5. #26022054	C₂	semidirect
6. #26022052	C₂	semidirect
7. #26022009	C₂	semidirect
9. C₄×#15667951	C₂	semidirect
10. #346×D₈₉	C₂	semidirect
11. C₈₉×#2164	C₂	semidirect
197. #2164	D₈₉	semidirect
204. C₈₉	#33421	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
#26022059	2
#26022057	2
#26022054	2
#26022052	2
#26022009	2
#26021864	2
C₄×#15667951	2
#346×D₈₉	2
C₈₉×#2164	2
#33421	89