Subgroups — #36537573

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031373	C₂	1	1	yes	maximal, characteristic
2	#26031177	C₂	1	1	yes	maximal, characteristic
3	#26028965	C₂	1	1	yes	maximal, characteristic
4	#26019796	C₂	1	4	yes	maximal, complemented
5	#26019672	C₂	1	4	yes	maximal, complemented
6	#26019000	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018937	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018924	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	—	2	1	no	—
11	#19152148	C₂²	1	1	yes	characteristic
12	#19152001	—	2	1	no	—
13	#19151972	—	2	1	no	—
14	#19151948	—	2	1	no	—
15	#19151946	—	2	1	no	—
16	#19151911	—	2	2	no	—
17	#19151875	C₂²	1	4	yes	—
18	#19151800	C₂²	1	4	yes	—
19	#19151737	—	2	2	no	—
20	#19151677	—	2	1	no	—
21	#19151677	C₂²	1	2	yes	—
22	C₈×D₃₅₆	—	2	1	no	—
23	C₈×D₃₅₆	C₂²	1	2	yes	—
24	#19151651	C₂²	1	1	yes	characteristic
25	#19151651	C₂²	1	1	yes	characteristic
26	#19151622	C₂²	1	1	yes	characteristic
27	#19151564	—	2	1	no	—
28	#19151351	C₂²	1	2	yes	—
29	#19151349	C₂²	1	2	yes	—
30	#19151344	C₂²	1	4	yes	—
31	#19151344	C₂²	1	4	yes	—
32	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
34	#487×C₈₉	—	2	1	no	—
35	#487×C₈₉	—	2	1	no	—
36	#487×C₈₉	C₂²	1	2	yes	—
37	#487×C₈₉	C₂²	1	2	yes	—
38	C₂×#103×C₈₉	—	2	1	no	—
39	C₂×#103×C₈₉	—	2	1	no	—
40	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	—	2	1	no	—
42	C₂×#13083561	—	2	4	no	—
43	#15668018	—	4	4	no	—
44	C₂×#13083558	—	2	1	no	—
45	C₂×#13083557	—	2	2	no	—
46	C₂×C₈×D₈₉	—	2	2	no	—
47	#15668012	D₄	1	1	yes	characteristic
48	#15668009	D₄	1	1	yes	characteristic
49	#15668000	—	2	1	no	—
50	#15667986	—	2	1	no	—
51	#15667986	D₄	1	2	yes	—
52	C₄×D₃₅₆	—	2	1	no	—
53	C₄×D₃₅₆	C₂³	1	2	yes	—
54	#15667945	—	2	4	no	—
55	C₄⋊#10623494	—	2	1	no	—
56	C₄⋊#10623494	C₂³	1	2	yes	—
57	C₄⋊#10623494	C₂³	1	2	yes	—
58	C₄⋊#10623494	C₂³	1	2	yes	—
59	#15667936	—	2	4	no	—
60	#15667932	—	2	4	no	—
61	C₂²⋊#10623494	—	2	1	no	—
62	C₂²⋊#10623494	—	2	1	no	—
63	#15667924	—	2	2	no	—
64	#15667924	—	2	2	no	—
65	C₄×#10623494	—	2	1	no	—
66	C₄×#10623494	—	2	1	no	—
67	C₂²×Q₈×C₈₉	—	2	1	no	—
68	C₂²×Q₈×C₈₉	—	2	1	no	—
69	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄⋊Q₈×C₈₉	—	2	1	no	—
73	C₄⋊Q₈×C₈₉	—	2	1	no	—
74	C₄⋊Q₈×C₈₉	—	2	1	no	—
75	C₄⋊Q₈×C₈₉	—	2	1	no	—
76	C₄⋊Q₈×C₈₉	—	2	1	no	—
77	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
78	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
84	#103×C₈₉	—	2	2	no	—
85	#103×C₈₉	—	2	2	no	—
86	#103×C₈₉	—	2	2	no	—
87	#103×C₈₉	—	2	2	no	—
88	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
89	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	#13083574	—	4	2	no	—
93	#13083574	—	4	2	no	—
94	C₂×Dic₁₇₈	—	2	1	no	—
95	C₂×D₃₅₆	—	2	1	no	—
96	C₂×D₃₅₆	—	2	1	no	—
97	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
98	C₂×C₂²⋊D₈₉	—	2	2	no	—
99	C₂×C₄×D₈₉	—	2	2	no	—
100	#13083561	—	8	4	no	—
101	#13083558	—	4	2	no	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₂×#10623494	—	2	2	no	—
106	C₈×D₈₉	—	4	4	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
109	#13083551	—	2	2	no	—
110	#13083551	—	2	2	no	—
111	C₂×Q₈×C₈₉	—	2	2	no	—
112	C₂×Q₈×C₈₉	—	2	2	no	—
113	C₂×Q₈×C₈₉	—	4	1	no	—
114	C₂×Q₈×C₈₉	—	4	1	no	—
115	C₂×Q₈×C₈₉	—	4	2	no	—
116	C₂×Q₈×C₈₉	—	4	2	no	—
117	C₂×Q₈×C₈₉	—	8	2	no	—
118	C₂²×C₄×C₈₉	—	4	1	no	—
119	C₂²×C₄×C₈₉	—	4	1	no	—
120	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
123	C₂×C₈×C₈₉	—	2	2	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	—	2	2	no	—
132	C₄⋊C₄×C₈₉	—	2	2	no	—
133	C₄⋊C₄×C₈₉	—	4	1	no	—
134	C₄⋊C₄×C₈₉	—	4	1	no	—
135	C₄⋊C₄×C₈₉	—	4	1	no	—
136	C₄⋊C₄×C₈₉	—	4	1	no	—
137	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₈₉	#41	1	1	yes	characteristic
140	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
141	C₂³×C₈₉	#121	1	1	yes	characteristic
142	C₂²×D₈₉	—	2	2	no	—
143	Q₈×C₈₉	—	4	2	no	—
144	Q₈×C₈₉	—	4	2	no	—
145	Q₈×C₈₉	—	4	4	no	—
146	Q₈×C₈₉	—	8	2	no	—
147	Q₈×C₈₉	—	8	2	no	—
148	C₂×C₄×C₈₉	—	2	2	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	4	2	no	—
151	C₂×C₄×C₈₉	—	4	2	no	—
152	C₂×C₄×C₈₉	—	8	2	no	—
153	C₂×C₄×C₈₉	—	8	2	no	—
154	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
155	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
156	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
157	C₂×C₄×C₈₉	#135	1	2	yes	—
158	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
159	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
162	C₂²⋊D₈₉	—	4	4	no	—
163	C₂×Dic₈₉	—	2	2	no	—
164	D₃₅₆	—	2	4	no	—
165	D₃₅₆	—	4	2	no	—
166	D₃₅₆	—	4	2	no	—
167	C₄×D₈₉	—	4	2	no	—
168	C₄×D₈₉	—	4	2	no	—
169	Dic₁₇₈	—	4	2	no	—
170	C₈×C₈₉	—	2	1	no	—
171	C₈×C₈₉	—	2	1	no	—
172	C₈×C₈₉	—	2	1	no	—
173	C₈×C₈₉	—	2	1	no	—
174	#10623494	—	4	2	no	—
175	#10623494	—	4	2	no	—
176	#10623494	—	4	2	no	—
177	C₂²×C₈₉	—	2	1	no	—
178	C₂²×C₈₉	—	2	1	no	—
179	C₂²×C₈₉	#463	1	2	yes	—
180	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
181	C₂×D₈₉	—	4	2	no	—
182	C₂×D₈₉	—	4	2	no	—
183	C₂×D₈₉	—	4	2	no	—
184	C₄×C₈₉	—	2	1	no	—
185	C₄×C₈₉	—	2	1	no	—
186	C₄×C₈₉	—	8	2	no	—
187	C₄×C₈₉	—	8	2	no	—
188	C₄×C₈₉	#463	1	2	yes	—
189	C₄×C₈₉	#473	1	2	yes	—
190	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
191	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₄×C₈₉	#577	1	2	yes	—
193	Dic₈₉	—	4	2	no	—
194	#19197	—	89	1	no	maximal, Sylow
195	C₂×C₈₉	—	2	2	no	—
196	C₂×C₈₉	C₂×#463	1	1	yes	characteristic
197	C₂×C₈₉	#3057	1	1	yes	characteristic
198	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
199	D₈₉	—	8	2	no	—
200	#3226	—	89	1	no	—
201	#3206	—	89	1	no	—
202	C₂×#487	D₈₉	1	1	yes	characteristic, complemented
203	#2931	—	89	1	no	—
204	#1699	—	89	4	no	—
205	#1679	—	89	4	no	—
206	#1564	—	89	1	no	—
207	#1549	—	89	1	no	—
208	#1546	—	89	1	no	—
209	C₈₉	#19197	1	1	yes	characteristic, complemented, Sylow
210	C₂×#131	—	178	1	no	—
211	#532	—	89	1	no	—
212	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
213	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
214	#500	—	178	1	no	—
215	#492	—	178	1	no	—
216	#487	—	2	1	no	—
217	#487	—	2	1	no	—
218	#487	—	178	1	no	—
219	#487	C₂×D₈₉	1	2	yes	—
220	#487	C₂×D₈₉	1	2	yes	—
221	#486	—	178	1	no	—
222	#480	—	178	2	no	—
223	#474	—	89	4	no	—
224	#461	—	89	4	no	—
225	#448	—	178	2	no	—
226	#436	—	89	2	no	—
227	#436	—	178	1	no	—
228	C₈×D₄	—	89	2	no	—
229	C₈×D₄	—	178	1	no	—
230	#431	—	89	1	no	—
231	#431	—	89	1	no	—
232	#423	—	89	1	no	—
233	C₂×#103	—	2	1	no	—
234	C₂×#103	—	2	1	no	—
235	#408	—	178	1	no	—
236	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
237	#335	—	89	2	no	—
238	#334	—	89	2	no	—
239	#332	—	89	4	no	—
240	#332	—	89	4	no	—
241	C₂×#41	—	178	1	no	—
242	C₂²×Q₈	—	2	1	no	—
243	C₂²×Q₈	—	2	1	no	—
244	C₂×SD₁₆	—	178	4	no	—
245	#131	—	356	4	no	—
246	C₂×M₁₆	—	178	1	no	—
247	C₂×M₁₆	—	178	2	no	—
248	C₂²×C₈	—	178	2	no	—
249	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
250	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	—	2	1	no	—
256	C₄⋊Q₈	—	2	1	no	—
257	C₄⋊Q₈	—	89	1	no	—
258	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
259	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
260	#127	—	89	1	no	—
261	#124	—	178	1	no	—
262	#121	—	89	2	no	—
263	#121	—	178	1	no	—
264	C₄×D₄	—	89	2	no	—
265	C₄×D₄	—	178	1	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	1	no	—
270	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
271	#107	—	178	4	no	—
272	C₄⋊C₈	—	89	2	no	—
273	C₄⋊C₈	—	89	2	no	—
274	C₄⋊C₈	—	89	2	no	—
275	C₄⋊C₈	—	178	1	no	—
276	#103	—	2	2	no	—
277	#103	—	2	2	no	—
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#103	—	178	4	no	—
281	#102	—	178	4	no	—
282	C₂²⋊C₈	—	178	1	no	—
283	C₂²⋊C₈	—	178	1	no	—
284	C₂²⋊C₈	—	178	2	no	—
285	C₂²⋊C₈	—	178	2	no	—
286	C₄×C₈	—	178	1	no	—
287	C₄×C₈	—	178	1	no	—
288	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
289	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
291	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
292	#41	—	356	2	no	—
293	#41	—	356	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	4	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	—	8	2	no	—
301	C₂×Q₈	—	178	1	no	—
302	C₂×D₄	—	89	2	no	—
303	C₂×D₄	—	178	2	no	—
304	C₂×D₄	—	178	1	no	—
305	C₂×D₄	—	178	1	no	—
306	C₂²×C₄	—	4	1	no	—
307	C₂²×C₄	—	4	1	no	—
308	C₂²×C₄	—	178	2	no	—
309	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
311	C₂²×C₄	#13083573	1	1	yes	characteristic
312	SD₁₆	—	712	4	no	—
313	M₁₆	—	356	4	no	—
314	M₁₆	—	356	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	356	4	no	—
318	C₂×C₈	—	178	2	no	—
319	C₂×C₈	—	178	2	no	—
320	C₂×C₈	—	178	2	no	—
321	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
322	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
323	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
324	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	4	1	no	—
330	C₄⋊C₄	—	4	1	no	—
331	C₄⋊C₄	—	4	1	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₄⋊C₄	—	89	2	no	—
334	C₄⋊C₄	—	178	1	no	—
335	C₂²⋊C₄	—	178	2	no	—
336	C₂²⋊C₄	—	178	2	no	—
337	C₄²	D₄×D₈₉	1	1	yes	characteristic
338	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
339	C₄²	#13083573	1	1	yes	characteristic
340	C₄²	C₂³×D₈₉	1	1	yes	characteristic
341	C₂³	—	178	2	no	—
342	C₂³	#15667983	1	1	yes	characteristic
343	Q₈	—	4	2	no	—
344	Q₈	—	4	4	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	8	2	no	—
347	Q₈	—	8	2	no	—
348	Q₈	—	356	2	no	—
349	D₄	—	356	4	no	—
350	D₄	—	356	2	no	—
351	D₄	—	356	2	no	—
352	D₄	—	178	4	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	4	2	no	—
357	C₂×C₄	—	8	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	356	2	no	—
360	C₂×C₄	—	356	2	no	—
361	C₂×C₄	—	178	2	no	—
362	C₂×C₄	#15667983	1	1	yes	characteristic
363	C₂×C₄	#15667983	1	1	yes	characteristic
364	C₂×C₄	#15667983	1	1	yes	characteristic
365	C₂×C₄	#15668032	1	2	yes	—
366	C₂×C₄	#15668037	1	2	yes	—
367	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
368	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
369	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	2	1	no	—
373	C₈	—	2	1	no	—
374	C₈	—	356	2	no	—
375	C₈	—	356	2	no	—
376	C₈	—	356	2	no	—
377	C₂²	—	2	1	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	356	2	no	—
380	C₂²	—	356	2	no	—
381	C₂²	—	356	2	no	—
382	C₂²	#19151810	1	2	yes	—
383	C₂²	C₂×#15667983	1	1	yes	characteristic
384	C₄	—	2	1	no	—
385	C₄	—	2	1	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	—	356	2	no	—
389	C₄	#19151810	1	2	yes	—
390	C₄	#19151870	1	2	yes	—
391	C₄	C₂×#15668032	1	1	yes	characteristic
392	C₄	C₂×#15668037	1	1	yes	characteristic
393	C₄	#19152404	1	2	yes	—
394	C₂	—	2	2	no	—
395	C₂	—	712	2	no	—
396	C₂	C₂×#19151810	1	1	yes	characteristic
397	C₂	#26029887	1	1	yes	characteristic
398	C₂	C₂×#19151870	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#487×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#121
prime radical	#19152148	C₂²
hypercenter	C₂×#487	D₈₉
hypocenter	C₈₉	#19197
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
4. #26019796	C₂	semidirect
5. #26019672	C₂	semidirect
6. #26019000	C₂	semidirect
7. #26018937	C₂	semidirect
8. #26018924	C₂	semidirect
9. C₂×#487×C₈₉	C₂	semidirect
202. C₂×#487	D₈₉	semidirect
209. C₈₉	#19197	semidirect

Maximal subgroups

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

subgroup	index
#26031373	2
#26031177	2
#26028965	2
#26019796	2
#26019672	2
#26019000	2
#26018937	2
#26018924	2
C₂×#487×C₈₉	2
#19197	89