Subgroups — #36524972

Order 22784.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030825	C₂	1	1	yes	maximal, characteristic
2	#26029229	C₂	1	2	yes	maximal, complemented
3	#26019149	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019139	C₂	1	2	yes	maximal, complemented
5	#26019136	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×#13083561	C₂	1	2	yes	maximal, complemented
7	C₈×D₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019014	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018711	C₂	1	2	yes	maximal, complemented
10	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152135	C₂²	1	2	yes	—
13	#19151999	C₄	1	1	yes	characteristic
14	#19151983	C₄	1	1	yes	characteristic
15	#19151954	C₄	1	2	yes	—
16	#19151836	C₄	1	1	yes	characteristic
17	#19151835	C₄	1	1	yes	characteristic
18	#19151830	C₄	1	2	yes	—
19	#19151729	C₂²	1	2	yes	—
20	C₄×#13083561	C₂²	1	2	yes	—
21	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
22	#19151675	C₂²	1	2	yes	—
23	#19151668	—	2	2	no	—
24	C₈×D₃₅₆	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	2	yes	—
26	#19151658	—	2	2	no	—
27	#19151654	C₂²	1	2	yes	—
28	#19151552	—	2	2	no	—
29	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
32	#19151351	C₂²	1	1	yes	characteristic
33	#19151348	C₂²	1	2	yes	—
34	Q₈⋊#10623494	C₂²	1	2	yes	—
35	#19151327	C₂²	1	2	yes	—
36	#19151325	C₂²	1	2	yes	—
37	C₈×#10623494	C₂²	1	1	yes	characteristic
38	C₈×#10623494	C₂²	1	1	yes	characteristic
39	C₈×#10623494	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#335×C₈₉	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₉	C₂²	1	2	yes	—
44	C₈²×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083561	C₂×C₄	1	2	yes	—
46	C₂×D₇₁₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#13083557	—	2	2	no	—
48	C₂×C₈×D₈₉	—	2	1	no	—
49	C₂×C₈×D₈₉	—	2	2	no	—
50	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
53	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
54	#15668007	—	2	2	no	—
55	#15668003	C₂×C₄	1	2	yes	—
56	#15668001	C₂×C₄	1	2	yes	—
57	#15667997	—	2	2	no	—
58	#15667988	C₂×C₄	1	2	yes	—
59	#15667987	C₂×C₄	1	2	yes	—
60	C₄×D₃₅₆	—	2	2	no	—
61	C₄×D₃₅₆	C₂³	1	2	yes	—
62	#15667969	—	2	2	no	—
63	C₄²×D₈₉	C₂³	1	1	yes	characteristic
64	#15667946	C₂×C₄	1	1	yes	characteristic
65	#15667943	C₂×C₄	1	2	yes	—
66	C₄⋊#10623494	—	2	2	no	—
67	C₄⋊#10623494	C₂³	1	2	yes	—
68	#15667936	C₂×C₄	1	2	yes	—
69	#15667934	C₂×C₄	1	2	yes	—
70	#15667932	C₂×C₄	1	2	yes	—
71	#15667924	—	2	2	no	—
72	#15667924	—	2	2	no	—
73	#15667924	—	2	2	no	—
74	#15667923	—	2	2	no	—
75	#15667922	—	2	2	no	—
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₄×#10623494	C₂³	1	1	yes	characteristic
78	C₄×#10623494	C₂³	1	1	yes	characteristic
79	C₈×Dic₈₉	—	2	1	no	—
80	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₈₉	D₄	1	1	yes	characteristic
83	C₈×Dic₈₉	D₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
86	#107×C₈₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
89	#103×C₈₉	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₈₉	—	2	2	no	—
91	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×D₃₅₆	—	2	2	no	—
95	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₈₉	—	2	2	no	—
97	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₈₉	#41	1	1	yes	characteristic
100	#13083561	—	2	4	no	—
101	D₇₁₂	—	2	2	no	—
102	#13083557	—	4	4	no	—
103	C₂×#10623494	—	2	1	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
106	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
109	C₈×D₈₉	—	2	4	no	—
110	C₈×D₈₉	—	2	2	no	—
111	C₈×D₈₉	—	2	2	no	—
112	C₈×D₈₉	—	2	2	no	—
113	C₈×D₈₉	—	2	2	no	—
114	C₈×D₈₉	—	4	4	no	—
115	C₄⋊Dic₈₉	—	2	2	no	—
116	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
117	#13083553	—	2	2	no	—
118	#13083553	—	2	2	no	—
119	#13083551	—	2	2	no	—
120	#13083551	—	2	2	no	—
121	#13083551	—	2	2	no	—
122	#13083551	—	2	2	no	—
123	#13083551	—	2	2	no	—
124	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₈₉	#41	1	1	yes	characteristic
127	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
128	Q₁₆×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	2	no	—
131	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₈₉	—	2	2	no	—
136	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
137	C₄²×C₈₉	—	2	2	no	—
138	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₂²×D₈₉	—	2	2	no	—
140	C₂²×D₈₉	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₈₉	—	2	2	no	—
142	Q₈×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₈₉	—	2	2	no	—
149	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₈₉	C₄×D₄	1	1	yes	characteristic
152	D₃₅₆	—	2	2	no	—
153	D₃₅₆	—	2	2	no	—
154	D₃₅₆	—	4	4	no	—
155	C₄×D₈₉	—	2	2	no	—
156	C₄×D₈₉	—	2	1	no	—
157	C₄×D₈₉	—	2	1	no	—
158	C₄×D₈₉	—	2	1	no	—
159	C₄×D₈₉	—	2	1	no	—
160	C₄×D₈₉	—	4	2	no	—
161	C₄×D₈₉	—	4	2	no	—
162	C₈×C₈₉	—	2	2	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	2	1	no	—
165	C₈×C₈₉	—	4	2	no	—
166	C₈×C₈₉	#131	1	2	yes	—
167	C₈×C₈₉	#135	1	2	yes	—
168	#10623494	—	2	2	no	—
169	#10623494	—	2	1	no	—
170	#10623494	—	2	1	no	—
171	#10623494	—	4	2	no	—
172	#10623494	#131	1	2	yes	—
173	#10623494	#135	1	2	yes	—
174	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₈₉	—	2	1	no	—
176	C₂×D₈₉	—	2	1	no	—
177	C₂×D₈₉	—	4	2	no	—
178	C₂×D₈₉	—	4	2	no	—
179	C₂×D₈₉	—	4	2	no	—
180	C₂×D₈₉	#443	1	2	yes	—
181	C₄×C₈₉	—	2	1	no	—
182	C₄×C₈₉	—	4	2	no	—
183	C₄×C₈₉	—	4	2	no	—
184	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
185	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
186	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
187	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
188	Dic₈₉	—	2	1	no	—
189	Dic₈₉	—	2	1	no	—
190	Dic₈₉	—	2	1	no	—
191	Dic₈₉	—	4	2	no	—
192	#17959	—	89	1	no	maximal, Sylow
193	C₂×C₈₉	#2931	1	1	yes	characteristic
194	C₂×C₈₉	C₄×#135	1	1	yes	characteristic
195	C₂×C₈₉	C₂×#443	1	1	yes	characteristic
196	D₈₉	—	2	2	no	—
197	D₈₉	—	8	2	no	—
198	#3168	—	89	1	no	—
199	#2966	—	89	2	no	—
200	#1591	—	89	1	no	—
201	#1590	—	89	2	no	—
202	#1589	—	89	1	no	—
203	C₈×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
204	C₈×SD₁₆	—	89	2	no	—
205	C₈×D₈	—	89	1	no	—
206	#1568	—	89	1	no	—
207	#1494	—	89	2	no	—
208	C₂×C₈²	—	89	1	no	—
209	C₈₉	#17959	1	1	yes	characteristic, complemented, Sylow
210	#529	—	89	2	no	—
211	#499	—	89	1	no	—
212	#495	—	89	1	no	—
213	#488	—	89	2	no	—
214	#467	—	89	1	no	—
215	#466	—	89	1	no	—
216	#465	—	89	2	no	—
217	#446	—	89	2	no	—
218	C₈×Q₈	C₂×D₈₉	1	2	yes	—
219	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
220	C₄×SD₁₆	—	89	2	no	—
221	C₄×D₈	—	89	1	no	—
222	#436	—	89	2	no	—
223	#435	—	178	2	no	—
224	C₈×D₄	—	89	2	no	—
225	C₈×D₄	—	178	2	no	—
226	#433	—	178	2	no	—
227	#432	—	89	2	no	—
228	#405	—	178	2	no	—
229	C₂×C₄×C₈	—	89	1	no	—
230	C₂×C₄×C₈	—	89	1	no	—
231	C₂×C₄×C₈	—	89	1	no	—
232	#335	—	89	1	no	—
233	#335	C₂×D₈₉	1	1	yes	characteristic
234	#334	—	89	2	no	—
235	Q₈⋊C₈	—	89	2	no	—
236	Q₈⋊C₈	C₂×D₈₉	1	2	yes	—
237	D₄⋊C₈	—	89	2	no	—
238	D₄⋊C₈	—	89	2	no	—
239	C₈²	—	89	1	no	—
240	C₈²	—	89	1	no	—
241	C₈²	—	89	1	no	—
242	C₈²	C₂×D₈₉	1	1	yes	characteristic
243	C₂×Q₁₆	C₄×D₈₉	1	1	yes	characteristic
244	C₂×SD₁₆	—	89	2	no	—
245	C₂×D₈	—	89	1	no	—
246	C₂×M₁₆	—	178	2	no	—
247	C₂²×C₈	—	89	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	89	1	no	—
250	C₂²×C₈	—	89	1	no	—
251	C₂²×C₈	—	178	1	no	—
252	C₂²×C₈	—	178	2	no	—
253	#126	—	178	2	no	—
254	#125	—	89	2	no	—
255	#124	—	89	2	no	—
256	#123	—	178	2	no	—
257	C₂²⋊Q₈	—	89	2	no	—
258	#121	—	89	2	no	—
259	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
260	C₄×D₄	—	89	2	no	—
261	C₄×D₄	—	178	2	no	—
262	#117	—	178	2	no	—
263	C₂×C₄²	—	89	1	no	—
264	#107	—	89	1	no	—
265	#107	C₄×D₈₉	1	1	yes	characteristic
266	#106	—	89	2	no	—
267	C₄⋊C₈	—	2	2	no	—
268	C₄⋊C₈	—	89	2	no	—
269	C₄⋊C₈	—	178	2	no	—
270	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
271	#103	—	89	2	no	—
272	#103	C₄×D₈₉	1	2	yes	—
273	#102	—	89	2	no	—
274	#102	—	89	2	no	—
275	C₂²⋊C₈	—	178	2	no	—
276	C₂²⋊C₈	—	178	2	no	—
277	C₂²⋊C₈	—	178	2	no	—
278	#97	—	178	2	no	—
279	#97	—	178	2	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	89	1	no	—
282	C₄×C₈	—	89	1	no	—
283	C₄×C₈	—	89	1	no	—
284	C₄×C₈	—	89	1	no	—
285	C₄×C₈	—	89	1	no	—
286	C₄×C₈	—	89	1	no	—
287	C₄×C₈	—	89	1	no	—
288	C₄×C₈	—	178	1	no	—
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	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
293	C₂×D₄	—	89	2	no	—
294	C₂×D₄	—	178	2	no	—
295	C₂²×C₄	—	89	1	no	—
296	C₂²×C₄	—	89	1	no	—
297	C₂²×C₄	—	89	1	no	—
298	C₂²×C₄	—	178	2	no	—
299	Q₁₆	—	2	2	no	—
300	SD₁₆	—	178	4	no	—
301	D₈	—	178	2	no	—
302	M₁₆	—	356	4	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	89	1	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	89	1	no	—
309	C₂×C₈	—	356	4	no	—
310	C₂×C₈	—	178	2	no	—
311	C₂×C₈	—	178	2	no	—
312	C₂×C₈	—	178	4	no	—
313	C₂×C₈	—	178	2	no	—
314	C₂×C₈	—	178	1	no	—
315	C₂×C₈	—	178	2	no	—
316	C₂×C₈	—	178	2	no	—
317	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
319	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
320	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	89	2	no	—
323	C₄⋊C₄	—	178	2	no	—
324	C₄⋊C₄	—	178	2	no	—
325	C₄⋊C₄	—	178	2	no	—
326	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
327	C₂²⋊C₄	—	178	2	no	—
328	C₂²⋊C₄	—	178	2	no	—
329	C₂²⋊C₄	—	178	2	no	—
330	C₂²⋊C₄	—	178	2	no	—
331	C₂²⋊C₄	—	178	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	89	1	no	—
334	C₄²	—	89	1	no	—
335	C₄²	—	89	1	no	—
336	C₄²	C₂³×D₈₉	1	1	yes	characteristic
337	C₂³	—	89	1	no	—
338	C₂³	—	178	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	D₄	—	356	4	no	—
342	D₄	—	178	2	no	—
343	D₄	—	178	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	89	1	no	—
348	C₂×C₄	—	89	1	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	178	2	no	—
352	C₂×C₄	—	178	2	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
356	C₂×C₄	—	178	1	no	—
357	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
358	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
359	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
360	C₈	—	2	2	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	2	no	—
364	C₈	—	89	2	no	—
365	C₈	—	89	2	no	—
366	C₈	—	356	2	no	—
367	C₈	—	178	2	no	—
368	C₈	—	178	1	no	—
369	C₈	—	178	1	no	—
370	C₈	#15668018	1	2	yes	—
371	C₈	#15668035	1	2	yes	—
372	C₂²	—	89	2	no	—
373	C₂²	—	356	2	no	—
374	C₂²	—	356	2	no	—
375	C₂²	—	356	2	no	—
376	C₂²	—	178	1	no	—
377	C₂²	—	178	1	no	—
378	C₂²	C₄×D₄×D₈₉	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	4	2	no	—
381	C₄	—	4	2	no	—
382	C₄	—	356	2	no	—
383	C₄	—	178	1	no	—
384	C₄	—	178	1	no	—
385	C₄	—	178	1	no	—
386	C₄	#131×D₈₉	1	1	yes	characteristic
387	C₄	C₂×#15668018	1	1	yes	characteristic
388	C₄	#135×D₈₉	1	1	yes	characteristic
389	C₄	C₂×#15668035	1	1	yes	characteristic
390	C₂	—	178	2	no	—
391	C₂	—	712	2	no	—
392	C₂	#26028959	1	1	yes	characteristic
393	C₂	C₄×#15668035	1	1	yes	characteristic
394	C₂	#443×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19151835	C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×Q₁₆	D₈₉
hypocenter	C₈₉	#17959
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26029229	C₂	semidirect
3. #26019149	C₂	semidirect
4. #26019139	C₂	semidirect
5. #26019136	C₂	semidirect
6. C₈×#13083561	C₂	semidirect
7. C₈×D₇₁₂	C₂	semidirect
8. #26019014	C₂	semidirect
9. #26018711	C₂	semidirect
10. C₈²×D₈₉	C₂	semidirect
11. C₈×Q₁₆×C₈₉	C₂	semidirect
203. C₈×Q₁₆	D₈₉	semidirect
209. C₈₉	#17959	semidirect

Maximal subgroups

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

subgroup	index
#26030825	2
#26029229	2
#26019149	2
#26019139	2
#26019136	2
C₈×#13083561	2
C₈×D₇₁₂	2
#26019014	2
#26018711	2
C₈²×D₈₉	2
C₈×Q₁₆×C₈₉	2
#17959	89