Subgroups — #36524583

Order 22784.

586 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	#26030292	C₂	1	1	yes	maximal, characteristic
2	#26029346	C₂	1	1	yes	maximal, characteristic
3	#26029332	C₂	1	1	yes	maximal, characteristic
4	#26019125	C₂	1	2	yes	maximal, complemented
5	#26019104	C₂	1	2	yes	maximal, complemented
6	#26019072	C₂	1	2	yes	maximal, complemented
7	C₈×Dic₃₅₆	C₂	1	2	yes	maximal, complemented
8	#26019032	C₂	1	1	yes	maximal, characteristic
9	#26018700	C₂	1	1	yes	maximal, characteristic
10	#26018674	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1453	C₂	1	1	yes	maximal, characteristic
12	#19152155	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
14	#19151934	C₄	1	2	yes	—
15	#19151894	C₄	1	2	yes	—
16	#19151809	C₄	1	2	yes	—
17	#19151756	C₄	1	2	yes	—
18	#19151728	C₂²	1	2	yes	—
19	#19151727	—	2	2	no	—
20	C₈×Dic₁₇₈	C₂²	1	2	yes	—
21	C₈×Dic₁₇₈	C₂²	1	2	yes	—
22	C₈×Dic₁₇₈	C₂²	1	2	yes	—
23	#19151722	—	2	2	no	—
24	C₄×Dic₃₅₆	C₂²	1	2	yes	—
25	C₄×Dic₃₅₆	C₂²	1	2	yes	—
26	#19151656	—	2	2	no	—
27	#19151655	C₂²	1	1	yes	characteristic
28	#19151651	C₂²	1	1	yes	characteristic
29	#19151648	—	2	2	no	—
30	#19151351	C₂²	1	2	yes	—
31	#19151349	C₂²	1	2	yes	—
32	#19151330	C₂²	1	2	yes	—
33	#19151330	C₂²	1	2	yes	—
34	#19151330	C₂²	1	2	yes	—
35	#19151330	C₂²	1	2	yes	—
36	#431×C₈₉	C₂²	1	1	yes	characteristic
37	#431×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	#322×C₈₉	C₂²	1	1	yes	characteristic
40	#322×C₈₉	C₂²	1	1	yes	characteristic
41	#322×C₈₉	C₂²	1	1	yes	characteristic
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂²×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
44	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
45	C₂×Dic₃₅₆	C₂×C₄	1	2	yes	—
46	#15668012	C₂×C₄	1	1	yes	characteristic
47	#15668012	C₂×C₄	1	1	yes	characteristic
48	#15668004	—	2	1	no	—
49	#15667991	—	2	1	no	—
50	#15667991	C₂×C₄	1	2	yes	—
51	C₄×Dic₁₇₈	—	2	1	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	C₄×Dic₁₇₈	—	2	1	no	—
54	C₄×Dic₁₇₈	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₇₈	C₂³	1	2	yes	—
56	C₄×Dic₁₇₈	C₂³	1	2	yes	—
57	#15667971	—	2	1	no	—
58	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
59	C₂×#13083553	—	2	1	no	—
60	C₂×C₄×Dic₈₉	—	2	1	no	—
61	#15667946	C₂×C₄	1	2	yes	—
62	#15667946	C₂×C₄	1	2	yes	—
63	C₄⋊#10623494	C₂³	1	2	yes	—
64	C₄⋊#10623494	C₂³	1	2	yes	—
65	#15667941	—	2	2	no	—
66	#15667941	—	2	2	no	—
67	#15667941	—	2	2	no	—
68	#15667941	—	2	2	no	—
69	#15667937	C₂×C₄	1	2	yes	—
70	#15667937	C₂×C₄	1	2	yes	—
71	#15667937	C₂×C₄	1	2	yes	—
72	#15667937	C₂×C₄	1	2	yes	—
73	C₂²⋊#10623494	—	2	1	no	—
74	C₂²⋊#10623494	—	2	1	no	—
75	#15667923	—	2	1	no	—
76	#15667922	—	2	2	no	—
77	C₄×#10623494	—	2	1	no	—
78	C₈×Dic₈₉	—	2	2	no	—
79	C₈×Dic₈₉	—	2	2	no	—
80	C₈×Dic₈₉	—	2	2	no	—
81	C₂×M₁₆×C₈₉	—	2	1	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
85	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
87	#97×C₈₉	—	2	1	no	—
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
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×Dic₁₇₈	—	2	1	no	—
95	C₂×Dic₁₇₈	—	2	4	no	—
96	C₂×Dic₁₇₈	—	2	1	no	—
97	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
98	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
99	C₂²×Dic₈₉	—	2	1	no	—
100	C₂²×Dic₈₉	—	2	1	no	—
101	Dic₃₅₆	—	2	4	no	—
102	Dic₃₅₆	—	2	4	no	—
103	C₂×#10623494	—	2	2	no	—
104	C₂×#10623494	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	1	no	—
106	C₄⋊Dic₈₉	—	2	1	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	—	2	1	no	—
109	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
111	#13083553	—	2	2	no	—
112	#13083553	—	2	2	no	—
113	#13083553	—	2	2	no	—
114	#13083553	—	2	2	no	—
115	C₂²⋊Dic₈₉	—	2	2	no	—
116	C₄×Dic₈₉	—	2	2	no	—
117	C₄×Dic₈₉	—	2	2	no	—
118	C₄×Dic₈₉	—	2	2	no	—
119	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
122	M₁₆×C₈₉	—	4	2	no	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	1	no	—
125	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
128	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
129	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₈₉	#41	1	1	yes	characteristic
132	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	—	2	2	no	—
135	C₂×C₄×C₈₉	—	2	2	no	—
136	C₂×C₄×C₈₉	—	4	1	no	—
137	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₈₉	—	2	4	no	—
144	C₂×Dic₈₉	—	2	2	no	—
145	C₂×Dic₈₉	—	2	2	no	—
146	C₂×Dic₈₉	—	2	2	no	—
147	C₂×Dic₈₉	—	2	2	no	—
148	C₂×Dic₈₉	—	8	1	no	—
149	Dic₁₇₈	—	2	4	no	—
150	Dic₁₇₈	—	2	2	no	—
151	Dic₁₇₈	—	2	2	no	—
152	Dic₁₇₈	—	2	2	no	—
153	Dic₁₇₈	—	8	1	no	—
154	Dic₁₇₈	#131	1	4	yes	—
155	C₈×C₈₉	—	2	2	no	—
156	C₈×C₈₉	—	2	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	—	2	2	no	—
159	C₈×C₈₉	—	2	2	no	—
160	C₈×C₈₉	—	4	1	no	—
161	#10623494	—	4	2	no	—
162	#10623494	—	4	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	—	2	1	no	—
171	C₄×C₈₉	C₂×#131	1	2	yes	—
172	C₄×C₈₉	#578	1	2	yes	—
173	Dic₈₉	—	2	4	no	—
174	Dic₈₉	—	4	2	no	—
175	Dic₈₉	—	4	2	no	—
176	Dic₈₉	—	4	2	no	—
177	#17925	—	89	1	no	maximal, Sylow
178	C₂×C₈₉	—	4	1	no	—
179	C₂×C₈₉	#2929	1	1	yes	characteristic
180	C₂×C₈₉	#2950	1	1	yes	characteristic
181	C₂×C₈₉	#2956	1	1	yes	characteristic
182	#3104	—	89	1	no	—
183	#2985	—	89	1	no	—
184	#2982	—	89	1	no	—
185	#1587	—	89	2	no	—
186	#1584	—	89	2	no	—
187	#1579	—	89	2	no	—
188	C₈×Q₁₆	—	89	2	no	—
189	#1572	—	89	1	no	—
190	#1491	—	89	1	no	—
191	#1485	—	89	1	no	—
192	#1453	D₈₉	1	1	yes	characteristic
193	C₈₉	#17925	1	1	yes	characteristic, complemented, Sylow
194	#533	—	89	1	no	—
195	C₂×C₄×Q₈	—	89	1	no	—
196	#484	—	89	2	no	—
197	#477	—	89	2	no	—
198	#462	—	89	2	no	—
199	#451	—	89	2	no	—
200	#446	—	89	2	no	—
201	#446	—	178	2	no	—
202	C₈×Q₈	—	89	2	no	—
203	C₈×Q₈	—	89	2	no	—
204	C₈×Q₈	—	89	2	no	—
205	C₈×Q₈	—	178	2	no	—
206	C₄×Q₁₆	—	89	2	no	—
207	C₄×Q₁₆	—	89	2	no	—
208	#433	—	178	2	no	—
209	#432	—	89	1	no	—
210	#431	—	89	1	no	—
211	#431	—	178	2	no	—
212	#431	C₂×D₈₉	1	1	yes	characteristic
213	#431	C₂×D₈₉	1	1	yes	characteristic
214	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
215	#335	—	89	2	no	—
216	#334	—	89	2	no	—
217	Q₈⋊C₈	—	89	2	no	—
218	Q₈⋊C₈	—	89	2	no	—
219	Q₈⋊C₈	—	89	2	no	—
220	Q₈⋊C₈	—	89	2	no	—
221	#322	C₂×D₈₉	1	1	yes	characteristic
222	#322	C₂×D₈₉	1	1	yes	characteristic
223	#322	C₂×D₈₉	1	1	yes	characteristic
224	C₈²	C₂×D₈₉	1	1	yes	characteristic
225	C₂²×Q₈	—	89	1	no	—
226	C₂×Q₁₆	—	89	2	no	—
227	C₂×Q₁₆	—	89	2	no	—
228	C₂×M₁₆	—	2	1	no	—
229	C₄⋊Q₈	—	89	1	no	—
230	C₄⋊Q₈	—	89	1	no	—
231	#125	—	178	1	no	—
232	C₂²⋊Q₈	—	89	2	no	—
233	C₂²⋊Q₈	—	178	1	no	—
234	C₄×Q₈	—	89	2	no	—
235	C₄×Q₈	—	89	2	no	—
236	C₄×Q₈	—	89	2	no	—
237	C₄×Q₈	—	178	1	no	—
238	C₄×Q₈	—	178	1	no	—
239	C₄×Q₈	—	178	1	no	—
240	#117	—	178	1	no	—
241	C₂×C₄⋊C₄	—	89	1	no	—
242	C₂×C₄⋊C₄	—	178	1	no	—
243	C₂×C₄²	—	178	1	no	—
244	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
245	#107	—	89	2	no	—
246	#107	—	89	2	no	—
247	C₄⋊C₈	—	89	2	no	—
248	C₄⋊C₈	—	89	2	no	—
249	C₄⋊C₈	—	178	2	no	—
250	C₄⋊C₈	—	178	2	no	—
251	C₄⋊C₈	—	178	2	no	—
252	C₄⋊C₈	—	178	2	no	—
253	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
254	C₄⋊C₈	D₃₅₆	1	2	yes	—
255	#103	—	89	2	no	—
256	#103	—	89	2	no	—
257	#103	—	89	2	no	—
258	#103	—	89	2	no	—
259	C₂²⋊C₈	—	178	1	no	—
260	C₂²⋊C₈	—	178	1	no	—
261	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
262	C₂²⋊C₈	D₃₅₆	1	2	yes	—
263	#97	—	2	1	no	—
264	#97	—	178	2	no	—
265	#97	—	178	1	no	—
266	C₄×C₈	—	178	2	no	—
267	C₄×C₈	—	178	2	no	—
268	C₄×C₈	—	178	2	no	—
269	C₄×C₈	—	178	1	no	—
270	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
272	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
276	C₂×Q₈	—	89	2	no	—
277	C₂×Q₈	—	89	2	no	—
278	C₂×Q₈	—	178	4	no	—
279	C₂×Q₈	—	178	1	no	—
280	C₂×Q₈	—	178	1	no	—
281	C₂²×C₄	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
284	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
285	C₂²×C₄	#13083574	1	1	yes	characteristic
286	Q₁₆	—	178	4	no	—
287	Q₁₆	—	178	4	no	—
288	M₁₆	—	4	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	1	no	—
291	C₂×C₈	—	178	2	no	—
292	C₂×C₈	—	178	2	no	—
293	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
294	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
295	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
296	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
297	C₄⋊C₄	—	89	2	no	—
298	C₄⋊C₄	—	89	2	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₄⋊C₄	—	178	1	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₂²⋊C₄	—	178	2	no	—
308	C₄²	—	178	2	no	—
309	C₄²	—	178	2	no	—
310	C₄²	—	178	2	no	—
311	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
312	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
313	C₄²	#13083574	1	1	yes	characteristic
314	C₄²	C₂³×D₈₉	1	1	yes	characteristic
315	C₂³	C₄×D₃₅₆	1	1	yes	characteristic
316	Q₈	—	89	4	no	—
317	Q₈	—	178	4	no	—
318	Q₈	—	178	2	no	—
319	Q₈	—	178	2	no	—
320	Q₈	—	178	2	no	—
321	Q₈	—	712	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	178	2	no	—
326	C₂×C₄	—	178	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	4	no	—
329	C₂×C₄	—	178	2	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
332	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
333	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
334	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
335	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
336	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
337	C₈	—	2	2	no	—
338	C₈	—	2	2	no	—
339	C₈	—	2	2	no	—
340	C₈	—	2	2	no	—
341	C₈	—	2	2	no	—
342	C₈	—	4	1	no	—
343	C₈	—	356	2	no	—
344	C₈	—	356	2	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	2	1	no	—
348	C₂²	C₂×C₄×D₃₅₆	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	356	2	no	—
354	C₄	—	356	2	no	—
355	C₄	—	356	2	no	—
356	C₄	—	178	4	no	—
357	C₄	#131×D₈₉	1	2	yes	—
358	C₄	#19152415	1	2	yes	—
359	C₂	—	4	1	no	—
360	C₂	#26028948	1	1	yes	characteristic
361	C₂	#26029121	1	1	yes	characteristic
362	C₂	#26029163	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1453	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂³×C₈₉	C₄×D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1453	D₈₉
hypocenter	C₈₉	#17925
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
4. #26019125	C₂	semidirect
5. #26019104	C₂	semidirect
6. #26019072	C₂	semidirect
7. C₈×Dic₃₅₆	C₂	semidirect
193. C₈₉	#17925	semidirect

Maximal subgroups

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

subgroup	index
#26030292	2
#26029346	2
#26029332	2
#26019125	2
#26019104	2
#26019072	2
C₈×Dic₃₅₆	2
#26019032	2
#26018700	2
#26018674	2
C₈₉×#1453	2
#17925	89