Subgroups — #36523687

Order 22784.

588 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	#26029218	C₂	1	1	yes	maximal, characteristic
2	#26028913	C₂	1	2	yes	maximal
3	#26019042	C₂	1	2	yes	maximal, complemented
4	#26019036	C₂	1	2	yes	maximal, complemented
5	#26019030	C₂	1	2	yes	maximal, complemented
6	#26019016	C₂	1	2	yes	maximal, complemented
7	#26019015	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019001	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13083574	C₂²	1	1	yes	characteristic
11	#19151728	C₄	1	1	yes	characteristic
12	#19151728	C₄	1	1	yes	characteristic
13	#19151677	C₄	1	1	yes	characteristic
14	#19151677	C₄	1	1	yes	characteristic
15	#19151669	C₄	1	2	yes	—
16	#19151669	C₄	1	2	yes	—
17	#19151657	—	2	2	no	—
18	#19151656	—	2	2	no	—
19	#19151655	C₂²	1	2	yes	—
20	#19151654	C₂²	1	2	yes	—
21	#19151651	C₂²	1	1	yes	characteristic
22	#19151650	C₂²	1	2	yes	—
23	#19151649	C₂²	1	2	yes	—
24	#19151620	C₂²	1	2	yes	—
25	C₄⋊C₈×D₈₉	C₂²	1	2	yes	—
26	#19151563	—	2	2	no	—
27	#19151552	—	2	2	no	—
28	C₄×#13083557	C₂²	1	2	yes	—
29	#19151351	C₂²	1	2	yes	—
30	#19151350	C₂²	1	2	yes	—
31	#19151350	C₂²	1	2	yes	—
32	#19151349	C₂²	1	1	yes	characteristic
33	#19151349	C₂²	1	1	yes	characteristic
34	#19151348	C₂²	1	2	yes	—
35	#19151348	C₂²	1	2	yes	—
36	#431×C₈₉	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
38	#335×C₈₉	C₂²	1	2	yes	—
39	#334×C₈₉	C₂²	1	1	yes	characteristic
40	#334×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083574	D₄	1	1	yes	characteristic
42	C₂×#13083557	—	2	2	no	—
43	C₂×#13083557	C₂×C₄	1	2	yes	—
44	C₂×#13083557	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₈₉	—	2	2	no	—
46	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
47	C₄×Dic₁₇₈	Q₈	1	1	yes	characteristic
48	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
49	C₄×D₃₅₆	Q₈	1	1	yes	characteristic
50	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
51	C₄×C₂²⋊D₈₉	Q₈	1	2	yes	—
52	#15667971	D₄	1	1	yes	characteristic
53	#15667969	D₄	1	2	yes	—
54	C₄²×D₈₉	C₂³	1	2	yes	—
55	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#10623494	C₂³	1	2	yes	—
58	C₄⋊#10623494	C₂³	1	2	yes	—
59	#15667941	—	2	2	no	—
60	#15667941	—	2	2	no	—
61	#15667941	C₂×C₄	1	2	yes	—
62	#15667941	C₂×C₄	1	2	yes	—
63	C₂²⋊#10623494	—	2	2	no	—
64	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
65	#15667924	—	2	2	no	—
66	#15667924	—	2	2	no	—
67	#15667924	C₂×C₄	1	2	yes	—
68	#15667924	C₂×C₄	1	2	yes	—
69	#15667923	—	2	2	no	—
70	#15667922	—	2	2	no	—
71	#15667922	C₂×C₄	1	2	yes	—
72	#15667922	C₂×C₄	1	2	yes	—
73	C₄×#10623494	C₂³	1	2	yes	—
74	C₈×Dic₈₉	—	2	2	no	—
75	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
79	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
80	C₂²⋊C₈×C₈₉	—	2	2	no	—
81	C₄×C₈×C₈₉	—	2	2	no	—
82	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
83	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	#13083574	—	4	2	no	—
87	C₂×Dic₁₇₈	C₄⋊C₄	1	1	yes	characteristic
88	C₂×D₃₅₆	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₂²⋊D₈₉	C₄⋊C₄	1	2	yes	—
90	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
92	C₂×C₄×D₈₉	C₂×Q₈	1	2	yes	—
93	#13083557	—	2	4	no	—
94	#13083557	—	4	2	no	—
95	#13083557	—	4	4	no	—
96	C₂×#10623494	—	2	2	no	—
97	C₂×#10623494	—	2	2	no	—
98	C₂×#10623494	C₂²×C₄	1	2	yes	—
99	C₂×#10623494	C₂²×C₄	1	2	yes	—
100	C₈×D₈₉	—	4	4	no	—
101	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
102	C₄⋊Dic₈₉	C₄⋊C₄	1	1	yes	characteristic
103	#13083553	C₄⋊C₄	1	2	yes	—
104	#13083553	C₄⋊C₄	1	2	yes	—
105	C₂²⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
106	#13083551	C₄⋊C₄	1	2	yes	—
107	#13083551	C₄⋊C₄	1	2	yes	—
108	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
109	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
110	C₄×Dic₈₉	C₂×Q₈	1	2	yes	—
111	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₂×C₈×C₈₉	—	2	2	no	—
117	C₂×C₈×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
122	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
125	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	—	4	1	no	—
132	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×Dic₈₉	C₂×C₄⋊C₄	1	2	yes	—
142	D₃₅₆	—	4	1	no	—
143	C₄×D₈₉	—	2	2	no	—
144	C₄×D₈₉	—	2	2	no	—
145	C₄×D₈₉	—	2	2	no	—
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	4	2	no	—
148	Dic₁₇₈	—	4	1	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	—	2	1	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	—	4	2	no	—
153	C₈×C₈₉	—	4	2	no	—
154	C₈×C₈₉	#131	1	2	yes	—
155	#10623494	—	2	2	no	—
156	#10623494	—	2	2	no	—
157	#10623494	—	4	2	no	—
158	#10623494	—	4	2	no	—
159	C₂²×C₈₉	—	2	1	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	C₂²×C₄⋊C₄	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	1	no	—
167	C₄×C₈₉	—	2	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	2	1	no	—
170	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
171	C₄×C₈₉	#568	1	1	yes	characteristic
172	C₄×C₈₉	#575	1	2	yes	—
173	Dic₈₉	—	2	2	no	—
174	Dic₈₉	—	2	2	no	—
175	Dic₈₉	—	2	2	no	—
176	#17831	—	89	1	no	maximal, Sylow
177	C₂×C₈₉	—	4	1	no	—
178	C₂×C₈₉	#2906	1	1	yes	characteristic
179	C₂×C₈₉	#2911	1	1	yes	characteristic
180	C₂×C₈₉	#2916	1	1	yes	characteristic
181	D₈₉	—	4	2	no	—
182	#2965	—	89	1	no	—
183	#2925	—	89	2	no	—
184	#1574	—	89	2	no	—
185	#1573	—	89	2	no	—
186	#1572	—	89	2	no	—
187	#1569	—	89	2	no	—
188	#1568	—	89	1	no	—
189	#1565	—	89	2	no	—
190	#1564	D₈₉	1	1	yes	characteristic, complemented
191	C₈₉	#17831	1	1	yes	characteristic, complemented, Sylow
192	C₄×#41	—	89	1	no	—
193	#446	—	89	1	no	—
194	#446	—	89	1	no	—
195	#436	—	89	1	no	—
196	#436	—	89	1	no	—
197	#435	—	89	2	no	—
198	#435	—	89	2	no	—
199	#433	—	178	2	no	—
200	#433	—	178	2	no	—
201	#432	—	89	2	no	—
202	#432	—	89	2	no	—
203	#431	—	89	1	no	—
204	#431	—	89	2	no	—
205	#431	—	89	2	no	—
206	#431	C₂×D₈₉	1	2	yes	—
207	#423	—	89	2	no	—
208	C₂×C₄⋊C₈	—	89	2	no	—
209	#408	—	178	2	no	—
210	#405	—	178	2	no	—
211	C₄×M₁₆	—	89	2	no	—
212	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
213	#335	—	89	2	no	—
214	#335	—	89	2	no	—
215	#335	—	89	2	no	—
216	#335	C₂×D₈₉	1	2	yes	—
217	#334	—	89	2	no	—
218	#334	—	89	2	no	—
219	#334	—	89	1	no	—
220	#334	—	89	1	no	—
221	#334	C₂×D₈₉	1	1	yes	characteristic
222	#334	C₂×D₈₉	1	1	yes	characteristic
223	C₂×#41	—	89	1	no	—
224	C₂×M₁₆	—	89	2	no	—
225	C₂×M₁₆	—	89	2	no	—
226	C₂×M₁₆	—	178	2	no	—
227	C₂²×C₈	—	178	2	no	—
228	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
229	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×Q₈	—	89	1	no	—
231	C₄×Q₈	—	89	1	no	—
232	C₄×D₄	—	89	2	no	—
233	C₄×D₄	—	89	2	no	—
234	C₄×D₄	—	89	1	no	—
235	C₄×D₄	—	89	1	no	—
236	#117	—	89	1	no	—
237	#117	—	89	2	no	—
238	C₂×C₄²	—	89	2	no	—
239	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
240	C₄⋊C₈	—	89	2	no	—
241	C₄⋊C₈	—	89	2	no	—
242	C₄⋊C₈	—	89	1	no	—
243	C₄⋊C₈	—	89	1	no	—
244	C₄⋊C₈	—	89	2	no	—
245	C₄⋊C₈	—	89	2	no	—
246	C₄⋊C₈	—	178	2	no	—
247	C₄⋊C₈	—	178	2	no	—
248	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
249	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
250	C₂²⋊C₈	—	2	2	no	—
251	C₂²⋊C₈	—	89	2	no	—
252	C₂²⋊C₈	—	89	2	no	—
253	C₂²⋊C₈	—	89	2	no	—
254	C₂²⋊C₈	—	178	2	no	—
255	C₂²⋊C₈	—	178	2	no	—
256	C₂²⋊C₈	—	178	2	no	—
257	#97	—	89	2	no	—
258	#97	—	89	2	no	—
259	#97	—	178	2	no	—
260	#97	—	178	2	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	—	89	2	no	—
263	C₄×C₈	—	178	2	no	—
264	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
265	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	#41	—	356	2	no	—
269	C₂×Q₈	—	89	1	no	—
270	C₂×D₄	—	89	2	no	—
271	C₂×D₄	—	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	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
277	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
278	M₁₆	—	356	2	no	—
279	M₁₆	—	356	4	no	—
280	M₁₆	—	178	4	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	89	2	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	356	4	no	—
288	C₂×C₈	—	178	2	no	—
289	C₂×C₈	—	178	2	no	—
290	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
294	C₄⋊C₄	—	89	2	no	—
295	C₄⋊C₄	—	89	2	no	—
296	C₄⋊C₄	—	89	1	no	—
297	C₄⋊C₄	—	89	1	no	—
298	C₂²⋊C₄	—	89	2	no	—
299	C₂²⋊C₄	—	89	2	no	—
300	C₂²⋊C₄	—	89	2	no	—
301	C₄²	—	89	2	no	—
302	C₄²	—	89	2	no	—
303	C₄²	—	89	2	no	—
304	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
305	C₄²	D₄×D₈₉	1	1	yes	characteristic
306	C₄²	Q₈×D₈₉	1	1	yes	characteristic
307	C₄²	C₂³×D₈₉	1	1	yes	characteristic
308	C₂³	—	89	2	no	—
309	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
310	Q₈	—	356	1	no	—
311	D₄	—	356	1	no	—
312	D₄	—	356	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	89	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	178	2	no	—
322	C₂×C₄	—	178	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
326	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
327	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	4	2	no	—
335	C₈	—	4	2	no	—
336	C₈	—	356	2	no	—
337	C₈	—	356	2	no	—
338	C₈	—	178	2	no	—
339	C₈	—	178	2	no	—
340	C₈	#15668018	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	178	2	no	—
345	C₂²	—	178	2	no	—
346	C₂²	—	178	2	no	—
347	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	178	2	no	—
353	C₄	—	178	2	no	—
354	C₄	—	178	2	no	—
355	C₄	C₂×#15668018	1	1	yes	characteristic
356	C₄	#19152364	1	1	yes	characteristic
357	C₄	#19152395	1	2	yes	—
358	C₂	—	4	1	no	—
359	C₂	—	356	2	no	—
360	C₂	#26028817	1	1	yes	characteristic
361	C₂	#26028840	1	1	yes	characteristic
362	C₂	#26028860	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1564	C₂
socle	C₂²×C₈₉	C₂²×C₄⋊C₄
prime radical	C₂×#13083574	D₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#1564	D₈₉
hypocenter	C₈₉	#17831
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #26019042	C₂	semidirect
4. #26019036	C₂	semidirect
5. #26019030	C₂	semidirect
6. #26019016	C₂	semidirect
7. #26019015	C₂	semidirect
8. #26019001	C₂	semidirect
9. C₈₉×#1564	C₂	semidirect
190. #1564	D₈₉	semidirect
191. C₈₉	#17831	semidirect

Maximal subgroups

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

subgroup	index
#26029218	2
#26028913	2
#26019042	2
#26019036	2
#26019030	2
#26019016	2
#26019015	2
#26019001	2
C₈₉×#1564	2
#17831	89