Subgroups — #36445486

Order 22784.

400 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020003	C₂	1	1	yes	maximal, characteristic, complemented
2	#26018840	C₂	1	1	yes	maximal, characteristic
3	#26018738	C₂	1	1	yes	maximal, characteristic
4	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018150	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018144	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2966	C₂	1	1	yes	maximal, characteristic
8	#19151651	C₂²	1	1	yes	characteristic
9	C₄×#13083558	C₂²	1	1	yes	characteristic
10	C₂×#15667923	C₂²	1	1	yes	characteristic
11	#19151444	C₂²	1	1	yes	characteristic
12	#19151352	—	2	1	no	—
13	#19151352	—	2	1	no	—
14	#19151340	C₄	1	2	yes	complemented
15	#19151338	C₄	1	2	yes	complemented
16	#19151335	C₄	1	2	yes	—
17	#19151332	C₄	1	2	yes	—
18	#19151320	—	2	1	no	—
19	C₈×#10623494	—	2	1	no	—
20	#529×C₈₉	C₂²	1	1	yes	characteristic
21	#446×C₈₉	C₄	1	1	yes	characteristic, complemented
22	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic, complemented
23	#436×C₈₉	C₄	1	1	yes	characteristic, complemented
24	#435×C₈₉	—	2	1	no	—
25	C₈×D₄×C₈₉	—	2	1	no	—
26	C₈×D₄×C₈₉	C₄	1	1	yes	characteristic, complemented
27	#433×C₈₉	—	2	1	no	—
28	#432×C₈₉	C₂²	1	1	yes	characteristic
29	#405×C₈₉	—	2	1	no	—
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083558	—	2	1	no	—
32	C₂²×#10623494	—	2	1	no	—
33	C₂×C₄×Dic₈₉	—	2	1	no	—
34	C₄⋊#10623494	—	2	1	no	—
35	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
36	#15667923	—	2	1	no	—
37	#15667923	C₂×C₄	1	2	yes	—
38	#15667923	C₂×C₄	1	2	yes	—
39	C₄×#10623494	—	2	1	no	—
40	C₄×#10623494	—	2	1	no	—
41	#15667919	—	2	1	no	—
42	#15667919	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₈₉	—	2	1	no	—
44	C₂²×C₈×C₈₉	—	2	1	no	—
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	#126×C₈₉	—	2	1	no	—
48	#125×C₈₉	C₂×C₄	1	1	yes	characteristic
49	#124×C₈₉	C₂×C₄	1	1	yes	characteristic
50	#123×C₈₉	—	2	1	no	—
51	C₂²⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
52	#121×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₄×D₄×C₈₉	—	2	1	no	—
55	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
56	#117×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂²⋊C₈×C₈₉	—	2	1	no	—
63	C₂²⋊C₈×C₈₉	—	2	1	no	—
64	C₂²⋊C₈×C₈₉	—	2	1	no	—
65	#97×C₈₉	—	2	1	no	—
66	#97×C₈₉	—	2	1	no	—
67	C₄×C₈×C₈₉	—	2	1	no	—
68	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×Dic₈₉	—	2	2	no	—
73	#13083558	—	2	4	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	—	4	1	no	—
78	C₂×#10623494	—	4	1	no	—
79	C₄×Dic₈₉	—	4	1	no	—
80	C₄×Dic₈₉	—	4	1	no	—
81	C₂×Q₈×C₈₉	C₄²	1	1	yes	characteristic
82	C₂×D₄×C₈₉	—	2	1	no	—
83	C₂×D₄×C₈₉	C₄²	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	M₁₆×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	4	1	no	—
93	C₂×C₈×C₈₉	—	4	2	no	—
94	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₄²	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂²⋊C₄×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	#41	1	1	yes	characteristic
114	C₂³×C₈₉	—	2	1	no	—
115	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
116	Q₈×C₈₉	—	4	1	no	—
117	D₄×C₈₉	—	4	1	no	—
118	D₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	4	1	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	4	1	no	—
142	C₈×C₈₉	—	4	1	no	—
143	#10623494	—	2	4	no	—
144	#10623494	—	4	2	no	—
145	#10623494	—	4	2	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₈₉	#443	1	2	yes	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	#421	1	1	yes	characteristic
162	C₄×C₈₉	#421	1	1	yes	characteristic
163	Dic₈₉	—	8	2	no	—
164	#8250	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	2	no	—
166	C₂×C₈₉	—	8	1	no	—
167	C₂×C₈₉	#1757	1	1	yes	characteristic
168	C₂×C₈₉	#1759	1	1	yes	characteristic
169	C₂×C₈₉	#1764	1	1	yes	characteristic
170	#2966	D₈₉	1	1	yes	characteristic
171	#1746	—	89	1	no	—
172	#1524	—	89	1	no	—
173	#1500	—	89	1	no	—
174	C₈×M₁₆	—	89	1	no	—
175	#1302	—	89	1	no	—
176	#1300	—	89	1	no	—
177	C₈₉	#8250	1	1	yes	characteristic, complemented, Sylow
178	#529	C₂×D₈₉	1	1	yes	characteristic
179	#446	Dic₈₉	1	1	yes	characteristic, complemented
180	C₈×Q₈	Dic₈₉	1	1	yes	characteristic, complemented
181	#436	Dic₈₉	1	1	yes	characteristic, complemented
182	#435	—	2	1	no	—
183	C₈×D₄	—	2	1	no	—
184	C₈×D₄	Dic₈₉	1	1	yes	characteristic, complemented
185	#433	—	2	1	no	—
186	#432	C₂×D₈₉	1	1	yes	characteristic
187	#431	—	89	1	no	—
188	#405	—	2	1	no	—
189	C₄×M₁₆	—	89	1	no	—
190	C₂×#97	—	89	1	no	—
191	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
192	#376	—	89	1	no	—
193	#336	—	178	1	no	—
194	#336	—	178	1	no	—
195	#330	—	89	2	no	—
196	#329	—	89	2	no	—
197	#328	—	89	2	no	—
198	#327	—	89	2	no	—
199	#322	—	178	1	no	—
200	C₈²	—	178	1	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	178	1	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	178	1	no	—
205	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
207	#126	—	2	1	no	—
208	#125	C₄×D₈₉	1	1	yes	characteristic
209	#124	C₄×D₈₉	1	1	yes	characteristic
210	#123	—	2	1	no	—
211	C₂²⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
212	#121	C₄×D₈₉	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₈₉	1	1	yes	characteristic
214	C₄×D₄	—	2	1	no	—
215	C₄×D₄	C₂×Dic₈₉	1	1	yes	characteristic
216	#117	—	2	1	no	—
217	C₂×C₄²	—	178	1	no	—
218	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₄⋊C₈	—	178	1	no	—
222	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
223	C₄⋊C₈	C₂×Dic₈₉	1	1	yes	characteristic
224	C₂²⋊C₈	—	2	1	no	—
225	C₂²⋊C₈	—	2	1	no	—
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	89	2	no	—
228	#97	—	2	1	no	—
229	#97	—	2	1	no	—
230	#97	—	89	2	no	—
231	#97	—	89	2	no	—
232	#97	—	178	1	no	—
233	C₄×C₈	—	2	1	no	—
234	C₄×C₈	—	178	1	no	—
235	C₄×C₈	—	178	1	no	—
236	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
237	C₄×C₈	C₂×Dic₈₉	1	1	yes	characteristic
238	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
239	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
240	#95	—	89	2	no	—
241	#95	—	178	1	no	—
242	C₂×Q₈	C₄×Dic₈₉	1	1	yes	characteristic
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	C₄×Dic₈₉	1	1	yes	characteristic
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	178	2	no	—
247	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
249	C₂²×C₄	#13083574	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	178	4	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	356	1	no	—
258	C₂×C₈	—	356	1	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	—	178	2	no	—
261	C₂×C₈	—	178	2	no	—
262	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
263	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
271	C₄⋊C₄	C₄×Dic₈₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₄²	—	2	1	no	—
278	C₄²	—	356	1	no	—
279	C₄²	—	356	1	no	—
280	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
281	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
283	C₄²	#13083574	1	1	yes	characteristic
284	C₂³	—	2	1	no	—
285	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
286	Q₈	—	4	1	no	—
287	D₄	—	4	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	356	2	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
302	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
303	C₂×C₄	#15667971	1	1	yes	characteristic
304	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
305	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	4	1	no	—
312	C₈	—	4	1	no	—
313	C₈	—	356	2	no	—
314	C₈	—	356	2	no	—
315	C₈	—	178	4	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
322	C₂²	#19151713	1	2	yes	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	712	2	no	—
332	C₄	#19151615	1	1	yes	characteristic
333	C₄	#19151615	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	8	1	no	—
336	C₂	#26020052	1	1	yes	characteristic
337	C₂	#26020062	1	1	yes	characteristic
338	C₂	#26020087	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊Dic₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#2966	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#121×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#2966	D₈₉
hypocenter	C₈₉	#8250
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2966

Complements

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

N	H ≅ G/N	type
1. #26020003	C₂	semidirect
4. C₈×#13083558	C₂	semidirect
5. #26018150	C₂	semidirect
6. #26018144	C₂	semidirect
14. #19151340	C₄	semidirect
15. #19151338	C₄	semidirect
21. #446×C₈₉	C₄	semidirect
22. C₈×Q₈×C₈₉	C₄	semidirect
23. #436×C₈₉	C₄	semidirect
26. C₈×D₄×C₈₉	C₄	semidirect
177. C₈₉	#8250	semidirect
179. #446	Dic₈₉	semidirect
180. C₈×Q₈	Dic₈₉	semidirect
181. #436	Dic₈₉	semidirect
184. C₈×D₄	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020003	2
#26018840	2
#26018738	2
C₈×#13083558	2
#26018150	2
#26018144	2
C₈₉×#2966	2
#8250	89