Subgroups — #36452705

Order 22784.

450 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	#26021046	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667932	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151325	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018093	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
10	C₂×#15667932	C₄	1	1	yes	characteristic
11	C₂×#15667932	C₄	1	1	yes	characteristic
12	C₂×#15667924	—	2	1	no	—
13	C₂×#15667924	—	2	1	no	—
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	#19151360	C₄	1	1	yes	characteristic
17	#19151360	C₄	1	1	yes	characteristic
18	#19151325	—	2	2	no	—
19	#19151320	—	2	2	no	—
20	C₂×C₄⋊C₈×C₈₉	—	2	1	no	—
21	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
24	#336×C₈₉	—	2	1	no	—
25	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×D₈₉	—	2	1	no	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	—	2	1	no	—
29	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
30	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
31	C₄×D₃₅₆	—	2	1	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₄×D₃₅₆	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
36	C₂×#13083551	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	#15667932	—	2	2	no	—
40	#15667932	C₈	1	4	yes	complemented
41	#15667924	—	4	2	no	—
42	#15667924	—	4	2	no	—
43	C₄×#10623494	—	2	1	no	—
44	C₄×#10623494	—	2	1	no	—
45	C₄×#10623494	—	2	1	no	—
46	C₄×#10623494	D₄	1	2	yes	—
47	C₂²×C₈×C₈₉	—	2	1	no	—
48	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	—	4	2	no	—
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	#95×C₈₉	—	2	1	no	—
60	C₂³×D₈₉	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	4	1	no	—
69	C₂×C₄×D₈₉	—	4	1	no	—
70	C₂×C₄×D₈₉	—	4	1	no	—
71	C₂×C₄×D₈₉	—	4	1	no	—
72	C₂×C₄×D₈₉	—	4	1	no	—
73	C₂×C₄×D₈₉	—	4	1	no	—
74	C₂×#10623494	—	2	1	no	—
75	C₂×#10623494	—	2	1	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	2	2	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₂×#10623494	—	4	1	no	—
82	C₂×#10623494	—	4	1	no	—
83	C₂×#10623494	—	4	1	no	—
84	C₂×#10623494	C₂×C₈	1	2	yes	—
85	C₂×#10623494	M₁₆	1	2	yes	—
86	C₄⋊Dic₈₉	—	2	1	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	#13083551	—	4	1	no	—
89	#13083551	—	4	1	no	—
90	C₂²×C₄×C₈₉	—	2	1	no	—
91	C₂²×C₄×C₈₉	—	2	1	no	—
92	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	4	1	no	—
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
108	C₄²×C₈₉	—	4	1	no	—
109	C₄²×C₈₉	—	4	1	no	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	#41	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₄²×C₈₉	#41	1	1	yes	characteristic
114	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	2	no	—
116	C₂²×D₈₉	—	4	1	no	—
117	C₂²×D₈₉	—	4	1	no	—
118	C₂²×D₈₉	—	4	1	no	—
119	C₂²×D₈₉	—	4	1	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	—	2	2	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	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	1	no	—
130	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#126	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
139	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
140	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
144	C₂×Dic₈₉	—	4	1	no	—
145	C₂×Dic₈₉	—	4	1	no	—
146	C₂×Dic₈₉	—	4	1	no	—
147	D₃₅₆	—	2	2	no	—
148	D₃₅₆	—	2	2	no	—
149	D₃₅₆	—	4	1	no	—
150	D₃₅₆	—	4	1	no	—
151	C₄×D₈₉	—	8	1	no	—
152	C₄×D₈₉	—	8	1	no	—
153	C₄×D₈₉	—	8	1	no	—
154	C₄×D₈₉	—	8	1	no	—
155	C₈×C₈₉	—	4	1	no	—
156	C₈×C₈₉	—	4	1	no	—
157	C₈×C₈₉	—	8	1	no	—
158	#10623494	—	2	2	no	—
159	#10623494	—	2	2	no	—
160	#10623494	—	4	2	no	—
161	#10623494	—	8	1	no	—
162	C₂²×C₈₉	#388	1	1	yes	characteristic
163	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₈₉	#436	1	1	yes	characteristic
165	C₂²×C₈₉	#440	1	1	yes	characteristic
166	C₂²×C₈₉	#442	1	1	yes	characteristic
167	C₂²×C₈₉	#444	1	1	yes	characteristic
168	C₂²×C₈₉	#444	1	1	yes	characteristic
169	C₂×D₈₉	—	4	2	no	—
170	C₂×D₈₉	—	4	2	no	—
171	C₂×D₈₉	—	4	2	no	—
172	C₂×D₈₉	—	8	1	no	—
173	C₂×D₈₉	—	8	1	no	—
174	C₂×D₈₉	—	8	1	no	—
175	C₂×D₈₉	—	8	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	8	1	no	—
180	C₄×C₈₉	#431	1	1	yes	characteristic
181	C₄×C₈₉	#433	1	1	yes	characteristic
182	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
183	C₄×C₈₉	#435	1	1	yes	characteristic
184	C₄×C₈₉	#468	1	1	yes	characteristic
185	C₄×C₈₉	#469	1	1	yes	characteristic
186	C₄×C₈₉	#489	1	1	yes	characteristic
187	C₄×C₈₉	#490	1	1	yes	characteristic
188	Dic₈₉	—	8	1	no	—
189	#9556	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	SD₁₆⋊C₈	1	1	yes	characteristic
191	C₂×C₈₉	#1580	1	1	yes	characteristic
192	C₂×C₈₉	#1592	1	1	yes	characteristic
193	C₂×C₈₉	#1594	1	1	yes	characteristic
194	C₂×C₈₉	#1923	1	1	yes	characteristic
195	C₂×C₈₉	#1936	1	1	yes	characteristic
196	C₂×C₈₉	#1938	1	1	yes	characteristic
197	D₈₉	—	8	2	no	—
198	#1954	—	89	1	no	—
199	#1939	D₈₉	1	1	yes	characteristic, complemented
200	C₄×#102	—	89	1	no	—
201	C₂×D₄⋊C₈	—	89	1	no	—
202	C₂×#322	—	89	1	no	—
203	#1279	—	89	1	no	—
204	#1277	—	89	1	no	—
205	C₈₉	#9556	1	1	yes	characteristic, complemented, Sylow
206	C₂×C₄×D₄	—	89	1	no	—
207	C₂×C₄⋊C₈	—	2	1	no	—
208	C₂×C₄⋊C₈	—	89	1	no	—
209	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
210	C₂×#102	—	89	1	no	—
211	C₂×#102	—	89	1	no	—
212	C₂×C₂²⋊C₈	—	178	1	no	—
213	C₂×C₂²⋊C₈	—	178	1	no	—
214	C₂×C₄×C₈	—	89	1	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
217	C₄×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
218	#340	—	89	1	no	—
219	#340	—	89	1	no	—
220	#336	—	2	1	no	—
221	D₄⋊C₈	—	178	2	no	—
222	#322	—	178	2	no	—
223	C₂²×D₄	—	89	1	no	—
224	C₂³×C₄	—	178	1	no	—
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	89	1	no	—
227	C₂²×C₈	—	89	1	no	—
228	C₂²×C₈	—	178	1	no	—
229	C₂²×C₈	—	178	1	no	—
230	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
232	C₄×D₄	—	178	1	no	—
233	C₄×D₄	—	178	1	no	—
234	C₄×D₄	—	178	1	no	—
235	C₄×D₄	—	178	1	no	—
236	C₂×C₄⋊C₄	—	89	1	no	—
237	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₂²⋊C₄	—	178	1	no	—
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
242	C₄⋊C₈	—	2	1	no	—
243	C₄⋊C₈	—	2	1	no	—
244	C₄⋊C₈	—	4	2	no	—
245	C₄⋊C₈	—	178	1	no	—
246	C₄⋊C₈	—	178	1	no	—
247	#102	—	89	4	no	—
248	#102	—	178	2	no	—
249	C₂²⋊C₈	—	356	2	no	—
250	C₂²⋊C₈	—	356	2	no	—
251	C₄×C₈	—	2	1	no	—
252	C₄×C₈	—	2	1	no	—
253	C₄×C₈	—	89	2	no	—
254	C₄×C₈	—	178	1	no	—
255	C₄×C₈	—	178	1	no	—
256	C₄×C₈	—	178	1	no	—
257	#95	—	2	1	no	—
258	C₂⁴	—	178	1	no	—
259	C₂×D₄	—	89	2	no	—
260	C₂×D₄	—	178	1	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	—	178	1	no	—
264	C₂×D₄	—	178	1	no	—
265	C₂²×C₄	—	2	1	no	—
266	C₂²×C₄	—	2	1	no	—
267	C₂²×C₄	—	356	1	no	—
268	C₂²×C₄	—	356	1	no	—
269	C₂²×C₄	—	356	1	no	—
270	C₂²×C₄	—	356	1	no	—
271	C₂²×C₄	—	356	1	no	—
272	C₂²×C₄	—	356	1	no	—
273	C₂²×C₄	—	178	1	no	—
274	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₂²×C₄	#13083574	1	1	yes	characteristic
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	2	1	no	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	4	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	—	89	2	no	—
287	C₂×C₈	—	89	2	no	—
288	C₂×C₈	—	356	1	no	—
289	C₂×C₈	—	356	1	no	—
290	C₂×C₈	—	356	1	no	—
291	C₂×C₈	—	356	1	no	—
292	C₂×C₈	—	356	1	no	—
293	C₂×C₈	—	178	2	no	—
294	C₂×C₈	—	178	1	no	—
295	C₂×C₈	—	178	1	no	—
296	C₂×C₈	—	178	1	no	—
297	C₂×C₈	—	178	1	no	—
298	C₄⋊C₄	—	2	1	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
304	C₂²⋊C₄	—	356	1	no	—
305	C₂²⋊C₄	—	356	1	no	—
306	C₄²	—	4	1	no	—
307	C₄²	—	4	1	no	—
308	C₄²	D₄×D₈₉	1	1	yes	characteristic
309	C₄²	#13083573	1	1	yes	characteristic
310	C₄²	#13083575	1	1	yes	characteristic
311	C₄²	#13083575	1	1	yes	characteristic
312	C₂³	—	356	1	no	—
313	C₂³	—	356	1	no	—
314	C₂³	—	356	1	no	—
315	C₂³	—	356	1	no	—
316	C₂³	—	356	1	no	—
317	C₂³	—	356	1	no	—
318	C₂³	—	178	2	no	—
319	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
320	D₄	—	356	1	no	—
321	D₄	—	356	1	no	—
322	D₄	—	178	2	no	—
323	D₄	—	178	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	356	1	no	—
333	C₂×C₄	—	356	1	no	—
334	C₂×C₄	—	356	1	no	—
335	C₂×C₄	—	712	1	no	—
336	C₂×C₄	—	712	1	no	—
337	C₂×C₄	—	712	1	no	—
338	C₂×C₄	—	712	1	no	—
339	C₂×C₄	#15667970	1	1	yes	characteristic
340	C₂×C₄	#15667974	1	1	yes	characteristic
341	C₂×C₄	#15667984	1	1	yes	characteristic
342	C₂×C₄	#15667997	1	1	yes	characteristic
343	C₂×C₄	#15668001	1	1	yes	characteristic
344	C₂×C₄	#15668007	1	1	yes	characteristic
345	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
346	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
347	C₂×C₄	#15668018	1	1	yes	characteristic
348	C₂×C₄	#15668020	1	1	yes	characteristic
349	C₂×C₄	#15668037	1	1	yes	characteristic
350	C₂×C₄	#15668037	1	1	yes	characteristic
351	C₂×C₄	#15668038	1	1	yes	characteristic
352	C₂×C₄	#15668043	1	1	yes	characteristic
353	C₈	—	4	1	no	—
354	C₈	—	4	1	no	—
355	C₈	—	8	1	no	—
356	C₈	—	356	2	no	—
357	C₈	—	178	2	no	—
358	C₈	—	178	2	no	—
359	C₈	—	712	1	no	—
360	C₂²	—	356	2	no	—
361	C₂²	—	356	2	no	—
362	C₂²	—	356	2	no	—
363	C₂²	—	712	1	no	—
364	C₂²	—	712	1	no	—
365	C₂²	—	712	1	no	—
366	C₂²	—	712	1	no	—
367	C₂²	#19151488	1	1	yes	characteristic
368	C₂²	C₈×C₂²⋊D₈₉	1	1	yes	characteristic
369	C₂²	#19151674	1	1	yes	characteristic
370	C₂²	#19151696	1	1	yes	characteristic
371	C₂²	#19151708	1	1	yes	characteristic
372	C₂²	#19151718	1	1	yes	characteristic
373	C₂²	#19151718	1	1	yes	characteristic
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	8	1	no	—
378	C₄	—	712	1	no	—
379	C₄	#19151650	1	1	yes	characteristic
380	C₄	#19151658	1	1	yes	characteristic
381	C₄	#19151663	1	1	yes	characteristic
382	C₄	#19151668	1	1	yes	characteristic
383	C₄	#19151846	1	1	yes	characteristic
384	C₄	#19151849	1	1	yes	characteristic
385	C₄	#19151960	1	1	yes	characteristic
386	C₄	#19151964	1	1	yes	characteristic
387	C₂	—	712	2	no	—
388	C₂	#26019062	1	1	yes	characteristic
389	C₂	#26019074	1	1	yes	characteristic
390	C₂	#26019152	1	1	yes	characteristic
391	C₂	#26019164	1	1	yes	characteristic
392	C₂	#26020857	1	1	yes	characteristic
393	C₂	#26020948	1	1	yes	characteristic
394	C₂	#26020962	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1939	C₂
socle	C₂³×C₈₉	C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1939	D₈₉
hypocenter	C₈₉	#9556
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1939

Complements

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

N	H ≅ G/N	type
4. C₂×#19151320	C₂	semidirect
5. #26018093	C₂	semidirect
6. #26018086	C₂	semidirect
7. C₈₉×#1939	C₂	semidirect
40. #15667932	C₈	semidirect
199. #1939	D₈₉	semidirect
205. C₈₉	#9556	semidirect

Maximal subgroups

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

subgroup	index
#26021046	2
C₄×#15667932	2
C₂×#19151325	2
C₂×#19151320	2
#26018093	2
#26018086	2
C₈₉×#1939	2
#9556	89