Subgroups — #36445274

Order 22784.

550 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#15667933	C₂	1	1	yes	maximal, characteristic
3	C₂×D₄⋊#10623494	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018091	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018086	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#436×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
9	C₂×#15667933	C₄	1	2	yes	—
10	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
11	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₈₉	—	2	1	no	—
13	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151360	C₄	1	2	yes	—
15	#19151352	—	2	1	no	—
16	D₄⋊#10623494	C₄	1	4	yes	complemented
17	#19151319	—	2	2	no	—
18	C₂²×M₁₆×C₈₉	—	2	1	no	—
19	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
20	#436×C₈₉	—	2	2	no	—
21	#436×C₈₉	C₄	1	4	yes	complemented
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×#10623494	—	2	1	no	—
26	C₂²×#10623494	—	2	1	no	—
27	C₂²×#10623494	C₂×C₄	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₈₉	—	2	1	no	—
30	C₄⋊#10623494	—	2	1	no	—
31	C₄⋊#10623494	C₂×C₄	1	2	yes	—
32	#15667933	—	2	4	no	—
33	C₄×#10623494	—	2	1	no	—
34	C₄×#10623494	—	2	1	no	—
35	C₄×#10623494	—	2	1	no	—
36	C₄×#10623494	C₂×C₄	1	2	yes	—
37	C₈×Dic₈₉	—	4	2	no	—
38	#15667919	—	2	1	no	—
39	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂³×C₄×C₈₉	—	2	1	no	—
41	C₂×M₁₆×C₈₉	—	2	4	no	—
42	C₂×M₁₆×C₈₉	—	4	2	no	—
43	C₂×M₁₆×C₈₉	—	4	2	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	C₄×D₄×C₈₉	—	2	1	no	—
48	C₄×D₄×C₈₉	—	2	1	no	—
49	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
50	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₉	—	2	1	no	—
55	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
56	C₂²⋊C₈×C₈₉	—	2	4	no	—
57	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₂²×Dic₈₉	—	2	2	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	2	2	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	2	2	no	—
65	C₂×#10623494	—	4	2	no	—
66	C₂×#10623494	—	4	1	no	—
67	C₂×#10623494	—	4	1	no	—
68	C₄⋊Dic₈₉	—	2	2	no	—
69	C₄⋊Dic₈₉	M₁₆	1	4	yes	complemented
70	C₄×Dic₈₉	—	4	1	no	—
71	C₄×Dic₈₉	—	4	1	no	—
72	C₂⁴×C₈₉	—	2	1	no	—
73	C₂×D₄×C₈₉	—	2	1	no	—
74	C₂×D₄×C₈₉	—	2	2	no	—
75	C₂×D₄×C₈₉	—	2	2	no	—
76	C₂×D₄×C₈₉	C₄²	1	2	yes	—
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	—	4	1	no	—
81	C₂²×C₄×C₈₉	—	4	1	no	—
82	C₂²×C₄×C₈₉	—	4	1	no	—
83	C₂²×C₄×C₈₉	—	4	1	no	—
84	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
87	M₁₆×C₈₉	—	4	8	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	4	1	no	—
92	C₂×C₈×C₈₉	—	4	1	no	—
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
94	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₈₉	D₈	1	2	yes	—
96	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
97	C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₄⋊C₄×C₈₉	C₄²	1	2	yes	—
99	C₂²⋊C₄×C₈₉	—	2	2	no	—
100	C₂²⋊C₄×C₈₉	—	2	2	no	—
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₈₉	#41	1	1	yes	characteristic
105	C₂³×C₈₉	—	2	2	no	—
106	C₂³×C₈₉	—	4	2	no	—
107	C₂³×C₈₉	—	4	1	no	—
108	C₂³×C₈₉	—	4	1	no	—
109	C₂³×C₈₉	—	4	2	no	—
110	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
111	D₄×C₈₉	—	2	2	no	—
112	D₄×C₈₉	—	2	2	no	—
113	D₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	2	2	no	—
115	C₂×C₄×C₈₉	—	2	2	no	—
116	C₂×C₄×C₈₉	—	4	1	no	—
117	C₂×C₄×C₈₉	—	4	1	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
129	C₂×C₄×C₈₉	#131	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
134	C₂×Dic₈₉	—	2	4	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	C₂×Dic₈₉	—	4	2	no	—
137	C₈×C₈₉	—	2	2	no	—
138	C₈×C₈₉	—	4	2	no	—
139	C₈×C₈₉	#102	1	4	yes	complemented
140	#10623494	—	4	2	no	—
141	#10623494	—	4	2	no	—
142	#10623494	—	4	2	no	—
143	C₂²×C₈₉	—	4	2	no	—
144	C₂²×C₈₉	—	4	2	no	—
145	C₂²×C₈₉	—	4	2	no	—
146	C₂²×C₈₉	—	8	2	no	—
147	C₂²×C₈₉	—	8	2	no	—
148	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₈₉	#435	1	2	yes	—
150	C₂²×C₈₉	C₄×D₈	1	1	yes	characteristic
151	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
152	C₂²×C₈₉	#444	1	2	yes	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	4	2	no	—
156	C₄×C₈₉	C₄×M₁₆	1	1	yes	characteristic
157	C₄×C₈₉	#405	1	1	yes	characteristic
158	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₈₉	#416	1	1	yes	characteristic
160	C₄×C₈₉	#436	1	2	yes	—
161	C₄×C₈₉	C₄×D₈	1	1	yes	characteristic
162	C₄×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
163	Dic₈₉	—	4	4	no	—
164	#8210	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	8	2	no	—
166	C₂×C₈₉	#1581	1	2	yes	—
167	C₂×C₈₉	#1583	1	2	yes	—
168	C₂×C₈₉	#1753	1	1	yes	characteristic
169	C₂×C₈₉	C₄×#102	1	1	yes	characteristic
170	C₂×C₈₉	#1765	1	1	yes	characteristic
171	C₂×#436	D₈₉	1	1	yes	characteristic
172	C₈×C₄⋊C₄	—	89	1	no	—
173	C₄×#102	—	89	1	no	—
174	C₂×D₄⋊C₈	—	89	1	no	—
175	C₂×#322	—	89	1	no	—
176	#1279	—	89	1	no	—
177	#1277	—	89	1	no	—
178	C₈₉	#8210	1	1	yes	characteristic, complemented, Sylow
179	C₂²×M₁₆	—	2	1	no	—
180	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
181	#436	—	2	2	no	—
182	#436	Dic₈₉	1	4	yes	complemented
183	C₂×C₄⋊C₈	—	89	1	no	—
184	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
185	C₂×#102	—	89	2	no	—
186	C₂×C₂²⋊C₈	—	2	1	no	—
187	C₂×C₄×C₈	—	89	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	C₂×C₄×C₈	—	178	1	no	—
190	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
191	C₄×C₄⋊C₄	—	89	1	no	—
192	#340	—	89	2	no	—
193	#336	—	178	1	no	—
194	D₄⋊C₈	—	89	4	no	—
195	#322	—	178	2	no	—
196	C₂²×D₄	C₄×D₈₉	1	1	yes	characteristic
197	C₂³×C₄	—	2	1	no	—
198	C₂×M₁₆	—	2	4	no	—
199	C₂×M₁₆	—	4	2	no	—
200	C₂×M₁₆	—	4	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	89	2	no	—
203	C₂²×C₈	—	178	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	C₄×D₄	—	2	1	no	—
208	C₄×D₄	—	2	1	no	—
209	C₄×D₄	C₄×D₈₉	1	2	yes	—
210	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
211	C₂×C₄⋊C₄	—	89	2	no	—
212	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	2	1	no	—
214	C₂×C₄²	—	178	1	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	89	2	no	—
218	C₄⋊C₈	—	178	1	no	—
219	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
220	#102	—	178	4	no	—
221	C₂²⋊C₈	—	2	4	no	—
222	C₄×C₈	—	89	2	no	—
223	C₄×C₈	—	356	2	no	—
224	C₄×C₈	—	178	1	no	—
225	C₄×C₈	—	178	1	no	—
226	C₄×C₈	—	178	1	no	—
227	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
228	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
229	#95	—	178	1	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	C₄×Dic₈₉	1	2	yes	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	178	2	no	—
243	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
245	C₂²×C₄	#13083574	1	1	yes	characteristic
246	M₁₆	—	4	8	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	356	2	no	—
253	C₂×C₈	—	356	1	no	—
254	C₂×C₈	—	356	1	no	—
255	C₂×C₈	—	178	2	no	—
256	C₂×C₈	—	178	2	no	—
257	C₂×C₈	—	178	2	no	—
258	C₂×C₈	—	178	2	no	—
259	C₂×C₈	—	178	2	no	—
260	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
261	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
262	C₂×C₈	#13083559	1	2	yes	—
263	C₂×C₈	#13083562	1	2	yes	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	89	4	no	—
266	C₄⋊C₄	—	178	2	no	—
267	C₄⋊C₄	C₄×Dic₈₉	1	2	yes	—
268	C₂²⋊C₄	—	2	2	no	—
269	C₂²⋊C₄	—	2	2	no	—
270	C₄²	—	356	1	no	—
271	C₄²	—	356	1	no	—
272	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
273	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
274	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₄²	#13083574	1	1	yes	characteristic
276	C₂³	—	2	2	no	—
277	C₂³	—	4	2	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
282	D₄	—	2	2	no	—
283	D₄	—	2	2	no	—
284	D₄	—	4	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	178	4	no	—
296	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
298	C₂×C₄	#15667971	1	1	yes	characteristic
299	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
303	C₂×C₄	#15668020	1	2	yes	—
304	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
306	C₂×C₄	#15668032	1	1	yes	characteristic
307	C₂×C₄	#15668032	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	356	2	no	—
311	C₈	—	356	2	no	—
312	C₈	—	356	2	no	—
313	C₈	#15667933	1	4	yes	complemented
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	C₄×C₂²⋊Dic₈₉	1	1	yes	characteristic
320	C₂²	#19151667	1	2	yes	—
321	C₂²	C₄×#13083559	1	1	yes	characteristic
322	C₂²	C₄×#13083562	1	1	yes	characteristic
323	C₂²	#19151718	1	2	yes	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	2	no	—
327	C₄	—	356	4	no	—
328	C₄	M₁₆×Dic₈₉	1	1	yes	characteristic
329	C₄	#19151556	1	1	yes	characteristic
330	C₄	C₂×#15667933	1	1	yes	characteristic
331	C₄	#19151594	1	1	yes	characteristic
332	C₄	#19151674	1	2	yes	—
333	C₄	C₄×#13083559	1	1	yes	characteristic
334	C₄	C₄×#13083562	1	1	yes	characteristic
335	C₂	—	8	2	no	—
336	C₂	#26019079	1	2	yes	—
337	C₂	#26019093	1	2	yes	—
338	C₂	#26020033	1	1	yes	characteristic
339	C₂	C₄×#15667933	1	1	yes	characteristic
340	C₂	#26020094	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#436×C₈₉	C₂
socle	C₂³×C₈₉	C₄×D₄
prime radical	C₂²×D₄×C₈₉	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#436	D₈₉
hypocenter	C₈₉	#8210
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#436

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₈₉	C₂	semidirect
4. C₂×#19151319	C₂	semidirect
5. #26018091	C₂	semidirect
6. #26018086	C₂	semidirect
16. D₄⋊#10623494	C₄	semidirect
21. #436×C₈₉	C₄	semidirect
69. C₄⋊Dic₈₉	M₁₆	semidirect
139. C₈×C₈₉	#102	semidirect
178. C₈₉	#8210	semidirect
182. #436	Dic₈₉	semidirect
313. C₈	#15667933	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₈₉	2
C₄×#15667933	2
C₂×D₄⋊#10623494	2
C₂×#19151319	2
#26018091	2
#26018086	2
C₂×#436×C₈₉	2
#8210	89