Subgroups — #36441018

Order 22784.

490 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020232	C₂	1	1	yes	maximal, characteristic
2	#26018482	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#1797	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄⋊Dic₈₉	—	2	1	no	—
5	C₂×C₄⋊#10623494	—	2	1	no	—
6	C₂×C₂²⋊#10623494	—	2	1	no	—
7	C₂×#15667919	—	2	1	no	—
8	C₂×#15667919	C₄	1	2	yes	—
9	#19151353	C₄	1	2	yes	complemented
10	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊C₈×C₈₉	C₄	1	2	yes	complemented
12	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
13	#336×C₈₉	—	2	1	no	—
14	C₂³×Dic₈₉	—	2	2	no	—
15	C₂²×#10623494	—	2	2	no	—
16	C₂×C₄⋊Dic₈₉	—	2	4	no	—
17	C₂×C₄⋊Dic₈₉	—	4	2	no	—
18	C₂×C₄⋊Dic₈₉	—	4	1	no	—
19	C₂×C₄⋊Dic₈₉	—	4	1	no	—
20	C₄⋊#10623494	—	4	2	no	—
21	C₂²⋊#10623494	—	4	1	no	—
22	C₂²⋊#10623494	—	4	1	no	—
23	#15667919	—	4	2	no	—
24	#15667919	—	4	1	no	—
25	#15667919	—	4	1	no	—
26	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
28	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₉	—	2	2	no	—
30	C₂×C₄²×C₈₉	—	4	2	no	—
31	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₈₉	Q₈	1	1	yes	characteristic
35	C₄⋊C₈×C₈₉	—	4	2	no	—
36	C₂²⋊C₈×C₈₉	—	4	1	no	—
37	C₂²⋊C₈×C₈₉	—	4	1	no	—
38	C₂²×Dic₈₉	—	2	4	no	—
39	C₂²×Dic₈₉	—	4	2	no	—
40	C₂²×Dic₈₉	—	4	2	no	—
41	C₂²×Dic₈₉	—	4	2	no	—
42	C₂²×Dic₈₉	—	4	2	no	—
43	C₂²×Dic₈₉	—	4	2	no	—
44	C₂²×Dic₈₉	—	4	2	no	—
45	C₂×#10623494	—	4	2	no	—
46	C₂×#10623494	—	4	2	no	—
47	C₂×#10623494	—	4	2	no	—
48	C₄⋊Dic₈₉	—	4	4	no	—
49	C₄⋊Dic₈₉	—	4	4	no	—
50	C₂⁴×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	—	2	1	no	—
55	C₂²×C₄×C₈₉	—	4	1	no	—
56	C₂²×C₄×C₈₉	—	4	1	no	—
57	C₂²×C₄×C₈₉	—	4	1	no	—
58	C₂²×C₄×C₈₉	—	4	1	no	—
59	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
68	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
70	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	Q₁₆	1	1	yes	characteristic
73	C₂×C₈×C₈₉	—	4	2	no	—
74	C₂×C₈×C₈₉	—	4	2	no	—
75	C₂×C₈×C₈₉	—	4	2	no	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	4	1	no	—
79	C₄²×C₈₉	—	4	1	no	—
80	C₂³×C₈₉	—	2	1	no	—
81	C₂³×C₈₉	—	2	1	no	—
82	C₂³×C₈₉	—	2	1	no	—
83	C₂³×C₈₉	—	2	1	no	—
84	C₂³×C₈₉	#95	1	1	yes	characteristic
85	C₂³×C₈₉	#100	1	1	yes	characteristic
86	C₂³×C₈₉	#101	1	1	yes	characteristic
87	C₂³×C₈₉	#102	1	1	yes	characteristic
88	C₂³×C₈₉	#102	1	1	yes	characteristic
89	C₂³×C₈₉	#103	1	1	yes	characteristic
90	C₂³×C₈₉	#103	1	1	yes	characteristic
91	C₂×C₄×C₈₉	—	2	1	no	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	1	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	1	no	—
102	C₂×C₄×C₈₉	—	2	1	no	—
103	C₂×C₄×C₈₉	—	2	1	no	—
104	C₂×C₄×C₈₉	—	2	1	no	—
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	4	1	no	—
108	C₂×C₄×C₈₉	—	4	1	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	—	4	1	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	#99	1	2	yes	—
114	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#103	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#104	1	2	yes	—
119	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#107	1	1	yes	characteristic
123	C₂×Dic₈₉	—	4	4	no	—
124	C₂×Dic₈₉	—	4	4	no	—
125	C₂×Dic₈₉	—	4	4	no	—
126	C₂×Dic₈₉	—	8	2	no	—
127	C₂×Dic₈₉	—	8	2	no	—
128	C₂×Dic₈₉	—	8	2	no	—
129	C₂×Dic₈₉	—	8	2	no	—
130	C₈×C₈₉	—	8	2	no	—
131	#10623494	—	8	2	no	—
132	C₂²×C₈₉	—	2	1	no	—
133	C₂²×C₈₉	—	2	1	no	—
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	#331	1	1	yes	characteristic
145	C₂²×C₈₉	#332	1	1	yes	characteristic
146	C₂²×C₈₉	#332	1	1	yes	characteristic
147	C₂²×C₈₉	#333	1	1	yes	characteristic
148	C₂²×C₈₉	H(3,ℤ/4)	1	1	yes	characteristic
149	C₂²×C₈₉	#339	1	1	yes	characteristic
150	C₂²×C₈₉	#340	1	1	yes	characteristic
151	C₂²×C₈₉	#340	1	1	yes	characteristic
152	C₂²×C₈₉	#342	1	1	yes	characteristic
153	C₂²×C₈₉	#343	1	1	yes	characteristic
154	C₂²×C₈₉	#344	1	1	yes	characteristic
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	4	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	#327	1	2	yes	—
164	C₄×C₈₉	#328	1	2	yes	—
165	Dic₈₉	—	8	4	no	—
166	#7039	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	1	no	—
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	#1285	1	1	yes	characteristic
172	C₂×C₈₉	#1286	1	1	yes	characteristic
173	C₂×C₈₉	#1289	1	1	yes	characteristic
174	C₂×C₈₉	#1294	1	1	yes	characteristic
175	C₂×C₈₉	#1296	1	1	yes	characteristic
176	C₂×C₈₉	#1303	1	1	yes	characteristic
177	C₂×C₈₉	#1305	1	1	yes	characteristic
178	#1797	—	89	1	no	—
179	#1797	D₈₉	1	1	yes	characteristic
180	C₄⋊#95	—	89	1	no	—
181	C₈₉	#7039	1	1	yes	characteristic, complemented, Sylow
182	C₂²×C₄⋊C₄	—	178	1	no	—
183	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
184	C₂×C₄⋊C₈	—	178	1	no	—
185	C₂×C₄⋊C₈	Dic₈₉	1	2	yes	complemented
186	C₂×C₂²⋊C₈	—	2	1	no	—
187	C₂×C₂²⋊C₈	—	178	1	no	—
188	C₂×#95	—	89	2	no	—
189	C₂×#95	—	178	1	no	—
190	#336	—	2	1	no	—
191	#336	—	89	2	no	—
192	C₂³×C₄	—	178	2	no	—
193	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
194	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
195	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	178	2	no	—
198	C₂×C₄⋊C₄	—	356	2	no	—
199	C₂×C₄⋊C₄	—	356	1	no	—
200	C₂×C₄⋊C₄	—	356	1	no	—
201	C₂×C₄⋊C₄	—	178	4	no	—
202	C₂×C₄²	—	4	2	no	—
203	C₂×C₄²	Dic₁₇₈	1	1	yes	characteristic
204	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
205	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
206	C₂×C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
207	C₄⋊C₈	—	4	2	no	—
208	C₄⋊C₈	—	356	2	no	—
209	C₂²⋊C₈	—	4	1	no	—
210	C₂²⋊C₈	—	4	1	no	—
211	C₂²⋊C₈	—	356	1	no	—
212	C₂²⋊C₈	—	356	1	no	—
213	#95	—	356	2	no	—
214	#95	—	356	1	no	—
215	#95	—	356	1	no	—
216	C₂⁴	#13083551	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	356	2	no	—
226	C₂²×C₄	—	356	2	no	—
227	C₂²×C₄	—	356	2	no	—
228	C₂²×C₄	—	356	2	no	—
229	C₂²×C₄	—	356	2	no	—
230	C₂²×C₄	—	356	2	no	—
231	C₂²×C₄	—	178	4	no	—
232	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
233	C₂²×C₄	#13083551	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
235	C₂²×C₄	#13083553	1	1	yes	characteristic
236	C₂²×C₄	#13083553	1	1	yes	characteristic
237	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
238	C₂²×C₄	#13083559	1	1	yes	characteristic
239	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
240	C₂²×C₄	#13083561	1	1	yes	characteristic
241	C₂²×C₄	#13083562	1	1	yes	characteristic
242	C₂²×C₄	#13083563	1	1	yes	characteristic
243	C₂²×C₄	#13083563	1	1	yes	characteristic
244	C₂²×C₄	#13083564	1	1	yes	characteristic
245	C₂²×C₄	Dic₃₅₆	1	1	yes	characteristic
246	C₂×C₈	—	4	2	no	—
247	C₂×C₈	—	4	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	356	2	no	—
250	C₂×C₈	—	356	2	no	—
251	C₂×C₈	—	356	2	no	—
252	C₄⋊C₄	—	356	4	no	—
253	C₄⋊C₄	—	356	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	4	1	no	—
257	C₄²	—	4	1	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	2	1	no	—
260	C₂³	—	2	1	no	—
261	C₂³	—	2	1	no	—
262	C₂³	#15667919	1	1	yes	characteristic
263	C₂³	#15667928	1	1	yes	characteristic
264	C₂³	#15667930	1	1	yes	characteristic
265	C₂³	#15667932	1	1	yes	characteristic
266	C₂³	#15667934	1	1	yes	characteristic
267	C₂³	#15667935	1	1	yes	characteristic
268	C₂³	#15667937	1	1	yes	characteristic
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	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	—	356	4	no	—
292	C₂×C₄	—	356	4	no	—
293	C₂×C₄	—	356	4	no	—
294	C₂×C₄	—	712	2	no	—
295	C₂×C₄	—	712	2	no	—
296	C₂×C₄	—	712	2	no	—
297	C₂×C₄	—	712	2	no	—
298	C₂×C₄	#15667927	1	2	yes	—
299	C₂×C₄	#15667933	1	1	yes	characteristic
300	C₂×C₄	#15667934	1	1	yes	characteristic
301	C₂×C₄	#15667936	1	1	yes	characteristic
302	C₂×C₄	#15667937	1	1	yes	characteristic
303	C₂×C₄	#15667939	1	2	yes	—
304	C₂×C₄	#15667943	1	1	yes	characteristic
305	C₂×C₄	#15667944	1	1	yes	characteristic
306	C₂×C₄	#15667945	1	1	yes	characteristic
307	C₂×C₄	#15667946	1	1	yes	characteristic
308	C₈	—	8	2	no	—
309	C₈	—	712	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	#19151341	1	1	yes	characteristic
323	C₂²	#19151343	1	1	yes	characteristic
324	C₂²	#19151345	1	1	yes	characteristic
325	C₂²	#19151346	1	1	yes	characteristic
326	C₂²	#19151355	1	1	yes	characteristic
327	C₂²	#19151357	1	1	yes	characteristic
328	C₂²	#19151360	1	1	yes	characteristic
329	C₂²	#19151361	1	1	yes	characteristic
330	C₂²	#19151365	1	1	yes	characteristic
331	C₂²	#19151366	1	1	yes	characteristic
332	C₂²	#19151370	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	712	4	no	—
342	C₄	#19151332	1	2	yes	—
343	C₄	#19151335	1	2	yes	—
344	C₂	—	2	1	no	—
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	—	2	1	no	—
348	C₂	#26018109	1	1	yes	characteristic
349	C₂	#26018112	1	1	yes	characteristic
350	C₂	#26018120	1	1	yes	characteristic
351	C₂	#26018130	1	1	yes	characteristic
352	C₂	#26018136	1	1	yes	characteristic
353	C₂	#26018154	1	1	yes	characteristic
354	C₂	#26018157	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667919
commutator	C₂²×C₄×C₈₉	C₄²
Frattini	C₂²×C₄²	C₂×D₈₉
Fitting	C₈₉×#1797	C₂
socle	C₂³×C₈₉	#95
prime radical	C₂⁴×C₈₉	C₂²⋊C₄
cosocle	C₂²×C₄²×C₈₉	C₂²
hypercenter	#1797	D₈₉
hypocenter	C₈₉	#7039
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ #1797

Complements

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

N	H ≅ G/N	type
9. #19151353	C₄	semidirect
11. C₂×C₄⋊C₈×C₈₉	C₄	semidirect
181. C₈₉	#7039	semidirect
185. C₂×C₄⋊C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020232	2
#26018482	2
C₈₉×#1797	2
#7039	89