Subgroups — #36520106

Order 22784.

718 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	#26029213	C₂	1	1	yes	maximal, characteristic
2	C₈×#13083574	C₂	1	1	yes	maximal, characteristic
3	#26028897	C₂	1	1	yes	maximal, characteristic
4	#26018551	C₂	1	2	yes	maximal, complemented
5	#26018546	C₂	1	2	yes	maximal, complemented
6	#26018545	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018543	C₂	1	2	yes	maximal, complemented
8	#26018540	C₂	1	2	yes	maximal, complemented
9	#26018539	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083574	C₂²	1	1	yes	characteristic
13	#19151725	C₄	1	2	yes	—
14	C₈×Dic₁₇₈	C₄	1	2	yes	—
15	#19151675	C₄	1	2	yes	—
16	#19151666	C₄	1	4	yes	—
17	C₈×D₃₅₆	C₄	1	2	yes	—
18	C₈×C₂²⋊D₈₉	C₄	1	4	yes	—
19	#19151660	C₄	1	4	yes	—
20	#19151651	C₂²	1	1	yes	characteristic
21	#19151649	C₂²	1	2	yes	—
22	#19151649	C₂²	1	2	yes	—
23	#19151649	C₂²	1	2	yes	—
24	#19151649	C₂²	1	2	yes	—
25	#19151626	C₄	1	2	yes	—
26	#19151566	C₄	1	2	yes	—
27	C₄×#13083557	C₂²	1	2	yes	—
28	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₈₉	C₂²	1	2	yes	—
31	#19151320	C₂²	1	1	yes	characteristic
32	#19151320	C₂²	1	1	yes	characteristic
33	#19151320	C₂²	1	1	yes	characteristic
34	#19151320	C₂²	1	1	yes	characteristic
35	#19151320	C₂²	1	2	yes	—
36	#19151320	C₂²	1	2	yes	—
37	#19151320	C₂²	1	2	yes	—
38	#19151319	C₂²	1	2	yes	—
39	#431×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
42	#322×C₈₉	C₂²	1	2	yes	—
43	C₈²×C₈₉	C₂²	1	2	yes	—
44	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
45	C₂×#13083557	C₂×C₄	1	4	yes	—
46	C₂²×#10623494	C₂×C₄	1	2	yes	—
47	C₂²×#10623494	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
49	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
50	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
51	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
52	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
53	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
54	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
55	#15667971	C₂×C₄	1	1	yes	characteristic
56	#15667969	C₂×C₄	1	2	yes	—
57	C₄²×D₈₉	C₂³	1	2	yes	—
58	C₄⋊#10623494	C₂×C₄	1	2	yes	—
59	#15667941	C₂×C₄	1	4	yes	—
60	#15667941	C₂×C₄	1	4	yes	—
61	#15667941	C₂×C₄	1	4	yes	—
62	#15667941	C₂×C₄	1	4	yes	—
63	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	4	yes	—
65	#15667924	C₂×C₄	1	4	yes	—
66	#15667924	C₂×C₄	1	4	yes	—
67	#15667924	C₂×C₄	1	4	yes	—
68	#15667922	C₂×C₄	1	4	yes	—
69	C₄×#10623494	C₂×C₄	1	2	yes	—
70	C₄×#10623494	C₂×C₄	1	2	yes	—
71	C₄×#10623494	C₂³	1	1	yes	characteristic
72	C₄×#10623494	C₂³	1	1	yes	characteristic
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	2	yes	—
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
77	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
78	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
81	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
82	#97×C₈₉	C₂×C₄	1	2	yes	—
83	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	2	yes	—
87	C₄×C₈×C₈₉	C₂³	1	2	yes	—
88	#13083574	—	2	4	no	—
89	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
90	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
91	C₂×C₂²⋊D₈₉	C₄²	1	2	yes	—
92	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
93	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
95	#13083557	—	4	4	no	—
96	C₂×#10623494	—	2	4	no	—
97	C₂×#10623494	—	4	2	no	—
98	C₂×#10623494	C₂²×C₄	1	4	yes	—
99	C₂×#10623494	C₂²×C₄	1	2	yes	—
100	C₂×#10623494	C₂²×C₄	1	2	yes	—
101	C₂×#10623494	C₂²×C₄	1	2	yes	—
102	C₂×#10623494	C₂²×C₄	1	2	yes	—
103	C₈×D₈₉	—	4	4	no	—
104	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
105	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
106	#13083553	C₄²	1	2	yes	—
107	#13083553	C₄²	1	2	yes	—
108	C₂²⋊Dic₈₉	C₄²	1	2	yes	—
109	#13083551	C₄²	1	2	yes	—
110	#13083551	C₄²	1	2	yes	—
111	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
112	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
113	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₈₉	—	2	4	no	—
118	C₂×C₈×C₈₉	—	4	2	no	—
119	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
121	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
123	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
128	C₂²×D₈₉	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₈₉	—	2	2	no	—
130	C₂×C₄×C₈₉	—	2	2	no	—
131	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#131	1	4	yes	—
135	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
138	C₂²⋊D₈₉	—	2	4	no	—
139	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
140	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
141	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
142	D₃₅₆	—	2	2	no	—
143	C₄×D₈₉	—	2	4	no	—
144	C₄×D₈₉	—	2	4	no	—
145	C₄×D₈₉	—	2	4	no	—
146	Dic₁₇₈	—	2	2	no	—
147	C₈×C₈₉	—	2	4	no	—
148	C₈×C₈₉	—	2	2	no	—
149	C₈×C₈₉	—	2	2	no	—
150	C₈×C₈₉	—	2	4	no	—
151	#10623494	—	2	2	no	—
152	#10623494	—	2	2	no	—
153	#10623494	—	2	4	no	—
154	#10623494	—	2	2	no	—
155	#10623494	—	2	2	no	—
156	C₂²×C₈₉	—	2	1	no	—
157	C₂²×C₈₉	—	2	1	no	—
158	C₂²×C₈₉	#405	1	2	yes	—
159	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	2	2	no	—
162	C₂×D₈₉	—	2	2	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	—	2	1	no	—
165	C₄×C₈₉	#405	1	2	yes	—
166	C₄×C₈₉	C₂×#131	1	2	yes	—
167	C₄×C₈₉	#568	1	2	yes	—
168	C₄×C₈₉	#568	1	2	yes	—
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	2	2	no	—
171	Dic₈₉	—	2	2	no	—
172	#17329	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	2	2	no	—
174	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
175	C₂×C₈₉	#2875	1	1	yes	characteristic
176	C₂×C₈₉	#2875	1	1	yes	characteristic
177	D₈₉	—	4	2	no	—
178	#2965	—	89	1	no	—
179	C₈×#41	—	89	1	no	—
180	#2922	—	89	1	no	—
181	#1454	—	89	2	no	—
182	#1453	—	89	2	no	—
183	#1452	—	89	1	no	—
184	#1452	—	89	2	no	—
185	#1451	—	89	1	no	—
186	#1451	—	89	2	no	—
187	C₈×M₁₆	D₈₉	1	1	yes	characteristic, complemented
188	C₂×#322	—	89	1	no	—
189	C₈₉	#17329	1	1	yes	characteristic, complemented, Sylow
190	C₄×#41	—	89	1	no	—
191	#446	—	89	2	no	—
192	C₈×Q₈	—	89	2	no	—
193	#436	—	89	2	no	—
194	#435	—	89	4	no	—
195	C₈×D₄	—	89	4	no	—
196	C₈×D₄	—	89	2	no	—
197	#433	—	89	4	no	—
198	#431	—	89	1	no	—
199	#431	—	89	2	no	—
200	#431	—	89	2	no	—
201	#431	—	89	2	no	—
202	#431	—	89	2	no	—
203	#431	C₂×D₈₉	1	1	yes	characteristic
204	#424	—	89	2	no	—
205	#408	—	89	2	no	—
206	C₄×M₁₆	—	89	2	no	—
207	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
208	C₂×C₄×C₈	—	89	1	no	—
209	C₂×C₄×C₈	—	89	1	no	—
210	C₂×C₄×C₈	—	89	2	no	—
211	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
212	#322	—	89	1	no	—
213	#322	—	89	1	no	—
214	#322	—	89	1	no	—
215	#322	—	89	1	no	—
216	#322	—	89	2	no	—
217	#322	—	89	2	no	—
218	#322	—	89	2	no	—
219	#322	—	89	2	no	—
220	#322	C₂×D₈₉	1	2	yes	—
221	C₈²	C₂×D₈₉	1	2	yes	—
222	C₂×#41	—	89	1	no	—
223	C₂×M₁₆	—	89	4	no	—
224	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	—	89	4	no	—
228	C₂²×C₈	C₄×D₈₉	1	2	yes	—
229	C₄×Q₈	—	89	1	no	—
230	C₄×Q₈	—	89	1	no	—
231	C₄×D₄	—	89	2	no	—
232	C₄×D₄	—	89	2	no	—
233	C₄×D₄	—	89	1	no	—
234	C₄×D₄	—	89	1	no	—
235	#117	—	89	1	no	—
236	#117	—	89	2	no	—
237	C₂×C₄²	—	89	2	no	—
238	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
239	C₄⋊C₈	—	89	4	no	—
240	C₄⋊C₈	—	89	4	no	—
241	C₄⋊C₈	—	89	4	no	—
242	C₄⋊C₈	—	89	4	no	—
243	C₄⋊C₈	—	89	2	no	—
244	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
245	C₂²⋊C₈	—	89	2	no	—
246	C₂²⋊C₈	—	89	4	no	—
247	C₂²⋊C₈	—	89	4	no	—
248	C₂²⋊C₈	—	89	4	no	—
249	C₂²⋊C₈	—	89	4	no	—
250	C₂²⋊C₈	C₄×D₈₉	1	2	yes	—
251	#97	—	89	4	no	—
252	#97	C₄×D₈₉	1	2	yes	—
253	C₄×C₈	—	89	4	no	—
254	C₄×C₈	—	89	2	no	—
255	C₄×C₈	—	89	2	no	—
256	C₄×C₈	—	89	1	no	—
257	C₄×C₈	—	89	1	no	—
258	C₄×C₈	—	89	1	no	—
259	C₄×C₈	—	89	1	no	—
260	C₄×C₈	—	89	2	no	—
261	C₄×C₈	C₄×D₈₉	1	2	yes	—
262	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₈₉	1	2	yes	—
264	C₄×C₈	C₂²×D₈₉	1	2	yes	—
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	#41	—	178	4	no	—
267	C₂×Q₈	—	89	1	no	—
268	C₂×D₄	—	89	2	no	—
269	C₂×D₄	—	89	1	no	—
270	C₂²×C₄	—	89	2	no	—
271	C₂²×C₄	—	89	2	no	—
272	C₂²×C₄	—	89	2	no	—
273	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
276	M₁₆	—	2	4	no	—
277	M₁₆	—	356	4	no	—
278	C₂×C₈	—	4	2	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	2	no	—
281	C₂×C₈	—	89	2	no	—
282	C₂×C₈	—	89	2	no	—
283	C₂×C₈	—	89	4	no	—
284	C₂×C₈	—	356	4	no	—
285	C₂×C₈	—	356	2	no	—
286	C₂×C₈	—	178	4	no	—
287	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
288	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
290	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
291	C₄⋊C₄	—	89	2	no	—
292	C₄⋊C₄	—	89	2	no	—
293	C₄⋊C₄	—	89	1	no	—
294	C₄⋊C₄	—	89	1	no	—
295	C₂²⋊C₄	—	89	2	no	—
296	C₂²⋊C₄	—	89	2	no	—
297	C₂²⋊C₄	—	89	2	no	—
298	C₄²	—	89	2	no	—
299	C₄²	—	89	2	no	—
300	C₄²	—	89	2	no	—
301	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
302	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
303	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
304	C₄²	C₂³×D₈₉	1	1	yes	characteristic
305	C₂³	—	89	2	no	—
306	C₂³	C₄²×D₈₉	1	1	yes	characteristic
307	Q₈	—	178	2	no	—
308	D₄	—	178	4	no	—
309	D₄	—	178	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	178	4	no	—
316	C₂×C₄	—	178	4	no	—
317	C₂×C₄	—	178	4	no	—
318	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
319	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
320	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
321	C₂×C₄	#15668018	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
325	C₈	—	2	4	no	—
326	C₈	—	2	4	no	—
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	178	2	no	—
330	C₈	—	178	2	no	—
331	C₈	—	178	2	no	—
332	C₈	—	178	2	no	—
333	C₈	—	178	4	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	178	2	no	—
337	C₂²	—	178	2	no	—
338	C₂²	—	178	2	no	—
339	C₂²	#19151552	1	2	yes	—
340	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	178	2	no	—
344	C₄	—	178	2	no	—
345	C₄	—	178	2	no	—
346	C₄	#19151552	1	2	yes	—
347	C₄	C₂×#15668018	1	2	yes	—
348	C₄	#19152363	1	2	yes	—
349	C₄	#19152364	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	356	2	no	—
352	C₂	C₂×#19151552	1	1	yes	characteristic
353	C₂	#26028679	1	1	yes	characteristic
354	C₂	#26028681	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂²×C₄²
Fitting	C₈×M₁₆×C₈₉	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₈×M₁₆	D₈₉
hypocenter	C₈₉	#17329
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄² ◁ C₈×M₁₆

Complements

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

N	H ≅ G/N	type
4. #26018551	C₂	semidirect
5. #26018546	C₂	semidirect
6. #26018545	C₂	semidirect
7. #26018543	C₂	semidirect
8. #26018540	C₂	semidirect
9. #26018539	C₂	semidirect
10. C₂×#19151320	C₂	semidirect
11. C₈×M₁₆×C₈₉	C₂	semidirect
187. C₈×M₁₆	D₈₉	semidirect
189. C₈₉	#17329	semidirect

Maximal subgroups

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

subgroup	index
#26029213	2
C₈×#13083574	2
#26028897	2
#26018551	2
#26018546	2
#26018545	2
#26018543	2
#26018540	2
#26018539	2
C₂×#19151320	2
C₈×M₁₆×C₈₉	2
#17329	89