Subgroups — #36712570

Order 22784.

466 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152027	C₂	1	1	yes	maximal, characteristic, complemented
2	#508×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022572	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022569	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022568	C₂	1	2	yes	maximal, complemented
6	#26022565	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022563	C₂	1	2	yes	maximal, complemented
8	#26022560	C₂	1	2	yes	maximal, complemented
9	#26022558	C₂	1	2	yes	maximal, complemented
10	M₆₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2263	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	—	2	1	no	—
13	#135×D₈₉	—	2	1	no	—
14	#19152031	C₂²	1	1	yes	characteristic, complemented
15	#19152030	C₂²	1	1	yes	characteristic
16	#19152029	—	2	1	no	—
17	#19152028	—	2	1	no	—
18	#19152027	—	2	1	no	—
19	#19152027	—	2	1	no	—
20	#19152027	C₂²	1	2	yes	—
21	#19152027	C₂²	1	1	yes	characteristic, complemented
22	#19152027	C₂²	1	2	yes	complemented
23	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
24	Q₃₂×D₈₉	C₂²	1	1	yes	characteristic, complemented
25	C₂×#15667956	—	2	1	no	—
26	SD₃₂×D₈₉	—	2	1	no	—
27	D₁₆×D₈₉	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
29	Dic₁₄₂₄	C₂²	1	2	yes	complemented
30	#19151438	C₂²	1	2	yes	complemented
31	#19151437	C₂²	1	2	yes	complemented
32	#19151436	C₂²	1	2	yes	complemented
33	#19151436	C₂²	1	2	yes	complemented
34	#19151433	C₂²	1	2	yes	complemented
35	#19151432	C₂²	1	1	yes	characteristic, complemented
36	#19151431	C₂²	1	2	yes	complemented
37	C₃₂×D₈₉	C₂²	1	2	yes	complemented
38	#508×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	SD₆₄×C₈₉	C₂²	1	2	yes	complemented
41	D₃₂×C₈₉	C₂²	1	2	yes	complemented
42	M₆₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	C₂×#13083573	—	4	1	no	—
44	#41×D₈₉	—	4	1	no	—
45	#15668035	—	2	1	no	—
46	#15668034	—	2	1	no	—
47	#15668033	—	4	1	no	—
48	#15668032	—	4	1	no	—
49	#15668031	—	2	2	no	—
50	#15668031	—	2	2	no	—
51	#15668031	—	2	1	no	—
52	#15668031	—	4	1	no	—
53	#15668031	—	4	1	no	—
54	C₂×Dic₃₅₆	—	2	1	no	—
55	Q₁₆×D₈₉	—	2	1	no	—
56	C₂×#13083562	—	4	1	no	—
57	SD₁₆×D₈₉	—	4	1	no	—
58	D₈×D₈₉	—	2	1	no	—
59	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
60	Dic₇₁₂	—	2	1	no	—
61	Dic₇₁₂	C₂³	1	1	yes	characteristic
62	Dic₇₁₂	C₂³	1	2	yes	—
63	#15667958	—	2	1	no	—
64	#15667957	—	2	1	no	—
65	#15667956	—	2	1	no	—
66	#15667956	—	2	2	no	—
67	#15667956	—	2	2	no	—
68	#15667955	—	2	1	no	—
69	D₁₄₂₄	C₂³	1	1	yes	characteristic
70	#15667953	—	2	1	no	—
71	C₂×#13083549	D₄	1	1	yes	characteristic
72	C₁₆×D₈₉	D₄	1	1	yes	characteristic
73	C₁₆×D₈₉	D₄	1	1	yes	characteristic
74	C₁₆×D₈₉	C₂³	1	1	yes	characteristic
75	C₁₆×D₈₉	C₂³	1	1	yes	characteristic
76	#15667918	C₂³	1	2	yes	—
77	#135×C₈₉	—	2	1	no	—
78	C₂×D₈×C₈₉	—	2	1	no	—
79	Q₃₂×C₈₉	C₂³	1	1	yes	characteristic
80	SD₃₂×C₈₉	—	2	1	no	—
81	D₁₆×C₈₉	—	2	1	no	—
82	D₁₆×C₈₉	C₂³	1	2	yes	—
83	D₁₆×C₈₉	C₂³	1	1	yes	characteristic
84	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
85	C₃₂×C₈₉	C₂³	1	2	yes	—
86	#13083575	—	4	1	no	—
87	#13083574	—	4	1	no	—
88	#13083574	—	8	1	no	—
89	#13083573	—	4	2	no	—
90	#13083573	—	4	2	no	—
91	#13083573	—	4	1	no	—
92	#13083573	—	8	1	no	—
93	#13083573	—	8	1	no	—
94	#13083573	—	8	1	no	—
95	C₂×Dic₁₇₈	—	4	1	no	—
96	Q₈×D₈₉	—	4	1	no	—
97	C₂×C₂²⋊D₈₉	—	8	1	no	—
98	D₄×D₈₉	—	4	1	no	—
99	D₄×D₈₉	—	8	1	no	—
100	C₂²×Dic₈₉	—	8	1	no	—
101	C₂×C₄×D₈₉	—	8	1	no	—
102	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
103	Dic₃₅₆	—	2	1	no	—
104	Dic₃₅₆	—	2	2	no	—
105	Dic₃₅₆	—	4	1	no	—
106	#13083564	—	4	1	no	—
107	#13083563	—	4	1	no	—
108	#13083562	—	4	1	no	—
109	#13083562	—	4	2	no	—
110	#13083562	—	4	2	no	—
111	#13083561	—	4	1	no	—
112	D₇₁₂	—	2	1	no	—
113	#13083559	—	4	1	no	—
114	C₂×#10623494	D₈	1	1	yes	characteristic
115	C₈×D₈₉	D₈	1	1	yes	characteristic
116	C₈×D₈₉	D₈	1	1	yes	characteristic
117	C₈×D₈₉	C₂×D₄	1	1	yes	characteristic
118	C₈×D₈₉	C₂×D₄	1	1	yes	characteristic
119	#13083549	C₂×D₄	1	1	yes	characteristic
120	#13083549	C₂×D₄	1	1	yes	characteristic
121	#41×C₈₉	—	4	1	no	—
122	C₂×D₄×C₈₉	—	4	1	no	—
123	Q₁₆×C₈₉	—	2	1	no	—
124	SD₁₆×C₈₉	—	4	1	no	—
125	D₈×C₈₉	—	2	1	no	—
126	D₈×C₈₉	—	2	2	no	—
127	D₈×C₈₉	—	4	1	no	—
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₁₆×C₈₉	C₂⁴	1	1	yes	characteristic
131	C₂³×C₈₉	—	8	1	no	—
132	C₂²×D₈₉	—	8	1	no	—
133	C₂²×D₈₉	D₁₆	1	1	yes	characteristic
134	Q₈×C₈₉	—	4	1	no	—
135	D₄×C₈₉	—	4	2	no	—
136	D₄×C₈₉	—	4	1	no	—
137	D₄×C₈₉	—	8	1	no	—
138	D₄×C₈₉	—	8	1	no	—
139	C₂×C₄×C₈₉	—	8	1	no	—
140	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
141	C₂²⋊D₈₉	—	8	1	no	—
142	C₂²⋊D₈₉	—	8	1	no	—
143	C₂²⋊D₈₉	—	8	2	no	—
144	C₂²⋊D₈₉	—	8	2	no	—
145	C₂²⋊D₈₉	—	8	1	no	—
146	C₂×Dic₈₉	—	8	1	no	—
147	C₂×Dic₈₉	—	8	1	no	—
148	C₂×Dic₈₉	—	8	2	no	—
149	C₂×Dic₈₉	—	8	2	no	—
150	C₂×Dic₈₉	D₁₆	1	1	yes	characteristic
151	D₃₅₆	—	4	1	no	—
152	D₃₅₆	—	8	1	no	—
153	C₄×D₈₉	—	8	1	no	—
154	C₄×D₈₉	—	8	1	no	—
155	C₄×D₈₉	—	8	1	no	—
156	C₄×D₈₉	D₁₆	1	1	yes	characteristic
157	C₄×D₈₉	D₁₆	1	1	yes	characteristic
158	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
159	C₄×D₈₉	C₂×D₈	1	1	yes	characteristic
160	Dic₁₇₈	—	4	2	no	—
161	Dic₁₇₈	—	4	1	no	—
162	Dic₁₇₈	—	8	1	no	—
163	Dic₁₇₈	—	8	1	no	—
164	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
165	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
166	#10623494	C₂×D₈	1	1	yes	characteristic
167	#10623494	C₂×D₈	1	1	yes	characteristic
168	C₂²×C₈₉	—	8	2	no	—
169	C₂²×C₈₉	—	8	1	no	—
170	C₂²×C₈₉	—	16	2	no	—
171	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
172	C₂×D₈₉	—	4	1	no	—
173	C₂×D₈₉	—	8	1	no	—
174	C₂×D₈₉	—	16	2	no	—
175	C₂×D₈₉	C₂×D₁₆	1	1	yes	characteristic
176	C₂×D₈₉	C₂×D₁₆	1	1	yes	characteristic
177	C₄×C₈₉	—	8	1	no	—
178	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
179	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
180	Dic₈₉	—	8	1	no	—
181	Dic₈₉	—	8	2	no	—
182	Dic₈₉	C₂×D₁₆	1	1	yes	characteristic
183	Dic₈₉	C₂×D₁₆	1	1	yes	characteristic
184	#33981	—	89	1	no	maximal, Sylow
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	16	1	no	—
187	C₂×C₈₉	—	16	2	no	—
188	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
189	D₈₉	—	2	1	no	—
190	D₈₉	—	2	1	no	—
191	D₈₉	—	16	1	no	—
192	C₂×#508	—	89	1	no	—
193	C₂×#508	—	89	1	no	—
194	#2264	—	89	1	no	—
195	#2264	—	89	2	no	—
196	#2264	—	89	1	no	—
197	#2263	—	89	1	no	—
198	#2263	—	89	2	no	—
199	#2263	D₈₉	1	1	yes	characteristic, complemented
200	#2262	—	89	2	no	—
201	#2262	—	89	2	no	—
202	C₂×M₆₄	—	89	1	no	—
203	C₈₉	#33981	1	1	yes	characteristic, complemented, Sylow
204	C₂×#135	—	178	1	no	—
205	C₂×#135	—	178	1	no	—
206	#508	—	89	1	no	—
207	#508	—	89	2	no	—
208	#508	—	89	2	no	—
209	#508	—	89	1	no	—
210	#508	—	89	1	no	—
211	#508	—	178	1	no	—
212	#508	—	178	1	no	—
213	#508	—	178	1	no	—
214	#508	—	178	1	no	—
215	#508	C₂×D₈₉	1	1	yes	characteristic
216	C₂×Q₃₂	—	89	1	no	—
217	C₂×Q₃₂	—	89	1	no	—
218	C₂×SD₃₂	—	178	1	no	—
219	C₂×SD₃₂	—	178	1	no	—
220	C₂×D₁₆	—	89	1	no	—
221	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
222	C₂²×C₁₆	—	89	1	no	—
223	Q₆₄	—	89	2	no	—
224	Q₆₄	—	89	2	no	—
225	SD₆₄	—	89	2	no	—
226	SD₆₄	—	89	2	no	—
227	SD₆₄	—	89	2	no	—
228	SD₆₄	C₂×D₈₉	1	2	yes	complemented
229	D₃₂	—	89	2	no	—
230	D₃₂	C₂×D₈₉	1	2	yes	complemented
231	M₆₄	—	89	1	no	—
232	M₆₄	—	89	2	no	—
233	M₆₄	C₂×D₈₉	1	1	yes	characteristic, complemented
234	C₂×C₃₂	—	89	2	no	—
235	C₂×#41	—	356	1	no	—
236	C₂×#41	—	356	1	no	—
237	#135	—	2	1	no	—
238	#135	—	356	1	no	—
239	#135	—	356	1	no	—
240	#135	—	356	1	no	—
241	#135	—	356	1	no	—
242	#135	—	178	2	no	—
243	#135	—	178	2	no	—
244	#135	—	178	1	no	—
245	#135	—	178	1	no	—
246	#135	—	178	1	no	—
247	C₂×Q₁₆	—	178	1	no	—
248	C₂×Q₁₆	—	178	1	no	—
249	C₂×SD₁₆	—	356	1	no	—
250	C₂×SD₁₆	—	356	1	no	—
251	C₂×D₈	—	2	1	no	—
252	C₂×D₈	—	178	1	no	—
253	C₂²×C₈	—	89	1	no	—
254	Q₃₂	—	89	1	no	—
255	Q₃₂	—	89	2	no	—
256	Q₃₂	—	178	1	no	—
257	Q₃₂	—	178	1	no	—
258	Q₃₂	C₂²×D₈₉	1	1	yes	characteristic
259	SD₃₂	—	2	1	no	—
260	SD₃₂	—	178	2	no	—
261	SD₃₂	—	178	2	no	—
262	SD₃₂	—	178	1	no	—
263	SD₃₂	—	178	1	no	—
264	SD₃₂	—	178	1	no	—
265	D₁₆	—	2	1	no	—
266	D₁₆	—	89	1	no	—
267	D₁₆	—	178	1	no	—
268	D₁₆	C₂²×D₈₉	1	2	yes	—
269	D₁₆	C₂²×D₈₉	1	1	yes	characteristic
270	C₂×C₁₆	—	89	1	no	—
271	C₂×C₁₆	—	89	1	no	—
272	C₂×C₁₆	—	89	1	no	—
273	C₂×C₁₆	—	89	1	no	—
274	C₂×C₁₆	—	89	1	no	—
275	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
276	C₃₂	—	89	2	no	—
277	C₃₂	C₂²×D₈₉	1	2	yes	—
278	#41	—	4	1	no	—
279	#41	—	356	2	no	—
280	#41	—	356	2	no	—
281	#41	—	356	1	no	—
282	#41	—	356	1	no	—
283	#41	—	356	1	no	—
284	#41	—	712	1	no	—
285	#41	—	712	1	no	—
286	#41	—	712	1	no	—
287	#41	—	712	1	no	—
288	C₂×Q₈	—	356	1	no	—
289	C₂×Q₈	—	356	1	no	—
290	C₂×D₄	—	4	1	no	—
291	C₂×D₄	—	356	1	no	—
292	C₂×D₄	—	712	1	no	—
293	C₂×D₄	—	712	1	no	—
294	C₂²×C₄	—	89	1	no	—
295	C₂²×C₄	—	712	1	no	—
296	C₂²×C₄	—	712	1	no	—
297	Q₁₆	—	2	1	no	—
298	Q₁₆	—	356	1	no	—
299	Q₁₆	—	356	1	no	—
300	Q₁₆	—	178	2	no	—
301	Q₁₆	—	178	1	no	—
302	SD₁₆	—	4	1	no	—
303	SD₁₆	—	356	2	no	—
304	SD₁₆	—	356	2	no	—
305	SD₁₆	—	356	1	no	—
306	SD₁₆	—	356	1	no	—
307	SD₁₆	—	356	1	no	—
308	D₈	—	2	2	no	—
309	D₈	—	2	1	no	—
310	D₈	—	4	1	no	—
311	D₈	—	356	1	no	—
312	D₈	—	178	1	no	—
313	C₂×C₈	—	89	1	no	—
314	C₂×C₈	—	89	1	no	—
315	C₂×C₈	—	89	1	no	—
316	C₂×C₈	—	89	1	no	—
317	C₂×C₈	—	89	1	no	—
318	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
319	C₁₆	—	89	1	no	—
320	C₁₆	—	89	1	no	—
321	C₁₆	D₄×D₈₉	1	1	yes	characteristic
322	C₁₆	C₂³×D₈₉	1	1	yes	characteristic
323	C₂³	—	8	1	no	—
324	C₂³	—	89	1	no	—
325	C₂³	—	712	1	no	—
326	Q₈	—	4	1	no	—
327	Q₈	—	356	2	no	—
328	Q₈	—	356	1	no	—
329	Q₈	—	712	1	no	—
330	Q₈	—	712	1	no	—
331	D₄	—	4	2	no	—
332	D₄	—	4	1	no	—
333	D₄	—	8	1	no	—
334	D₄	—	8	1	no	—
335	D₄	—	356	1	no	—
336	D₄	—	712	1	no	—
337	D₄	—	712	1	no	—
338	D₄	—	712	1	no	—
339	D₄	—	712	2	no	—
340	D₄	—	712	2	no	—
341	D₄	—	712	1	no	—
342	C₂×C₄	—	8	1	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	89	1	no	—
348	C₂×C₄	—	712	1	no	—
349	C₂×C₄	—	712	1	no	—
350	C₂×C₄	—	712	1	no	—
351	C₂×C₄	—	712	2	no	—
352	C₂×C₄	—	712	2	no	—
353	C₂×C₄	—	712	1	no	—
354	C₂×C₄	—	712	1	no	—
355	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
356	C₈	—	89	1	no	—
357	C₈	—	89	1	no	—
358	C₈	D₈×D₈₉	1	1	yes	characteristic
359	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
360	C₂²	—	8	1	no	—
361	C₂²	—	8	2	no	—
362	C₂²	—	16	2	no	—
363	C₂²	—	89	1	no	—
364	C₂²	—	89	1	no	—
365	C₂²	—	356	1	no	—
366	C₂²	—	1424	2	no	—
367	C₂²	—	712	1	no	—
368	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
369	C₄	—	8	1	no	—
370	C₄	—	89	1	no	—
371	C₄	—	89	1	no	—
372	C₄	—	712	1	no	—
373	C₄	—	712	2	no	—
374	C₄	D₁₆×D₈₉	1	1	yes	characteristic
375	C₄	C₂×D₈×D₈₉	1	1	yes	characteristic
376	C₂	—	2	1	no	—
377	C₂	—	16	2	no	—
378	C₂	—	16	1	no	—
379	C₂	—	1424	1	no	—
380	C₂	—	178	1	no	—
381	C₂	—	178	1	no	—
382	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₁₆×D₈₉
commutator, cosocle	C₁₆×C₈₉	C₂⁴
Frattini	C₁₆	C₂³×D₈₉
Fitting	C₈₉×#2263	C₂
socle	C₂×C₈₉	C₂²×D₁₆
hypercenter	#2263	D₈₉
hypocenter	C₈₉	#33981
LCS	G ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #2263

Complements

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

N	H ≅ G/N	type
1. C₂×#19152027	C₂	semidirect
2. #508×D₈₉	C₂	semidirect
3. #26022572	C₂	semidirect
4. #26022569	C₂	semidirect
5. #26022568	C₂	semidirect
6. #26022565	C₂	semidirect
7. #26022563	C₂	semidirect
8. #26022560	C₂	semidirect
9. #26022558	C₂	semidirect
10. M₆₄×D₈₉	C₂	semidirect
11. C₈₉×#2263	C₂	semidirect
14. #19152031	C₂²	semidirect
21. #19152027	C₂²	semidirect
22. #19152027	C₂²	semidirect
23. C₂×Dic₇₁₂	C₂²	semidirect
24. Q₃₂×D₈₉	C₂²	semidirect
27. D₁₆×D₈₉	C₂²	semidirect
29. Dic₁₄₂₄	C₂²	semidirect
30. #19151438	C₂²	semidirect
31. #19151437	C₂²	semidirect
32. #19151436	C₂²	semidirect
33. #19151436	C₂²	semidirect
34. #19151433	C₂²	semidirect
35. #19151432	C₂²	semidirect
36. #19151431	C₂²	semidirect
37. C₃₂×D₈₉	C₂²	semidirect
39. C₂×D₁₆×C₈₉	C₂²	semidirect
40. SD₆₄×C₈₉	C₂²	semidirect
41. D₃₂×C₈₉	C₂²	semidirect
42. M₆₄×C₈₉	C₂²	semidirect
199. #2263	D₈₉	semidirect
203. C₈₉	#33981	semidirect
221. C₂×D₁₆	C₂×D₈₉	semidirect
228. SD₆₄	C₂×D₈₉	semidirect
230. D₃₂	C₂×D₈₉	semidirect
233. M₆₄	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152027	2
#508×D₈₉	2
#26022572	2
#26022569	2
#26022568	2
#26022565	2
#26022563	2
#26022560	2
#26022558	2
M₆₄×D₈₉	2
C₈₉×#2263	2
#33981	89