Subgroups — #36707782

Order 22784.

660 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029128	C₂	1	1	yes	maximal, characteristic
2	#26029120	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022123	C₂	1	2	yes	maximal, complemented
4	#26022117	C₂	1	2	yes	maximal, complemented
5	Q₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
6	SD₃₂×Dic₈₉	C₂	1	2	yes	maximal, complemented
7	#26022040	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021959	C₂	1	1	yes	maximal, characteristic
9	C₂×#19151402	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021851	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#510×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152081	—	2	1	no	—
13	#19152076	—	2	1	no	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	#19151705	—	2	2	no	—
16	#19151699	—	2	2	no	—
17	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
18	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
19	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
20	D₈×Dic₈₉	C₂²	1	2	yes	—
21	#19151636	C₂²	1	1	yes	characteristic
22	C₂×#15667946	C₂²	1	1	yes	characteristic
23	#19151607	—	2	1	no	—
24	#19151599	—	2	1	no	—
25	#19151553	C₂²	1	1	yes	characteristic
26	#19151426	C₂²	1	2	yes	—
27	#19151424	C₂²	1	2	yes	—
28	#19151402	C₂²	1	2	yes	—
29	#19151402	C₂²	1	2	yes	—
30	#19151402	C₂²	1	1	yes	characteristic
31	#19151402	C₂²	1	1	yes	characteristic
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151399	C₂²	1	1	yes	characteristic
34	#19151373	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#510×C₈₉	C₄	1	8	yes	complemented
39	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	—
40	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
41	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#13083558	D₄	1	1	yes	characteristic
43	Q₈×Dic₈₉	—	2	2	no	—
44	Q₈×Dic₈₉	—	2	2	no	—
45	Q₈×Dic₈₉	—	2	2	no	—
46	Q₈×Dic₈₉	—	4	1	no	—
47	Q₈×Dic₈₉	—	4	1	no	—
48	D₄×Dic₈₉	—	2	2	no	—
49	D₄×Dic₈₉	—	4	2	no	—
50	#15667971	—	2	1	no	—
51	#15667971	—	4	1	no	—
52	#15667971	—	4	2	no	—
53	#15667971	D₄	1	1	yes	characteristic
54	C₂×C₄⋊Dic₈₉	—	2	1	no	—
55	C₂×C₄⋊Dic₈₉	—	4	1	no	—
56	C₂×#13083549	—	2	2	no	—
57	C₂×#13083549	C₂³	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	#15667946	—	2	1	no	—
60	#15667946	C₂³	1	2	yes	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667944	—	2	1	no	—
64	#15667936	—	2	2	no	—
65	#15667936	—	2	2	no	—
66	#15667936	—	2	2	no	—
67	#15667933	—	2	2	no	—
68	#15667922	D₄	1	2	yes	—
69	C₈×Dic₈₉	C₂³	1	2	yes	—
70	C₂×#41×C₈₉	—	2	1	no	—
71	C₂²×Q₈×C₈₉	—	2	1	no	—
72	#135×C₈₉	—	2	2	no	—
73	#135×C₈₉	C₂×C₄	1	4	yes	—
74	C₂×Q₁₆×C₈₉	—	2	1	no	—
75	C₂×Q₁₆×C₈₉	—	2	2	no	—
76	C₂×Q₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
77	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
78	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×SD₁₆×C₈₉	—	2	1	no	—
80	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	Q₃₂×C₈₉	C₂×C₄	1	8	yes	complemented
83	SD₃₂×C₈₉	C₂×C₄	1	8	yes	complemented
84	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
85	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
86	C₂²×Dic₈₉	—	4	1	no	—
87	C₂²×Dic₈₉	—	4	2	no	—
88	#13083558	—	2	2	no	—
89	C₂×#10623494	C₂×D₄	1	2	yes	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	4	1	no	—
94	C₄⋊Dic₈₉	—	4	2	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
98	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
99	C₂²⋊Dic₈₉	—	4	1	no	—
100	C₂²⋊Dic₈₉	—	4	2	no	—
101	C₂²⋊Dic₈₉	D₈	1	2	yes	—
102	C₄×Dic₈₉	—	4	1	no	—
103	C₄×Dic₈₉	—	4	2	no	—
104	C₄×Dic₈₉	—	4	2	no	—
105	C₄×Dic₈₉	—	4	2	no	—
106	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
107	#13083549	—	2	1	no	—
108	#13083549	—	2	1	no	—
109	#41×C₈₉	—	2	4	no	—
110	#41×C₈₉	—	4	2	no	—
111	C₂×Q₈×C₈₉	—	2	4	no	—
112	C₂×Q₈×C₈₉	—	2	1	no	—
113	C₂×Q₈×C₈₉	—	2	2	no	—
114	C₂×Q₈×C₈₉	—	4	1	no	—
115	C₂×Q₈×C₈₉	—	4	2	no	—
116	C₂×D₄×C₈₉	—	2	1	no	—
117	C₂×D₄×C₈₉	—	4	1	no	—
118	C₂²×C₄×C₈₉	—	4	1	no	—
119	C₂²×C₄×C₈₉	—	4	1	no	—
120	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	Q₁₆×C₈₉	—	2	4	no	—
122	Q₁₆×C₈₉	—	2	2	no	—
123	Q₁₆×C₈₉	—	2	1	no	—
124	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
125	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
126	SD₁₆×C₈₉	—	2	4	no	—
127	D₈×C₈₉	—	2	1	no	—
128	D₈×C₈₉	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₉	#41	1	2	yes	—
132	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
134	C₂³×C₈₉	—	4	1	no	—
135	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
136	Q₈×C₈₉	—	2	2	no	—
137	Q₈×C₈₉	—	2	4	no	—
138	Q₈×C₈₉	—	4	2	no	—
139	Q₈×C₈₉	—	4	1	no	—
140	Q₈×C₈₉	—	4	4	no	—
141	D₄×C₈₉	—	2	2	no	—
142	D₄×C₈₉	—	4	4	no	—
143	D₄×C₈₉	—	4	1	no	—
144	C₂×C₄×C₈₉	—	4	1	no	—
145	C₂×C₄×C₈₉	—	4	2	no	—
146	C₂×C₄×C₈₉	—	4	4	no	—
147	C₂×C₄×C₈₉	—	4	4	no	—
148	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
149	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#135	1	2	yes	—
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₈₉	—	4	1	no	—
153	C₂×Dic₈₉	—	4	1	no	—
154	C₂×Dic₈₉	—	4	2	no	—
155	C₂×Dic₈₉	—	16	2	no	—
156	C₂×Dic₈₉	—	16	1	no	—
157	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
158	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
159	C₈×C₈₉	C₄×D₄	1	2	yes	—
160	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
161	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
162	#10623494	—	2	2	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	4	2	no	—
166	C₂²×C₈₉	—	8	1	no	—
167	C₂²×C₈₉	—	8	1	no	—
168	C₂²×C₈₉	C₄×D₈	1	2	yes	—
169	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
170	C₄×C₈₉	—	4	2	no	—
171	C₄×C₈₉	—	4	4	no	—
172	C₄×C₈₉	C₄×D₈	1	2	yes	—
173	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
175	Dic₈₉	—	2	2	no	—
176	Dic₈₉	—	2	2	no	—
177	Dic₈₉	—	8	1	no	—
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	8	1	no	—
180	#33455	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	—	2	2	no	—
182	C₂×C₈₉	—	8	2	no	—
183	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
184	C₂×C₈₉	#3416	1	1	yes	characteristic
185	C₂×C₈₉	#3417	1	1	yes	characteristic
186	C₂×#510	D₈₉	1	1	yes	characteristic
187	#2951	—	89	1	no	—
188	#2950	—	89	1	no	—
189	#2176	—	89	2	no	—
190	#2175	—	89	2	no	—
191	C₄×Q₃₂	—	89	2	no	—
192	C₄×SD₃₂	—	89	2	no	—
193	#2159	—	89	1	no	—
194	#2138	—	89	1	no	—
195	C₂×#358	—	89	1	no	—
196	#2108	—	89	1	no	—
197	C₈₉	#33455	1	1	yes	characteristic, complemented, Sylow
198	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
199	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
200	#520	—	178	1	no	—
201	#519	—	178	1	no	—
202	#510	Dic₈₉	1	8	yes	complemented
203	C₂×Q₃₂	C₂×D₈₉	1	2	yes	—
204	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
205	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
206	C₂²×C₁₆	—	89	1	no	—
207	#441	—	178	2	no	—
208	#440	—	178	2	no	—
209	C₄×Q₁₆	—	89	2	no	—
210	C₄×Q₁₆	—	89	2	no	—
211	C₄×Q₁₆	—	89	2	no	—
212	C₄×D₈	—	89	2	no	—
213	#427	—	89	1	no	—
214	C₂×#107	—	89	1	no	—
215	#419	—	178	1	no	—
216	#417	—	178	1	no	—
217	#405	—	89	1	no	—
218	#367	—	89	2	no	—
219	#366	—	89	2	no	—
220	#358	—	89	2	no	—
221	#358	—	89	2	no	—
222	#358	—	89	1	no	—
223	#358	—	89	1	no	—
224	#357	—	89	1	no	—
225	#357	—	89	1	no	—
226	#346	—	89	2	no	—
227	C₄×C₁₆	—	89	2	no	—
228	C₂×#41	—	2	1	no	—
229	C₂²×Q₈	—	2	1	no	—
230	#135	—	2	2	no	—
231	#135	C₂×Dic₈₉	1	4	yes	—
232	C₂×Q₁₆	—	2	1	no	—
233	C₂×Q₁₆	—	2	2	no	—
234	C₂×Q₁₆	C₂×Dic₈₉	1	4	yes	complemented
235	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
236	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
237	C₂×SD₁₆	—	2	1	no	—
238	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₂×M₁₆	—	89	1	no	—
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₄×Q₈	—	356	1	no	—
242	C₄×Q₈	—	356	1	no	—
243	C₄×Q₈	—	178	2	no	—
244	C₄×Q₈	—	178	2	no	—
245	C₄×Q₈	—	178	2	no	—
246	C₄×D₄	—	356	2	no	—
247	C₄×D₄	—	178	2	no	—
248	#117	—	89	1	no	—
249	#117	—	356	2	no	—
250	#117	—	356	1	no	—
251	#117	—	178	1	no	—
252	C₂×C₄⋊C₄	—	356	1	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	Q₃₂	C₂×Dic₈₉	1	8	yes	complemented
255	SD₃₂	C₂×Dic₈₉	1	8	yes	complemented
256	M₃₂	C₂×Dic₈₉	1	4	yes	complemented
257	C₂×C₁₆	—	89	1	no	—
258	C₂×C₁₆	—	89	1	no	—
259	C₂×C₁₆	—	178	2	no	—
260	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
261	#107	—	89	1	no	—
262	#107	—	89	1	no	—
263	#107	—	89	2	no	—
264	#107	—	178	1	no	—
265	#106	—	178	1	no	—
266	#103	—	178	2	no	—
267	#103	—	178	2	no	—
268	#103	—	178	2	no	—
269	#102	—	178	2	no	—
270	#97	—	89	2	no	—
271	C₄×C₈	—	89	2	no	—
272	#41	—	2	4	no	—
273	#41	—	4	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	1	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	2	no	—
279	C₂×D₄	—	2	1	no	—
280	C₂×D₄	—	4	1	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	356	2	no	—
284	C₂²×C₄	—	356	1	no	—
285	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
286	Q₁₆	—	2	4	no	—
287	Q₁₆	—	2	2	no	—
288	Q₁₆	—	2	1	no	—
289	Q₁₆	C₂²×Dic₈₉	1	4	yes	—
290	Q₁₆	C₂²×Dic₈₉	1	2	yes	—
291	SD₁₆	—	2	4	no	—
292	D₈	—	2	1	no	—
293	D₈	C₂²×Dic₈₉	1	2	yes	—
294	M₁₆	—	178	2	no	—
295	C₂×C₈	—	89	2	no	—
296	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
297	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	#13083573	1	2	yes	—
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	89	1	no	—
301	C₄⋊C₄	—	89	1	no	—
302	C₄⋊C₄	—	356	2	no	—
303	C₄⋊C₄	—	356	2	no	—
304	C₄⋊C₄	—	356	2	no	—
305	C₄⋊C₄	—	356	1	no	—
306	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	—	178	1	no	—
308	C₄⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	89	2	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₂²⋊C₄	—	356	1	no	—
312	C₄²	—	89	2	no	—
313	C₄²	—	356	2	no	—
314	C₄²	—	356	2	no	—
315	C₄²	—	356	2	no	—
316	C₄²	—	356	1	no	—
317	C₁₆	—	178	1	no	—
318	C₁₆	—	178	1	no	—
319	C₁₆	C₂²×Dic₈₉	1	4	yes	—
320	C₂³	—	4	1	no	—
321	C₂³	D₈×D₈₉	1	1	yes	characteristic
322	Q₈	—	2	2	no	—
323	Q₈	—	2	4	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	4	2	no	—
326	Q₈	—	4	1	no	—
327	D₄	—	2	2	no	—
328	D₄	—	4	4	no	—
329	D₄	—	4	1	no	—
330	C₂×C₄	—	4	4	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	4	no	—
334	C₂×C₄	—	89	2	no	—
335	C₂×C₄	—	89	2	no	—
336	C₂×C₄	—	356	1	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	1424	2	no	—
340	C₂×C₄	—	1424	1	no	—
341	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
342	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
343	C₂×C₄	#15668031	1	2	yes	—
344	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
345	C₈	—	178	2	no	—
346	C₈	D₄×Dic₈₉	1	2	yes	—
347	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
348	C₈	C₂×#13083573	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	4	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	—	8	1	no	—
354	C₂²	D₈×Dic₈₉	1	2	yes	—
355	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
356	C₄	—	4	2	no	—
357	C₄	—	4	4	no	—
358	C₄	—	178	2	no	—
359	C₄	—	178	2	no	—
360	C₄	—	712	2	no	—
361	C₄	—	712	1	no	—
362	C₄	—	712	1	no	—
363	C₄	D₈×Dic₈₉	1	2	yes	—
364	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
365	C₄	C₂×#15668031	1	1	yes	characteristic
366	C₂	—	2	2	no	—
367	C₂	—	8	2	no	—
368	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic
369	C₂	#26033397	1	1	yes	characteristic
370	C₂	#26033404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#510×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#135×C₈₉	C₂²
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#510	D₈₉
hypocenter	C₈₉	#33455
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#510

Complements

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

N	H ≅ G/N	type
2. #26029120	C₂	semidirect
3. #26022123	C₂	semidirect
4. #26022117	C₂	semidirect
5. Q₃₂×Dic₈₉	C₂	semidirect
6. SD₃₂×Dic₈₉	C₂	semidirect
7. #26022040	C₂	semidirect
9. C₂×#19151402	C₂	semidirect
10. #26021851	C₂	semidirect
38. #510×C₈₉	C₄	semidirect
76. C₂×Q₁₆×C₈₉	C₂×C₄	semidirect
82. Q₃₂×C₈₉	C₂×C₄	semidirect
83. SD₃₂×C₈₉	C₂×C₄	semidirect
84. M₃₂×C₈₉	C₂×C₄	semidirect
197. C₈₉	#33455	semidirect
202. #510	Dic₈₉	semidirect
234. C₂×Q₁₆	C₂×Dic₈₉	semidirect
254. Q₃₂	C₂×Dic₈₉	semidirect
255. SD₃₂	C₂×Dic₈₉	semidirect
256. M₃₂	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029128	2
#26029120	2
#26022123	2
#26022117	2
Q₃₂×Dic₈₉	2
SD₃₂×Dic₈₉	2
#26022040	2
#26021959	2
C₂×#19151402	2
#26021851	2
C₂×#510×C₈₉	2
#33455	89