Subgroups — #36545955

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032601	C₂	1	1	yes	maximal, characteristic
2	#26032562	C₂	1	2	yes	maximal, complemented
3	#26019861	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019829	C₂	1	2	yes	maximal, complemented
5	#26019648	C₂	1	1	yes	maximal, characteristic, complemented
6	#26019646	C₂	1	1	yes	maximal, characteristic, complemented
7	#26019466	C₂	1	2	yes	maximal, complemented
8	#26019402	C₂	1	2	yes	maximal, complemented
9	#26019137	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3380	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152253	C₂²	1	2	yes	—
13	#19152003	C₂²	1	1	yes	characteristic
14	#19151950	C₂²	1	2	yes	—
15	#19151948	C₂²	1	2	yes	—
16	#19151928	—	2	2	no	—
17	#19151910	—	2	2	no	—
18	#19151885	C₂²	1	2	yes	—
19	#19151878	C₂²	1	2	yes	—
20	#19151854	—	2	2	no	—
21	#19151831	—	2	1	no	—
22	#19151808	C₂²	1	2	yes	—
23	#19151758	—	2	2	no	—
24	C₄×#13083561	C₂²	1	1	yes	characteristic
25	#19151676	C₂²	1	2	yes	—
26	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
27	#107×D₈₉	—	2	1	no	—
28	#19151597	—	2	2	no	—
29	#19151351	C₂²	1	1	yes	characteristic
30	#19151348	C₂²	1	1	yes	characteristic
31	#19151347	C₂²	1	2	yes	—
32	#19151343	C₂²	1	2	yes	—
33	Q₈⋊#10623494	C₂²	1	1	yes	characteristic
34	D₄⋊#10623494	C₂²	1	2	yes	—
35	#19151325	C₂²	1	1	yes	characteristic
36	C₈×#10623494	C₂²	1	1	yes	characteristic
37	#554×C₈₉	C₂²	1	2	yes	—
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#499×C₈₉	—	2	1	no	—
40	#478×C₈₉	—	2	2	no	—
41	#476×C₈₉	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₈₉	—	2	1	no	—
44	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#13083574	—	2	2	no	—
46	C₂×#13083564	—	2	2	no	—
47	C₂×#13083561	D₄	1	1	yes	characteristic
48	C₂×#13083558	—	2	2	no	—
49	C₂×C₈×D₈₉	—	2	1	no	—
50	#15668012	C₂³	1	2	yes	—
51	#15667994	—	4	2	no	—
52	#15667991	—	2	2	no	—
53	#15667983	—	2	2	no	—
54	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
55	C₄⋊C₄×D₈₉	—	2	2	no	—
56	#15667946	—	2	1	no	—
57	#15667946	—	2	1	no	—
58	#15667945	—	2	2	no	—
59	#15667943	—	2	2	no	—
60	#15667943	D₄	1	1	yes	characteristic
61	C₄⋊#10623494	C₂³	1	1	yes	characteristic
62	C₄⋊#10623494	C₂³	1	2	yes	—
63	#15667936	—	4	2	no	—
64	#15667936	D₄	1	1	yes	characteristic
65	#15667935	—	2	2	no	—
66	#15667935	—	2	2	no	—
67	#15667935	—	2	2	no	—
68	#15667933	—	2	2	no	—
69	#15667932	—	2	2	no	—
70	#15667932	D₄	1	1	yes	characteristic
71	C₂²⋊#10623494	—	2	2	no	—
72	#15667924	—	2	1	no	—
73	C₄×#10623494	C₂³	1	1	yes	characteristic
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
76	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
77	#125×C₈₉	—	2	2	no	—
78	C₂²⋊Q₈×C₈₉	—	2	2	no	—
79	C₂²⋊Q₈×C₈₉	—	4	2	no	—
80	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₈₉	—	2	2	no	—
82	C₄×D₄×C₈₉	C₂³	1	2	yes	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	#107×C₈₉	—	2	1	no	—
85	#107×C₈₉	—	2	1	no	—
86	#107×C₈₉	D₄	1	1	yes	characteristic
87	#106×C₈₉	—	2	2	no	—
88	C₄⋊C₈×C₈₉	—	2	1	no	—
89	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
90	#102×C₈₉	—	2	2	no	—
91	#102×C₈₉	D₄	1	2	yes	—
92	C₄×C₈×C₈₉	—	2	1	no	—
93	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
94	#13083574	—	4	4	no	—
95	#13083574	—	8	2	no	—
96	C₂×Dic₁₇₈	—	2	2	no	—
97	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₈₉	—	4	2	no	—
99	C₂×C₄×D₈₉	—	2	1	no	—
100	C₂×C₄×D₈₉	—	4	2	no	—
101	#13083564	—	8	2	no	—
102	#13083561	—	2	2	no	—
103	#13083558	—	8	2	no	—
104	C₂×#10623494	—	2	1	no	—
105	C₂×#10623494	—	2	2	no	—
106	C₂×#10623494	—	2	1	no	—
107	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
108	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
109	C₈×D₈₉	—	4	2	no	—
110	C₄⋊Dic₈₉	—	2	2	no	—
111	C₄⋊Dic₈₉	—	2	2	no	—
112	C₄⋊Dic₈₉	C₂×D₄	1	1	yes	characteristic
113	#13083553	—	4	2	no	—
114	C₂²⋊Dic₈₉	—	4	2	no	—
115	#13083551	—	2	1	no	—
116	C₂×Q₈×C₈₉	—	2	2	no	—
117	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₈₉	—	2	2	no	—
120	C₂²×C₄×C₈₉	—	4	2	no	—
121	D₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	1	no	—
123	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	4	2	no	—
129	C₄⋊C₄×C₈₉	—	4	2	no	—
130	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₈₉	—	2	2	no	—
133	C₂²⋊C₄×C₈₉	—	4	2	no	—
134	C₄²×C₈₉	—	2	1	no	—
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	—	2	2	no	—
137	C₂²×D₈₉	—	2	1	no	—
138	Q₈×C₈₉	—	2	1	no	—
139	Q₈×C₈₉	—	2	1	no	—
140	Q₈×C₈₉	—	8	2	no	—
141	D₄×C₈₉	—	2	2	no	—
142	D₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	1	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	2	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	16	2	no	—
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²⋊D₈₉	—	8	4	no	—
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	4	2	no	—
156	D₃₅₆	—	2	1	no	—
157	D₃₅₆	#135	1	2	yes	—
158	C₄×D₈₉	—	4	1	no	—
159	C₄×D₈₉	—	4	1	no	—
160	C₄×D₈₉	—	8	4	no	—
161	Dic₁₇₈	—	4	2	no	—
162	Dic₁₇₈	—	4	2	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	4	1	no	—
165	C₈×C₈₉	#137	1	2	yes	—
166	#10623494	—	2	1	no	—
167	#10623494	—	2	1	no	—
168	#10623494	—	4	1	no	—
169	#10623494	—	4	1	no	—
170	#10623494	—	4	2	no	—
171	C₂²×C₈₉	—	4	2	no	—
172	C₂²×C₈₉	—	4	2	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	D₄²	1	1	yes	characteristic
175	C₂×D₈₉	—	2	2	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	—	4	1	no	—
178	C₄×C₈₉	—	2	1	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	—	4	1	no	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	—	8	2	no	—
183	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
184	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
185	C₄×C₈₉	#576	1	1	yes	characteristic
186	C₄×C₈₉	#578	1	1	yes	characteristic
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	—	8	2	no	—
189	#19903	—	89	1	no	maximal, Sylow
190	C₂×C₈₉	—	8	2	no	—
191	C₂×C₈₉	#3276	1	1	yes	characteristic
192	C₂×C₈₉	#3280	1	1	yes	characteristic
193	C₂×C₈₉	#3290	1	1	yes	characteristic
194	D₈₉	—	4	2	no	—
195	#3380	D₈₉	1	1	yes	characteristic, complemented
196	#3334	—	89	1	no	—
197	#3331	—	89	2	no	—
198	#1710	—	89	1	no	—
199	#1703	—	89	2	no	—
200	#1675	—	89	1	no	—
201	#1674	—	89	1	no	—
202	#1644	—	89	2	no	—
203	#1635	—	89	2	no	—
204	#1589	—	89	1	no	—
205	C₈×SD₁₆	—	89	1	no	—
206	C₈₉	#19903	1	1	yes	characteristic, complemented, Sylow
207	#554	C₂×D₈₉	1	2	yes	—
208	#547	—	89	2	no	—
209	#500	—	89	1	no	—
210	#500	C₂×D₈₉	1	1	yes	characteristic
211	#499	—	2	1	no	—
212	#487	—	89	2	no	—
213	#487	—	89	2	no	—
214	#483	—	178	2	no	—
215	#480	—	178	2	no	—
216	#478	—	2	2	no	—
217	#476	—	89	2	no	—
218	#476	C₂×D₈₉	1	2	yes	—
219	#474	—	89	2	no	—
220	#470	—	178	2	no	—
221	#466	—	178	1	no	—
222	#462	—	89	2	no	—
223	#452	—	178	2	no	—
224	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
225	C₄×SD₁₆	—	89	1	no	—
226	C₄×D₈	—	2	1	no	—
227	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
228	#436	—	89	2	no	—
229	C₈×D₄	—	89	1	no	—
230	C₂×#107	—	178	1	no	—
231	#417	—	178	2	no	—
232	#335	—	89	1	no	—
233	#334	—	89	1	no	—
234	#333	—	89	2	no	—
235	#332	—	89	2	no	—
236	Q₈⋊C₈	—	89	1	no	—
237	D₄⋊C₈	—	89	2	no	—
238	D₄⋊C₈	—	89	1	no	—
239	C₈²	—	89	1	no	—
240	C₂×#41	—	178	2	no	—
241	C₂×Q₁₆	—	178	2	no	—
242	C₂×SD₁₆	—	89	1	no	—
243	C₂×D₈	C₂²⋊D₈₉	1	1	yes	characteristic
244	C₂×M₁₆	—	178	2	no	—
245	C₂²×C₈	—	178	1	no	—
246	C₄⋊Q₈	—	89	2	no	—
247	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
248	#125	—	2	2	no	—
249	#123	—	356	2	no	—
250	C₂²⋊Q₈	—	2	2	no	—
251	C₂²⋊Q₈	—	4	2	no	—
252	C₂²⋊Q₈	—	178	2	no	—
253	#121	—	178	2	no	—
254	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	89	1	no	—
257	C₄×D₄	C₂²×D₈₉	1	2	yes	—
258	C₂×C₄⋊C₄	—	2	2	no	—
259	C₂×C₄⋊C₄	—	178	2	no	—
260	#107	—	2	1	no	—
261	#107	—	2	1	no	—
262	#107	—	178	1	no	—
263	#107	—	178	1	no	—
264	#107	—	178	2	no	—
265	#107	C₂²⋊D₈₉	1	1	yes	characteristic
266	#106	—	2	2	no	—
267	#106	—	89	1	no	—
268	#106	—	178	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	89	2	no	—
271	C₄⋊C₈	—	89	1	no	—
272	C₄⋊C₈	C₂²×D₈₉	1	1	yes	characteristic
273	#103	—	89	1	no	—
274	#103	—	356	2	no	—
275	#103	—	178	2	no	—
276	#103	—	178	2	no	—
277	#103	—	178	2	no	—
278	#102	—	2	2	no	—
279	#102	—	89	1	no	—
280	#102	—	178	2	no	—
281	#102	—	178	2	no	—
282	#102	C₂²⋊D₈₉	1	2	yes	—
283	C₂²⋊C₈	—	178	2	no	—
284	C₂²⋊C₈	—	178	1	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	—	89	1	no	—
287	C₄×C₈	—	89	1	no	—
288	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
289	#41	—	356	4	no	—
290	#41	—	712	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	178	2	no	—
293	C₂×Q₈	D₄×D₈₉	1	1	yes	characteristic
294	C₂×D₄	—	89	1	no	—
295	C₂×D₄	—	356	2	no	—
296	C₂×D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	356	2	no	—
300	C₂²×C₄	—	178	1	no	—
301	Q₁₆	—	712	2	no	—
302	SD₁₆	—	178	2	no	—
303	D₈	—	2	2	no	—
304	M₁₆	—	712	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	89	1	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	356	2	no	—
309	C₂×C₈	—	178	2	no	—
310	C₂×C₈	—	178	1	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
313	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	4	2	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	89	1	no	—
320	C₄⋊C₄	—	356	2	no	—
321	C₄⋊C₄	—	178	2	no	—
322	C₄⋊C₄	—	178	2	no	—
323	C₄⋊C₄	D₄×D₈₉	1	1	yes	characteristic
324	C₄⋊C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	356	2	no	—
328	C₂²⋊C₄	—	178	1	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂³×D₈₉	1	1	yes	characteristic
331	C₂³	—	2	2	no	—
332	C₂³	—	178	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	8	2	no	—
336	Q₈	—	356	2	no	—
337	Q₈	—	356	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	89	2	no	—
341	D₄	—	178	1	no	—
342	D₄	—	712	4	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	4	2	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	16	2	no	—
350	C₂×C₄	—	356	1	no	—
351	C₂×C₄	—	356	1	no	—
352	C₂×C₄	—	356	2	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	712	4	no	—
355	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
357	C₂×C₄	#15668057	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	356	1	no	—
361	C₈	—	356	1	no	—
362	C₈	—	356	2	no	—
363	C₈	—	178	1	no	—
364	C₈	—	178	1	no	—
365	C₈	#15668043	1	2	yes	—
366	C₂²	—	4	2	no	—
367	C₂²	—	4	2	no	—
368	C₂²	—	4	2	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	356	1	no	—
371	C₂²	—	178	2	no	—
372	C₂²	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	2	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	8	2	no	—
378	C₄	—	356	1	no	—
379	C₄	—	712	2	no	—
380	C₄	#135×D₈₉	1	1	yes	characteristic
381	C₄	C₂×#15668043	1	1	yes	characteristic
382	C₄	#19152400	1	1	yes	characteristic
383	C₄	#19152413	1	1	yes	characteristic
384	C₂	—	8	2	no	—
385	C₂	—	356	2	no	—
386	C₂	#26031898	1	1	yes	characteristic
387	C₂	#26031931	1	1	yes	characteristic
388	C₂	#26032034	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3380	C₂
socle	C₂²×C₈₉	D₄²
prime radical	#26032601	C₂
hypercenter	#3380	D₈₉
hypocenter	C₈₉	#19903
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3380

Complements

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

N	H ≅ G/N	type
2. #26032562	C₂	semidirect
3. #26019861	C₂	semidirect
4. #26019829	C₂	semidirect
5. #26019648	C₂	semidirect
6. #26019646	C₂	semidirect
7. #26019466	C₂	semidirect
8. #26019402	C₂	semidirect
9. #26019137	C₂	semidirect
10. C₈×#13083561	C₂	semidirect
11. C₈₉×#3380	C₂	semidirect
195. #3380	D₈₉	semidirect
206. C₈₉	#19903	semidirect

Maximal subgroups

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

subgroup	index
#26032601	2
#26032562	2
#26019861	2
#26019829	2
#26019648	2
#26019646	2
#26019466	2
#26019402	2
#26019137	2
C₈×#13083561	2
C₈₉×#3380	2
#19903	89