Subgroups — #36709273

Order 22784.

588 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029900	C₂	1	1	yes	maximal, characteristic
2	#26028852	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022444	C₂	1	2	yes	maximal, complemented
4	#26022444	C₂	1	2	yes	maximal, complemented
5	#26022438	C₂	1	2	yes	maximal, complemented
6	#26022433	C₂	1	2	yes	maximal, complemented
7	#26022040	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021969	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151400	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
13	#19152162	—	2	1	no	—
14	#19152081	—	2	1	no	—
15	#19151851	—	2	2	no	—
16	#19151834	C₂²	1	2	yes	—
17	#19151834	C₂²	1	2	yes	—
18	#19151640	—	2	1	no	—
19	#19151640	—	2	2	no	—
20	#19151636	C₂²	1	1	yes	characteristic
21	#19151636	C₂²	1	1	yes	characteristic
22	#19151636	C₂²	1	2	yes	complemented
23	C₂×#15667946	C₂²	1	1	yes	characteristic
24	C₂×#15667946	C₂²	1	2	yes	complemented
25	#19151604	—	2	1	no	—
26	#19151426	C₂²	1	1	yes	characteristic, complemented
27	#19151426	C₂²	1	2	yes	complemented
28	#19151426	C₂²	1	1	yes	characteristic, complemented
29	#19151424	C₂²	1	1	yes	characteristic, complemented
30	#19151424	C₂²	1	2	yes	complemented
31	#19151424	C₂²	1	1	yes	characteristic, complemented
32	#19151400	C₂²	1	2	yes	—
33	#19151400	C₂²	1	2	yes	—
34	#19151400	C₂²	1	2	yes	—
35	#19151400	C₂²	1	2	yes	—
36	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
38	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
40	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
41	C₂²×D₃₅₆	—	2	1	no	—
42	C₂×D₇₁₂	—	2	1	no	—
43	C₂×D₇₁₂	—	2	2	no	—
44	C₂×D₇₁₂	—	2	2	no	—
45	C₂×D₇₁₂	C₂³	1	2	yes	—
46	#15668008	—	4	1	no	—
47	#15668001	—	4	1	no	—
48	#15667986	—	2	2	no	—
49	#15667986	—	2	2	no	—
50	#15667980	—	4	2	no	—
51	Q₈×Dic₈₉	—	4	1	no	—
52	D₄×Dic₈₉	—	4	2	no	—
53	D₄×Dic₈₉	—	4	1	no	—
54	#15667971	—	2	1	no	—
55	#15667971	—	2	1	no	—
56	#15667971	—	2	2	no	—
57	C₂×C₄⋊Dic₈₉	—	2	1	no	—
58	C₂×C₄⋊Dic₈₉	—	2	2	no	—
59	#15667946	—	2	1	no	—
60	#15667946	—	2	2	no	—
61	#15667946	C₂³	1	2	yes	—
62	#15667946	C₂³	1	1	yes	characteristic
63	#15667946	C₂³	1	1	yes	characteristic
64	#15667946	C₂³	1	1	yes	characteristic
65	#15667946	C₂³	1	1	yes	characteristic
66	#15667944	—	2	1	no	—
67	#15667944	—	2	1	no	—
68	#15667944	—	2	2	no	—
69	#15667934	—	2	2	no	—
70	#15667934	—	2	2	no	—
71	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
72	#131×C₈₉	—	2	2	no	—
73	#131×C₈₉	—	2	2	no	—
74	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
75	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
77	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	M₃₂×C₈₉	—	2	2	no	—
79	C₂×C₁₆×C₈₉	—	2	2	no	—
80	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
82	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
83	C₂³×D₈₉	—	4	1	no	—
84	C₂×D₃₅₆	—	2	2	no	—
85	C₂×D₃₅₆	—	4	2	no	—
86	C₂×D₃₅₆	—	4	2	no	—
87	C₂×D₃₅₆	—	4	1	no	—
88	C₂×C₂²⋊D₈₉	—	4	2	no	—
89	C₂×C₂²⋊D₈₉	—	4	2	no	—
90	C₂²×Dic₈₉	—	4	2	no	—
91	C₂²×Dic₈₉	—	4	1	no	—
92	D₇₁₂	—	2	2	no	—
93	D₇₁₂	—	2	2	no	—
94	D₇₁₂	—	4	2	no	—
95	C₄⋊Dic₈₉	—	2	1	no	—
96	C₄⋊Dic₈₉	—	2	1	no	—
97	C₄⋊Dic₈₉	—	2	1	no	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	4	2	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	C₂²⋊Dic₈₉	—	4	2	no	—
103	C₂²⋊Dic₈₉	—	4	1	no	—
104	C₂²⋊Dic₈₉	—	4	1	no	—
105	#13083551	—	4	2	no	—
106	#13083551	—	4	2	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₄×Dic₈₉	—	4	1	no	—
109	C₄×Dic₈₉	—	4	1	no	—
110	#41×C₈₉	—	2	2	no	—
111	#41×C₈₉	—	2	2	no	—
112	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
113	C₂×D₄×C₈₉	D₈	1	2	yes	—
114	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	M₁₆×C₈₉	—	2	2	no	—
118	M₁₆×C₈₉	—	2	4	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	2	yes	—
124	C₂×C₈×C₈₉	#41	1	2	yes	—
125	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	1	no	—
127	C₁₆×C₈₉	—	2	1	no	—
128	C₁₆×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	C₂×D₈	1	2	yes	—
130	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂²×D₈₉	—	4	2	no	—
132	C₂²×D₈₉	—	8	2	no	—
133	C₂²×D₈₉	—	16	1	no	—
134	C₂²×D₈₉	—	16	1	no	—
135	Q₈×C₈₉	—	2	2	no	—
136	D₄×C₈₉	—	2	4	no	—
137	D₄×C₈₉	—	2	2	no	—
138	C₂×C₄×C₈₉	—	2	2	no	—
139	C₂×C₄×C₈₉	—	2	2	no	—
140	C₂×C₄×C₈₉	#123	1	2	yes	—
141	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	#137	1	2	yes	—
144	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₈₉	—	16	2	no	—
146	C₂²⋊D₈₉	—	16	2	no	—
147	C₂×Dic₈₉	—	4	1	no	—
148	C₂×Dic₈₉	—	4	1	no	—
149	C₂×Dic₈₉	—	4	1	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	4	1	no	—
152	C₂×Dic₈₉	—	16	1	no	—
153	C₂×Dic₈₉	—	16	2	no	—
154	D₃₅₆	—	4	2	no	—
155	D₃₅₆	—	4	2	no	—
156	D₃₅₆	—	8	2	no	—
157	C₈×C₈₉	—	2	2	no	—
158	C₈×C₈₉	#123	1	2	yes	—
159	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	2	2	no	—
164	C₂²×C₈₉	—	2	2	no	—
165	C₂²×C₈₉	—	2	2	no	—
166	C₂²×C₈₉	#480	1	2	yes	—
167	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₂×D₈₉	—	16	4	no	—
172	C₄×C₈₉	—	2	2	no	—
173	C₄×C₈₉	#480	1	2	yes	—
174	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
175	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
176	Dic₈₉	—	8	2	no	—
177	Dic₈₉	—	8	1	no	—
178	Dic₈₉	—	8	1	no	—
179	Dic₈₉	—	8	1	no	—
180	Dic₈₉	—	8	1	no	—
181	#33650	—	89	1	no	maximal, Sylow
182	C₂×C₈₉	—	2	2	no	—
183	C₂×C₈₉	—	4	2	no	—
184	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
185	C₂×C₈₉	#3415	1	1	yes	characteristic
186	C₂×C₈₉	#3416	1	1	yes	characteristic
187	D₈₉	—	16	2	no	—
188	#3058	—	89	1	no	—
189	#2913	—	89	1	no	—
190	#2234	—	89	2	no	—
191	#2234	—	89	2	no	—
192	#2233	—	89	2	no	—
193	#2232	—	89	2	no	—
194	#2159	—	89	1	no	—
195	#2158	—	89	1	no	—
196	#2140	—	89	1	no	—
197	C₂×#357	—	89	1	no	—
198	#2113	D₈₉	1	1	yes	characteristic, complemented
199	C₈₉	#33650	1	1	yes	characteristic, complemented, Sylow
200	C₂²×D₈	—	89	1	no	—
201	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
202	#535	—	178	1	no	—
203	#520	—	178	1	no	—
204	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
205	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
206	#469	—	178	2	no	—
207	#466	—	89	2	no	—
208	#466	—	89	2	no	—
209	#428	—	178	1	no	—
210	#428	—	178	2	no	—
211	#427	—	89	1	no	—
212	#427	—	89	1	no	—
213	#427	—	89	2	no	—
214	C₂×#107	—	89	1	no	—
215	C₂×#107	—	89	2	no	—
216	#418	—	178	1	no	—
217	#367	—	89	1	no	—
218	#367	—	89	1	no	—
219	#367	—	89	2	no	—
220	#366	—	89	1	no	—
221	#366	—	89	1	no	—
222	#366	—	89	2	no	—
223	#357	—	89	2	no	—
224	#357	—	89	2	no	—
225	#357	—	89	2	no	—
226	#357	—	89	2	no	—
227	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
228	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
229	C₂×#41	D₃₅₆	1	1	yes	characteristic
230	C₂²×D₄	—	178	1	no	—
231	C₂×D₈	—	89	2	no	—
232	C₂×D₈	—	178	1	no	—
233	C₂×D₈	—	178	2	no	—
234	C₂×D₈	—	178	2	no	—
235	#131	—	2	2	no	—
236	#131	—	2	2	no	—
237	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
238	C₂×M₁₆	D₃₅₆	1	2	yes	—
239	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	#127	—	356	1	no	—
242	#124	—	356	1	no	—
243	#121	—	178	2	no	—
244	#121	—	178	2	no	—
245	#120	—	356	2	no	—
246	C₄×Q₈	—	356	1	no	—
247	C₄×D₄	—	356	2	no	—
248	C₄×D₄	—	356	1	no	—
249	#117	—	178	2	no	—
250	#117	—	178	1	no	—
251	#117	—	178	1	no	—
252	C₂×C₄⋊C₄	—	178	2	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	M₃₂	—	2	2	no	—
255	C₂×C₁₆	—	2	2	no	—
256	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
258	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
259	#107	—	89	2	no	—
260	#107	—	89	1	no	—
261	#107	—	89	1	no	—
262	#107	—	89	1	no	—
263	#107	—	89	1	no	—
264	#107	—	178	2	no	—
265	#107	—	178	1	no	—
266	#106	—	178	2	no	—
267	#106	—	178	1	no	—
268	#106	—	178	1	no	—
269	#102	—	178	2	no	—
270	#102	—	178	2	no	—
271	C₂⁴	—	356	1	no	—
272	#41	—	2	2	no	—
273	#41	—	2	2	no	—
274	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
275	C₂×D₄	—	356	2	no	—
276	C₂×D₄	—	356	2	no	—
277	C₂×D₄	—	356	2	no	—
278	C₂×D₄	—	356	1	no	—
279	C₂×D₄	—	356	2	no	—
280	C₂×D₄	—	178	2	no	—
281	C₂×D₄	D₇₁₂	1	2	yes	—
282	C₂×D₄	D₇₁₂	1	1	yes	characteristic
283	C₂²×C₄	—	356	2	no	—
284	C₂²×C₄	—	356	1	no	—
285	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
286	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
287	D₈	—	356	2	no	—
288	D₈	—	178	2	no	—
289	D₈	—	178	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
295	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
296	C₂×C₈	#13083573	1	2	yes	—
297	C₂×C₈	#13083573	1	2	yes	—
298	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
299	C₄⋊C₄	—	356	2	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	178	1	no	—
302	C₄⋊C₄	—	178	1	no	—
303	C₄⋊C₄	—	178	1	no	—
304	C₄⋊C₄	—	178	2	no	—
305	C₄⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	356	2	no	—
307	C₂²⋊C₄	—	356	1	no	—
308	C₂²⋊C₄	—	356	1	no	—
309	C₂²⋊C₄	—	356	2	no	—
310	C₂²⋊C₄	—	356	2	no	—
311	C₄²	—	356	2	no	—
312	C₄²	—	356	1	no	—
313	C₄²	—	356	1	no	—
314	C₁₆	—	2	1	no	—
315	C₁₆	—	2	1	no	—
316	C₁₆	—	2	2	no	—
317	C₂³	—	356	2	no	—
318	C₂³	—	1424	1	no	—
319	C₂³	—	1424	1	no	—
320	C₂³	—	712	2	no	—
321	C₂³	C₂×D₇₁₂	1	2	yes	—
322	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
323	Q₈	—	2	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	2	2	no	—
326	D₄	—	356	2	no	—
327	D₄	—	356	2	no	—
328	D₄	—	1424	2	no	—
329	D₄	—	1424	2	no	—
330	D₄	—	712	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	356	1	no	—
334	C₂×C₄	—	356	1	no	—
335	C₂×C₄	—	356	1	no	—
336	C₂×C₄	—	356	2	no	—
337	C₂×C₄	—	356	1	no	—
338	C₂×C₄	—	1424	1	no	—
339	C₂×C₄	—	1424	2	no	—
340	C₂×C₄	#15667996	1	2	yes	—
341	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
342	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
343	C₂×C₄	#15668045	1	2	yes	—
344	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	#15667996	1	2	yes	—
347	C₈	C₂×#13083573	1	1	yes	characteristic
348	C₈	C₂×#13083573	1	1	yes	characteristic
349	C₂²	—	2	1	no	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	1424	4	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	—	712	2	no	—
358	C₂²	#19151912	1	2	yes	—
359	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
360	C₄	—	2	2	no	—
361	C₄	—	712	2	no	—
362	C₄	—	712	1	no	—
363	C₄	—	712	1	no	—
364	C₄	—	712	1	no	—
365	C₄	—	712	1	no	—
366	C₄	#19151912	1	2	yes	—
367	C₄	C₂×#15667996	1	1	yes	characteristic
368	C₄	C₂×#15668045	1	1	yes	characteristic
369	C₂	—	2	2	no	—
370	C₂	—	4	2	no	—
371	C₂	—	1424	2	no	—
372	C₂	C₂×#19151912	1	1	yes	characteristic
373	C₂	#26033393	1	1	yes	characteristic
374	C₂	#26033400	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2113	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26029900	C₂
hypercenter	#2113	D₈₉
hypocenter	C₈₉	#33650
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ #2113

Complements

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

N	H ≅ G/N	type
2. #26028852	C₂	semidirect
3. #26022444	C₂	semidirect
4. #26022444	C₂	semidirect
5. #26022438	C₂	semidirect
6. #26022433	C₂	semidirect
7. #26022040	C₂	semidirect
8. #26022036	C₂	semidirect
9. #26021969	C₂	semidirect
10. C₂×#19151400	C₂	semidirect
11. C₈₉×#2113	C₂	semidirect
22. #19151636	C₂²	semidirect
24. C₂×#15667946	C₂²	semidirect
26. #19151426	C₂²	semidirect
27. #19151426	C₂²	semidirect
28. #19151426	C₂²	semidirect
29. #19151424	C₂²	semidirect
30. #19151424	C₂²	semidirect
31. #19151424	C₂²	semidirect
39. C₂²⋊C₁₆×C₈₉	C₂²	semidirect
40. C₂²⋊C₁₆×C₈₉	C₂²	semidirect
198. #2113	D₈₉	semidirect
199. C₈₉	#33650	semidirect
227. C₂²⋊C₁₆	C₂×D₈₉	semidirect
228. C₂²⋊C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029900	2
#26028852	2
#26022444	2
#26022438	2
#26022433	2
#26022040	2
#26022036	2
#26021969	2
C₂×#19151400	2
C₈₉×#2113	2
#33650	89