Finite Group Databasebeta

← #21427555

Subgroups — #21427555

Order 7552.

800 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14273695C211yesmaximal, characteristic
2C2×C4×C22⋊D59C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×D59C211yesmaximal, characteristic, complemented
4#16837337C211yesmaximal, characteristic, complemented
5#16837330C211yesmaximal, characteristic, complemented
6#16837318C211yesmaximal, characteristic, complemented
7#16837316C211yesmaximal, characteristic, complemented
8#16837295C211yesmaximal, characteristic, complemented
9#16837292C211yesmaximal, characteristic, complemented
10#16837281C211yesmaximal, characteristic, complemented
11#16837278C211yesmaximal, characteristic, complemented
12#16837273C211yesmaximal, characteristic, complemented
13#16837259C211yesmaximal, characteristic, complemented
14#16837257C211yesmaximal, characteristic, complemented
15C2×#123×C59C211yesmaximal, characteristic, complemented
16C24×D59C2211yescharacteristic
17C22×C22⋊D59C2211yescharacteristic
18C22×C22⋊D59C2211yescharacteristic
19C22×C4×D59C2211yescharacteristic
20C22×C4×D59C2211yescharacteristic
21#14273695—24no—
22C4×C22⋊D59—24no—
23C2×C4⋊Dic59C2211yescharacteristic, complemented
24C2×Dic59⋊C4C2211yescharacteristic, complemented
25C2×Dic59⋊C4C2211yescharacteristic, complemented
26C2×C22⋊Dic59C2211yescharacteristic
27C2×C22⋊Dic59C2211yescharacteristic
28C2×C22⋊Dic59C2211yescharacteristic
29C2×C22⋊Dic59C2211yescharacteristic, complemented
30C2×C22⋊Dic59C2211yescharacteristic, complemented
31C2×(C2×D59)⋊C4C2211yescharacteristic
32C2×(C2×D59)⋊C4C2211yescharacteristic
33C2×(C2×D59)⋊C4C2211yescharacteristic
34C2×(C2×D59)⋊C4C2211yescharacteristic
35C2×(C2×D59)⋊C4C2211yescharacteristic
36C2×(C2×D59)⋊C4C2211yescharacteristic
37C2×(C2×D59)⋊C4C2211yescharacteristic
38C2×(C2×D59)⋊C4C2211yescharacteristic
39C22⋊C4×D59—24no—
40C2×C4×Dic59C2211yescharacteristic, complemented
41C59⋊#95C2211yescharacteristic, complemented
42C59⋊#95C2211yescharacteristic, complemented
43C59⋊#95C2211yescharacteristic, complemented
44C59⋊#95C2211yescharacteristic, complemented
45C59⋊#95C2211yescharacteristic, complemented
46C59⋊#95C2211yescharacteristic, complemented
47C22×D4×C59C2211yescharacteristic
48C23×C4×C59C2211yescharacteristic, complemented
49#123×C59—24no—
50C2×C4⋊C4×C59C2211yescharacteristic, complemented
51C2×C4⋊C4×C59C2211yescharacteristic, complemented
52C2×C22⋊C4×C59C2211yescharacteristic
53C2×C22⋊C4×C59C2211yescharacteristic
54C2×C22⋊C4×C59C2211yescharacteristic, complemented
55C24×C59C2311yescharacteristic
56C24×C59C2311yescharacteristic
57C23×D59—22no—
58C23×D59—22no—
59C23×D59—22no—
60C23×D59—22no—
61C23×D59—22no—
62C23×D59—41no—
63C23×D59—41no—
64C23×D59C2311yescharacteristic
65C23×D59C2311yescharacteristic
66C2×D4×C59—22no—
67C2×D4×C59—22no—
68C2×D4×C59—41no—
69C22×C4×C59—22no—
70C22×C4×C59—22no—
71C22×C4×C59—41no—
72C22×C4×C59C2311yescharacteristic
73C22×C4×C59C2311yescharacteristic
74C22×C4×C59C2311yescharacteristic
75C22×C4×C59C2311yescharacteristic
76C22×C4×C59C2311yescharacteristic
77C2×C22⋊D59—22no—
78C2×C22⋊D59—22no—
79C2×C22⋊D59—22no—
80C2×C22⋊D59—22no—
81C2×C22⋊D59—41no—
82C2×C22⋊D59D414yes—
83C22×Dic59C2311yescharacteristic
84C22×Dic59C2311yescharacteristic
85C22×Dic59C2311yescharacteristic
86C22×Dic59C2311yescharacteristic
87C22×Dic59C2311yescharacteristic
88C22×Dic59C2311yescharacteristic
89C2×C4×D59—22no—
90C2×C4×D59—22no—
91C2×C4×D59—22no—
92C2×C4×D59—22no—
93C2×C4×D59—41no—
94C2×C4×D59—41no—
95C4⋊C4×C59—22no—
96C4⋊C4×C59—22no—
97C22⋊C4×C59—22no—
98C22⋊C4×C59—22no—
99C22⋊C4×C59D414yescomplemented
100C22⋊Dic59—22no—
101C22⋊Dic59—22no—
102C22⋊Dic59—22no—
103C22⋊Dic59—41no—
104C22⋊Dic59—41no—
105(C2×D59)⋊C4—22no—
106(C2×D59)⋊C4—22no—
107(C2×D59)⋊C4—22no—
108(C2×D59)⋊C4—22no—
109(C2×D59)⋊C4—22no—
110(C2×D59)⋊C4—22no—
111(C2×D59)⋊C4—22no—
112(C2×D59)⋊C4—41no—
113C4⋊Dic59—41no—
114Dic59⋊C4—22no—
115Dic59⋊C4—41no—
116C4×Dic59—22no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—21no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—21no—
128C23×C59—21no—
129C23×C59C2×D412yes—
130C23×C59C2×D412yes—
131C23×C59C2411yescharacteristic
132C22×D59—21no—
133C22×D59—21no—
134C22×D59—21no—
135C22×D59—21no—
136C22×D59—28no—
137C22×D59—21no—
138C22×D59—21no—
139C22×D59—21no—
140C22×D59—21no—
141C22×D59—21no—
142C22×D59—21no—
143C22×D59—44no—
144C22×D59—44no—
145C22×D59—44no—
146C22×D59—44no—
147C22×D59—44no—
148C22×D59—44no—
149C22×D59C2×D412yes—
150C22×D59#4112yes—
151C22×D59#4112yes—
152C22×D59#4112yes—
153D4×C59—44no—
154C2×C4×C59—21no—
155C2×C4×C59—21no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—21no—
167C2×C4×C59—44no—
168C2×C4×C59C2×D412yes—
169C2×C4×C59C2×D412yes—
170C22⋊D59—28no—
171C22⋊D59—44no—
172C2×Dic59—21no—
173C2×Dic59—21no—
174C2×Dic59—21no—
175C2×Dic59—21no—
176C2×Dic59—21no—
177C2×Dic59—21no—
178C2×Dic59—21no—
179C2×Dic59—21no—
180C2×Dic59—21no—
181C2×Dic59—21no—
182C2×Dic59—21no—
183C2×Dic59—21no—
184C2×Dic59—21no—
185C2×Dic59—21no—
186C2×Dic59—21no—
187C2×Dic59—21no—
188C2×Dic59—21no—
189C2×Dic59C2×D412yes—
190C4×D59—44no—
191C4×D59—44no—
192C22×C59—22no—
193C22×C59—22no—
194C22×C59—22no—
195C22×C59—22no—
196C22×C59—22no—
197C22×C59—22no—
198C22×C59—41no—
199C22×C59—41no—
200C22×C59—41no—
201C22×C59—41no—
202C22×C59—41no—
203C22×C59—41no—
204C22×C59—41no—
205C22×C59—41no—
206C22×C59C22×D411yescharacteristic
207C22×C59C22×D411yescharacteristic
208C22×C59C2×#4111yescharacteristic
209C22×C59C2×#4111yescharacteristic
210C22×C59C2×#4111yescharacteristic
211C22×C59H(5,𝔽2)11yescharacteristic
212C22×C59H(5,𝔽2)11yescharacteristic
213C2×D59—22no—
214C2×D59—22no—
215C2×D59—22no—
216C2×D59—22no—
217C2×D59—22no—
218C2×D59—22no—
219C2×D59—22no—
220C2×D59—22no—
221C2×D59—22no—
222C2×D59—416no—
223C2×D59—41no—
224C2×D59—41no—
225C2×D59—41no—
226C2×D59—41no—
227C2×D59#12314yescomplemented
228C4×C59—22no—
229C4×C59—22no—
230C4×C59—41no—
231C4×C59—41no—
232C4×C59—41no—
233Dic59—22no—
234Dic59—41no—
235Dic59—41no—
236Dic59—41no—
237Dic59—41no—
238Dic59—41no—
239#2418—591nomaximal, Sylow
240C2×C59—42no—
241C2×C59—42no—
242C2×C59C2×#12311yescharacteristic
243C2×C59#52511yescharacteristic
244C2×C59#53811yescharacteristic
245C2×C59D4211yescharacteristic
246C2×C59#54611yescharacteristic
247C2×C59#54611yescharacteristic
248C2×C59#55111yescharacteristic
249D59—24no—
250D59—42no—
251C2×#123D5911yescharacteristic, complemented
252C2×#120—591no—
253C2×C4×D4—591no—
254C22×C22⋊C4—591no—
255#399—591no—
256#397—591no—
257#394—591no—
258#394—591no—
259#388—591no—
260#388—591no—
261#386—591no—
262#386—591no—
263#385—591no—
264#380—591no—
265#379—591no—
266C59#241811yescharacteristic, complemented, Sylow
267C25—591no—
268C22×D4—591no—
269C22×D4—591no—
270C22×D4C2×D5911yescharacteristic
271C23×C4—591no—
272C23×C4—591no—
273C23×C4C2×D5911yescharacteristic, complemented
274#123—24no—
275#120—1184no—
276C4×D4—1184no—
277C2×C4⋊C4—591no—
278C2×C4⋊C4—591no—
279C2×C4⋊C4—591no—
280C2×C4⋊C4C2×D5911yescharacteristic, complemented
281C2×C4⋊C4C2×D5911yescharacteristic, complemented
282C2×C22⋊C4—591no—
283C2×C22⋊C4—591no—
284C2×C22⋊C4—591no—
285C2×C22⋊C4—591no—
286C2×C22⋊C4—591no—
287C2×C22⋊C4—591no—
288C2×C22⋊C4—591no—
289C2×C22⋊C4—591no—
290C2×C22⋊C4—591no—
291C2×C22⋊C4—591no—
292C2×C22⋊C4—591no—
293C2×C22⋊C4—591no—
294C2×C22⋊C4—591no—
295C2×C22⋊C4—1184no—
296C2×C22⋊C4C2×D5911yescharacteristic
297C2×C22⋊C4C2×D5911yescharacteristic
298C2×C22⋊C4C2×D5911yescharacteristic, complemented
299C2×C42—591no—
300#95—591no—
301#95—591no—
302#95—591no—
303#95—591no—
304#95—591no—
305#95—591no—
306C24—591no—
307C24—591no—
308C24—1182no—
309C24—1182no—
310C24—1182no—
311C24—1182no—
312C24—1182no—
313C24—2361no—
314C24—2361no—
315C24C22×D5911yescharacteristic
316C24C22×D5911yescharacteristic
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—41no—
320C2×D4—594no—
321C2×D4—1182no—
322C2×D4—1182no—
323C2×D4—1182no—
324C2×D4—1182no—
325C2×D4—2361no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—41no—
329C22×C4—591no—
330C22×C4—591no—
331C22×C4—591no—
332C22×C4—591no—
333C22×C4—591no—
334C22×C4—591no—
335C22×C4—1182no—
336C22×C4—1182no—
337C22×C4—1182no—
338C22×C4—1182no—
339C22×C4—2361no—
340C22×C4—2361no—
341C22×C4C22×D5911yescharacteristic
342C22×C4C22×D5911yescharacteristic
343C22×C4C22×D5911yescharacteristic
344C22×C4C22×D5911yescharacteristic
345C22×C4C22×D5911yescharacteristic
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—1182no—
349C4⋊C4—2361no—
350C4⋊C4—2361no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—1182no—
354C22⋊C4—1182no—
355C22⋊C4—1182no—
356C22⋊C4—1182no—
357C22⋊C4—1182no—
358C22⋊C4—1182no—
359C22⋊C4—1182no—
360C22⋊C4—1182no—
361C22⋊C4—1182no—
362C22⋊C4—1182no—
363C22⋊C4—2361no—
364C22⋊C4—2361no—
365C22⋊C4—2361no—
366C22⋊C4C22⋊D5914yescomplemented
367C42—1182no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—592no—
381C23—592no—
382C23—592no—
383C23—592no—
384C23—1188no—
385C23—1181no—
386C23—1181no—
387C23—1181no—
388C23—1181no—
389C23—1181no—
390C23—1181no—
391C23—1181no—
392C23—1181no—
393C23—1181no—
394C23—1181no—
395C23—2364no—
396C23—2364no—
397C23—2364no—
398C23—2364no—
399C23—2364no—
400C23—2364no—
401C23D4×D5912yes—
402C23C2×C22⋊D5912yes—
403C23C23×D5911yescharacteristic
404D4—44no—
405D4—1188no—
406D4—2364no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—44no—
421C2×C4—592no—
422C2×C4—1181no—
423C2×C4—1181no—
424C2×C4—1181no—
425C2×C4—1181no—
426C2×C4—1181no—
427C2×C4—1181no—
428C2×C4—1181no—
429C2×C4—1181no—
430C2×C4—1181no—
431C2×C4—1181no—
432C2×C4—1181no—
433C2×C4—1181no—
434C2×C4—1181no—
435C2×C4—1181no—
436C2×C4—1181no—
437C2×C4—1181no—
438C2×C4—1181no—
439C2×C4—2364no—
440C2×C4—2364no—
441C2×C4C2×C22⋊D5912yes—
442C2×C4C2×C22⋊D5912yes—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—594no—
458C22—1182no—
459C22—1182no—
460C22—1182no—
461C22—1182no—
462C22—1182no—
463C22—1182no—
464C22—1182no—
465C22—1182no—
466C22—1182no—
467C22—23616no—
468C22—2361no—
469C22—2361no—
470C22—2361no—
471C22—2361no—
472C22C2×D4×D5911yescharacteristic
473C22C22×C22⋊D5911yescharacteristic
474C22#41×D5911yescharacteristic
475C22#41×D5911yescharacteristic
476C22#41×D5911yescharacteristic
477C22#1427377211yescharacteristic
478C22#1427377311yescharacteristic
479C4—22no—
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—1182no—
485C4—2361no—
486C4—2361no—
487C4—2361no—
488C4—2361no—
489C4—2361no—
490C2—42no—
491C2—42no—
492C2—1184no—
493C2—2362no—
494C2#123×D5911yescharacteristic
495C2#1683790911yescharacteristic
496C2#1683798811yescharacteristic
497C2D4×C22⋊D5911yescharacteristic
498C2#1683804011yescharacteristic
499C2#1683804411yescharacteristic
500C2#1683808011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC2×#123×C59C2
hypercenterC2×#123D59
hypocenterC59#2418
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D59C2semidirect
3. C2×C22⋊C4×D59C2semidirect
4. #16837337C2semidirect
5. #16837330C2semidirect
6. #16837318C2semidirect
7. #16837316C2semidirect
8. #16837295C2semidirect
9. #16837292C2semidirect
10. #16837281C2semidirect
11. #16837278C2semidirect
12. #16837273C2semidirect
13. #16837259C2semidirect
14. #16837257C2semidirect
15. C2×#123×C59C2semidirect
23. C2×C4⋊Dic59C22semidirect
24. C2×Dic59⋊C4C22semidirect
25. C2×Dic59⋊C4C22semidirect
29. C2×C22⋊Dic59C22semidirect
30. C2×C22⋊Dic59C22semidirect
40. C2×C4×Dic59C22semidirect
41. C59⋊#95C22semidirect
42. C59⋊#95C22semidirect
43. C59⋊#95C22semidirect
44. C59⋊#95C22semidirect
45. C59⋊#95C22semidirect
46. C59⋊#95C22semidirect
48. C23×C4×C59C22semidirect
50. C2×C4⋊C4×C59C22semidirect
51. C2×C4⋊C4×C59C22semidirect
54. C2×C22⋊C4×C59C22semidirect
99. C22⋊C4×C59D4semidirect
227. C2×D59#123semidirect
251. C2×#123D59semidirect
266. C59#2418semidirect
273. C23×C4C2×D59semidirect
280. C2×C4⋊C4C2×D59semidirect
281. C2×C4⋊C4C2×D59semidirect
298. C2×C22⋊C4C2×D59semidirect
366. C22⋊C4C22⋊D59semidirect