Finite Group Databasebeta

← #28588484

Subgroups — #28588484

Order 15104.

1774 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D59C211yesmaximal, characteristic
2C2×#16837364C211yesmaximal, characteristic
3#21426478C211yesmaximal, characteristic
4#21424185C214yesmaximal, complemented
5#21424183C211yesmaximal, characteristic, complemented
6#21424182C211yesmaximal, characteristic, complemented
7#21424180C211yesmaximal, characteristic, complemented
8#21423814C214yesmaximal, complemented
9C59×#1823C211yesmaximal, characteristic, complemented
10C23×C4×D59—21no—
11C23×C4×D59C2211yescharacteristic
12C22×#11745659C2212yes—
13C2×#14273685C2212yes—
14C22×C4⋊Dic59C2211yescharacteristic
15C22×Dic59⋊C4—21no—
16C2×C4⋊C4×D59—21no—
17C2×C4⋊C4×D59—21no—
18C2×C4⋊C4×D59—24no—
19C2×C4⋊C4×D59C418yescomplemented
20C2×C4⋊C4×D59C2212yes—
21C2×C4⋊C4×D59C2212yes—
22#16837364—24no—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×(C2×D59)⋊C8C2212yes—
25#16837264—21no—
26#16837262C2211yescharacteristic
27#16837259C2212yes—
28#16837257—21no—
29C22⋊C4×Dic59—21no—
30#16837249—21no—
31#16837165C2214yescomplemented
32#16837164C2214yescomplemented
33#16837164C2214yescomplemented
34C22×C4⋊C4×C59C2211yescharacteristic
35C2×C22⋊C8×C59C2211yescharacteristic
36C59×#380C2211yescharacteristic
37C59×#340C2214yescomplemented
38C24×D59D411yescharacteristic
39C23×Dic59—21no—
40C23×Dic59D411yescharacteristic
41C22×C4×D59—28no—
42C22×C4×D59—28no—
43C22×C4×D59—22no—
44C22×C4×D59—22no—
45C22×C4×D59—44no—
46C22×C4×D59—44no—
47C22×C4×D59C2×C414yes—
48C22×C4×D59D411yescharacteristic
49C22×C4×D59D411yescharacteristic
50C22×C4×D59D414yes—
51C22×C4×D59C2311yescharacteristic
52C22×C4×D59C2311yescharacteristic
53C2×#11745659—24no—
54C2×#11745659—28no—
55C22×C59⋊C8C2312yes—
56#14273685—28no—
57C2×C4⋊Dic59—21no—
58C2×C4⋊Dic59—22no—
59C2×C4⋊Dic59C2×C414yes—
60C2×C4⋊Dic59C2312yes—
61C2×C4⋊Dic59C2312yes—
62C2×Dic59⋊C4—28no—
63C2×Dic59⋊C4—22no—
64C2×Dic59⋊C4—22no—
65C4⋊C4×D59—28no—
66C4⋊C4×D59—28no—
67C4⋊C4×D59—24no—
68C4⋊C4×D59—24no—
69C4⋊C4×D59C2×C418yes—
70C4⋊C4×D59C2×C418yes—
71C2×C22⋊Dic59—21no—
72C2×(C2×D59)⋊C4—22no—
73C2×(C2×D59)⋊C4—22no—
74C2×C4×Dic59—22no—
75C22⋊(C59⋊C8)—22no—
76(C2×D59)⋊C8—24no—
77C59⋊#95—21no—
78C59⋊#95—22no—
79C23×C4×C59—21no—
80C23×C4×C59C2311yescharacteristic
81C22×C8×C59C2312yes—
82C2×C4⋊C4×C59—21no—
83C2×C4⋊C4×C59—22no—
84C2×C4⋊C4×C59C2×C414yes—
85C2×C4⋊C4×C59C2312yes—
86C2×C4⋊C4×C59C2312yes—
87C2×C22⋊C4×C59—21no—
88C22⋊C8×C59—22no—
89#95×C59—21no—
90C24×C59C2×D411yescharacteristic
91C23×D59—22no—
92C23×D59—22no—
93C23×D59—24no—
94C23×D59—24no—
95C23×D59C22⋊C414yes—
96C23×D59C2×D411yescharacteristic
97C23×D59C2×D411yescharacteristic
98C22×C4×C59—24no—
99C22×C4×C59—24no—
100C22×C4×C59—22no—
101C22×C4×C59—22no—
102C22×C4×C59—42no—
103C22×C4×C59—42no—
104C22×C4×C59C22×C412yes—
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D412yes—
107C22×C4×C59C2411yescharacteristic
108C22×Dic59—24no—
109C22×Dic59—24no—
110C22×Dic59—22no—
111C22×Dic59—22no—
112C22×Dic59—42no—
113C22×Dic59—42no—
114C22×Dic59C22⋊C414yes—
115C22×Dic59C2×D411yescharacteristic
116C22×Dic59C2×D411yescharacteristic
117C2×C4×D59—24no—
118C2×C4×D59—24no—
119C2×C4×D59—24no—
120C2×C4×D59—216no—
121C2×C4×D59—216no—
122C2×C4×D59—28no—
123C2×C4×D59—28no—
124C2×C4×D59—44no—
125C2×C4×D59—44no—
126C2×C4×D59—44no—
127C2×C4×D59—44no—
128C2×C4×D59—432no—
129C2×C4×D59C22⋊C414yes—
130C2×C4×D59C22⋊C414yes—
131C2×C4×D59C22×C412yes—
132C2×C4×D59C22×C412yes—
133C2×C4×D59C2×D412yes—
134C2×C4×D59C2×D412yes—
135C2×C4×D59#4118yes—
136C2×C8×C59—22no—
137C2×C8×C59—24no—
138C4⋊C4×C59—24no—
139C4⋊C4×C59—22no—
140C4⋊C4×C59—24no—
141C4⋊C4×C59C22×C414yes—
142C22⋊C4×C59—42no—
143C22⋊Dic59—42no—
144(C2×D59)⋊C4—44no—
145(C2×D59)⋊C4—44no—
146C4⋊Dic59—24no—
147C4⋊Dic59—22no—
148C4⋊Dic59—24no—
149C4⋊Dic59C22×C414yes—
150Dic59⋊C4—28no—
151Dic59⋊C4—28no—
152C4×Dic59—44no—
153C2×C59⋊C8—22no—
154C2×C59⋊C8—24no—
155#11745659—48no—
156C23×C59—22no—
157C23×C59—22no—
158C23×C59C2×C22⋊C412yes—
159C23×C59AGL(1,ℤ/8)12yes—
160C23×C59AGL(1,ℤ/8)12yes—
161C23×C59C22×D411yescharacteristic
162C22×D59—216no—
163C22×D59—216no—
164C22×D59—22no—
165C22×D59—21no—
166C22×D59—21no—
167C22×D59—21no—
168C22×D59—21no—
169C22×D59—22no—
170C22×D59—28no—
171C22×D59—28no—
172C22×D59—28no—
173C22×D59C2×C22⋊C412yes—
174C22×D59C2×C22⋊C412yes—
175C22×D59#12314yes—
176C22×D59#13714yes—
177C2×C4×C59—22no—
178C2×C4×C59—24no—
179C2×C4×C59—24no—
180C2×C4×C59—24no—
181C2×C4×C59—42no—
182C2×C4×C59—42no—
183C2×C4×C59—48no—
184C2×C4×C59—42no—
185C2×C4×C59—42no—
186C2×C4×C59—42no—
187C2×C4×C59C2×C22⋊C412yes—
188C2×C4×C59C23×C411yescharacteristic
189C2×C4×C59C22×D411yescharacteristic
190C2×C4×C59C2×#4114yes—
191C2×Dic59—22no—
192C2×Dic59—28no—
193C2×Dic59—24no—
194C2×Dic59—42no—
195C2×Dic59—42no—
196C2×Dic59—48no—
197C2×Dic59—42no—
198C2×Dic59—42no—
199C2×Dic59—42no—
200C2×Dic59C2×C22⋊C412yes—
201C2×Dic59C2×C22⋊C412yes—
202C2×Dic59#12314yes—
203C4×D59—28no—
204C4×D59—24no—
205C4×D59—24no—
206C4×D59—28no—
207C4×D59—416no—
208C4×D59—416no—
209C4×D59#11718yes—
210C4×D59#12318yes—
211C8×C59—42no—
212C59⋊C8—42no—
213#20700—591nomaximal, Sylow
214C22×C59—24no—
215C22×C59—24no—
216C22×C59—22no—
217C22×C59—22no—
218C22×C59#41814yes—
219C22×C59C22×C22⋊C411yescharacteristic
220C22×C59C2×#12312yes—
221C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
222C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
223C22×C59C2×#13712yes—
224C2×D59—232no—
225C2×D59—24no—
226C2×D59—24no—
227C2×D59—24no—
228C2×D59—24no—
229C2×D59—22no—
230C2×D59—22no—
231C2×D59—22no—
232C2×D59—22no—
233C2×D59#38114yes—
234C2×D59#41914yes—
235C4×C59—22no—
236C4×C59—44no—
237C4×C59—82no—
238C4×C59C2×#11712yes—
239C4×C59C2×#12312yes—
240Dic59—22no—
241Dic59—44no—
242Dic59—82no—
243Dic59#38114yes—
244C23×C4⋊C4—591no—
245C2×#407—591no—
246#2280—591no—
247#1824—594no—
248#1823—591no—
249#1823—591no—
250#1823—591no—
251#1823D5911yescharacteristic, complemented
252#1736—594no—
253C2×C59—24no—
254C2×C59C2×#38111yescharacteristic
255C2×C59C2×#41811yescharacteristic
256C2×C59C2×#41911yescharacteristic
257C2×C59#309212yes—
258C2×C59#309212yes—
259D59—24no—
260D59—24no—
261C24×C4—591no—
262C24×C4—1181no—
263C22×M16—592no—
264C2×#117—592no—
265C22×C4⋊C4—591no—
266C22×C4⋊C4—598no—
267C22×C4⋊C4—592no—
268C22×C4⋊C4—592no—
269C22×C4⋊C4—1181no—
270C22×C4⋊C4—1181no—
271C22×C4⋊C4—1184no—
272C22×C4⋊C4—1181no—
273C22×C4⋊C4C2×D5911yescharacteristic
274#407—1184no—
275C2×C22⋊C8—591no—
276C2×C22⋊C8—592no—
277C2×C22⋊C8C2×D5911yescharacteristic
278#381—1181no—
279#380—591no—
280#380—592no—
281#380C2×D5911yescharacteristic
282#379—1181no—
283C4×C22⋊C4—1181no—
284C4×C22⋊C4—1181no—
285#340—594no—
286#340—594no—
287#340—594no—
288#340C2×D5914yescomplemented
289C59#2070011yescharacteristic, complemented, Sylow
290C25—591no—
291C23×C4—21no—
292C23×C4—591no—
293C23×C4—591no—
294C23×C4—594no—
295C23×C4—591no—
296C23×C4—591no—
297C23×C4—594no—
298C23×C4—591no—
299C23×C4—1188no—
300C23×C4—1188no—
301C23×C4—1182no—
302C23×C4—1182no—
303C23×C4—1181no—
304C23×C4—2364no—
305C23×C4—2364no—
306C23×C4C22×D5911yescharacteristic
307C2×M16—1184no—
308C2×M16—1188no—
309C22×C8—592no—
310C22×C8C22×D5912yes—
311#117—1188no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—594no—
315C2×C4⋊C4—592no—
316C2×C4⋊C4—598no—
317C2×C4⋊C4—598no—
318C2×C4⋊C4—592no—
319C2×C4⋊C4—1188no—
320C2×C4⋊C4—1188no—
321C2×C4⋊C4—1184no—
322C2×C4⋊C4—1184no—
323C2×C4⋊C4—1188no—
324C2×C4⋊C4—1182no—
325C2×C4⋊C4—1182no—
326C2×C4⋊C4—1181no—
327C2×C4⋊C4—1182no—
328C2×C4⋊C4C4×D5914yes—
329C2×C4⋊C4C22×D5912yes—
330C2×C4⋊C4C22×D5912yes—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—1182no—
333C2×C22⋊C4—1182no—
334C2×C22⋊C4—1181no—
335C2×C42—1182no—
336C22⋊C8—22no—
337C22⋊C8—1184no—
338C22⋊C8—1182no—
339#95—21no—
340#95—1182no—
341#95—1181no—
342C24—591no—
343C24—591no—
344C24—594no—
345C24—1184no—
346C24—1184no—
347C24—1182no—
348C24—1182no—
349C24D4×D5911yescharacteristic
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—591no—
357C22×C4—592no—
358C22×C4—592no—
359C22×C4—598no—
360C22×C4—594no—
361C22×C4—591no—
362C22×C4—594no—
363C22×C4—594no—
364C22×C4—592no—
365C22×C4—592no—
366C22×C4—1184no—
367C22×C4—11816no—
368C22×C4—11816no—
369C22×C4—1184no—
370C22×C4—1188no—
371C22×C4—1188no—
372C22×C4—1184no—
373C22×C4—1184no—
374C22×C4—1184no—
375C22×C4—1182no—
376C22×C4—1182no—
377C22×C4—23632no—
378C22×C4—2364no—
379C22×C4—2364no—
380C22×C4—2364no—
381C22×C4—2364no—
382C22×C4—2362no—
383C22×C4—2362no—
384C22×C4C2×C4×D5912yes—
385C22×C4D4×D5911yescharacteristic
386C22×C4D4×D5912yes—
387C22×C4C23×D5911yescharacteristic
388M16—2368no—
389C2×C8—22no—
390C2×C8—24no—
391C2×C8—1182no—
392C2×C8—1184no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—594no—
397C4⋊C4—1188no—
398C4⋊C4—1188no—
399C4⋊C4—1184no—
400C4⋊C4—1182no—
401C4⋊C4—1184no—
402C4⋊C4C2×C4×D5914yes—
403C22⋊C4—42no—
404C22⋊C4—2364no—
405C22⋊C4—2364no—
406C22⋊C4—2362no—
407C42—2364no—
408C23—22no—
409C23—22no—
410C23—594no—
411C23—594no—
412C23—592no—
413C23—592no—
414C23—1188no—
415C23—1188no—
416C23—11816no—
417C23—11816no—
418C23—1182no—
419C23—1188no—
420C23—1181no—
421C23—1181no—
422C23—1181no—
423C23—1181no—
424C23—1182no—
425C23C22⋊C4×D5912yes—
426C23#1427375112yes—
427C23#1427375412yes—
428C23C2×D4×D5911yescharacteristic
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—48no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—594no—
440C2×C4—592no—
441C2×C4—598no—
442C2×C4—598no—
443C2×C4—592no—
444C2×C4—1188no—
445C2×C4—1188no—
446C2×C4—1188no—
447C2×C4—1182no—
448C2×C4—1184no—
449C2×C4—1184no—
450C2×C4—1184no—
451C2×C4—23616no—
452C2×C4—23616no—
453C2×C4—2362no—
454C2×C4—2362no—
455C2×C4—2362no—
456C2×C4—2368no—
457C2×C4—2362no—
458C2×C4—2362no—
459C2×C4C22⋊C4×D5912yes—
460C2×C4C22×C4×D5911yescharacteristic
461C2×C4C2×D4×D5911yescharacteristic
462C2×C4#41×D5914yes—
463C8—42no—
464C8—2362no—
465C22—22no—
466C22—22no—
467C22—24no—
468C22—24no—
469C22—594no—
470C22—594no—
471C22—11832no—
472C22—1182no—
473C22—1182no—
474C22—1182no—
475C22—1182no—
476C22—1184no—
477C22—1184no—
478C22—1184no—
479C22—1184no—
480C22#1683740514yes—
481C22C2×C22⋊C4×D5911yescharacteristic
482C22#123×D5912yes—
483C22C2×#1427375111yescharacteristic
484C22C2×#1427375411yescharacteristic
485C22#137×D5912yes—
486C4—22no—
487C4—44no—
488C4—82no—
489C4—594no—
490C4—1182no—
491C4—4722no—
492C4—2364no—
493C4#117×D5912yes—
494C4#123×D5912yes—
495C2—24no—
496C2—1184no—
497C2—1184no—
498C2#381×D5911yescharacteristic
499C2C2×#1683740511yescharacteristic
500C2#419×D5911yescharacteristic
501C2#2143401512yes—
502C2#2143402012yes—
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1823C2
socleC23×C59C22×D4
prime radicalC24×D59D4
cosocleC22×C4×C59C24
hypercenter#1823D59
hypocenterC59#20700
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. #21424185C2semidirect
5. #21424183C2semidirect
6. #21424182C2semidirect
7. #21424180C2semidirect
8. #21423814C2semidirect
9. C59×#1823C2semidirect
19. C2×C4⋊C4×D59C4semidirect
31. #16837165C22semidirect
32. #16837164C22semidirect
33. #16837164C22semidirect
37. C59×#340C22semidirect
251. #1823D59semidirect
288. #340C2×D59semidirect
289. C59#20700semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D592
C2×#168373642
#214264782
#214241852
#214241832
#214241822
#214241802
#214238142
C59×#18232
#2070059