Finite Group Databasebeta

← #28485173

Subgroups — #28485173

Order 15104.

504 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433152C211yesmaximal, characteristic, complemented
2#21422600C211yesmaximal, characteristic, complemented
3C2×#16837136C211yesmaximal, characteristic, complemented
4#21422456C211yesmaximal, characteristic, complemented
5#21421984C211yesmaximal, characteristic, complemented
6#21421938C211yesmaximal, characteristic, complemented
7C59×#3188C211yesmaximal, characteristic
8#16837474—21no—
9#16837474—21no—
10D4×C59⋊C8—21no—
11D4×C59⋊C8—21no—
12#16837463—21no—
13#16837369—21no—
14#16837366C2211yescharacteristic
15C2×C22⋊(C59⋊C8)C2211yescharacteristic
16C2×C22⋊(C59⋊C8)C2211yescharacteristic
17C2×C59⋊#95C2211yescharacteristic
18#16837139—21no—
19#16837136—21no—
20#16837136—21no—
21#16837136—21no—
22#16837126—21no—
23#16837126—21no—
24C59×#525C2211yescharacteristic
25C2×#121×C59C2211yescharacteristic
26C59×#484C411yescharacteristic, complemented
27C59×#482—21no—
28C59×#481C411yescharacteristic, complemented
29C59×#466—21no—
30C59×#465—21no—
31C59×#450C411yescharacteristic, complemented
32C59×#449—21no—
33C59×#447C411yescharacteristic, complemented
34C2×C22⋊C8×C59C2211yescharacteristic
35C23×Dic59—21no—
36C2×#11745664—21no—
37C2×#11745664—21no—
38C22×C59⋊C8—21no—
39C22×C59⋊C8—21no—
40C22×C59⋊C8—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)—21no—
47C22⋊(C59⋊C8)—21no—
48C22⋊(C59⋊C8)—21no—
49C22⋊(C59⋊C8)D411yescharacteristic
50C22⋊(C59⋊C8)D411yescharacteristic
51C22⋊(C59⋊C8)D411yescharacteristic
52C22⋊(C59⋊C8)D411yescharacteristic
53#14273638—21no—
54C4×C59⋊C8—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C2×#41×C59D411yescharacteristic
58C22×D4×C59—21no—
59C22×D4×C59C2×C411yescharacteristic
60C23×C4×C59C2311yescharacteristic
61C2×SD16×C59—21no—
62C2×D8×C59—21no—
63C22×C8×C59—21no—
64#123×C59—21no—
65C22⋊Q8×C59C2×C411yescharacteristic
66C22⋊Q8×C59D411yescharacteristic
67#121×C59—21no—
68#121×C59—21no—
69#121×C59—21no—
70#121×C59—21no—
71#121×C59C2×C411yescharacteristic
72#121×C59D411yescharacteristic
73#120×C59—21no—
74C4×D4×C59—21no—
75C4×D4×C59—21no—
76#117×C59D411yescharacteristic
77C2×C4⋊C4×C59C2×C411yescharacteristic
78C2×C22⋊C4×C59—21no—
79#107×C59—21no—
80#106×C59—21no—
81#103×C59—21no—
82#102×C59—21no—
83#102×C59—21no—
84#102×C59—21no—
85C22⋊C8×C59—21no—
86C22⋊C8×C59C2×C411yescharacteristic
87C22⋊C8×C59C2×C411yescharacteristic
88C24×C59—21no—
89C24×C59#4111yescharacteristic
90#41×C59—81no—
91C2×Q8×C59C22⋊C411yescharacteristic
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—21no—
97C2×D4×C59—41no—
98C2×D4×C59—41no—
99C2×D4×C59—81no—
100C2×D4×C59—81no—
101C2×D4×C59—81no—
102C2×D4×C59C22⋊C411yescharacteristic
103C22×C4×C59—21no—
104C22×C4×C59—21no—
105C22×C4×C59—81no—
106C22×C4×C59C22×C411yescharacteristic
107C22×C4×C59C2×D411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D411yescharacteristic
110C22×C4×C59C2×D411yescharacteristic
111C22×C4×C59#4111yescharacteristic
112C22×Dic59—22no—
113C22×Dic59—41no—
114C22×Dic59—41no—
115C22×Dic59—81no—
116C22×Dic59—81no—
117SD16×C59—81no—
118D8×C59—81no—
119C2×C8×C59—21no—
120C2×C8×C59—21no—
121C2×C8×C59—81no—
122C4⋊C4×C59—21no—
123C4⋊C4×C59—21no—
124C4⋊C4×C59—21no—
125C4⋊C4×C59—21no—
126C4⋊C4×C59C22⋊C411yescharacteristic
127C4⋊C4×C59C22⋊C411yescharacteristic
128C22⋊C4×C59—21no—
129C22⋊C4×C59—21no—
130C22⋊C4×C59—21no—
131C22⋊C4×C59—21no—
132C22⋊C4×C59—21no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—41no—
135C42×C59—21no—
136#11745664—81no—
137#11745664—81no—
138C2×C59⋊C8—21no—
139C2×C59⋊C8—21no—
140C2×C59⋊C8—21no—
141C2×C59⋊C8—21no—
142C2×C59⋊C8—41no—
143C2×C59⋊C8—81no—
144C2×C59⋊C8—81no—
145C2×C59⋊C8—81no—
146C23×C59—21no—
147C23×C59—21no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—41no—
152C23×C59—41no—
153C23×C59—41no—
154C23×C59—41no—
155C23×C59—81no—
156C23×C59—81no—
157C23×C59C4×D411yescharacteristic
158C23×C59#12111yescharacteristic
159C23×C59#12311yescharacteristic
160Q8×C59—41no—
161D4×C59—41no—
162D4×C59—41no—
163D4×C59—41no—
164D4×C59—41no—
165D4×C59—41no—
166D4×C59—41no—
167D4×C59—41no—
168D4×C59—81no—
169D4×C59—81no—
170D4×C59—82no—
171D4×C59—81no—
172D4×C59—81no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—81no—
186C2×C4×C59C2×C22⋊C411yescharacteristic
187C2×C4×C59C4×D411yescharacteristic
188C2×C4×C59#12011yescharacteristic
189C2×C4×C59#12111yescharacteristic
190C2×Dic59—42no—
191C2×Dic59—81no—
192C2×Dic59—82no—
193C2×Dic59—81no—
194C2×Dic59—82no—
195C8×C59—41no—
196C8×C59—41no—
197C59⋊C8—41no—
198C59⋊C8—41no—
199C59⋊C8—41no—
200C59⋊C8—41no—
201C59⋊C8—81no—
202#9264—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59—21no—
210C22×C59—41no—
211C22×C59—41no—
212C22×C59—41no—
213C22×C59—41no—
214C22×C59—41no—
215C22×C59—41no—
216C22×C59—41no—
217C22×C59—81no—
218C22×C59—81no—
219C22×C59—81no—
220C22×C59—161no—
221C22×C59—161no—
222C22×C59#38611yescharacteristic
223C22×C59#44311yescharacteristic
224C22×C59#44411yescharacteristic
225C22×C59#47111yescharacteristic
226C22×C59#47211yescharacteristic
227C4×C59—21no—
228C4×C59—21no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—41no—
232C4×C59—41no—
233C4×C59—41no—
234C4×C59—81no—
235Dic59—82no—
236#3188D5911yescharacteristic
237#2977—591no—
238#1504—591no—
239C2×#327—591no—
240#1471—591no—
241#1298—591no—
242#1280—591no—
243C2×C59—21no—
244C2×C59—21no—
245C2×C59—21no—
246C2×C59—21no—
247C2×C59—41no—
248C2×C59—81no—
249C2×C59—161no—
250C2×C59#187611yescharacteristic
251C2×C59#190711yescharacteristic
252C2×C59#191311yescharacteristic
253#525C2×D5911yescharacteristic
254C2×#121C2×D5911yescharacteristic
255#484Dic5911yescharacteristic, complemented
256#482—21no—
257#481Dic5911yescharacteristic, complemented
258#466—21no—
259#465—21no—
260#450Dic5911yescharacteristic, complemented
261#449—21no—
262#447Dic5911yescharacteristic, complemented
263#435—1181no—
264#435—1181no—
265C8×D4—1181no—
266C8×D4—1181no—
267#433—1181no—
268#408—1181no—
269#407—591no—
270C2×C22⋊C8—591no—
271C2×C22⋊C8—591no—
272C2×C22⋊C8C2×D5911yescharacteristic
273C2×#95—591no—
274#328—1181no—
275#327—1181no—
276#327—1181no—
277#327—1181no—
278#323—1181no—
279#323—1181no—
280C59#926411yescharacteristic, complemented, Sylow
281C2×#41C22⋊D5911yescharacteristic
282C22×D4—21no—
283C22×D4C2×Dic5911yescharacteristic
284C23×C4—1181no—
285C23×C4C22×D5911yescharacteristic
286C2×SD16—21no—
287C2×D8—21no—
288C2×M16—1181no—
289C2×M16—1181no—
290C22×C8—21no—
291C22×C8—1181no—
292C22×C8—1181no—
293C22×C8—1181no—
294#123—21no—
295C22⋊Q8C2×Dic5911yescharacteristic
296C22⋊Q8C22⋊D5911yescharacteristic
297#121—21no—
298#121—21no—
299#121—21no—
300#121—21no—
301#121C2×Dic5911yescharacteristic
302#121C22⋊D5911yescharacteristic
303#120—21no—
304C4×D4—21no—
305C4×D4—21no—
306#117C22⋊D5911yescharacteristic
307C2×C4⋊C4C2×Dic5911yescharacteristic
308C2×C22⋊C4—21no—
309#107—21no—
310#106—21no—
311C4⋊C8—1181no—
312C4⋊C8—1181no—
313#103—21no—
314#102—21no—
315#102—21no—
316#102—21no—
317C22⋊C8—21no—
318C22⋊C8—591no—
319C22⋊C8—591no—
320C22⋊C8—591no—
321C22⋊C8—591no—
322C22⋊C8—1181no—
323C22⋊C8—1181no—
324C22⋊C8—1181no—
325C22⋊C8—1181no—
326C22⋊C8—1181no—
327C22⋊C8—1181no—
328C22⋊C8C2×Dic5911yescharacteristic
329C22⋊C8C2×Dic5911yescharacteristic
330#97—1181no—
331C4×C8—1181no—
332#95—1181no—
333#95—1181no—
334C24—21no—
335C24#1174568711yescharacteristic
336#41—81no—
337C2×Q8C22⋊Dic5911yescharacteristic
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4—81no—
348C2×D4C22⋊Dic5911yescharacteristic
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—81no—
352C22×C4—1182no—
353C22×C4—4721no—
354C22×C4—4721no—
355C22×C4—2361no—
356C22×C4—2361no—
357C22×C4D4×D5911yescharacteristic
358C22×C4D4×D5911yescharacteristic
359C22×C4#1174568711yescharacteristic
360C22×C4C22×Dic5911yescharacteristic
361C22×C4C2×C22⋊D5911yescharacteristic
362C22×C4C2×C22⋊D5911yescharacteristic
363SD16—81no—
364D8—81no—
365M16—4721no—
366M16—4721no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—81no—
370C2×C8—1181no—
371C2×C8—1181no—
372C2×C8—1181no—
373C2×C8—1181no—
374C2×C8—4721no—
375C2×C8—4721no—
376C2×C8—4721no—
377C2×C8—2361no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4C22⋊Dic5911yescharacteristic
383C4⋊C4C22⋊Dic5911yescharacteristic
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—21no—
387C22⋊C4—21no—
388C22⋊C4—21no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C42—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—81no—
402C23—81no—
403C23D4×Dic5911yescharacteristic
404C23#1427369911yescharacteristic
405C23#1427370811yescharacteristic
406Q8—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—41no—
412D4—41no—
413D4—41no—
414D4—81no—
415D4—81no—
416D4—82no—
417D4—81no—
418D4—81no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—4721no—
433C2×C4—4722no—
434C2×C4—4721no—
435C2×C4—4722no—
436C2×C4—2362no—
437C2×C4C2×C22⋊Dic5911yescharacteristic
438C2×C4D4×Dic5911yescharacteristic
439C2×C4#1427369511yescharacteristic
440C2×C4#1427369911yescharacteristic
441C8—41no—
442C8—41no—
443C8—4721no—
444C8—2361no—
445C8—2361no—
446C8—2361no—
447C8—2361no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—161no—
466C22—161no—
467C22#1683728211yescharacteristic
468C22#1683751911yescharacteristic
469C22#1683752411yescharacteristic
470C22#1683766311yescharacteristic
471C22#1683767011yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—41no—
477C4—41no—
478C4—41no—
479C4—81no—
480C4—4722no—
481C2—21no—
482C2—21no—
483C2—21no—
484C2—21no—
485C2—41no—
486C2—81no—
487C2—161no—
488C2#2142442511yescharacteristic
489C2#2142460611yescharacteristic
490C2#2142463911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
Fitting, prime radicalC59×#3188C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenter#3188D59
hypocenterC59#9264
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3188

Complements

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

NH ≅ G/Ntype
1. #21433152C2semidirect
2. #21422600C2semidirect
3. C2×#16837136C2semidirect
4. #21422456C2semidirect
5. #21421984C2semidirect
6. #21421938C2semidirect
26. C59×#484C4semidirect
28. C59×#481C4semidirect
31. C59×#450C4semidirect
33. C59×#447C4semidirect
255. #484Dic59semidirect
257. #481Dic59semidirect
260. #450Dic59semidirect
262. #447Dic59semidirect
280. C59#9264semidirect

Maximal subgroups

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

subgroupindex
#214331522
#214226002
C2×#168371362
#214224562
#214219842
#214219382
C59×#31882
#926459