Finite Group Databasebeta

← #28745595

Subgroups — #28745595

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435839C211yesmaximal, characteristic, complemented
2#21426330C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426257C211yesmaximal, characteristic, complemented
6#21426190C211yesmaximal, characteristic, complemented
7#21426180C211yesmaximal, characteristic, complemented
8#21426151C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic, complemented
10#21426012C211yesmaximal, characteristic, complemented
11C4×#14273671C211yesmaximal, characteristic, complemented
12D4×C59⋊C16C211yesmaximal, characteristic, complemented
13#21425801C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3339C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18#16838061—21no—
19C2×#14273671C2211yescharacteristic
20C2×#14273671C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837625—21no—
28#16837566—21no—
29#16837543—21no—
30C4×Dic236C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837205C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#553—21no—
49C59×#499C2211yescharacteristic
50C59×#482—21no—
51C59×#481—21no—
52C59×#478—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×#11745685—21no—
60C22×Dic118—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236—22no—
64C2×Dic236—21no—
65C2×Dic236—41no—
66C2×Dic236C2311yescharacteristic
67C2×Dic236C2311yescharacteristic
68C2×#11745660—21no—
69C2×#11745660—21no—
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273713—41no—
73#14273706—21no—
74#14273705—41no—
75#14273701—21no—
76#14273699—41no—
77C4×Dic118—21no—
78(C22⋊D59)⋊C4—41no—
79#14273671—22no—
80#14273671—41no—
81#14273668—41no—
82C2×C59⋊C16—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273649—21no—
92C2×D8×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95#126×C59—41no—
96#125×C59—21no—
97#123×C59—41no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—41no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104#117×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#106×C59—21no—
111#106×C59—21no—
112C4⋊C8×C59D411yescharacteristic
113#102×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C2×D4×C59—41no—
122C2×D4×C59D811yescharacteristic
123C22×C4×C59—41no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C2×C22⋊D59—41no—
128C2×C22⋊D59—41no—
129C22×Dic59—41no—
130#11745685—42no—
131#11745685—81no—
132C2×D236—21no—
133C2×C4×D59—41no—
134C2×Dic118—21no—
135C2×Dic118—21no—
136C2×Dic118—42no—
137C2×Dic118—41no—
138C2×Dic118—81no—
139D8×C59—21no—
140D8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2×D411yescharacteristic
148C2×C8×C59C2411yescharacteristic
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59D811yescharacteristic
156C22⋊C4×C59—41no—
157C22⋊C4×C59—41no—
158C22⋊C4×C59—41no—
159C22⋊C4×C59—41no—
160C22⋊C4×C59D811yescharacteristic
161C22⋊C4×C59D811yescharacteristic
162C42×C59—41no—
163C42×C59C2×D411yescharacteristic
164C22⋊Dic59—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168C4⋊Dic59—21no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172Dic236—21no—
173Dic236—21no—
174Dic236—21no—
175Dic236—42no—
176Dic236#4112yes—
177D472—21no—
178D472—21no—
179#11745660—42no—
180#11745660—81no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189D4×C59—22no—
190D4×C59—41no—
191D4×C59—41no—
192D4×C59—82no—
193D4×C59—161no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—41no—
201C2×C4×C59—161no—
202C2×C4×C59—161no—
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C2×D811yescharacteristic
205C2×C4×C59C2×D811yescharacteristic
206C2×C4×C59C2×D811yescharacteristic
207C2×C4×C59C22×D411yescharacteristic
208C22⋊D59—82no—
209C22⋊D59—82no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—82no—
214D236—41no—
215D236—41no—
216C4×D59—82no—
217Dic118—22no—
218Dic118—41no—
219Dic118—41no—
220Dic118—41no—
221Dic118—82no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227#33868—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—81no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59#50812yes—
237C22×C59C22×D811yescharacteristic
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—81no—
241C4×C59—21no—
242C4×C59—21no—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—81no—
246C4×C59—81no—
247C4×C59—81no—
248C4×C59#54611yescharacteristic
249C4×C59#57711yescharacteristic
250Dic59—42no—
251Dic59—81no—
252Dic59—81no—
253#3339D5911yescharacteristic, complemented
254#3285—591no—
255#2242—591no—
256#2237—591no—
257#2233—591no—
258#2229—591no—
259#2217—591no—
260#2215—591no—
261#2210—591no—
262#2186—591no—
263#2185—591no—
264C4×SD32—591no—
265D4×C16—591no—
266#2138—591no—
267C2×#358—591no—
268C2×C59—22no—
269C2×C59—41no—
270C2×C59—161no—
271C2×C59#329411yescharacteristic
272C2×C59C2×#50811yescharacteristic
273C2×C59#341611yescharacteristic
274D59—161no—
275C2×#135—591no—
276C22×Q16—591no—
277#553—21no—
278#548—1181no—
279C2×SD32—591no—
280C2×SD32—591no—
281C2×D16—591no—
282C22×C16—591no—
283C22×C16—591no—
284#499C2×D5911yescharacteristic
285#495—591no—
286#482—21no—
287#481—21no—
288#478—21no—
289#467—591no—
290#466—591no—
291#466C2×D5911yescharacteristic
292#466C2×D5911yescharacteristic
293#464—1181no—
294#452—1181no—
295#448—1181no—
296C4×Q16—591no—
297C4×D8C2×D5911yescharacteristic
298C8×D4C2×D5911yescharacteristic
299#427C2×D5911yescharacteristic
300C2×#107C2×D5911yescharacteristic
301#367—591no—
302#367—591no—
303#366—591no—
304C4⋊C16—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#358—591no—
310#358—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315C22⋊C16—591no—
316C22⋊C16—591no—
317C4×C16—591no—
318C59#3386811yescharacteristic, complemented, Sylow
319C2×#41—1181no—
320C22×Q8—1181no—
321#135—1182no—
322#135—2361no—
323C2×Q16—591no—
324C2×Q16—591no—
325C2×Q16—1182no—
326C2×Q16—1181no—
327C2×Q16—2361no—
328C2×SD16—1181no—
329C2×SD16—1181no—
330C2×D8—591no—
331C2×D8C22×D5911yescharacteristic
332C22×C8C22×D5911yescharacteristic
333C22×C8C22×D5911yescharacteristic
334#126—41no—
335#125—21no—
336#124—1181no—
337#124—2361no—
338#123—41no—
339C22⋊Q8—1181no—
340C22⋊Q8—2361no—
341#121—21no—
342#121—21no—
343#121—41no—
344#121—1181no—
345#121—2361no—
346C4×Q8—1181no—
347C4×D4—21no—
348C4×D4—41no—
349C4×D4—2361no—
350C4×D4C22⋊D5911yescharacteristic
351#117—21no—
352C2×C4⋊C4—21no—
353SD32—1182no—
354SD32—2361no—
355D16—2361no—
356C2×C16—591no—
357C2×C16—591no—
358C2×C16—591no—
359C2×C16—591no—
360C2×C16—2361no—
361C2×C16—2361no—
362#107—21no—
363#107—591no—
364#107C22×D5911yescharacteristic
365#107C22×D5911yescharacteristic
366#107C22×D5911yescharacteristic
367#106—21no—
368#106—21no—
369C4⋊C8C22⋊D5911yescharacteristic
370#103—1181no—
371#103—1181no—
372#102—21no—
373#102—21no—
374#102—21no—
375#102—1181no—
376C22⋊C8C22⋊D5911yescharacteristic
377C22⋊C8C22⋊D5911yescharacteristic
378C4×C8C22×D5911yescharacteristic
379#41—4721no—
380#41—2362no—
381C2×Q8—1181no—
382C2×Q8—1181no—
383C2×Q8—4721no—
384C2×Q8—2362no—
385C2×Q8—2361no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—1181no—
390C2×D4—2361no—
391C2×D4—2361no—
392C2×D4#1174566911yescharacteristic
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—2361no—
396C22×C4—2361no—
397C22×C4C2×C22⋊D5911yescharacteristic
398C22×C4C2×C22⋊D5911yescharacteristic
399Q16—592no—
400Q16—1181no—
401Q16—1181no—
402Q16—1181no—
403Q16—2362no—
404SD16—4721no—
405SD16—2362no—
406D8—21no—
407D8—21no—
408D8—1181no—
409D8—1181no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×C22⋊D5911yescharacteristic
415C2×C8C2×C22⋊D5911yescharacteristic
416C2×C8C2×C22⋊D5911yescharacteristic
417C2×C8C23×D5911yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—1181no—
425C4⋊C4—2361no—
426C4⋊C4—2361no—
427C4⋊C4#1174566911yescharacteristic
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2361no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4#1174566911yescharacteristic
437C22⋊C4#1174566911yescharacteristic
438C42—41no—
439C42—2361no—
440C42C2×C22⋊D5911yescharacteristic
441C16—1181no—
442C16—1181no—
443C16—1181no—
444C16—1181no—
445C23—41no—
446C23—2361no—
447C23C2×#1174566911yescharacteristic
448C23C2×#1174566911yescharacteristic
449Q8—1182no—
450Q8—4722no—
451Q8—2361no—
452Q8—2361no—
453Q8—2361no—
454D4—22no—
455D4—41no—
456D4—41no—
457D4—82no—
458D4—161no—
459D4—4722no—
460D4—4722no—
461D4—2361no—
462D4—2361no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—161no—
471C2×C4—161no—
472C2×C4—4722no—
473C2×C4—4722no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4C2×#1174566911yescharacteristic
478C2×C4C2×#1174566911yescharacteristic
479C2×C4C2×#1174566911yescharacteristic
480C2×C4C2×#1174566911yescharacteristic
481C2×C4C22×C22⋊D5911yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8#41×D5911yescharacteristic
486C8#1427377311yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—4721no—
496C22—4721no—
497C22—4721no—
498C22#1683783212yes—
499C22C22×#1174566911yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—4721no—
508C4—4721no—
509C4—2362no—
510C4#1683804011yescharacteristic
511C4#1683821011yescharacteristic
512C2—22no—
513C2—41no—
514C2—161no—
515C2—9441no—
516C2#2143590311yescharacteristic
517C2C2×#1683783211yescharacteristic
518C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3339C2
socleC22×C59C22×D8
hypercenter#3339D59
hypocenterC59#33868
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3339

Complements

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

NH ≅ G/Ntype
1. #21435839C2semidirect
2. #21426330C2semidirect
3. #21426304C2semidirect
4. #21426281C2semidirect
5. #21426257C2semidirect
6. #21426190C2semidirect
7. #21426180C2semidirect
8. #21426151C2semidirect
9. #21426017C2semidirect
10. #21426012C2semidirect
11. C4×#14273671C2semidirect
12. D4×C59⋊C16C2semidirect
13. #21425801C2semidirect
14. C2×#16837205C2semidirect
15. C59×#3339C2semidirect
253. #3339D59semidirect
318. C59#33868semidirect

Maximal subgroups

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

subgroupindex
#214358392
#214263302
#214263042
#214262812
#214262572
#214261902
#214261802
#214261512
#214260172
#214260122
C4×#142736712
D4×C59⋊C162
#214258012
C2×#168372052
C59×#33392
#3386859