Finite Group Databasebeta

← #28482734

Subgroups — #28482734

Order 15104.

702 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426796C211yesmaximal, characteristic
2#21424413C211yesmaximal, characteristic
3C2×#16837137C211yesmaximal, characteristic
4#21422359C211yesmaximal, characteristic, complemented
5#21421974C211yesmaximal, characteristic, complemented
6#21421938C211yesmaximal, characteristic, complemented
7C59×#1852C211yesmaximal, characteristic, complemented
8C2×#14273701C2211yescharacteristic
9C2×(C22⋊D59)⋊C4—21no—
10C22×Dic59⋊C4—21no—
11C2×#14273647—21no—
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13#16837292—21no—
14#16837285—21no—
15#16837280—21no—
16#16837279—21no—
17#16837273—21no—
18C2×C59⋊#95—21no—
19C2×C59⋊#95C2211yescharacteristic
20C2×C59⋊#95C2211yescharacteristic
21#16837164—21no—
22#16837137C414yescomplemented
23C22×C4⋊C4×C59C2211yescharacteristic
24C2×#107×C59—21no—
25C2×C22⋊C8×C59C2211yescharacteristic
26C59×#380C2211yescharacteristic
27C59×#340—21no—
28C22×D236D411yescharacteristic
29C22×C22⋊D59—21no—
30C23×Dic59—21no—
31C23×Dic59D411yescharacteristic
32C23×Dic59D411yescharacteristic
33C22×C4×D59—21no—
34C22×C59⋊C8—21no—
35#14273701—21no—
36#14273701—21no—
37#14273701C2×C412yes—
38#14273701Q812yes—
39(C22⋊D59)⋊C4—42no—
40(C22⋊D59)⋊C4—42no—
41C2×C4⋊Dic59D411yescharacteristic
42C2×Dic59⋊C4—21no—
43C2×Dic59⋊C4—42no—
44C2×Dic59⋊C4—42no—
45C2×Dic59⋊C4—42no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C2×(C2×D59)⋊C4—21no—
49C2×(C2×D59)⋊C4—21no—
50C2×C4×Dic59—21no—
51#14273647—42no—
52C22⋊(C59⋊C8)—21no—
53C22⋊(C59⋊C8)—21no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—21no—
59C59⋊#95—42no—
60C59⋊#95C2×C412yescomplemented
61C23×C4×C59—21no—
62C23×C4×C59C2311yescharacteristic
63C22×C8×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—42no—
67C2×C4⋊C4×C59D411yescharacteristic
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C22⋊C4×C59—21no—
70#107×C59—42no—
71C22⋊C8×C59—21no—
72C22⋊C8×C59C2×C412yescomplemented
73#95×C59—21no—
74C24×C59C2×D411yescharacteristic
75C23×D59—21no—
76C22×C4×C59—22no—
77C22×C4×C59—21no—
78C22×C4×C59—21no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59—41no—
85C22×C4×C59—41no—
86C22×C4×C59—41no—
87C22×C4×C59C22×C411yescharacteristic
88C22×C4×C59C2×D411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C22×C4×C59C2×Q811yescharacteristic
91C22×C4×C59#4111yescharacteristic
92C22×C4×C59#4111yescharacteristic
93C2×C22⋊D59—22no—
94C2×C22⋊D59—22no—
95C2×C22⋊D59—41no—
96C2×C22⋊D59—41no—
97C2×C22⋊D59—41no—
98C2×C22⋊D59—41no—
99C22×Dic59—22no—
100C22×Dic59—22no—
101C22×Dic59—22no—
102C22×Dic59—21no—
103C22×Dic59—21no—
104C22×Dic59—21no—
105C22×Dic59—21no—
106C22×Dic59—21no—
107C22×Dic59—21no—
108C22×Dic59—41no—
109C22×Dic59—41no—
110C22×Dic59—41no—
111C22×Dic59—41no—
112C22×Dic59—41no—
113C22×Dic59—41no—
114C22×Dic59—41no—
115C22×Dic59—41no—
116C22×Dic59D812yes—
117C2×D236—21no—
118C2×D236—41no—
119C2×D236—41no—
120C2×D236C4⋊C412yescomplemented
121C2×C4×D59—42no—
122C2×C4×D59—81no—
123C2×C4×D59—81no—
124C2×C8×C59—22no—
125C2×C8×C59—41no—
126C2×C8×C59—41no—
127C4⋊C4×C59—22no—
128C4⋊C4×C59—22no—
129C4⋊C4×C59—41no—
130C4⋊C4×C59—41no—
131C4⋊C4×C59—41no—
132C4⋊C4×C59—41no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—41no—
135(C2×D59)⋊C4—22no—
136(C2×D59)⋊C4—22no—
137(C2×D59)⋊C4—42no—
138(C2×D59)⋊C4—81no—
139(C2×D59)⋊C4—81no—
140C4⋊Dic59—21no—
141C4⋊Dic59C4⋊C412yescomplemented
142Dic59⋊C4—42no—
143Dic59⋊C4—48no—
144C4×Dic59—42no—
145C2×C59⋊C8—41no—
146C2×C59⋊C8—41no—
147C2×C59⋊C8—41no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59C4×D411yescharacteristic
153C23×C59#12011yescharacteristic
154C23×C59C22⋊Q811yescharacteristic
155C23×C59C2×D811yescharacteristic
156C23×C59#13511yescharacteristic
157C23×C59AGL(1,ℤ/8)11yescharacteristic
158C23×C59#13711yescharacteristic
159C22×D59—22no—
160C22×D59—41no—
161C22×D59—41no—
162C22×D59—41no—
163C22×D59—41no—
164C22×D59—41no—
165C22×D59—41no—
166C2×C4×C59—21no—
167C2×C4×C59—21no—
168C2×C4×C59—21no—
169C2×C4×C59—21no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—41no—
173C2×C4×C59—41no—
174C2×C4×C59—42no—
175C2×C4×C59—42no—
176C2×C4×C59—41no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—42no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—81no—
184C2×C4×C59—81no—
185C2×C4×C59—81no—
186C2×C4×C59—81no—
187C2×C4×C59C2×C4⋊C411yescharacteristic
188C2×C4×C59C4×D411yescharacteristic
189C2×C4×C59C22⋊Q811yescharacteristic
190C2×C4×C59#12311yescharacteristic
191C22⋊D59—42no—
192C22⋊D59—42no—
193C22⋊D59—42no—
194C22⋊D59—42no—
195C2×Dic59—24no—
196C2×Dic59—22no—
197C2×Dic59—22no—
198C2×Dic59—42no—
199C2×Dic59—42no—
200C2×Dic59—42no—
201C2×Dic59—42no—
202C2×Dic59—42no—
203C2×Dic59—42no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—42no—
209C2×Dic59—42no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—41no—
213C2×Dic59—81no—
214C2×Dic59—81no—
215D236—42no—
216D236—81no—
217C4×D59—84no—
218C8×C59—42no—
219C59⋊C8—81no—
220#8866—591nomaximal, Sylow
221C22×C59—21no—
222C22×C59—21no—
223C22×C59—21no—
224C22×C59—21no—
225C22×C59—21no—
226C22×C59—21no—
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59#38511yescharacteristic
234C22×C59C4×D811yescharacteristic
235C22×C59#44011yescharacteristic
236C22×C59#44711yescharacteristic
237C22×C59#45211yescharacteristic
238C22×C59#45311yescharacteristic
239C22×C59#45611yescharacteristic
240C22×C59B(4,𝔽2)11yescharacteristic
241C22×C59#45811yescharacteristic
242C22×C59#47411yescharacteristic
243C22×C59#47811yescharacteristic
244C2×D59—42no—
245C2×D59—42no—
246C2×D59—42no—
247C2×D59—81no—
248C2×D59—81no—
249C2×D59—81no—
250C2×D59—81no—
251C4×C59—21no—
252C4×C59—21no—
253C4×C59—21no—
254C4×C59—21no—
255C4×C59—81no—
256C4×C59—82no—
257C4×C59—81no—
258Dic59—44no—
259Dic59—42no—
260Dic59—42no—
261Dic59—81no—
262#2325—591no—
263#1874—591no—
264#1852D5911yescharacteristic, complemented
265C2×#327—591no—
266#1444—591no—
267#1294—591no—
268#1280—591no—
269C2×C59—21no—
270C2×C59—21no—
271C2×C59—21no—
272C2×C59—21no—
273C2×C59#159511yescharacteristic
274C2×C59#160311yescharacteristic
275C2×C59#161011yescharacteristic
276C2×C59#161811yescharacteristic
277C2×C59#185911yescharacteristic
278C2×C59#186211yescharacteristic
279C2×C59#186711yescharacteristic
280D59—82no—
281C2×#121—591no—
282C2×C4×D4—1181no—
283C22×C4⋊C4—1181no—
284C22×C4⋊C4C2×D5911yescharacteristic
285C2×#107—21no—
286C2×#102—1181no—
287C2×C22⋊C8—591no—
288C2×C22⋊C8C2×D5911yescharacteristic
289#388—1181no—
290#387—1181no—
291#386—1181no—
292#386—1181no—
293#385—1181no—
294#380C2×D5911yescharacteristic
295C2×#95—591no—
296C2×#95—591no—
297C2×#95—1181no—
298#340—21no—
299#340—1181no—
300#327—594no—
301C59#886611yescharacteristic, complemented, Sylow
302C22×D4—591no—
303C22×D4—1181no—
304C23×C4—21no—
305C23×C4—591no—
306C23×C4—591no—
307C23×C4—1181no—
308C23×C4—1181no—
309C23×C4C22×D5911yescharacteristic
310C22×C8—21no—
311C22×C8—1181no—
312#121—592no—
313#121—592no—
314#121—1181no—
315#121—1181no—
316C4×D4—2362no—
317C4×D4—2362no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—591no—
322C2×C4⋊C4—1181no—
323C2×C4⋊C4—2362no—
324C2×C4⋊C4—2362no—
325C2×C4⋊C4—2362no—
326C2×C4⋊C4C22⋊D5911yescharacteristic
327C2×C4⋊C4C22⋊D5911yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1181no—
330C2×C22⋊C4—1181no—
331C2×C22⋊C4—1181no—
332C2×C22⋊C4—1181no—
333C2×C42—1181no—
334#107—42no—
335#102—2362no—
336C22⋊C8—21no—
337C22⋊C8—1181no—
338C22⋊C8—1181no—
339C22⋊C8C4×D5912yescomplemented
340#95—21no—
341#95—592no—
342#95—1181no—
343#95—1181no—
344#95—1181no—
345#95—1181no—
346#95—1181no—
347#95—2362no—
348C24—1181no—
349C24D4×D5911yescharacteristic
350C2×D4—592no—
351C2×D4—1182no—
352C2×D4—1182no—
353C2×D4—1181no—
354C2×D4—2361no—
355C2×D4—2361no—
356C2×D4—2361no—
357C2×D4—2361no—
358C2×D4—2361no—
359C2×D4—2361no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—592no—
372C22×C4—1181no—
373C22×C4—1182no—
374C22×C4—1182no—
375C22×C4—1182no—
376C22×C4—1181no—
377C22×C4—1181no—
378C22×C4—1181no—
379C22×C4—1181no—
380C22×C4—1181no—
381C22×C4—4721no—
382C22×C4—4721no—
383C22×C4—2362no—
384C22×C4—2361no—
385C22×C4—2361no—
386C22×C4—2361no—
387C22×C4—2361no—
388C22×C4—2361no—
389C22×C4—2361no—
390C22×C4—2361no—
391C22×C4—2361no—
392C22×C4C2×C4×D5911yescharacteristic
393C22×C4#1174568511yescharacteristic
394C22×C4D4×D5911yescharacteristic
395C22×C4Q8×D5911yescharacteristic
396C22×C4#1174568911yescharacteristic
397C22×C4C2×C22⋊D5911yescharacteristic
398C2×C8—22no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—2361no—
402C2×C8—2361no—
403C2×C8—2361no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—592no—
411C4⋊C4—1181no—
412C4⋊C4—2362no—
413C4⋊C4—2368no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—1182no—
417C22⋊C4—1182no—
418C22⋊C4—4721no—
419C22⋊C4—4721no—
420C22⋊C4—2362no—
421C42—2362no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—1182no—
427C23—2361no—
428C23—2361no—
429C23—2361no—
430C23—2361no—
431C23—2361no—
432C23—2361no—
433C23#1427368911yescharacteristic
434C23#1427369511yescharacteristic
435C23#1427370511yescharacteristic
436C23D8×D5911yescharacteristic
437C23#1427375011yescharacteristic
438C23#1427375111yescharacteristic
439C23#1427375911yescharacteristic
440D4—4721no—
441D4—2362no—
442D4—2362no—
443D4—2362no—
444D4—2362no—
445D4—2362no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—1184no—
468C2×C4—1182no—
469C2×C4—1182no—
470C2×C4—4724no—
471C2×C4—4721no—
472C2×C4—4721no—
473C2×C4—2362no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4—2362no—
478C2×C4—2362no—
479C2×C4—2362no—
480C2×C4—2362no—
481C2×C4—2362no—
482C2×C4—2362no—
483C2×C4—2361no—
484C2×C4—2361no—
485C2×C4—2361no—
486C2×C4—2361no—
487C2×C4—2362no—
488C2×C4C4⋊C4×D5911yescharacteristic
489C2×C4C4×C22⋊D5911yescharacteristic
490C2×C4#1427370311yescharacteristic
491C2×C4#1427371211yescharacteristic
492C8—42no—
493C8—4721no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—4721no—
507C22—4721no—
508C22—4721no—
509C22—4721no—
510C22—2362no—
511C22—2362no—
512C22—2362no—
513C22#1683727311yescharacteristic
514C22#1683748411yescharacteristic
515C22#1683750111yescharacteristic
516C22#1683753411yescharacteristic
517C22#1683756311yescharacteristic
518C22#1683756811yescharacteristic
519C22#1683758311yescharacteristic
520C22#1683758811yescharacteristic
521C22#1683759111yescharacteristic
522C22#1683768111yescharacteristic
523C22#1683770511yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—82no—
529C4—81no—
530C4—81no—
531C4—4721no—
532C4—2364no—
533C4—2362no—
534C4—2362no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2—4722no—
540C2#2142301211yescharacteristic
541C2#2142305811yescharacteristic
542C2#2142309711yescharacteristic
543C2#2142314311yescharacteristic
544C2#2142431911yescharacteristic
545C2#2142434011yescharacteristic
546C2#2142437511yescharacteristic
subgroupG/H
centerC23#14273695
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1852C2
socleC23×C59#120
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1852D59
hypocenterC59#8866
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
4. #21422359C2semidirect
5. #21421974C2semidirect
6. #21421938C2semidirect
7. C59×#1852C2semidirect
22. #16837137C4semidirect
60. C59⋊#95C2×C4semidirect
72. C22⋊C8×C59C2×C4semidirect
120. C2×D236C4⋊C4semidirect
141. C4⋊Dic59C4⋊C4semidirect
264. #1852D59semidirect
301. C59#8866semidirect
339. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
#214267962
#214244132
C2×#168371372
#214223592
#214219742
#214219382
C59×#18522
#886659