Finite Group Databasebeta

← #28484254

Subgroups — #28484254

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432378C211yesmaximal, characteristic
2C2×#16837137C212yesmaximal, complemented
3#21422319C211yesmaximal, characteristic, complemented
4#21421976C212yesmaximal, complemented
5C22×C22⋊C8×C59C211yesmaximal, characteristic, complemented
6C2×#14273701C411yescharacteristic, complemented
7C2×#14273701C411yescharacteristic, complemented
8C2×#14273701C2211yescharacteristic
9C2×#14273696—21no—
10C22×C22⋊Dic59—21no—
11#16837281—21no—
12#16837281—21no—
13#16837281—21no—
14#16837262C411yescharacteristic, complemented
15#16837262C411yescharacteristic, complemented
16#16837262C2211yescharacteristic
17C2×C59⋊#95—21no—
18C2×C59⋊#95C2212yes—
19#16837137—24no—
20C24×C4×C59C2211yescharacteristic
21C23×C8×C59—21no—
22C2×C22⋊C8×C59—21no—
23C2×C22⋊C8×C59—24no—
24C2×C22⋊C8×C59C2212yes—
25C22×D236C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C22×C22⋊D59—21no—
28C22×C22⋊D59—21no—
29C23×Dic59—22no—
30#14273701—22no—
31#14273701—22no—
32#14273701—42no—
33#14273701—42no—
34#14273696—42no—
35#14273696—42no—
36C2×C4⋊Dic59C2×C411yescharacteristic
37C2×C4⋊Dic59C2×C411yescharacteristic
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×C22⋊Dic59—24no—
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—42no—
44C2×C22⋊Dic59—42no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—22no—
52C59⋊#95—22no—
53C59⋊#95—42no—
54C25×C59D411yescharacteristic
55C23×C4×C59—22no—
56C23×C4×C59—22no—
57C23×C4×C59—42no—
58C23×C4×C59—42no—
59C23×C4×C59C2×C411yescharacteristic
60C23×C4×C59C2×C411yescharacteristic
61C23×C4×C59D411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59D411yescharacteristic
64C23×C4×C59C2311yescharacteristic
65C22×C8×C59—24no—
66C22×C8×C59—22no—
67C22×C8×C59—42no—
68C22×C8×C59—42no—
69C22⋊C8×C59—24no—
70C22⋊C8×C59—22no—
71C22⋊C8×C59D414yescomplemented
72C24×C59—22no—
73C24×C59—22no—
74C24×C59—22no—
75C24×C59—22no—
76C24×C59—41no—
77C24×C59—41no—
78C24×C59C22⋊C411yescharacteristic
79C24×C59C22⋊C411yescharacteristic
80C24×C59D812yes—
81C24×C59SD1612yes—
82C24×C59C2×D411yescharacteristic
83C23×D59—21no—
84C22×C4×C59—22no—
85C22×C4×C59—22no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—22no—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—41no—
97C22×C4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59—41no—
100C22×C4×C59—41no—
101C22×C4×C59—41no—
102C22×C4×C59—41no—
103C22×C4×C59—48no—
104C22×C4×C59—48no—
105C22×C4×C59C22⋊C411yescharacteristic
106C22×C4×C59C22⋊C411yescharacteristic
107C22×C4×C59C22×C411yescharacteristic
108C22×C4×C59C2×D411yescharacteristic
109C22×C4×C59C2×D412yes—
110C22×C4×C59#4112yes—
111C2×C22⋊D59—42no—
112C2×C22⋊D59—42no—
113C2×C22⋊D59—42no—
114C2×C22⋊D59—42no—
115C2×C22⋊D59—42no—
116C2×C22⋊D59—42no—
117C2×C22⋊D59—42no—
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C2×C22⋊D59—42no—
121C22×Dic59—24no—
122C22×Dic59—21no—
123C22×Dic59—21no—
124C22×Dic59—21no—
125C22×Dic59—42no—
126C22×Dic59—42no—
127C22×Dic59—42no—
128C22×Dic59—42no—
129C22×Dic59—42no—
130C22×Dic59—42no—
131C2×D236—22no—
132C2×D236—41no—
133C2×D236—41no—
134C2×C8×C59—24no—
135C2×C8×C59—48no—
136C2×C8×C59—42no—
137C2×C8×C59—42no—
138C22⋊Dic59—48no—
139C22⋊Dic59—42no—
140C22⋊Dic59—42no—
141C22⋊Dic59—41no—
142C22⋊Dic59—41no—
143(C2×D59)⋊C4—42no—
144(C2×D59)⋊C4—42no—
145(C2×D59)⋊C4—41no—
146(C2×D59)⋊C4—41no—
147C4⋊Dic59—22no—
148C4⋊Dic59—41no—
149C4⋊Dic59—41no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—21no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—24no—
159C23×C59—24no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—24no—
167C23×C59—24no—
168C23×C59—42no—
169C23×C59—42no—
170C23×C59—44no—
171C23×C59—44no—
172C23×C59—44no—
173C23×C59—44no—
174C23×C59#9912yes—
175C23×C59#10212yes—
176C23×C59#10212yes—
177C23×C59#10412yes—
178C23×C59C2×C22⋊C411yescharacteristic
179C23×C59#12112yes—
180C23×C59C2×D811yescharacteristic
181C23×C59C2×SD1611yescharacteristic
182C23×C59#13512yes—
183C22×D59—41no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59—41no—
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—41no—
190C2×C4×C59—24no—
191C2×C4×C59—24no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—42no—
202C2×C4×C59—42no—
203C2×C4×C59—44no—
204C2×C4×C59—44no—
205C2×C4×C59—44no—
206C2×C4×C59—42no—
207C2×C4×C59—42no—
208C2×C4×C59#9912yes—
209C2×C4×C59C4×D412yes—
210C2×C4×C59#12011yescharacteristic
211C2×C4×C59#12311yescharacteristic
212C22⋊D59—84no—
213C22⋊D59—84no—
214C22⋊D59—82no—
215C22⋊D59—82no—
216C2×Dic59—44no—
217C2×Dic59—44no—
218C2×Dic59—44no—
219C2×Dic59—41no—
220C2×Dic59—41no—
221C2×Dic59—41no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—82no—
229C2×Dic59—82no—
230C2×Dic59—82no—
231C2×Dic59—82no—
232D236—82no—
233C8×C59—44no—
234#9113—591nomaximal, Sylow
235C22×C59—22no—
236C22×C59—22no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—22no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—21no—
247C22×C59—22no—
248C22×C59—21no—
249C22×C59—22no—
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—48no—
253C22×C59—48no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59#32714yescomplemented
259C22×C59#38611yescharacteristic
260C22×C59C2×#9911yescharacteristic
261C22×C59C2×#9911yescharacteristic
262C22×C59C2×#10211yescharacteristic
263C22×C59#41611yescharacteristic
264C22×C59C2×#10411yescharacteristic
265C22×C59#42111yescharacteristic
266C22×C59#46512yes—
267C22×C59#46612yes—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C2×D59—81no—
274C2×D59—81no—
275C4×C59—21no—
276C4×C59—22no—
277C4×C59—22no—
278C4×C59—21no—
279C4×C59—41no—
280Dic59—84no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284C22×C22⋊C8D5911yescharacteristic, complemented
285#2847—591no—
286C2×#327—592no—
287C22⋊#95—591no—
288#1294—592no—
289C2×C59—22no—
290C2×C59—24no—
291C2×C59—22no—
292C2×C59—42no—
293C2×C59C2×#32711yescharacteristic
294C2×C59#150111yescharacteristic
295C2×C59#151611yescharacteristic
296C2×C59#151911yescharacteristic
297C2×C59#187111yescharacteristic
298C2×C59#189311yescharacteristic
299C2×C59#189811yescharacteristic
300D59—161no—
301C24×C4C2×D5911yescharacteristic
302C23×C8—21no—
303C2×#121—591no—
304C2×#121—591no—
305C2×#121—591no—
306C2×#120—1181no—
307C22×C22⋊C4—1181no—
308C2×C22⋊C8—21no—
309C2×C22⋊C8—24no—
310C2×C22⋊C8C2×D5912yes—
311#386—1181no—
312#386—1181no—
313#386—1181no—
314#380—591no—
315#380—591no—
316#380—591no—
317C2×#95—592no—
318C2×#95—1181no—
319#327—1184no—
320C59#911311yescharacteristic, complemented, Sylow
321C25D23611yescharacteristic
322C22×D4—591no—
323C22×D4—1181no—
324C22×D4—1181no—
325C22×D4—1181no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—42no—
329C23×C4—42no—
330C23×C4—1182no—
331C23×C4C4×D5911yescharacteristic
332C23×C4C4×D5911yescharacteristic
333C23×C4D23611yescharacteristic
334C23×C4C22⋊D5911yescharacteristic
335C23×C4C22⋊D5911yescharacteristic
336C23×C4C22×D5911yescharacteristic
337C22×C8—24no—
338C22×C8—22no—
339C22×C8—42no—
340C22×C8—42no—
341#121—1182no—
342#121—1182no—
343#121—2362no—
344#121—2362no—
345#120—2362no—
346#120—2362no—
347C2×C4⋊C4—591no—
348C2×C4⋊C4—591no—
349C2×C4⋊C4—591no—
350C2×C22⋊C4—1184no—
351C2×C22⋊C4—1181no—
352C2×C22⋊C4—1181no—
353C2×C22⋊C4—1181no—
354C2×C22⋊C4—1181no—
355C2×C22⋊C4—1181no—
356C2×C22⋊C4—1181no—
357C2×C22⋊C4—2362no—
358C2×C22⋊C4—2362no—
359C22⋊C8—24no—
360C22⋊C8—22no—
361C22⋊C8C22⋊D5914yescomplemented
362#95—1181no—
363#95—1181no—
364#95—1181no—
365#95—1182no—
366#95—1182no—
367#95—2362no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1181no—
375C24#1174566012yes—
376C24D47212yes—
377C24(C2×D59)⋊C411yescharacteristic
378C24(C2×D59)⋊C411yescharacteristic
379C24C2×D23611yescharacteristic
380C2×D4—1182no—
381C2×D4—2361no—
382C2×D4—2361no—
383C2×D4—2362no—
384C2×D4—2362no—
385C2×D4—2362no—
386C2×D4—2362no—
387C2×D4—2362no—
388C2×D4—2362no—
389C2×D4—2362no—
390C2×D4—2361no—
391C2×D4—2361no—
392C2×D4—2362no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—1184no—
415C22×C4—1181no—
416C22×C4—1181no—
417C22×C4—1181no—
418C22×C4—2362no—
419C22×C4—2362no—
420C22×C4—2362no—
421C22×C4—2362no—
422C22×C4—2362no—
423C22×C4—2362no—
424C22×C4(C2×D59)⋊C411yescharacteristic
425C22×C4(C2×D59)⋊C411yescharacteristic
426C22×C4C2×C4×D5911yescharacteristic
427C22×C4#1174568512yes—
428C22×C4C2×C22⋊D5911yescharacteristic
429C22×C4C2×C22⋊D5912yes—
430C2×C8—24no—
431C2×C8—48no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—1182no—
435C4⋊C4—2361no—
436C4⋊C4—2361no—
437C22⋊C4—2362no—
438C22⋊C4—2368no—
439C22⋊C4—2362no—
440C22⋊C4—2362no—
441C22⋊C4—2361no—
442C22⋊C4—2361no—
443C22⋊C4—2362no—
444C22⋊C4—2361no—
445C22⋊C4—2361no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—24no—
464C23—42no—
465C23—42no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—2361no—
471C23—2361no—
472C23—2361no—
473C23—2361no—
474C23—2361no—
475C23—2361no—
476C23—2361no—
477C23#1427364112yes—
478C23#1427364912yes—
479C23#1427364912yes—
480C23#1427365312yes—
481C23C2×(C2×D59)⋊C411yescharacteristic
482C23#1427370112yes—
483C23C2×D47211yescharacteristic
484C23C2×#1174566011yescharacteristic
485C23#1427374912yes—
486D4—4722no—
487D4—4724no—
488D4—4722no—
489D4—4724no—
490D4—4722no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—4722no—
510C2×C4—4722no—
511C2×C4—4722no—
512C2×C4—4722no—
513C2×C4—2364no—
514C2×C4—2364no—
515C2×C4—2364no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4#1427364112yes—
526C2×C4C4×C22⋊D5912yes—
527C2×C4#1427369611yescharacteristic
528C2×C4#1427371011yescharacteristic
529C8—44no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—22no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—21no—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—48no—
548C22—48no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—41no—
553C22—4721no—
554C22—4721no—
555C22—4721no—
556C22—4721no—
557C22—4721no—
558C22—4721no—
559C22—4721no—
560C22#1683713714yescomplemented
561C22#1683728111yescharacteristic
562C22C2×#1427364111yescharacteristic
563C22C2×#1427364111yescharacteristic
564C22C2×#1427364911yescharacteristic
565C22#1683739911yescharacteristic
566C22C2×#1427365311yescharacteristic
567C22#1683741811yescharacteristic
568C22#1683763112yes—
569C22#1683763812yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—4724no—
576C4—4721no—
577C4—4721no—
578C4—4721no—
579C2—22no—
580C2—24no—
581C2—22no—
582C2—42no—
583C2—9441no—
584C2C2×#1683713711yescharacteristic
585C2#2142258811yescharacteristic
586C2#2142265311yescharacteristic
587C2#2142266511yescharacteristic
588C2#2142439811yescharacteristic
589C2#2142452411yescharacteristic
590C2#2142454911yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC22×C22⋊C8×C59C2
socleC23×C59C2×C22⋊C4
prime radical#21432378C2
cosocleC23×C4×C59C23
hypercenterC22×C22⋊C8D59
hypocenterC59#9113
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#16837137C2semidirect
3. #21422319C2semidirect
4. #21421976C2semidirect
5. C22×C22⋊C8×C59C2semidirect
6. C2×#14273701C4semidirect
7. C2×#14273701C4semidirect
14. #16837262C4semidirect
15. #16837262C4semidirect
71. C22⋊C8×C59D4semidirect
258. C22×C59#327semidirect
284. C22×C22⋊C8D59semidirect
320. C59#9113semidirect
361. C22⋊C8C22⋊D59semidirect
560. C22#16837137semidirect

Maximal subgroups

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

subgroupindex
#214323782
C2×#168371372
#214223192
#214219762
C22×C22⋊C8×C592
#911359