Finite Group Databasebeta

← #28484272

Subgroups — #28484272

Order 15104.

1030 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837366C211yesmaximal, characteristic, complemented
2#21432378C211yesmaximal, characteristic
3C2×#16837135C212yesmaximal, complemented
4#21421975C212yesmaximal, complemented
5C59×C22⋊#95C211yesmaximal, characteristic, complemented
6C22×#11745664—21no—
7C2×#14273701C411yescharacteristic, complemented
8C2×#14273701C411yescharacteristic, complemented
9C2×#14273701C2211yescharacteristic
10C2×#14273696—21no—
11#16837366—24no—
12C2×C22⋊(C59⋊C8)—21no—
13C2×C22⋊(C59⋊C8)C2212yes—
14#16837281—21no—
15#16837281—21no—
16#16837281—21no—
17#16837262C411yescharacteristic, complemented
18#16837262C411yescharacteristic, complemented
19#16837262C2211yescharacteristic
20#16837135—24no—
21C24×C4×C59C2211yescharacteristic
22C22×C22⋊C4×C59—21no—
23C2×#95×C59—21no—
24C2×#95×C59C2212yes—
25C22×D236C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C22×C22⋊D59—21no—
28C22×C22⋊D59—21no—
29C2×#11745664—24no—
30C2×#11745664—42no—
31C2×#11745664—42no—
32C22×C59⋊C8—22no—
33#14273701—22no—
34#14273701—22no—
35#14273701—42no—
36#14273701—42no—
37#14273696—44no—
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C22⋊(C59⋊C8)—24no—
48C22⋊(C59⋊C8)—22no—
49C22⋊(C59⋊C8)D414yescomplemented
50C59⋊#95—21no—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C25×C59D411yescharacteristic
54C23×C4×C59—22no—
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
65C2×C22⋊C4×C59—24no—
66C2×C22⋊C4×C59—42no—
67C2×C22⋊C4×C59—42no—
68#95×C59—22no—
69#95×C59—22no—
70#95×C59—42no—
71C24×C59—22no—
72C24×C59—22no—
73C24×C59—21no—
74C24×C59—21no—
75C24×C59—22no—
76C24×C59—22no—
77C24×C59—41no—
78C24×C59—41no—
79C24×C59C22⋊C411yescharacteristic
80C24×C59C22⋊C411yescharacteristic
81C24×C59C2×D411yescharacteristic
82C23×D59—21no—
83C22×C4×C59—22no—
84C22×C4×C59—22no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—22no—
88C22×C4×C59—22no—
89C22×C4×C59—22no—
90C22×C4×C59—22no—
91C22×C4×C59—22no—
92C22×C4×C59—22no—
93C22×C4×C59—21no—
94C22×C4×C59—21no—
95C22×C4×C59—24no—
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×C59—42no—
106C22×C4×C59—42no—
107C22×C4×C59—42no—
108C22×C4×C59—42no—
109C22×C4×C59—42no—
110C22×C4×C59—42no—
111C22×C4×C59C22⋊C411yescharacteristic
112C22×C4×C59C22⋊C411yescharacteristic
113C22×C4×C59C22×C411yescharacteristic
114C22×C4×C59C2×D411yescharacteristic
115C22×C4×C59C2×D412yes—
116C22×C4×C59#4112yes—
117C2×C22⋊D59—42no—
118C2×C22⋊D59—42no—
119C2×C22⋊D59—42no—
120C2×C22⋊D59—42no—
121C2×C22⋊D59—42no—
122C2×C22⋊D59—42no—
123C2×C22⋊D59—42no—
124C2×C22⋊D59—42no—
125C2×C22⋊D59—41no—
126C2×C22⋊D59—41no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C2×D236—22no—
131C2×D236—41no—
132C2×D236—41no—
133C22⋊C4×C59—48no—
134C22⋊Dic59—42no—
135C22⋊Dic59—42no—
136C22⋊Dic59—42no—
137(C2×D59)⋊C4—42no—
138(C2×D59)⋊C4—42no—
139(C2×D59)⋊C4—41no—
140(C2×D59)⋊C4—41no—
141C4⋊Dic59—22no—
142C4⋊Dic59—41no—
143C4⋊Dic59—41no—
144#11745664—48no—
145C2×C59⋊C8—24no—
146C2×C59⋊C8—42no—
147C2×C59⋊C8—42no—
148C23×C59—21no—
149C23×C59—21no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—24no—
153C23×C59—24no—
154C23×C59—21no—
155C23×C59—21no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—21no—
159C23×C59—21no—
160C23×C59—21no—
161C23×C59—21no—
162C23×C59—21no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—44no—
167C23×C59—44no—
168C23×C59—44no—
169C23×C59—44no—
170C23×C59—42no—
171C23×C59—42no—
172C23×C59—42no—
173C23×C59—42no—
174C23×C59#9912yes—
175C23×C59#10412yes—
176C23×C59C2×C22⋊C411yescharacteristic
177C23×C59#12112yes—
178C23×C59AGL(1,ℤ/8)12yes—
179C23×C59AGL(1,ℤ/8)11yescharacteristic
180C23×C59#13711yescharacteristic
181C22×D59—41no—
182C22×D59—41no—
183C22×D59—41no—
184C22×D59—41no—
185C22×D59—41no—
186C22×D59—41no—
187C22×D59—41no—
188C2×C4×C59—24no—
189C2×C4×C59—24no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
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—42no—
200C2×C4×C59—42no—
201C2×C4×C59—44no—
202C2×C4×C59—44no—
203C2×C4×C59—44no—
204C2×C4×C59—42no—
205C2×C4×C59—42no—
206C2×C4×C59—44no—
207C2×C4×C59—44no—
208C2×C4×C59—44no—
209C2×C4×C59—82no—
210C2×C4×C59—82no—
211C2×C4×C59—82no—
212C2×C4×C59—82no—
213C2×C4×C59#9912yes—
214C2×C4×C59C4×D412yes—
215C2×C4×C59#12011yescharacteristic
216C2×C4×C59#12311yescharacteristic
217C22⋊D59—84no—
218C22⋊D59—84no—
219C22⋊D59—82no—
220C22⋊D59—82no—
221C2×Dic59—41no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230D236—82no—
231C59⋊C8—44no—
232#9117—591nomaximal, Sylow
233C22×C59—22no—
234C22×C59—22no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—22no—
238C22×C59—22no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—22no—
242C22×C59—22no—
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—22no—
247C22×C59—48no—
248C22×C59—48no—
249C22×C59—41no—
250C22×C59—41no—
251C22×C59—41no—
252C22×C59—41no—
253C22×C59—41no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59#38611yescharacteristic
258C22×C59C2×#9911yescharacteristic
259C22×C59C2×#9911yescharacteristic
260C22×C59#41711yescharacteristic
261C22×C59#41811yescharacteristic
262C22×C59C2×#10411yescharacteristic
263C22×C59#42111yescharacteristic
264C22×C59#46812yes—
265C22×C59#46912yes—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C2×D59—81no—
272C2×D59—81no—
273C4×C59—21no—
274C4×C59—22no—
275C4×C59—22no—
276C4×C59—21no—
277C4×C59—41no—
278C4×C59—84no—
279Dic59—81no—
280Dic59—81no—
281Dic59—81no—
282C2×#407—591no—
283#2847—591no—
284C2×#327—592no—
285C22⋊#95D5911yescharacteristic, complemented
286#1294—592no—
287C2×C59—22no—
288C2×C59—22no—
289C2×C59—42no—
290C2×C59—42no—
291C2×C59#150411yescharacteristic
292C2×C59#150711yescharacteristic
293C2×C59#151011yescharacteristic
294C2×C59#151311yescharacteristic
295C2×C59#187111yescharacteristic
296C2×C59#189511yescharacteristic
297C2×C59#189811yescharacteristic
298D59—161no—
299C24×C4C2×D5911yescharacteristic
300C22×M16—1181no—
301C2×#121—591no—
302C2×#121—591no—
303C2×#121—591no—
304C2×#120—1181no—
305C22×C22⋊C4—21no—
306#407—1184no—
307C2×C22⋊C8—592no—
308C2×C22⋊C8—1181no—
309#386—1181no—
310#386—1181no—
311#386—1181no—
312#380—591no—
313#380—591no—
314#380—591no—
315C2×#95—21no—
316C2×#95C2×D5912yes—
317#327—1184no—
318C59#911711yescharacteristic, complemented, Sylow
319C25C22⋊D5911yescharacteristic
320C22×D4—591no—
321C22×D4—1181no—
322C22×D4—1181no—
323C22×D4—1181no—
324C23×C4—22no—
325C23×C4—22no—
326C23×C4—22no—
327C23×C4—42no—
328C23×C4—42no—
329C23×C4C4×D5911yescharacteristic
330C23×C4C4×D5911yescharacteristic
331C23×C4D23611yescharacteristic
332C23×C4D23611yescharacteristic
333C23×C4C22⋊D5911yescharacteristic
334C23×C4C22×D5911yescharacteristic
335C2×M16—1184no—
336C2×M16—2362no—
337C2×M16—2362no—
338C22×C8—1182no—
339#121—1182no—
340#121—1182no—
341#121—2362no—
342#121—2362no—
343#120—2364no—
344C2×C4⋊C4—591no—
345C2×C4⋊C4—591no—
346C2×C4⋊C4—591no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—42no—
349C2×C22⋊C4—42no—
350C2×C22⋊C4—1181no—
351C2×C22⋊C4—1181no—
352C2×C22⋊C4—1181no—
353C2×C22⋊C4—1181no—
354C2×C22⋊C4—1181no—
355C2×C22⋊C4—1181no—
356C22⋊C8—594no—
357C22⋊C8—1184no—
358C22⋊C8—1182no—
359#95—22no—
360#95—22no—
361#95—42no—
362#95—1181no—
363#95—1181no—
364#95—1181no—
365C24—22no—
366C24—22no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—41no—
372C24—41no—
373C24—1181no—
374C24(C2×D59)⋊C411yescharacteristic
375C24(C2×D59)⋊C411yescharacteristic
376C24C2×C22⋊D5911yescharacteristic
377C2×D4—1182no—
378C2×D4—2361no—
379C2×D4—2361no—
380C2×D4—2362no—
381C2×D4—2362no—
382C2×D4—2362no—
383C2×D4—2362no—
384C2×D4—2362no—
385C2×D4—2362no—
386C2×D4—2362no—
387C2×D4—2362no—
388C2×D4—2361no—
389C2×D4—2361no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—48no—
411C22×C4—48no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—1181no—
419C22×C4—1181no—
420C22×C4—1181no—
421C22×C4(C2×D59)⋊C411yescharacteristic
422C22×C4(C2×D59)⋊C411yescharacteristic
423C22×C4C2×C4×D5911yescharacteristic
424C22×C4C2×D23611yescharacteristic
425C22×C4D4×D5912yes—
426C22×C4#1174568712yes—
427M16—2368no—
428C2×C8—1184no—
429C2×C8—2362no—
430C2×C8—2362no—
431C4⋊C4—1182no—
432C4⋊C4—2361no—
433C4⋊C4—2361no—
434C22⋊C4—48no—
435C22⋊C4—2362no—
436C22⋊C4—2362no—
437C22⋊C4—2362no—
438C22⋊C4—2362no—
439C22⋊C4—2362no—
440C22⋊C4—2361no—
441C22⋊C4—2361no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—2361no—
469C23—2361no—
470C23—2361no—
471C23—2361no—
472C23—2361no—
473C23—2361no—
474C23—2361no—
475C23#1427364112yes—
476C23#1427365512yes—
477C23C2×(C2×D59)⋊C411yescharacteristic
478C23#1427369912yes—
479C23#1427375212yes—
480C23#1427375311yescharacteristic
481C23#1427375811yescharacteristic
482D4—4722no—
483D4—4724no—
484D4—4724no—
485D4—4722no—
486D4—4722no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—2361no—
513C2×C4—2361no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4#1427364112yes—
522C2×C4(C22⋊D59)⋊C412yes—
523C2×C4#1427369711yescharacteristic
524C2×C4#1427371111yescharacteristic
525C8—2364no—
526C22—21no—
527C22—21no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—48no—
541C22—48no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—4721no—
555C22—4721no—
556C22—4721no—
557C22#1683727911yescharacteristic
558C22C2×#1427364111yescharacteristic
559C22C2×#1427364111yescharacteristic
560C22#1683740111yescharacteristic
561C22#1683740611yescharacteristic
562C22C2×#1427365511yescharacteristic
563C22#1683742011yescharacteristic
564C22#1683765012yes—
565C22#1683765312yes—
566C4—21no—
567C4—22no—
568C4—21no—
569C4—22no—
570C4—41no—
571C4—84no—
572C4—4721no—
573C4—4721no—
574C4—4721no—
575C2—22no—
576C2—22no—
577C2—42no—
578C2—42no—
579C2—9441no—
580C2#2142259811yescharacteristic
581C2#2142261111yescharacteristic
582C2#2142262411yescharacteristic
583C2#2142263711yescharacteristic
584C2#2142439611yescharacteristic
585C2#2142453211yescharacteristic
586C2#2142454711yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×C22⋊#95C2
socleC23×C59C2×C22⋊C4
prime radical#21432378C2
cosocleC23×C4×C59C23
hypercenterC22⋊#95D59
hypocenterC59#9117
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
1. C2×#16837366C2semidirect
3. C2×#16837135C2semidirect
4. #21421975C2semidirect
5. C59×C22⋊#95C2semidirect
7. C2×#14273701C4semidirect
8. C2×#14273701C4semidirect
17. #16837262C4semidirect
18. #16837262C4semidirect
49. C22⋊(C59⋊C8)D4semidirect
285. C22⋊#95D59semidirect
318. C59#9117semidirect

Maximal subgroups

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

subgroupindex
C2×#168373662
#214323782
C2×#168371352
#214219752
C59×C22⋊#952
#911759