Finite Group Databasebeta

← #28484252

Subgroups — #28484252

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
1C22×C22⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
2#21432378C211yesmaximal, characteristic
3C2×#16837135C212yesmaximal, complemented
4#21421975C212yesmaximal, complemented
5C59×C22⋊#95C211yesmaximal, characteristic, complemented
6C23×C59⋊C8—21no—
7C2×#14273701C411yescharacteristic, complemented
8C2×#14273701C411yescharacteristic, complemented
9C2×#14273701C2211yescharacteristic
10C2×#14273696—21no—
11C2×C22⋊(C59⋊C8)—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—
29C22×C59⋊C8—24no—
30C22×C59⋊C8—22no—
31C22×C59⋊C8—42no—
32C22×C59⋊C8—42no—
33#14273701—22no—
34#14273701—22no—
35#14273701—42no—
36#14273701—42no—
37#14273696—42no—
38#14273696—42no—
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×C4⋊Dic59C2×C411yescharacteristic
41C2×C4⋊Dic59C2×C411yescharacteristic
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C22⋊(C59⋊C8)—24no—
49C22⋊(C59⋊C8)—22no—
50C22⋊(C59⋊C8)D414yescomplemented
51C59⋊#95—21no—
52C59⋊#95—21no—
53C59⋊#95—21no—
54C25×C59D411yescharacteristic
55C23×C4×C59—22no—
56C23×C4×C59—22no—
57C23×C4×C59—22no—
58C23×C4×C59—42no—
59C23×C4×C59—42no—
60C23×C4×C59C2×C411yescharacteristic
61C23×C4×C59C2×C411yescharacteristic
62C23×C4×C59D411yescharacteristic
63C23×C4×C59D411yescharacteristic
64C23×C4×C59D411yescharacteristic
65C23×C4×C59C2311yescharacteristic
66C2×C22⋊C4×C59—24no—
67C2×C22⋊C4×C59—42no—
68C2×C22⋊C4×C59—42no—
69#95×C59—22no—
70#95×C59—22no—
71#95×C59—42no—
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—24no—
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—41no—
104C22×C4×C59—48no—
105C22×C4×C59—48no—
106C22×C4×C59—42no—
107C22×C4×C59—42no—
108C22×C4×C59—42no—
109C22×C4×C59—42no—
110C22×C4×C59—42no—
111C22×C4×C59—42no—
112C22×C4×C59C22⋊C411yescharacteristic
113C22×C4×C59C22⋊C411yescharacteristic
114C22×C4×C59C22×C411yescharacteristic
115C22×C4×C59C2×D411yescharacteristic
116C22×C4×C59C2×D412yes—
117C22×C4×C59#4112yes—
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—
127C2×C22⋊D59—42no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C2×D236—22no—
132C2×D236—41no—
133C2×D236—41no—
134C22⋊C4×C59—48no—
135C22⋊Dic59—42no—
136C22⋊Dic59—42no—
137C22⋊Dic59—41no—
138C22⋊Dic59—41no—
139(C2×D59)⋊C4—42no—
140(C2×D59)⋊C4—42no—
141(C2×D59)⋊C4—41no—
142(C2×D59)⋊C4—41no—
143C4⋊Dic59—22no—
144C4⋊Dic59—41no—
145C4⋊Dic59—41no—
146C2×C59⋊C8—24no—
147C2×C59⋊C8—48no—
148C2×C59⋊C8—42no—
149C2×C59⋊C8—42no—
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—44no—
209C2×C4×C59—44no—
210C2×C4×C59—44no—
211C2×C4×C59—82no—
212C2×C4×C59—82no—
213C2×C4×C59—82no—
214C2×C4×C59—82no—
215C2×C4×C59#9912yes—
216C2×C4×C59C4×D412yes—
217C2×C4×C59#12011yescharacteristic
218C2×C4×C59#12311yescharacteristic
219C22⋊D59—84no—
220C22⋊D59—84no—
221C22⋊D59—82no—
222C22⋊D59—82no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232D236—82no—
233C59⋊C8—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—
280C4×C59—84no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284C22×C22⋊C8—591no—
285#2847—591no—
286C2×#327—592no—
287C22⋊#95D5911yescharacteristic, complemented
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—1181no—
303C2×#121—591no—
304C2×#121—591no—
305C2×#121—591no—
306C2×#120—1181no—
307C22×C22⋊C4—21no—
308C2×C22⋊C8—592no—
309C2×C22⋊C8—1184no—
310C2×C22⋊C8—1181no—
311#386—1181no—
312#386—1181no—
313#386—1181no—
314#380—591no—
315#380—591no—
316#380—591no—
317C2×#95—21no—
318C2×#95C2×D5912yes—
319#327—1184no—
320C59#911311yescharacteristic, complemented, Sylow
321C25C22⋊D5911yescharacteristic
322C22×D4—591no—
323C22×D4—1181no—
324C22×D4—1181no—
325C22×D4—1181no—
326C23×C4—22no—
327C23×C4—22no—
328C23×C4—22no—
329C23×C4—42no—
330C23×C4—42no—
331C23×C4C4×D5911yescharacteristic
332C23×C4C4×D5911yescharacteristic
333C23×C4D23611yescharacteristic
334C23×C4D23611yescharacteristic
335C23×C4C22⋊D5911yescharacteristic
336C23×C4C22×D5911yescharacteristic
337C22×C8—1184no—
338C22×C8—1182no—
339C22×C8—2362no—
340C22×C8—2362no—
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—24no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—1181no—
354C2×C22⋊C4—1181no—
355C2×C22⋊C4—1181no—
356C2×C22⋊C4—1181no—
357C2×C22⋊C4—1181no—
358C2×C22⋊C4—1181no—
359C22⋊C8—594no—
360C22⋊C8—1184no—
361C22⋊C8—1182no—
362#95—22no—
363#95—22no—
364#95—42no—
365#95—1181no—
366#95—1181no—
367#95—1181no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—41no—
373C24—41no—
374C24—1181no—
375C24(C2×D59)⋊C411yescharacteristic
376C24(C2×D59)⋊C411yescharacteristic
377C24#1174566912yes—
378C24#1174567112yes—
379C24C2×C22⋊D5911yescharacteristic
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—24no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4(C2×D59)⋊C411yescharacteristic
425C22×C4(C2×D59)⋊C411yescharacteristic
426C22×C4C2×C4×D5911yescharacteristic
427C22×C4C2×D23611yescharacteristic
428C22×C4D4×D5912yes—
429C22×C4#1174568712yes—
430C2×C8—1184no—
431C2×C8—2368no—
432C2×C8—2362no—
433C2×C8—2362no—
434C4⋊C4—1182no—
435C4⋊C4—2361no—
436C4⋊C4—2361no—
437C22⋊C4—48no—
438C22⋊C4—2362no—
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—24no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
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#1427364712yes—
479C23#1427364712yes—
480C23#1427365512yes—
481C23C2×(C2×D59)⋊C411yescharacteristic
482C23#1427369912yes—
483C23C2×#1174566911yescharacteristic
484C23C2×#1174567111yescharacteristic
485C23#1427374712yes—
486D4—4722no—
487D4—4724no—
488D4—4722no—
489D4—4722no—
490D4—4724no—
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—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
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×C4(C22⋊D59)⋊C412yes—
527C2×C4#1427369711yescharacteristic
528C2×C4#1427371111yescharacteristic
529C8—2364no—
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#1683713514yescomplemented
561C22#1683727911yescharacteristic
562C22C2×#1427364111yescharacteristic
563C22C2×#1427364111yescharacteristic
564C22C2×#1427364711yescharacteristic
565C22#1683739611yescharacteristic
566C22C2×#1427365511yescharacteristic
567C22#1683742011yescharacteristic
568C22#1683763312yes—
569C22#1683763612yes—
570C4—21no—
571C4—22no—
572C4—21no—
573C4—22no—
574C4—41no—
575C4—84no—
576C4—4721no—
577C4—4721no—
578C4—4721no—
579C2—22no—
580C2—22no—
581C2—24no—
582C2—42no—
583C2—9441no—
584C2C2×#1683713511yescharacteristic
585C2#2142258511yescharacteristic
586C2#2142265011yescharacteristic
587C2#2142266311yescharacteristic
588C2#2142439611yescharacteristic
589C2#2142452211yescharacteristic
590C2#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#9113
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. C22×C22⋊(C59⋊C8)C2semidirect
3. C2×#16837135C2semidirect
4. #21421975C2semidirect
5. C59×C22⋊#95C2semidirect
7. C2×#14273701C4semidirect
8. C2×#14273701C4semidirect
17. #16837262C4semidirect
18. #16837262C4semidirect
50. C22⋊(C59⋊C8)D4semidirect
258. C22×C59#327semidirect
287. C22⋊#95D59semidirect
320. C59#9113semidirect
560. C22#16837135semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊(C59⋊C8)2
#214323782
C2×#168371352
#214219752
C59×C22⋊#952
#911359