Finite Group Databasebeta

← #28487830

Subgroups — #28487830

Order 15104.

650 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432399C211yesmaximal, characteristic
2#21424074C211yesmaximal, characteristic, complemented
3C2×#16837139C211yesmaximal, characteristic, complemented
4C2×#16837137C211yesmaximal, characteristic
5C2×#16837135C211yesmaximal, characteristic
6#21422321C211yesmaximal, characteristic, complemented
7C2×C59×#328C211yesmaximal, characteristic, complemented
8C2×#14273716C411yescharacteristic, complemented
9C2×#14273701C2211yescharacteristic
10C22×C4×Dic59C2211yescharacteristic
11C2×Dic59⋊C8—21no—
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13#16837309C411yescharacteristic, complemented
14#16837303C411yescharacteristic, complemented
15#16837292—21no—
16#16837292—21no—
17#16837279—21no—
18#16837279—21no—
19#16837267C411yescharacteristic, complemented
20C2×C59⋊#95—21no—
21C2×C59⋊#95—21no—
22C2×C59⋊#95C2211yescharacteristic
23#16837156—21no—
24#16837139—22no—
25#16837137—22no—
26#16837135—22no—
27C2×C22⋊Q8×C59C2211yescharacteristic
28C2×C22⋊C8×C59C2211yescharacteristic
29C2×#95×C59C2211yescharacteristic
30C59×#328—22no—
31C22×D236C2×C411yescharacteristic
32C22×C22⋊D59—21no—
33C23×Dic59—21no—
34C23×Dic59D411yescharacteristic
35C23×Dic59D411yescharacteristic
36C22×C59⋊C8—21no—
37#14273716—42no—
38#14273701—21no—
39#14273701—21no—
40#14273701—21no—
41#14273701—21no—
42C2×C4⋊Dic59C2×C411yescharacteristic
43C2×(C2×D59)⋊C4—21no—
44C2×(C2×D59)⋊C4—21no—
45C2×(C2×D59)⋊C4—21no—
46C2×(C2×D59)⋊C4—21no—
47C2×(C2×D59)⋊C4—21no—
48C2×C4×Dic59—42no—
49C2×C4×Dic59C2×C411yescharacteristic
50C2×C4×Dic59C2×C411yescharacteristic
51C2×C4×Dic59D411yescharacteristic
52C2×C4×Dic59D411yescharacteristic
53Dic59⋊C8—42no—
54C22⋊(C59⋊C8)—21no—
55C22⋊(C59⋊C8)—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—21no—
59C59⋊#95—21no—
60C59⋊#95—21no—
61C59⋊#95—42no—
62C59⋊#95—42no—
63C59⋊#95D412yes—
64C22×Q8×C59C2×C411yescharacteristic
65C23×C4×C59—21no—
66C23×C4×C59C2311yescharacteristic
67C22×C8×C59—21no—
68C22⋊Q8×C59—21no—
69C22⋊Q8×C59—21no—
70C22⋊Q8×C59—21no—
71C22⋊Q8×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C4⋊C4×C59C2×C411yescharacteristic
74C2×C22⋊C4×C59—21no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59—21no—
77#95×C59—21no—
78#95×C59D412yes—
79C24×C59C2×D411yescharacteristic
80C23×D59—21no—
81C2×Q8×C59—21no—
82C2×Q8×C59—21no—
83C2×Q8×C59—41no—
84C2×Q8×C59—41no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—22no—
88C22×C4×C59—41no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59C22×C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59#4111yescharacteristic
101C22×C4×C59#4111yescharacteristic
102C2×C22⋊D59—41no—
103C2×C22⋊D59—41no—
104C2×C22⋊D59—41no—
105C2×C22⋊D59—41no—
106C2×C22⋊D59—41no—
107C2×C22⋊D59—41no—
108C22×Dic59—21no—
109C22×Dic59—21no—
110C22×Dic59—21no—
111C22×Dic59—21no—
112C22×Dic59—22no—
113C22×Dic59—21no—
114C22×Dic59—22no—
115C22×Dic59—41no—
116C22×Dic59—41no—
117C22×Dic59—41no—
118C22×Dic59—41no—
119C22×Dic59—41no—
120C22×Dic59—41no—
121C22×Dic59—41no—
122C22×Dic59—41no—
123C22×Dic59—41no—
124C22×Dic59C22⋊C411yescharacteristic
125C22×Dic59C22⋊C411yescharacteristic
126C22×Dic59C22⋊C411yescharacteristic
127C22×Dic59C22⋊C411yescharacteristic
128C2×D236—21no—
129C2×D236—21no—
130C2×D236—41no—
131C2×D236—41no—
132C2×C8×C59—41no—
133C2×C8×C59—41no—
134C2×C8×C59—41no—
135C4⋊C4×C59—21no—
136C4⋊C4×C59—21no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C22⋊C4×C59—41no—
140C22⋊C4×C59—41no—
141(C2×D59)⋊C4—42no—
142(C2×D59)⋊C4—42no—
143(C2×D59)⋊C4—42no—
144(C2×D59)⋊C4—41no—
145(C2×D59)⋊C4—41no—
146(C2×D59)⋊C4—42no—
147C4⋊Dic59—21no—
148C4⋊Dic59—21no—
149C4×Dic59—22no—
150C4×Dic59—22no—
151C4×Dic59—41no—
152C4×Dic59—41no—
153C2×C59⋊C8—41no—
154C2×C59⋊C8—41no—
155C2×C59⋊C8—41no—
156C23×C59—21no—
157C23×C59—21no—
158C23×C59—21no—
159C23×C59—21no—
160C23×C59C2×C22⋊C411yescharacteristic
161C23×C59#12111yescharacteristic
162C23×C59#12111yescharacteristic
163C23×C59#13511yescharacteristic
164C23×C59#13511yescharacteristic
165C23×C59AGL(1,ℤ/8)11yescharacteristic
166C23×C59#13711yescharacteristic
167C22×D59—41no—
168C22×D59—41no—
169C22×D59—41no—
170C22×D59—41no—
171C22×D59—41no—
172C22×D59—41no—
173C22×D59—41no—
174Q8×C59—81no—
175Q8×C59—81no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—42no—
185C2×C4×C59—42no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—42no—
193C2×C4×C59—81no—
194C2×C4×C59—81no—
195C2×C4×C59—81no—
196C2×C4×C59—81no—
197C2×C4×C59C4×D411yescharacteristic
198C2×C4×C59C4×D411yescharacteristic
199C2×C4×C59#12411yescharacteristic
200C2×C4×C59#12711yescharacteristic
201C22⋊D59—81no—
202C22⋊D59—81no—
203C22⋊D59—81no—
204C22⋊D59—81no—
205C2×Dic59—21no—
206C2×Dic59—21no—
207C2×Dic59—21no—
208C2×Dic59—21no—
209C2×Dic59—21no—
210C2×Dic59—21no—
211C2×Dic59—21no—
212C2×Dic59—21no—
213C2×Dic59—21no—
214C2×Dic59—21no—
215C2×Dic59—21no—
216C2×Dic59—42no—
217C2×Dic59—42no—
218C2×Dic59—42no—
219C2×Dic59—42no—
220C2×Dic59—42no—
221C2×Dic59—42no—
222C2×Dic59—41no—
223C2×Dic59—41no—
224C2×Dic59—41no—
225C2×Dic59—42no—
226C2×Dic59—81no—
227C2×Dic59—81no—
228C2×Dic59—81no—
229C2×Dic59—81no—
230C2×Dic59#9912yes—
231D236—81no—
232D236—81no—
233C8×C59—81no—
234C59⋊C8—81no—
235#9716—591nomaximal, Sylow
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—21no—
242C22×C59—21no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—21no—
246C22×C59—21no—
247C22×C59—21no—
248C22×C59#39011yescharacteristic
249C22×C59C2×#9911yescharacteristic
250C22×C59#41011yescharacteristic
251C22×C59#41611yescharacteristic
252C22×C59#41711yescharacteristic
253C22×C59#42111yescharacteristic
254C22×C59#42111yescharacteristic
255C22×C59#46311yescharacteristic
256C22×C59#46311yescharacteristic
257C22×C59#46411yescharacteristic
258C22×C59#46411yescharacteristic
259C2×D59—81no—
260C2×D59—81no—
261C2×D59—81no—
262C2×D59—81no—
263C2×D59—81no—
264C2×D59—81no—
265C2×D59—81no—
266C4×C59—21no—
267C4×C59—21no—
268C4×C59—21no—
269C4×C59—21no—
270C4×C59—82no—
271C4×C59—81no—
272C4×C59—81no—
273Dic59—22no—
274Dic59—41no—
275Dic59—41no—
276Dic59—41no—
277Dic59—82no—
278Dic59—81no—
279#2850—591no—
280#1797—591no—
281C2×#328—591no—
282C2×#328D5911yescharacteristic, complemented
283C2×#327—591no—
284C2×#327—591no—
285#1434—591no—
286C2×C59—21no—
287C2×C59—21no—
288C2×C59—21no—
289C2×C59—21no—
290C2×C59#151211yescharacteristic
291C2×C59#151311yescharacteristic
292C2×C59#151611yescharacteristic
293C2×C59#152111yescharacteristic
294C2×C59#196311yescharacteristic
295C2×C59#196411yescharacteristic
296C2×C59#198311yescharacteristic
297D59—161no—
298C2×#124—591no—
299C2×C22⋊Q8C2×D5911yescharacteristic
300C2×#121—591no—
301C22×C42—591no—
302C2×C4⋊C8—1181no—
303C2×C22⋊C8—591no—
304C2×C22⋊C8C2×D5911yescharacteristic
305#391—591no—
306#390—591no—
307#388—1181no—
308#388—1181no—
309#386—1181no—
310#386—1181no—
311#382—591no—
312C2×#95—591no—
313C2×#95—1181no—
314C2×#95—1181no—
315C2×#95C2×D5911yescharacteristic
316#336—1181no—
317#328—22no—
318#328—1182no—
319#327—1182no—
320#327—1182no—
321C59#971611yescharacteristic, complemented, Sylow
322C22×Q8C4×D5911yescharacteristic
323C22×D4—591no—
324C22×D4—1181no—
325C23×C4—21no—
326C23×C4—591no—
327C23×C4—591no—
328C23×C4—1181no—
329C23×C4C22×D5911yescharacteristic
330C22×C8—21no—
331C22×C8—1181no—
332#124—2362no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—21no—
337#121—1181no—
338#121—1181no—
339#121—1181no—
340#121—1181no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—591no—
343C2×C4⋊C4C4×D5911yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—1181no—
346C2×C22⋊C4—1181no—
347C2×C22⋊C4—1181no—
348C2×C22⋊C4—1181no—
349C2×C22⋊C4—1181no—
350C2×C42—591no—
351C2×C42—591no—
352C2×C42—591no—
353C2×C42—591no—
354C2×C42—2362no—
355C4⋊C8—2362no—
356C22⋊C8—21no—
357C22⋊C8—21no—
358C22⋊C8—1181no—
359C22⋊C8—1181no—
360#95—21no—
361#95—592no—
362#95—1181no—
363#95—1181no—
364#95—1181no—
365#95—1181no—
366#95—1181no—
367#95—2362no—
368#95—2362no—
369#95C22⋊D5912yes—
370C24—1181no—
371C24D4×D5911yescharacteristic
372C2×Q8—21no—
373C2×Q8—21no—
374C2×Q8—41no—
375C2×Q8—41no—
376C2×D4—1181no—
377C2×D4—1181no—
378C2×D4—2361no—
379C2×D4—2361no—
380C2×D4—2361no—
381C2×D4—2361no—
382C2×D4—2361no—
383C2×D4—2361no—
384C2×D4—2361no—
385C2×D4—2361no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—591no—
398C22×C4—591no—
399C22×C4—591no—
400C22×C4—591no—
401C22×C4—1182no—
402C22×C4—1181no—
403C22×C4—1181no—
404C22×C4—1181no—
405C22×C4—1181no—
406C22×C4—1181no—
407C22×C4—1182no—
408C22×C4—2361no—
409C22×C4—2361no—
410C22×C4—2361no—
411C22×C4—2361no—
412C22×C4—2361no—
413C22×C4—2361no—
414C22×C4—2361no—
415C22×C4—2361no—
416C22×C4—2361no—
417C22×C4C2×C4×D5911yescharacteristic
418C22×C4#1174568511yescharacteristic
419C22×C4D4×D5911yescharacteristic
420C22×C4D4×D5911yescharacteristic
421C22×C4#1174568711yescharacteristic
422C22×C4C2×C22⋊D5911yescharacteristic
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—2361no—
427C2×C8—2361no—
428C2×C8—2361no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—1181no—
434C4⋊C4—1181no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—2362no—
438C22⋊C4—2362no—
439C22⋊C4—2362no—
440C22⋊C4—2362no—
441C22⋊C4—2361no—
442C22⋊C4—2361no—
443C42—1182no—
444C42—1182no—
445C42—2361no—
446C42—2361no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—2361no—
452C23—2361no—
453C23—2361no—
454C23—2361no—
455C23—2361no—
456C23—2361no—
457C23—2361no—
458C23C22⋊C4×D5911yescharacteristic
459C23#1427369911yescharacteristic
460C23#1427370011yescharacteristic
461C23#1427374811yescharacteristic
462C23#1427375011yescharacteristic
463C23#1427375411yescharacteristic
464C23#1427375911yescharacteristic
465Q8—81no—
466Q8—81no—
467D4—4721no—
468D4—4721no—
469D4—4721no—
470D4—4721no—
471D4—4721no—
472D4—4721no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—592no—
495C2×C4—1181no—
496C2×C4—1181no—
497C2×C4—1181no—
498C2×C4—1181no—
499C2×C4—1181no—
500C2×C4—1181no—
501C2×C4—1181no—
502C2×C4—1181no—
503C2×C4—1181no—
504C2×C4—1181no—
505C2×C4—1181no—
506C2×C4—4721no—
507C2×C4—4721no—
508C2×C4—4721no—
509C2×C4—4721no—
510C2×C4—2362no—
511C2×C4—2362no—
512C2×C4—2362no—
513C2×C4—2362no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2362no—
518C2×C4—2362no—
519C2×C4—2362no—
520C2×C4(C22⋊D59)⋊C411yescharacteristic
521C2×C4C4×C22⋊D5911yescharacteristic
522C2×C4#1427371311yescharacteristic
523C2×C4#1427372311yescharacteristic
524C8—81no—
525C8—4721no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—4721no—
539C22—4721no—
540C22—4721no—
541C22—4721no—
542C22—4721no—
543C22—4721no—
544C22—4721no—
545C22#1683730211yescharacteristic
546C22#99×D5911yescharacteristic
547C22#1683737411yescharacteristic
548C22#1683739711yescharacteristic
549C22#1683740211yescharacteristic
550C22#1683741711yescharacteristic
551C22#1683741711yescharacteristic
552C22#1683761911yescharacteristic
553C22#1683762011yescharacteristic
554C22#1683762611yescharacteristic
555C22#1683762711yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—1182no—
564C4—4722no—
565C4—4721no—
566C4—2361no—
567C4—2361no—
568C4—2361no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2—21no—
573C2—9441no—
574C2#2142263311yescharacteristic
575C2#2142263811yescharacteristic
576C2#2142265111yescharacteristic
577C2#2142267211yescharacteristic
578C2#2142492611yescharacteristic
579C2#2142493011yescharacteristic
580C2#2142502511yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#328C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenterC2×#328D59
hypocenterC59#9716
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
2. #21424074C2semidirect
3. C2×#16837139C2semidirect
6. #21422321C2semidirect
7. C2×C59×#328C2semidirect
8. C2×#14273716C4semidirect
13. #16837309C4semidirect
14. #16837303C4semidirect
19. #16837267C4semidirect
282. C2×#328D59semidirect
321. C59#9716semidirect

Maximal subgroups

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

subgroupindex
#214323992
#214240742
C2×#168371392
C2×#168371372
C2×#168371352
#214223212
C2×C59×#3282
#971659