Finite Group Databasebeta

← #28559323

Subgroups — #28559323

Order 15104.

622 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434691C211yesmaximal, characteristic
2#21433247C211yesmaximal, characteristic, complemented
3#21433171C211yesmaximal, characteristic, complemented
4#21423011C211yesmaximal, characteristic, complemented
5#21423000C211yesmaximal, characteristic, complemented
6#21422988C211yesmaximal, characteristic, complemented
7#21422982C211yesmaximal, characteristic, complemented
8#21422921C211yesmaximal, characteristic, complemented
9#21422910C211yesmaximal, characteristic, complemented
10C8×#11745660C211yesmaximal, characteristic, complemented
11C8×D472C211yesmaximal, characteristic, complemented
12#21422863C211yesmaximal, characteristic, complemented
13#21422557C211yesmaximal, characteristic, complemented
14#21422548C211yesmaximal, characteristic, complemented
15C8×M16×C59C211yesmaximal, characteristic, complemented
16#16837952C2211yescharacteristic
17#16837936C2211yescharacteristic
18#16837811C411yescharacteristic
19#16837791C411yescharacteristic
20#16837767C411yescharacteristic
21#16837762C411yescharacteristic
22#16837655C411yescharacteristic
23#16837655C411yescharacteristic
24#16837648C411yescharacteristic
25#16837648C411yescharacteristic
26#16837532C2211yescharacteristic
27C8×Dic118C2211yescharacteristic
28C4×#11745660C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×D472C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837481C2211yescharacteristic
33#16837481C2211yescharacteristic
34#16837481C2211yescharacteristic
35#16837477—21no—
36#16837476—21no—
37#16837472—21no—
38#16837472—21no—
39C8×D236C2211yescharacteristic
40C8×D236C2211yescharacteristic
41C8×D236C2211yescharacteristic
42#16837462—21no—
43#16837459C2211yescharacteristic
44#16837427—21no—
45#16837426C2211yescharacteristic
46#16837367—21no—
47#16837367—21no—
48#16837155C2211yescharacteristic
49#16837155C2211yescharacteristic
50#16837153C2211yescharacteristic
51#16837153C2211yescharacteristic
52#16837134C2211yescharacteristic
53#16837134C2211yescharacteristic
54#16837131C2211yescharacteristic
55#16837131C2211yescharacteristic
56#16837131C2211yescharacteristic
57#16837131C2211yescharacteristic
58#16837131C2211yescharacteristic
59#16837131C2211yescharacteristic
60C59×#431C2211yescharacteristic
61C4×M16×C59C2211yescharacteristic
62C2×C4×C8×C59C2211yescharacteristic
63C59×#322C2211yescharacteristic
64C59×#322C2211yescharacteristic
65C82×C59C2211yescharacteristic
66C82×C59C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745660C2×C411yescharacteristic
69C2×#11745660C2×C411yescharacteristic
70C2×D472C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×#11745664—21no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C2×#11745659—21no—
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79#14273727C2×C411yescharacteristic
80#14273724C2×C411yescharacteristic
81#14273721—21no—
82#14273719C2×C411yescharacteristic
83#14273715—21no—
84#14273715C2×C411yescharacteristic
85#14273710—21no—
86#14273706C2×C411yescharacteristic
87#14273701—21no—
88#14273701C2×C411yescharacteristic
89#14273701C2×C411yescharacteristic
90#14273701C2×C411yescharacteristic
91C4×Dic118C2311yescharacteristic
92C4×D236—21no—
93C4×D236C2311yescharacteristic
94C4×D236C2311yescharacteristic
95C4×D236C2311yescharacteristic
96C4×C22⋊D59—21no—
97#14273684—21no—
98#14273661C2×C411yescharacteristic
99#14273661C2×C411yescharacteristic
100#14273659C2×C411yescharacteristic
101#14273659C2×C411yescharacteristic
102C4⋊(C59⋊C8)—21no—
103C4⋊(C59⋊C8)C2311yescharacteristic
104C4⋊(C59⋊C8)C2311yescharacteristic
105C4⋊(C59⋊C8)C2311yescharacteristic
106C4⋊(C59⋊C8)C2311yescharacteristic
107Dic59⋊C8—21no—
108Dic59⋊C8—21no—
109#14273652C2×C411yescharacteristic
110#14273652C2×C411yescharacteristic
111#14273649C2×C411yescharacteristic
112#14273649C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114#14273649C2×C411yescharacteristic
115#14273649C2×C411yescharacteristic
116#14273649C2×C411yescharacteristic
117C22⋊(C59⋊C8)—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8—21no—
121(C2×D59)⋊C8—21no—
122(C2×D59)⋊C8—21no—
123(C2×D59)⋊C8—21no—
124#14273638—21no—
125#14273637—21no—
126C8×Dic59—21no—
127C2×M16×C59C2×C411yescharacteristic
128C2×M16×C59C2×C411yescharacteristic
129C22×C8×C59—21no—
130C2×C42×C59C2311yescharacteristic
131C4⋊C8×C59D411yescharacteristic
132C4⋊C8×C59D411yescharacteristic
133C22⋊C8×C59D411yescharacteristic
134C22⋊C8×C59D411yescharacteristic
135#97×C59C2×C411yescharacteristic
136#97×C59C2×C411yescharacteristic
137C4×C8×C59—21no—
138C4×C8×C59C2311yescharacteristic
139C4×C8×C59C2311yescharacteristic
140C4×C8×C59C2311yescharacteristic
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C22×C4×C59C22×C411yescharacteristic
145C22×C4×C59C2×D411yescharacteristic
146C22×C4×C59#4111yescharacteristic
147C2×C22⋊D59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150#11745685—81no—
151#11745685—81no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236C22×C411yescharacteristic
155C2×D236C22×C411yescharacteristic
156C2×D236C22×C411yescharacteristic
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×Dic118—21no—
161C2×Dic118C22×C411yescharacteristic
162M16×C59—22no—
163M16×C59—22no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—42no—
167C2×C8×C59C22×C411yescharacteristic
168C2×C8×C59C22×C411yescharacteristic
169C2×C8×C59C22×C411yescharacteristic
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C2×D411yescharacteristic
172C2×C8×C59C2×D411yescharacteristic
173C2×C8×C59C2×D411yescharacteristic
174C2×C8×C59C2×D411yescharacteristic
175C42×C59C22×C411yescharacteristic
176C42×C59C2×D411yescharacteristic
177C42×C59#4111yescharacteristic
178C42×C59C2411yescharacteristic
179C22⋊Dic59—21no—
180(C2×D59)⋊C4—21no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186(C2×D59)⋊C4—21no—
187(C2×D59)⋊C4—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59C22×C411yescharacteristic
190C4⋊Dic59C22×C411yescharacteristic
191C4⋊Dic59C22×C411yescharacteristic
192C4⋊Dic59C22×C411yescharacteristic
193Dic59⋊C4—21no—
194Dic59⋊C4—21no—
195Dic59⋊C4—21no—
196Dic59⋊C4—21no—
197C4×Dic59—21no—
198#11745664—81no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203D472—22no—
204D472—22no—
205#11745660—22no—
206#11745660—22no—
207#11745659—81no—
208#11745659—81no—
209#11745659—81no—
210C8×D59—81no—
211C8×D59—81no—
212C8×D59—81no—
213C23×C59C4×D411yescharacteristic
214C22×D59—21no—
215C22×D59—21no—
216C22×D59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—41no—
222C2×C4×C59C4×D411yescharacteristic
223C2×C4×C59C4×D411yescharacteristic
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C23×C411yescharacteristic
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C22⋊D59—81no—
229C22⋊D59—81no—
230C22⋊D59—81no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236D236—21no—
237D236—21no—
238D236—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236—81no—
243D236—81no—
244C4×D59—41no—
245C4×D59—41no—
246C4×D59—41no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250Dic118—21no—
251Dic118—21no—
252Dic118—81no—
253C8×C59—22no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59—21no—
257C8×C59—21no—
258C8×C59—21no—
259C8×C59—21no—
260C8×C59—41no—
261C8×C59#13112yes—
262C8×C59#13112yes—
263C59⋊C8—41no—
264C59⋊C8—41no—
265C59⋊C8—41no—
266C59⋊C8—41no—
267#17968—591nomaximal, Sylow
268C22×C59—21no—
269C22×C59—21no—
270C22×C59—21no—
271C22×C59C2×C4×D411yescharacteristic
272C2×D59—41no—
273C2×D59—41no—
274C2×D59—41no—
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C4×C59—21no—
282C4×C59—21no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59C2×#13111yescharacteristic
286C4×C59C2×#13111yescharacteristic
287C4×C59#57611yescharacteristic
288C4×C59#57711yescharacteristic
289Dic59—41no—
290Dic59—41no—
291Dic59—41no—
292Dic59—41no—
293Dic59—41no—
294#3171—591no—
295#2993—591no—
296#2981—591no—
297#1594—591no—
298#1592—591no—
299#1590—591no—
300#1589—591no—
301#1580—591no—
302SD16⋊C8—591no—
303C8×SD16—591no—
304C8×D8—591no—
305#1569—591no—
306#1494—591no—
307#1492—591no—
308C8×M16D5911yescharacteristic, complemented
309C2×C59—41no—
310C2×C59#293111yescharacteristic
311C2×C59#294611yescharacteristic
312C2×C59#295611yescharacteristic
313D59—81no—
314D59—81no—
315D59—81no—
316#532—591no—
317#529—591no—
318#501—591no—
319#496—591no—
320#490—591no—
321#489—591no—
322#469—591no—
323#469—591no—
324#468—591no—
325#468—591no—
326#446—591no—
327C8×Q8—591no—
328C4×SD16—591no—
329C4×SD16—591no—
330C4×D8—591no—
331C4×D8—591no—
332#436—591no—
333#436—591no—
334#436—591no—
335#436—1181no—
336#435—1181no—
337#435—1181no—
338#435—1181no—
339C8×D4—591no—
340C8×D4—591no—
341C8×D4—591no—
342#433—1181no—
343#432—591no—
344#431C2×D5911yescharacteristic
345#424—1181no—
346#423—591no—
347#408—1181no—
348#408—1181no—
349C4×M16C2×D5911yescharacteristic
350C2×C4×C8C2×D5911yescharacteristic
351#335—591no—
352#335—591no—
353#334—591no—
354#334—591no—
355Q8⋊C8—591no—
356Q8⋊C8—591no—
357D4⋊C8—591no—
358D4⋊C8—591no—
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363#322C2×D5911yescharacteristic
364#322C2×D5911yescharacteristic
365C82C2×D5911yescharacteristic
366C82C2×D5911yescharacteristic
367C59#1796811yescharacteristic, complemented, Sylow
368C2×#41—1181no—
369C2×SD16—591no—
370C2×SD16—591no—
371C2×D8—591no—
372C2×D8—591no—
373C2×M16—1181no—
374C2×M16—1181no—
375C2×M16—1181no—
376C2×M16—1181no—
377C2×M16C4×D5911yescharacteristic
378C2×M16C4×D5911yescharacteristic
379C22×C8—21no—
380C22×C8—1181no—
381C22×C8—1181no—
382C22×C8—1181no—
383C4⋊Q8—591no—
384#127—591no—
385#126—1181no—
386#125—591no—
387#124—591no—
388#124—1181no—
389#123—1181no—
390C22⋊Q8—591no—
391#121—591no—
392#121—591no—
393#121—591no—
394#121—1181no—
395C4×Q8—591no—
396C4×D4—591no—
397C4×D4—591no—
398C4×D4—591no—
399C4×D4—1181no—
400C4×D4—1181no—
401#117—1181no—
402C2×C42C22×D5911yescharacteristic
403#107—591no—
404#107—591no—
405#106—591no—
406#106—591no—
407C4⋊C8—591no—
408C4⋊C8—591no—
409C4⋊C8—591no—
410C4⋊C8—591no—
411C4⋊C8—1181no—
412C4⋊C8—1181no—
413C4⋊C8—1181no—
414C4⋊C8D23611yescharacteristic
415C4⋊C8D23611yescharacteristic
416#103—591no—
417#103—591no—
418#102—591no—
419#102—591no—
420#102—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424C22⋊C8—1181no—
425C22⋊C8—1181no—
426C22⋊C8—1181no—
427C22⋊C8—1181no—
428C22⋊C8—1181no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8D23611yescharacteristic
432C22⋊C8D23611yescharacteristic
433#97—1181no—
434#97—1181no—
435#97C4×D5911yescharacteristic
436#97C4×D5911yescharacteristic
437C4×C8—21no—
438C4×C8—1181no—
439C4×C8C22×D5911yescharacteristic
440C4×C8C22×D5911yescharacteristic
441C4×C8C22×D5911yescharacteristic
442C4×C8C22×D5911yescharacteristic
443C4×C8C22×D5911yescharacteristic
444C4×C8C22×D5911yescharacteristic
445#41—4721no—
446#41—4721no—
447C2×Q8—591no—
448C2×Q8—1181no—
449C2×D4—591no—
450C2×D4—591no—
451C2×D4—591no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4—1181no—
455C2×D4—1181no—
456C2×D4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4C2×C4×D5911yescharacteristic
461C22×C4C2×D23611yescharacteristic
462C22×C4#1174568511yescharacteristic
463SD16—1182no—
464SD16—1182no—
465D8—1182no—
466D8—1182no—
467M16—22no—
468M16—22no—
469M16—4721no—
470M16—4721no—
471M16—4721no—
472M16—4721no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—42no—
476C2×C8—1181no—
477C2×C8—1181no—
478C2×C8—1181no—
479C2×C8—1181no—
480C2×C8—4721no—
481C2×C8—4721no—
482C2×C8—4721no—
483C2×C8C2×C4×D5911yescharacteristic
484C2×C8C2×C4×D5911yescharacteristic
485C2×C8C2×C4×D5911yescharacteristic
486C2×C8C2×C4×D5911yescharacteristic
487C2×C8C2×D23611yescharacteristic
488C2×C8C2×D23611yescharacteristic
489C2×C8C2×D23611yescharacteristic
490C2×C8C2×D23611yescharacteristic
491C4⋊C4—591no—
492C4⋊C4—591no—
493C4⋊C4—591no—
494C4⋊C4—591no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C22⋊C4—1181no—
501C22⋊C4—1181no—
502C22⋊C4—1181no—
503C22⋊C4—1181no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C42—1181no—
510C42C2×C4×D5911yescharacteristic
511C42C2×D23611yescharacteristic
512C42#1174568511yescharacteristic
513C42C23×D5911yescharacteristic
514C23—1181no—
515C23—1181no—
516C23—1181no—
517C23C4×D23611yescharacteristic
518Q8—1181no—
519Q8—1181no—
520Q8—4721no—
521D4—1181no—
522D4—1181no—
523D4—1181no—
524D4—1181no—
525D4—1181no—
526D4—1181no—
527D4—4721no—
528D4—4721no—
529D4—4721no—
530D4—4721no—
531D4—4721no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4C4×D23611yescharacteristic
549C2×C4C4×D23611yescharacteristic
550C2×C4C4×D23611yescharacteristic
551C2×C4C22×C4×D5911yescharacteristic
552C2×C4C22×D23611yescharacteristic
553C2×C4C2×#1174568511yescharacteristic
554C8—22no—
555C8—21no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—2361no—
563C8—2361no—
564C8—2361no—
565C8—2361no—
566C8#1427373312yes—
567C8#1427373312yes—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—2361no—
572C22—2361no—
573C22—2361no—
574C22—2361no—
575C22—2361no—
576C22—2361no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22C2×C4×D23611yescharacteristic
581C4—21no—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—2361no—
586C4—2361no—
587C4—2361no—
588C4—2361no—
589C4—2361no—
590C4C2×#1427373311yescharacteristic
591C4C2×#1427373311yescharacteristic
592C4#1683820711yescharacteristic
593C4#1683821411yescharacteristic
594C2—41no—
595C2—4721no—
596C2—4721no—
597C2—4721no—
598C2#2143280911yescharacteristic
599C2#2143293011yescharacteristic
600C2#2143300711yescharacteristic
subgroupG/H
centerC2×C4C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC8×M16×C59C2
socleC22×C59C2×C4×D4
prime radical#21434691C2
cosocleC42×C59C24
hypercenterC8×M16D59
hypocenterC59#17968
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ #431 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #21433247C2semidirect
3. #21433171C2semidirect
4. #21423011C2semidirect
5. #21423000C2semidirect
6. #21422988C2semidirect
7. #21422982C2semidirect
8. #21422921C2semidirect
9. #21422910C2semidirect
10. C8×#11745660C2semidirect
11. C8×D472C2semidirect
12. #21422863C2semidirect
13. #21422557C2semidirect
14. #21422548C2semidirect
15. C8×M16×C59C2semidirect
308. C8×M16D59semidirect
367. C59#17968semidirect

Maximal subgroups

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

subgroupindex
#214346912
#214332472
#214331712
#214230112
#214230002
#214229882
#214229822
#214229212
#214229102
C8×#117456602
C8×D4722
#214228632
#214225572
#214225482
C8×M16×C592
#1796859