Finite Group Databasebeta

← #28559236

Subgroups — #28559236

Order 15104.

666 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434652C211yesmaximal, characteristic
2#21433027C211yesmaximal, characteristic, complemented
3#21432805C211yesmaximal, characteristic
4#21423000C211yesmaximal, characteristic, complemented
5#21422998C211yesmaximal, characteristic, complemented
6#21422996C211yesmaximal, characteristic, complemented
7#21422911C211yesmaximal, characteristic, complemented
8SD16⋊(C59⋊C8)C211yesmaximal, characteristic, complemented
9#21422907C211yesmaximal, characteristic, complemented
10#21422905C211yesmaximal, characteristic, complemented
11#21422853C211yesmaximal, characteristic, complemented
12#21422549C211yesmaximal, characteristic, complemented
13#21422545C211yesmaximal, characteristic
14#322×D59C211yesmaximal, characteristic, complemented
15C59×#1592C211yesmaximal, characteristic, complemented
16C2×#14273735—21no—
17#16837956C2211yescharacteristic
18#16837949C2211yescharacteristic
19#16837799C411yescharacteristic
20#16837773C411yescharacteristic
21#16837751C411yescharacteristic
22#16837747C411yescharacteristic
23#16837634C411yescharacteristic
24#16837632C411yescharacteristic
25#16837630C411yescharacteristic
26#16837628C411yescharacteristic
27#16837532C2211yescharacteristic
28#16837530—21no—
29Q8×C59⋊C8C2211yescharacteristic
30#16837526—21no—
31#16837526C2211yescharacteristic
32C4×#11745660C2211yescharacteristic
33C4×#11745671C2211yescharacteristic
34C4×#11745670C2211yescharacteristic
35#16837481C2211yescharacteristic
36#16837478—21no—
37D4×C59⋊C8C2211yescharacteristic
38#16837467C2211yescharacteristic
39(C22⋊D59)⋊C8—21no—
40#16837427—21no—
41#16837424C2211yescharacteristic
42#16837424C2211yescharacteristic
43#16837367—21no—
44#16837359—21no—
45C4×C8×D59C2211yescharacteristic
46C4×C8×D59C2211yescharacteristic
47C4×C8×D59C2211yescharacteristic
48#16837155C2211yescharacteristic
49#16837154C2211yescharacteristic
50#16837154C2211yescharacteristic
51#16837134C2211yescharacteristic
52Q8⋊(C59⋊C8)C2211yescharacteristic
53#16837132C2211yescharacteristic
54#16837131C2211yescharacteristic
55D4⋊(C59⋊C8)C2211yescharacteristic
56#16837129C2211yescharacteristic
57#16837124C2211yescharacteristic
58#16837124C2211yescharacteristic
59#16837123C2211yescharacteristic
60C59×#446C2211yescharacteristic
61C4×SD16×C59C2211yescharacteristic
62C59×#436C2211yescharacteristic
63C59×#335C2211yescharacteristic
64C59×Q8⋊C8C2211yescharacteristic
65C59×D4⋊C8C2211yescharacteristic
66C59×#322C2211yescharacteristic
67C2×#11745685—21no—
68C2×#11745660C2×C411yescharacteristic
69C2×#11745670C2×C411yescharacteristic
70C2×#11745671C2×C411yescharacteristic
71#14273735—44no—
72C2×#11745659—21no—
73C2×#11745659—21no—
74C2×#11745659—21no—
75C22×C59⋊C8—21no—
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78C2×C8×D59C2×C411yescharacteristic
79C2×C8×D59C2×C411yescharacteristic
80C2×C8×D59D411yescharacteristic
81C2×C8×D59D411yescharacteristic
82#14273726C2×C411yescharacteristic
83#14273726C2×C411yescharacteristic
84#14273725C2×C411yescharacteristic
85#14273723C2×C411yescharacteristic
86#14273717—21no—
87#14273713—21no—
88#14273707C2×C411yescharacteristic
89#14273705—21no—
90#14273703C2×C411yescharacteristic
91#14273702C2×C411yescharacteristic
92#14273700—21no—
93#14273698C2×C411yescharacteristic
94C4×Dic118—21no—
95C4×Dic118C2311yescharacteristic
96C4×D236C2311yescharacteristic
97C4×C22⋊D59—21no—
98#14273684—21no—
99C42×D59C2311yescharacteristic
100#14273659C2×C411yescharacteristic
101#14273658C2×C411yescharacteristic
102#14273658C2×C411yescharacteristic
103C4⋊(C59⋊C8)—21no—
104C4⋊(C59⋊C8)C2311yescharacteristic
105C4⋊(C59⋊C8)C2311yescharacteristic
106Dic59⋊C8—21no—
107Dic59⋊C8—21no—
108Dic59⋊C8—21no—
109Dic59⋊C8—21no—
110#14273652C2×C411yescharacteristic
111#14273651C2×C411yescharacteristic
112#14273650C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114#14273648C2×C411yescharacteristic
115#14273647C2×C411yescharacteristic
116C22⋊(C59⋊C8)—21no—
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119#14273637—21no—
120C4×C59⋊C8—21no—
121C4×C59⋊C8C2311yescharacteristic
122C4×C59⋊C8C2311yescharacteristic
123C4×C59⋊C8C2311yescharacteristic
124C8×Dic59—21no—
125C8×Dic59—21no—
126C8×Dic59C2×C411yescharacteristic
127C8×Dic59C2×C411yescharacteristic
128C8×Dic59D411yescharacteristic
129C8×Dic59D411yescharacteristic
130C2×SD16×C59C2×C411yescharacteristic
131C2×M16×C59—21no—
132C4×Q8×C59C2311yescharacteristic
133C4×D4×C59C2311yescharacteristic
134#106×C59C2×C411yescharacteristic
135C4⋊C8×C59—21no—
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138#103×C59C2×C411yescharacteristic
139#102×C59C2×C411yescharacteristic
140C22⋊C8×C59—21no—
141#97×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C4×C8×C59C2311yescharacteristic
145C2×Q8×C59C22×C411yescharacteristic
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C2×C22⋊D59—21no—
149C2×C22⋊D59—21no—
150C2×C22⋊D59—21no—
151#11745685—42no—
152#11745685—42no—
153C2×D236—21no—
154C2×D236C22×C411yescharacteristic
155C2×C4×D59—21no—
156C2×C4×D59C22×C411yescharacteristic
157C2×C4×D59C2×D411yescharacteristic
158C2×C4×D59#4111yescharacteristic
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118C22×C411yescharacteristic
162SD16×C59—22no—
163M16×C59—42no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59C22×C411yescharacteristic
168C2×C8×C59C22×C411yescharacteristic
169C2×C8×C59C2×D411yescharacteristic
170C2×C8×C59C2×D411yescharacteristic
171C4⋊C4×C59—21no—
172C4⋊C4×C59C22×C411yescharacteristic
173C4⋊C4×C59C22×C411yescharacteristic
174C22⋊C4×C59—21no—
175C42×C59—21no—
176C42×C59C2411yescharacteristic
177C22⋊Dic59—21no—
178#11745671—22no—
179#11745670—22no—
180(C2×D59)⋊C4—21no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59C22×C411yescharacteristic
186C4⋊Dic59C22×C411yescharacteristic
187Dic59⋊C4—21no—
188Dic59⋊C4—21no—
189Dic59⋊C4—21no—
190Dic59⋊C4—21no—
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193C4×Dic59—21no—
194C4×Dic59C22×C411yescharacteristic
195C4×Dic59C2×D411yescharacteristic
196C4×Dic59#4111yescharacteristic
197C2×C59⋊C8—21no—
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—42no—
201C2×C59⋊C8C22×C411yescharacteristic
202C2×C59⋊C8C22×C411yescharacteristic
203C2×C59⋊C8C2×D411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205#11745660—22no—
206#11745659—42no—
207#11745659—42no—
208#11745659—42no—
209C8×D59—22no—
210C8×D59—22no—
211C8×D59—22no—
212C8×D59—22no—
213C8×D59—42no—
214C8×D59—42no—
215C23×C59—21no—
216C22×D59—21no—
217C22×D59C4×D411yescharacteristic
218Q8×C59—21no—
219Q8×C59—21no—
220D4×C59—21no—
221D4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C22⋊D59—42no—
231C22⋊D59—42no—
232C22⋊D59—42no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59C4×D411yescharacteristic
237C2×Dic59C4×D411yescharacteristic
238C2×Dic59C4×D411yescharacteristic
239D236—21no—
240D236—21no—
241D236—42no—
242C4×D59—21no—
243C4×D59—21no—
244C4×D59—22no—
245C4×D59—41no—
246C4×D59—41no—
247C4×D59AGL(1,ℤ/8)12yes—
248C4×D59#13712yes—
249Dic118—21no—
250Dic118—21no—
251Dic118—42no—
252Dic118—42no—
253C8×C59—21no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59—41no—
257C8×C59—41no—
258C8×C59—41no—
259C8×C59#13112yescomplemented
260C59⋊C8—21no—
261C59⋊C8—21no—
262C59⋊C8—21no—
263C59⋊C8—41no—
264C59⋊C8—41no—
265C59⋊C8—41no—
266C59⋊C8#13112yescomplemented
267#17960—591nomaximal, Sylow
268C22×C59—41no—
269C22×C59—41no—
270C22×C59—41no—
271C22×C59C2×C4×D411yescharacteristic
272C2×D59—21no—
273C2×D59—21no—
274C2×D59—41no—
275C2×D59—41no—
276C2×D59—41no—
277C2×D59#44012yes—
278C4×C59—21no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—41no—
282C4×C59C2×#13111yescharacteristic
283C4×C59C2×#13111yescharacteristic
284C4×C59C2×AGL(1,ℤ/8)11yescharacteristic
285C4×C59C2×#13711yescharacteristic
286Dic59—21no—
287Dic59—41no—
288Dic59—41no—
289Dic59—41no—
290Dic59#44012yes—
291Dic59#44412yes—
292#3167—591no—
293#2960—591no—
294#2931—591no—
295#1592—591no—
296#1592—591no—
297#1592—591no—
298#1592D5911yescharacteristic, complemented
299SD16⋊C8—591no—
300SD16⋊C8—591no—
301SD16⋊C8—591no—
302SD16⋊C8—591no—
303#1567—591no—
304#1493—591no—
305#1492—591no—
306C2×#322—591no—
307C2×C59—81no—
308C2×C59#293111yescharacteristic
309C2×C59C2×#44011yescharacteristic
310C2×C59C2×#44411yescharacteristic
311D59—22no—
312D59—81no—
313C2×#131—1181no—
314#533—591no—
315#532—591no—
316#498—591no—
317#492—591no—
318#487—591no—
319#486—591no—
320#465—591no—
321#465—591no—
322#465—591no—
323#465—591no—
324#446—591no—
325#446—1181no—
326#446C2×D5911yescharacteristic
327C8×Q8—591no—
328C8×Q8—591no—
329C8×Q8—1181no—
330C4×SD16—591no—
331C4×SD16—591no—
332C4×SD16—591no—
333C4×SD16C2×D5911yescharacteristic
334#436—591no—
335#436—1181no—
336#436C2×D5911yescharacteristic
337C8×D4—591no—
338C8×D4—591no—
339C8×D4—1181no—
340#424—1181no—
341#423—591no—
342#423—591no—
343#408—1181no—
344#405—1181no—
345C2×C4×C8—591no—
346C2×C4×C8—591no—
347C2×C4×C8—591no—
348#335—591no—
349#335—591no—
350#335—591no—
351#335C2×D5911yescharacteristic
352Q8⋊C8—591no—
353Q8⋊C8—591no—
354Q8⋊C8—591no—
355Q8⋊C8C2×D5911yescharacteristic
356D4⋊C8—591no—
357D4⋊C8—591no—
358D4⋊C8—591no—
359D4⋊C8C2×D5911yescharacteristic
360#322—591no—
361#322—591no—
362#322—591no—
363#322C2×D5911yescharacteristic
364C59#1796011yescharacteristic, complemented, Sylow
365C2×#41—1181no—
366C2×SD16—591no—
367C2×SD16—591no—
368C2×SD16—591no—
369C2×SD16C4×D5911yescharacteristic
370#131—2364no—
371C2×M16—21no—
372C2×M16—1181no—
373C2×M16—1181no—
374C2×M16—1181no—
375C22×C8—591no—
376C22×C8—591no—
377C22×C8—591no—
378C22×C8—591no—
379C22×C8—1181no—
380C22×C8—1181no—
381C22×C8—1181no—
382C4⋊Q8—591no—
383C4⋊Q8—591no—
384C4⋊Q8—591no—
385#127—591no—
386#125—1181no—
387#124—1181no—
388C22⋊Q8—591no—
389C22⋊Q8—591no—
390C22⋊Q8—1181no—
391#121—591no—
392#121—591no—
393#121—1181no—
394C4×Q8—591no—
395C4×Q8—1181no—
396C4×Q8C22×D5911yescharacteristic
397C4×D4—591no—
398C4×D4—1181no—
399C4×D4C22×D5911yescharacteristic
400#117—1181no—
401C2×C42—591no—
402#106—591no—
403#106—591no—
404#106—591no—
405#106C4×D5911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—591no—
408C4⋊C8—591no—
409C4⋊C8—1181no—
410C4⋊C8—1181no—
411C4⋊C8—1181no—
412C4⋊C8—1181no—
413C4⋊C8—1181no—
414C4⋊C8C22×D5911yescharacteristic
415C4⋊C8C22×D5911yescharacteristic
416#103—591no—
417#103—591no—
418#103—591no—
419#103C4×D5911yescharacteristic
420#102—591no—
421#102—591no—
422#102—591no—
423#102C4×D5911yescharacteristic
424C22⋊C8—21no—
425C22⋊C8—1181no—
426C22⋊C8—1181no—
427C22⋊C8—1181no—
428#97—21no—
429#97—1181no—
430C4×C8—591no—
431C4×C8—591no—
432C4×C8—591no—
433C4×C8—591no—
434C4×C8—591no—
435C4×C8—591no—
436C4×C8—591no—
437C4×C8—1181no—
438C4×C8—1181no—
439C4×C8—1181no—
440C4×C8C22×D5911yescharacteristic
441C4×C8C22×D5911yescharacteristic
442C4×C8C22×D5911yescharacteristic
443#41—2362no—
444#41—2362no—
445C2×Q8—591no—
446C2×Q8—1181no—
447C2×Q8—1181no—
448C2×Q8C2×C4×D5911yescharacteristic
449C2×D4—591no—
450C2×D4—1181no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1181no—
454C2×D4C2×C4×D5911yescharacteristic
455C22×C4—21no—
456C22×C4—591no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—1181no—
460SD16—22no—
461SD16—1182no—
462SD16—1182no—
463SD16—1182no—
464M16—42no—
465M16—2362no—
466M16—2362no—
467M16—2362no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—591no—
472C2×C8—591no—
473C2×C8—591no—
474C2×C8—591no—
475C2×C8—1182no—
476C2×C8—1182no—
477C2×C8—1182no—
478C2×C8—1182no—
479C2×C8—1181no—
480C2×C8—1181no—
481C2×C8—1181no—
482C2×C8—2362no—
483C2×C8—2362no—
484C2×C8—2362no—
485C2×C8C2×C4×D5911yescharacteristic
486C2×C8C2×C4×D5911yescharacteristic
487C2×C8D4×D5911yescharacteristic
488C2×C8D4×D5911yescharacteristic
489C4⋊C4—21no—
490C4⋊C4—591no—
491C4⋊C4—591no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4C2×C4×D5911yescharacteristic
500C4⋊C4C2×C4×D5911yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—1181no—
503C22⋊C4—1181no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C42—21no—
508C42—591no—
509C42—591no—
510C42—591no—
511C42—1181no—
512C42C23×D5911yescharacteristic
513C23—21no—
514C23—591no—
515C23—1181no—
516Q8—21no—
517Q8—21no—
518Q8—1181no—
519Q8—1181no—
520Q8—2362no—
521Q8—2362no—
522D4—21no—
523D4—21no—
524D4—1181no—
525D4—1181no—
526D4—2362no—
527D4—2362no—
528D4—2362no—
529D4—2362no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—591no—
536C2×C4—591no—
537C2×C4—592no—
538C2×C4—592no—
539C2×C4—591no—
540C2×C4—1182no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4C22×C4×D5911yescharacteristic
549C2×C4C2×D4×D5911yescharacteristic
550C2×C4#41×D5911yescharacteristic
551C8—21no—
552C8—21no—
553C8—21no—
554C8—41no—
555C8—41no—
556C8—41no—
557C8—592no—
558C8—1181no—
559C8—1181no—
560C8—1181no—
561C8—2361no—
562C8—2361no—
563C8—2361no—
564C8#1427373512yescomplemented
565C22—41no—
566C22—41no—
567C22—41no—
568C22—592no—
569C22—1181no—
570C22—1181no—
571C22—2361no—
572C22—2361no—
573C22—2361no—
574C22C4×D4×D5911yescharacteristic
575C4—21no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—592no—
580C4—592no—
581C4—1181no—
582C4—2361no—
583C4—2361no—
584C4—2361no—
585C4#131×D5911yescharacteristic
586C4C2×#1427373511yescharacteristic
587C4AGL(1,ℤ/8)×D5911yescharacteristic
588C4#137×D5911yescharacteristic
589C2—81no—
590C2—1182no—
591C2—4721no—
592C2#2143280511yescharacteristic
593C2#440×D5911yescharacteristic
594C2#444×D5911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1592C2
socleC22×C59C2×C4×D4
prime radical#16837949C22
cosocleC42×C59C24
hypercenter#1592D59
hypocenterC59#17960
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1592

Complements

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

NH ≅ G/Ntype
2. #21433027C2semidirect
4. #21423000C2semidirect
5. #21422998C2semidirect
6. #21422996C2semidirect
7. #21422911C2semidirect
8. SD16⋊(C59⋊C8)C2semidirect
9. #21422907C2semidirect
10. #21422905C2semidirect
11. #21422853C2semidirect
12. #21422549C2semidirect
14. #322×D59C2semidirect
15. C59×#1592C2semidirect
259. C8×C59#131semidirect
266. C59⋊C8#131semidirect
298. #1592D59semidirect
364. C59#17960semidirect
564. C8#14273735semidirect

Maximal subgroups

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

subgroupindex
#214346522
#214330272
#214328052
#214230002
#214229982
#214229962
#214229112
SD16⋊(C59⋊C8)2
#214229072
#214229052
#214228532
#214225492
#214225452
#322×D592
C59×#15922
#1796059