Finite Group Databasebeta

← #28745162

Subgroups — #28745162

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837839C211yesmaximal, characteristic, complemented
2#21435837C211yesmaximal, characteristic, complemented
3D8×C22⋊D59C211yesmaximal, characteristic, complemented
4#21426361C211yesmaximal, characteristic, complemented
5#21426206C211yesmaximal, characteristic, complemented
6#21426182C211yesmaximal, characteristic, complemented
7#21426153C211yesmaximal, characteristic, complemented
8#21426134C211yesmaximal, characteristic, complemented
9#21426023C211yesmaximal, characteristic, complemented
10#21426018C211yesmaximal, characteristic, complemented
11#21426013C211yesmaximal, characteristic, complemented
12#21426008C211yesmaximal, characteristic, complemented
13#21425961C211yesmaximal, characteristic, complemented
14#21425921C211yesmaximal, characteristic, complemented
15C22×SD32×C59C211yesmaximal, characteristic, complemented
16C2×#14273750C2211yescharacteristic, complemented
17C2×D8×D59C2211yescharacteristic
18#16838063—21no—
19D4×C22⋊D59—21no—
20#16837839—24no—
21C2×#14273670C2211yescharacteristic, complemented
22C2×#14273670C2211yescharacteristic, complemented
23C2×D944C2211yescharacteristic, complemented
24C2×#14273668C2211yescharacteristic
25C2×#14273668C2211yescharacteristic, complemented
26C2×#14273666C2211yescharacteristic, complemented
27#16837787C2211yescharacteristic, complemented
28#16837777C2211yescharacteristic
29#16837640C2211yescharacteristic, complemented
30#16837638C2211yescharacteristic
31#16837638C2211yescharacteristic, complemented
32#16837635C2211yescharacteristic
33#16837626—21no—
34#16837595—21no—
35#16837563—21no—
36#16837541—21no—
37#16837536—21no—
38#16837534—21no—
39Q16×Dic59C2211yescharacteristic, complemented
40D8×Dic59C2211yescharacteristic
41C8×C22⋊D59C2211yescharacteristic
42#16837228C2211yescharacteristic, complemented
43Dic59⋊C16C2211yescharacteristic, complemented
44#16837206C2211yescharacteristic, complemented
45#16837206C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic, complemented
47#16837204C2211yescharacteristic, complemented
48#16837203C2211yescharacteristic
49#16837203C2211yescharacteristic, complemented
50C22⋊(C59⋊C16)C2211yescharacteristic, complemented
51(C2×D59)⋊C16C2211yescharacteristic, complemented
52#16837177C2211yescharacteristic, complemented
53C22×Q16×C59C2211yescharacteristic, complemented
54C22×D8×C59C2211yescharacteristic
55C2×SD32×C59—24no—
56C2×SD32×C59C2211yescharacteristic
57C2×SD32×C59C2211yescharacteristic
58C2×SD32×C59C2211yescharacteristic, complemented
59C2×SD32×C59C2211yescharacteristic, complemented
60C22×C16×C59C2211yescharacteristic, complemented
61C2×#11745689—21no—
62C22×C22⋊D59—41no—
63C2×D4×D59—21no—
64#14273750—22no—
65#14273750—41no—
66C2×#11745671—21no—
67C2×#11745671—21no—
68C2×D472C2311yescharacteristic
69C2×D472C2311yescharacteristic
70C2×#11745669—21no—
71C2×#11745669—21no—
72D8×D59—22no—
73D8×D59—41no—
74C2×C8×D59C2311yescharacteristic
75#14273723—21no—
76#14273716—21no—
77#14273716—41no—
78#14273703—21no—
79#14273703—41no—
80#14273701—21no—
81#14273701—21no—
82#14273701—41no—
83#14273699—41no—
84#14273698—21no—
85#14273696—41no—
86#14273695—41no—
87Q8×Dic59—21no—
88D4×Dic59—21no—
89C4×C22⋊D59—41no—
90C4×C22⋊D59D411yescharacteristic
91#14273670—22no—
92#14273670—41no—
93D944—22no—
94#14273668—22no—
95#14273668—41no—
96#14273666—22no—
97C2×C59⋊C16C2311yescharacteristic
98C2×C59⋊C16C2311yescharacteristic
99#14273661C2311yescharacteristic
100#14273661C2311yescharacteristic
101Dic59⋊C8D411yescharacteristic
102#14273651—21no—
103#14273651—21no—
104#14273649—21no—
105#14273649—21no—
106#14273648—21no—
107#14273648—21no—
108C22⋊(C59⋊C8)D411yescharacteristic
109(C2×D59)⋊C8D411yescharacteristic
110C8×Dic59C2311yescharacteristic
111C22×Q8×C59—21no—
112C22×D4×C59—21no—
113C2×Q16×C59—22no—
114C2×Q16×C59—21no—
115C2×Q16×C59—41no—
116C2×Q16×C59C2311yescharacteristic
117C2×Q16×C59C2311yescharacteristic
118C2×D8×C59—22no—
119C2×D8×C59—21no—
120C2×D8×C59—41no—
121C2×D8×C59C2311yescharacteristic
122C2×D8×C59C2311yescharacteristic
123C22×C8×C59C2311yescharacteristic
124SD32×C59—22no—
125SD32×C59—22no—
126SD32×C59—22no—
127SD32×C59D414yescomplemented
128C2×C16×C59—22no—
129C2×C16×C59C2311yescharacteristic
130C2×C16×C59C2311yescharacteristic
131C24×C59—41no—
132C23×D59—41no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—42no—
136C2×Q8×C59—41no—
137C2×Q8×C59—81no—
138C2×D4×C59—21no—
139C2×D4×C59—21no—
140C2×D4×C59—42no—
141C2×D4×C59—41no—
142C2×D4×C59—81no—
143C22×C4×C59—41no—
144C22×C4×C59C2×D411yescharacteristic
145C2×C22⋊D59—41no—
146C2×C22⋊D59—41no—
147C2×C22⋊D59—41no—
148C2×C22⋊D59—41no—
149C2×C22⋊D59—41no—
150C2×C22⋊D59—41no—
151C2×C22⋊D59—84no—
152C2×C22⋊D59D811yescharacteristic
153C22×Dic59—41no—
154#11745689—42no—
155#11745689—81no—
156D4×D59—42no—
157D4×D59—81no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—41no—
161C2×C4×D59—41no—
162C2×C4×D59C2×D411yescharacteristic
163Q16×C59—21no—
164Q16×C59—21no—
165Q16×C59—21no—
166Q16×C59—42no—
167Q16×C59C2×D412yescomplemented
168D8×C59—21no—
169D8×C59—21no—
170D8×C59—21no—
171D8×C59—42no—
172D8×C59C2×D412yes—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59C2×D411yescharacteristic
176C2×C8×C59C2411yescharacteristic
177C22⋊Dic59—41no—
178C22⋊Dic59—41no—
179C22⋊Dic59D811yescharacteristic
180#11745671—42no—
181#11745671—81no—
182#11745669—42no—
183#11745669—81no—
184(C2×D59)⋊C4—41no—
185(C2×D59)⋊C4—41no—
186(C2×D59)⋊C4—41no—
187(C2×D59)⋊C4—41no—
188(C2×D59)⋊C4—41no—
189(C2×D59)⋊C4D811yescharacteristic
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—41no—
193Dic59⋊C4—41no—
194Dic59⋊C4D811yescharacteristic
195C4×Dic59—41no—
196C4×Dic59C2×D411yescharacteristic
197C2×C59⋊C8C2×D411yescharacteristic
198C2×C59⋊C8C2×D411yescharacteristic
199D472—21no—
200D472—21no—
201D472—21no—
202D472—21no—
203C8×D59—21no—
204C8×D59—21no—
205C16×C59—21no—
206C16×C59C2×D412yescomplemented
207C59⋊C16—21no—
208C59⋊C16—21no—
209C23×C59—41no—
210C23×C59—41no—
211C23×C59—82no—
212C23×C59—82no—
213C23×C59—161no—
214C23×C59C2×D811yescharacteristic
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—82no—
218C22×D59—82no—
219C22×D59—161no—
220C22×D59C2×D811yescharacteristic
221Q8×C59—22no—
222Q8×C59—41no—
223Q8×C59—41no—
224Q8×C59—41no—
225Q8×C59—82no—
226D4×C59—22no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—41no—
230D4×C59—82no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—41no—
234C2×C4×C59—41no—
235C2×C4×C59—82no—
236C2×C4×C59C2×D811yescharacteristic
237C2×C4×C59C22×D411yescharacteristic
238C22⋊D59—22no—
239C22⋊D59—82no—
240C22⋊D59—82no—
241C22⋊D59—82no—
242C22⋊D59—82no—
243C22⋊D59—82no—
244C22⋊D59—161no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—82no—
248C2×Dic59C2×D811yescharacteristic
249C2×Dic59C2×D811yescharacteristic
250C2×Dic59C2×D811yescharacteristic
251D236—41no—
252D236—41no—
253D236—41no—
254D236—41no—
255D236—82no—
256C4×D59—21no—
257C4×D59—21no—
258C4×D59—82no—
259C8×C59—21no—
260C8×C59C22×D411yescharacteristic
261C8×C59H(5,𝔽2)11yescharacteristic
262C59⋊C8—21no—
263C59⋊C8—21no—
264#33831—591nomaximal, Sylow
265C22×C59—21no—
266C22×C59—21no—
267C22×C59—42no—
268C22×C59—81no—
269C22×C59—81no—
270C22×C59—81no—
271C22×C59—81no—
272C22×C59—81no—
273C22×C59—164no—
274C22×C59#50912yescomplemented
275C22×C59C22×D811yescharacteristic
276C2×D59—21no—
277C2×D59—21no—
278C2×D59—21no—
279C2×D59—81no—
280C2×D59—81no—
281C2×D59—81no—
282C2×D59—81no—
283C2×D59—81no—
284C2×D59—81no—
285C2×D59—164no—
286C4×C59—21no—
287C4×C59—42no—
288C4×C59—81no—
289C4×C59D4211yescharacteristic
290C4×C59#57611yescharacteristic
291Dic59—21no—
292Dic59—21no—
293Dic59—21no—
294Dic59—81no—
295Dic59—81no—
296C2×#509—591no—
297C22×SD32D5911yescharacteristic, complemented
298#3285—591no—
299D4×D8—591no—
300#2250—591no—
301#2220—591no—
302#2215—591no—
303#2210—591no—
304#2207—591no—
305#2187—591no—
306#2186—591no—
307#2185—591no—
308#2184—591no—
309#2175—591no—
310#2168—591no—
311C2×C59—22no—
312C2×C59—82no—
313C2×C59—161no—
314C2×C59D4×D811yescharacteristic
315C2×C59C2×#50911yescharacteristic
316C2×C59#341611yescharacteristic
317D59—41no—
318D59—161no—
319D59—161no—
320C2×#135—591no—
321C22×Q16C2×D5911yescharacteristic, complemented
322C22×D8—591no—
323C22×D8C2×D5911yescharacteristic
324#548—1181no—
325D42—1181no—
326#509—1184no—
327C2×SD32—24no—
328C2×SD32—591no—
329C2×SD32—591no—
330C2×SD32C2×D5911yescharacteristic
331C2×SD32C2×D5911yescharacteristic
332C2×SD32C2×D5911yescharacteristic, complemented
333C2×SD32C2×D5911yescharacteristic, complemented
334C2×D16—591no—
335C2×D16—591no—
336C2×D16—591no—
337C2×M32—591no—
338C22×C16C2×D5911yescharacteristic, complemented
339#495—591no—
340#493—591no—
341#467—591no—
342#466—591no—
343#466—591no—
344#466—591no—
345#464—1181no—
346#459—1181no—
347#452—1181no—
348#448—1181no—
349#447—1181no—
350#447—1181no—
351C4×Q16—591no—
352C4×D8—591no—
353C8×D4—591no—
354#366—591no—
355C4⋊C16—591no—
356#358—591no—
357#358—591no—
358#357—591no—
359#357—591no—
360#357—591no—
361#357—591no—
362C22⋊C16—591no—
363C22⋊C16—591no—
364#346—591no—
365C59#3383111yescharacteristic, complemented, Sylow
366C2×#41—1181no—
367C22×Q8—21no—
368C22×D4—21no—
369C22×D4—1181no—
370C22×D4—2361no—
371#135—1182no—
372#135—2361no—
373C2×Q16—22no—
374C2×Q16—21no—
375C2×Q16—41no—
376C2×Q16C22×D5911yescharacteristic
377C2×Q16C22×D5911yescharacteristic
378C2×SD16—1181no—
379C2×SD16—1181no—
380C2×D8—22no—
381C2×D8—21no—
382C2×D8—41no—
383C2×D8—591no—
384C2×D8—591no—
385C2×D8—1181no—
386C2×D8—1181no—
387C2×D8—1182no—
388C2×D8—2361no—
389C2×D8C22×D5911yescharacteristic
390C2×D8C22×D5911yescharacteristic
391C22×C8—591no—
392C22×C8C22×D5911yescharacteristic
393#127—1181no—
394#124—1181no—
395#124—2361no—
396C22⋊Q8—1181no—
397C22⋊Q8—2361no—
398#121—1181no—
399#121—1181no—
400#121—1181no—
401#121—2361no—
402#121—2361no—
403#120—2361no—
404#120—2361no—
405C4×Q8—1181no—
406C4×D4—591no—
407C4×D4—1181no—
408C4×D4—2361no—
409SD32—22no—
410SD32—22no—
411SD32—22no—
412SD32—1182no—
413SD32—2361no—
414SD32C22⋊D5914yescomplemented
415D16—1182no—
416D16—1182no—
417D16—2361no—
418M32—1182no—
419C2×C16—22no—
420C2×C16—591no—
421C2×C16—591no—
422C2×C16C22×D5911yescharacteristic
423C2×C16C22×D5911yescharacteristic
424#107—591no—
425#107—591no—
426C4⋊C8—591no—
427#103—1181no—
428#103—1181no—
429#102—1181no—
430#102—1181no—
431#102—1181no—
432#102—1181no—
433C22⋊C8—591no—
434C22⋊C8—591no—
435C4×C8—591no—
436C24—41no—
437C24—2361no—
438#41—4721no—
439#41—2362no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8—81no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—42no—
448C2×D4—41no—
449C2×D4—81no—
450C2×D4—591no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—4721no—
454C2×D4—4724no—
455C2×D4—2361no—
456C2×D4—2361no—
457C2×D4—2361no—
458C2×D4—2361no—
459C2×D4—2362no—
460C2×D4—2361no—
461C2×D4—2361no—
462C2×D4—2361no—
463C22×C4—41no—
464C22×C4—591no—
465C22×C4—2361no—
466C22×C4—2361no—
467C22×C4D4×D5911yescharacteristic
468Q16—21no—
469Q16—21no—
470Q16—21no—
471Q16—42no—
472Q16C2×C22⋊D5912yescomplemented
473SD16—4721no—
474SD16—2362no—
475D8—21no—
476D8—21no—
477D8—21no—
478D8—42no—
479D8—1181no—
480D8—1181no—
481D8—1181no—
482D8—1181no—
483D8—4721no—
484D8—2362no—
485D8C2×C22⋊D5912yes—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—591no—
489C2×C8—591no—
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8D4×D5911yescharacteristic
493C2×C8C23×D5911yescharacteristic
494C4⋊C4—591no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—2361no—
498C4⋊C4—2361no—
499C22⋊C4—591no—
500C22⋊C4—591no—
501C22⋊C4—2361no—
502C22⋊C4—2361no—
503C22⋊C4—2361no—
504C22⋊C4—2361no—
505C22⋊C4—2361no—
506C22⋊C4—2361no—
507C22⋊C4—2361no—
508C42—591no—
509C42—2361no—
510C16—21no—
511C16—1181no—
512C16—1181no—
513C16C2×C22⋊D5912yescomplemented
514C23—41no—
515C23—41no—
516C23—82no—
517C23—82no—
518C23—161no—
519C23—591no—
520C23—9441no—
521C23—4722no—
522C23—4722no—
523C23—2361no—
524C23—2361no—
525C23D8×D5911yescharacteristic
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—82no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—82no—
536D4—1182no—
537D4—9441no—
538D4—4722no—
539D4—4722no—
540D4—4722no—
541D4—4722no—
542D4—4722no—
543D4—4722no—
544D4—2361no—
545D4—2361no—
546D4—2361no—
547D4—2361no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—82no—
553C2×C4—591no—
554C2×C4—591no—
555C2×C4—591no—
556C2×C4—1181no—
557C2×C4—1181no—
558C2×C4—4722no—
559C2×C4—4722no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4D8×D5911yescharacteristic
563C2×C4C2×D4×D5911yescharacteristic
564C8—21no—
565C8—1181no—
566C8—1181no—
567C8C22×C22⋊D5911yescharacteristic
568C8#1427377211yescharacteristic
569C22—21no—
570C22—21no—
571C22—42no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—164no—
578C22—1181no—
579C22—1181no—
580C22—1181no—
581C22—9444no—
582C22—4721no—
583C22—4721no—
584C22—4721no—
585C22—4721no—
586C22—4721no—
587C22—4721no—
588C22#1683783912yescomplemented
589C22C2×D8×D5911yescharacteristic
590C4—21no—
591C4—42no—
592C4—81no—
593C4—1181no—
594C4—1181no—
595C4—1181no—
596C4—4721no—
597C4—4721no—
598C4D4×C22⋊D5911yescharacteristic
599C4#1683820311yescharacteristic
600C2—22no—
601C2—82no—
602C2—161no—
603C2—9441no—
604C2—9441no—
605C2—2361no—
606C2D8×C22⋊D5911yescharacteristic
607C2C2×#1683783911yescharacteristic
608C2#2143723811yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC22×SD32×C59C2
socleC22×C59C22×D8
hypercenterC22×SD32D59
hypocenterC59#33831
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×SD32

Complements

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

NH ≅ G/Ntype
1. C2×#16837839C2semidirect
2. #21435837C2semidirect
3. D8×C22⋊D59C2semidirect
4. #21426361C2semidirect
5. #21426206C2semidirect
6. #21426182C2semidirect
7. #21426153C2semidirect
8. #21426134C2semidirect
9. #21426023C2semidirect
10. #21426018C2semidirect
11. #21426013C2semidirect
12. #21426008C2semidirect
13. #21425961C2semidirect
14. #21425921C2semidirect
15. C22×SD32×C59C2semidirect
16. C2×#14273750C22semidirect
21. C2×#14273670C22semidirect
22. C2×#14273670C22semidirect
23. C2×D944C22semidirect
25. C2×#14273668C22semidirect
26. C2×#14273666C22semidirect
27. #16837787C22semidirect
29. #16837640C22semidirect
31. #16837638C22semidirect
39. Q16×Dic59C22semidirect
42. #16837228C22semidirect
43. Dic59⋊C16C22semidirect
44. #16837206C22semidirect
45. #16837206C22semidirect
46. #16837204C22semidirect
47. #16837204C22semidirect
49. #16837203C22semidirect
50. C22⋊(C59⋊C16)C22semidirect
51. (C2×D59)⋊C16C22semidirect
52. #16837177C22semidirect
53. C22×Q16×C59C22semidirect
58. C2×SD32×C59C22semidirect
59. C2×SD32×C59C22semidirect
60. C22×C16×C59C22semidirect
127. SD32×C59D4semidirect
167. Q16×C59C2×D4semidirect
206. C16×C59C2×D4semidirect
274. C22×C59#509semidirect
297. C22×SD32D59semidirect
321. C22×Q16C2×D59semidirect
332. C2×SD32C2×D59semidirect
333. C2×SD32C2×D59semidirect
338. C22×C16C2×D59semidirect
365. C59#33831semidirect
414. SD32C22⋊D59semidirect
472. Q16C2×C22⋊D59semidirect
513. C16C2×C22⋊D59semidirect
588. C22#16837839semidirect

Maximal subgroups

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

subgroupindex
C2×#168378392
#214358372
D8×C22⋊D592
#214263612
#214262062
#214261822
#214261532
#214261342
#214260232
#214260182
#214260132
#214260082
#214259612
#214259212
C22×SD32×C592
#3383159