Finite Group Databasebeta

← #28745536

Subgroups — #28745536

Order 15104.

568 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436465C211yesmaximal, characteristic, complemented
2#21435783C211yesmaximal, characteristic
3#21426328C211yesmaximal, characteristic, complemented
4#21426294C211yesmaximal, characteristic, complemented
5#21426288C211yesmaximal, characteristic, complemented
6#21426254C211yesmaximal, characteristic, complemented
7#21426192C211yesmaximal, characteristic, complemented
8#21426187C211yesmaximal, characteristic, complemented
9#21426159C211yesmaximal, characteristic, complemented
10#21426038C211yesmaximal, characteristic, complemented
11#21426027C211yesmaximal, characteristic, complemented
12C4×#14273672C211yesmaximal, characteristic, complemented
13#21425804C211yesmaximal, characteristic, complemented
14C2×#16837204C211yesmaximal, characteristic, complemented
15D4×C16×C59C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×D472C2211yescharacteristic
18#16838092—21no—
19#16838048—21no—
20C2×Dic472C2211yescharacteristic, complemented
21C2×#14273672C2211yescharacteristic
22C2×#14273672C2211yescharacteristic
23#16837804C2211yescharacteristic, complemented
24#16837786C2211yescharacteristic
25#16837744—21no—
26#16837733—21no—
27#16837710—21no—
28#16837643C2211yescharacteristic
29#16837643C2211yescharacteristic
30#16837643C2211yescharacteristic, complemented
31#16837638C2211yescharacteristic
32#16837618—21no—
33#16837555—21no—
34#16837549—21no—
35C4×Dic236C2211yescharacteristic, complemented
36C4×D472C2211yescharacteristic
37#16837440C2211yescharacteristic, complemented
38C2×#14273661C2211yescharacteristic
39#16837230C2211yescharacteristic, complemented
40#16837230C2211yescharacteristic, complemented
41#16837228C2211yescharacteristic, complemented
42#16837207C2211yescharacteristic, complemented
43#16837207C2211yescharacteristic, complemented
44#16837207C2211yescharacteristic, complemented
45#16837207C2211yescharacteristic, complemented
46#16837204C2211yescharacteristic
47#16837204C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837204C2211yescharacteristic
51#16837204C2211yescharacteristic
52C22×C16×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic, complemented
54C8×D4×C59C2211yescharacteristic
55C59×C4⋊C16C2211yescharacteristic, complemented
56C59×C22⋊C16C2211yescharacteristic
57C59×C22⋊C16C2211yescharacteristic, complemented
58C4×C16×C59C2211yescharacteristic, complemented
59C2×#11745685—21no—
60C22×D236—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236C2311yescharacteristic
64C2×Dic236C2311yescharacteristic
65C2×#11745660—21no—
66C2×#11745660—21no—
67C2×D472—22no—
68C2×D472—21no—
69C2×D472—41no—
70C2×D472C2311yescharacteristic
71C2×D472C2311yescharacteristic
72#14273720—41no—
73#14273719—21no—
74#14273717—41no—
75#14273715—21no—
76#14273712—41no—
77#14273708—41no—
78#14273706—21no—
79#14273706—21no—
80#14273706—21no—
81#14273701—21no—
82#14273700—41no—
83#14273695—41no—
84C4×Dic118—21no—
85C4×D236—21no—
86#14273686—21no—
87#14273683—41no—
88C2×C4⋊Dic59—21no—
89C22⋊C4×D59—41no—
90Dic472—41no—
91#14273672—22no—
92#14273672—41no—
93#14273661—21no—
94#14273661C2311yescharacteristic
95#14273661C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273659—21no—
99#14273659—21no—
100#14273652—21no—
101#14273652—21no—
102#14273652—21no—
103#14273652—21no—
104#14273649—21no—
105#14273649—21no—
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C4×D4×C59D411yescharacteristic
109C2×C16×C59—41no—
110C2×C16×C59—41no—
111C2×C16×C59C2311yescharacteristic
112C2×C16×C59C2311yescharacteristic
113C2×C16×C59C2311yescharacteristic
114C2×C16×C59C2311yescharacteristic
115C4⋊C8×C59D411yescharacteristic
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C23×D59—41no—
120C2×D4×C59D811yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C2×C22⋊D59—41no—
124C2×C22⋊D59—41no—
125C22×Dic59—41no—
126#11745685—42no—
127#11745685—81no—
128C2×D236—21no—
129C2×D236—21no—
130C2×D236—42no—
131C2×D236—41no—
132C2×D236—81no—
133C2×C4×D59—41no—
134C2×C4×D59—41no—
135C2×Dic118—21no—
136C2×Dic118—21no—
137C2×C8×C59—21no—
138C2×C8×C59—21no—
139C2×C8×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2×D411yescharacteristic
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2411yescharacteristic
145C4⋊C4×C59D811yescharacteristic
146C22⋊C4×C59D811yescharacteristic
147C22⋊C4×C59D811yescharacteristic
148C42×C59C2×D411yescharacteristic
149C22⋊Dic59—41no—
150C22⋊Dic59—41no—
151C22⋊Dic59—41no—
152C22⋊Dic59—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156(C2×D59)⋊C4—41no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—21no—
160C4⋊Dic59—21no—
161C4⋊Dic59—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168C4×Dic59—41no—
169C4×Dic59—41no—
170Dic236—21no—
171Dic236—21no—
172Dic236—21no—
173Dic236—21no—
174D472—21no—
175D472—21no—
176D472—21no—
177D472—42no—
178D472#4112yes—
179#11745660—42no—
180#11745660—81no—
181C16×C59—21no—
182C16×C59—21no—
183C16×C59—21no—
184C16×C59—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—82no—
190C22×D59—82no—
191C22×D59—161no—
192D4×C59—22no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—21no—
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C22×D411yescharacteristic
202C22⋊D59—82no—
203C22⋊D59—161no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—41no—
207C2×Dic59—41no—
208C2×Dic59—41no—
209C2×Dic59—41no—
210C2×Dic59—82no—
211D236—22no—
212D236—41no—
213D236—41no—
214D236—41no—
215D236—82no—
216C4×D59—82no—
217C4×D59—161no—
218Dic118—41no—
219Dic118—41no—
220Dic118—41no—
221Dic118—41no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227#33863—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59#50812yes—
234C22×C59C22×D811yescharacteristic
235C2×D59—42no—
236C2×D59—81no—
237C2×D59—81no—
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—81no—
241C2×D59—164no—
242C4×C59—21no—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—21no—
246C4×C59#54611yescharacteristic
247C4×C59#57711yescharacteristic
248Dic59—81no—
249Dic59—81no—
250Dic59—81no—
251Dic59—81no—
252Dic59—81no—
253Dic59—81no—
254#3336—591no—
255#3280—591no—
256#2242—591no—
257#2235—591no—
258#2234—591no—
259#2228—591no—
260#2217—591no—
261#2216—591no—
262#2211—591no—
263#2189—591no—
264#2187—591no—
265C4×SD32—591no—
266D4×C16D5911yescharacteristic, complemented
267#2138—591no—
268C2×#357—591no—
269C2×C59—22no—
270C2×C59—41no—
271C2×C59#329411yescharacteristic
272C2×C59C2×#50811yescharacteristic
273C2×C59#341611yescharacteristic
274D59—82no—
275D59—161no—
276C2×#135—591no—
277C22×D8—591no—
278#552—1181no—
279#546—1181no—
280C2×Q32—591no—
281C2×SD32—591no—
282C2×SD32—591no—
283C22×C16C2×D5911yescharacteristic
284C22×C16C2×D5911yescharacteristic, complemented
285#499—591no—
286#495—591no—
287#485—1181no—
288#483—1181no—
289#479—1181no—
290#467—591no—
291#467—591no—
292#467—591no—
293#466—591no—
294#463—1181no—
295#450—1181no—
296#449—1181no—
297C4×Q16—591no—
298C4×D8—591no—
299C8×D4C2×D5911yescharacteristic
300#427—591no—
301C2×#107—591no—
302#367—591no—
303#367—591no—
304#366—591no—
305C4⋊C16C2×D5911yescharacteristic, complemented
306#358—591no—
307#358—591no—
308#358—591no—
309#358—591no—
310#357—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315#357—591no—
316C22⋊C16C2×D5911yescharacteristic
317C22⋊C16C2×D5911yescharacteristic, complemented
318C4×C16C2×D5911yescharacteristic, complemented
319C59#3386311yescharacteristic, complemented, Sylow
320C2×#41—1181no—
321C22×D4—1181no—
322#135—1182no—
323#135—2361no—
324C2×Q16—591no—
325C2×Q16—591no—
326C2×SD16—1181no—
327C2×SD16—1181no—
328C2×D8—591no—
329C2×D8—591no—
330C2×D8—1182no—
331C2×D8—1181no—
332C2×D8—2361no—
333C22×C8C22×D5911yescharacteristic
334C22×C8C22×D5911yescharacteristic
335#126—2361no—
336#125—1181no—
337#125—2361no—
338#124—1181no—
339#123—2361no—
340#123—2361no—
341C22⋊Q8—1181no—
342C22⋊Q8—1181no—
343C22⋊Q8—1181no—
344#121—1181no—
345#121—2361no—
346#120—2361no—
347C4×Q8—1181no—
348C4×D4—1181no—
349C4×D4D23611yescharacteristic
350#117—1181no—
351#117—2361no—
352C2×C4⋊C4—1181no—
353C2×C22⋊C4—2361no—
354Q32—2361no—
355SD32—1182no—
356SD32—2361no—
357C2×C16—41no—
358C2×C16—41no—
359C2×C16C22×D5911yescharacteristic
360C2×C16C22×D5911yescharacteristic
361C2×C16C22×D5911yescharacteristic
362C2×C16C22×D5911yescharacteristic
363#107—591no—
364#107—591no—
365#107—591no—
366#107—591no—
367#107—1181no—
368#106—1181no—
369#106—1181no—
370C4⋊C8D23611yescharacteristic
371#103—1181no—
372#103—1181no—
373#103—1181no—
374#103—1181no—
375#102—1181no—
376#102—1181no—
377C22⋊C8D23611yescharacteristic
378C22⋊C8D23611yescharacteristic
379C4×C8C22×D5911yescharacteristic
380C24—2361no—
381#41—4721no—
382#41—2362no—
383C2×Q8—1181no—
384C2×Q8—1181no—
385C2×D4—1181no—
386C2×D4—1181no—
387C2×D4—4721no—
388C2×D4—2362no—
389C2×D4—2361no—
390C2×D4—2361no—
391C2×D4—2361no—
392C2×D4D47211yescharacteristic
393C22×C4—2361no—
394C22×C4—2361no—
395C22×C4—2361no—
396C22×C4C2×D23611yescharacteristic
397C22×C4C2×D23611yescharacteristic
398Q16—1181no—
399Q16—1181no—
400Q16—1181no—
401Q16—1181no—
402SD16—4721no—
403SD16—2362no—
404D8—592no—
405D8—1181no—
406D8—1181no—
407D8—1181no—
408D8—2362no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8C2×D23611yescharacteristic
414C2×C8C2×D23611yescharacteristic
415C2×C8C2×D23611yescharacteristic
416C2×C8C23×D5911yescharacteristic
417C4⋊C4—1181no—
418C4⋊C4—1181no—
419C4⋊C4—1181no—
420C4⋊C4—1181no—
421C4⋊C4—2361no—
422C4⋊C4—2361no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4—2361no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4D47211yescharacteristic
429C22⋊C4—2361no—
430C22⋊C4—2361no—
431C22⋊C4—2361no—
432C22⋊C4—2361no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4D47211yescharacteristic
438C22⋊C4D47211yescharacteristic
439C42—2361no—
440C42—2361no—
441C42C2×D23611yescharacteristic
442C16—21no—
443C16—21no—
444C16—21no—
445C16—21no—
446C23—9441no—
447C23—4722no—
448C23—4722no—
449C23—2361no—
450C23—2361no—
451C23C2×D47211yescharacteristic
452C23C2×D47211yescharacteristic
453Q8—2361no—
454Q8—2361no—
455Q8—2361no—
456Q8—2361no—
457D4—22no—
458D4—1182no—
459D4—9441no—
460D4—4722no—
461D4—4722no—
462D4—2361no—
463D4—2361no—
464D4—2361no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—9441no—
470C2×C4—4722no—
471C2×C4—4722no—
472C2×C4—2361no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4—2361no—
478C2×C4C2×D47211yescharacteristic
479C2×C4C2×D47211yescharacteristic
480C2×C4C2×D47211yescharacteristic
481C2×C4C2×D47211yescharacteristic
482C2×C4C22×D23611yescharacteristic
483C8—21no—
484C8—21no—
485C8—21no—
486C8#41×D5911yescharacteristic
487C8#1427377211yescharacteristic
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—9444no—
494C22—4721no—
495C22—4721no—
496C22—4721no—
497C22—4721no—
498C22—4721no—
499C22—2362no—
500C22#1683783412yes—
501C22C22×D47211yescharacteristic
502C4—21no—
503C4—21no—
504C4—21no—
505C4—21no—
506C4—4721no—
507C4—4721no—
508C4—4721no—
509C4—4721no—
510C4—4721no—
511C4—4721no—
512C4#1683804711yescharacteristic
513C4#1683821411yescharacteristic
514C2—22no—
515C2—41no—
516C2—9441no—
517C2—4722no—
518C2#2143591011yescharacteristic
519C2C2×#1683783411yescharacteristic
520C2#2143724411yescharacteristic
subgroupG/H
centerC22C22×D472
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingD4×C16×C59C2
socleC22×C59C22×D8
prime radical#21435783C2
hypercenterD4×C16D59
hypocenterC59#33863
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16

Complements

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

NH ≅ G/Ntype
1. #21436465C2semidirect
3. #21426328C2semidirect
4. #21426294C2semidirect
5. #21426288C2semidirect
6. #21426254C2semidirect
7. #21426192C2semidirect
8. #21426187C2semidirect
9. #21426159C2semidirect
10. #21426038C2semidirect
11. #21426027C2semidirect
12. C4×#14273672C2semidirect
13. #21425804C2semidirect
14. C2×#16837204C2semidirect
15. D4×C16×C59C2semidirect
20. C2×Dic472C22semidirect
23. #16837804C22semidirect
30. #16837643C22semidirect
35. C4×Dic236C22semidirect
37. #16837440C22semidirect
39. #16837230C22semidirect
40. #16837230C22semidirect
41. #16837228C22semidirect
42. #16837207C22semidirect
43. #16837207C22semidirect
44. #16837207C22semidirect
45. #16837207C22semidirect
53. C22×C16×C59C22semidirect
55. C59×C4⋊C16C22semidirect
57. C59×C22⋊C16C22semidirect
58. C4×C16×C59C22semidirect
266. D4×C16D59semidirect
284. C22×C16C2×D59semidirect
305. C4⋊C16C2×D59semidirect
317. C22⋊C16C2×D59semidirect
318. C4×C16C2×D59semidirect
319. C59#33863semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214357832
#214263282
#214262942
#214262882
#214262542
#214261922
#214261872
#214261592
#214260382
#214260272
C4×#142736722
#214258042
C2×#168372042
D4×C16×C592
#3386359