Finite Group Databasebeta

← #28745825

Subgroups — #28745825

Order 15104.

552 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436465C211yesmaximal, characteristic, complemented
2#21426337C211yesmaximal, characteristic, complemented
3#21426284C211yesmaximal, characteristic, complemented
4#21426279C211yesmaximal, characteristic, complemented
5#21426260C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic, complemented
7#21426190C211yesmaximal, characteristic, complemented
8#21426133C211yesmaximal, characteristic, complemented
9#21426036C211yesmaximal, characteristic, complemented
10#21426036C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14#21425803C211yesmaximal, characteristic, complemented
15C59×#3287C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837744—21no—
24#16837733—21no—
25#16837710—21no—
26#16837643C2211yescharacteristic
27#16837643C2211yescharacteristic
28C4×Dic236C2211yescharacteristic
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837203C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C2×#135×C59C2211yescharacteristic
50C59×#550—21no—
51C59×#493C2211yescharacteristic
52C59×#466C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#460—21no—
55C59×#449—21no—
56C59×#449—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273671—41no—
72#14273671—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273659—21no—
84#14273659—21no—
85#14273652—21no—
86#14273652—21no—
87#14273652—21no—
88C2×#41×C59—21no—
89C2×#41×C59—21no—
90#135×C59—22no—
91#135×C59—22no—
92#135×C59—41no—
93#135×C59—41no—
94C2×Q16×C59C2311yescharacteristic
95C2×SD16×C59—21no—
96C2×SD16×C59—21no—
97C2×D8×C59—21no—
98C2×D8×C59C2311yescharacteristic
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#127×C59—21no—
103#127×C59—41no—
104#121×C59—21no—
105#121×C59—21no—
106#121×C59—41no—
107#121×C59—41no—
108C4×Q8×C59—21no—
109C4×D4×C59—41no—
110C4×D4×C59D411yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#102×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119#41×C59—42no—
120#41×C59—42no—
121#41×C59—81no—
122#41×C59—81no—
123C2×Q8×C59—21no—
124C2×D4×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—41no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59—41no—
130C2×D4×C59—41no—
131C2×D4×C59—41no—
132C2×D4×C59D811yescharacteristic
133C22×C4×C59—41no—
134C22×C4×C59—41no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×Dic59—41no—
138C2×Dic118—21no—
139Q16×C59—21no—
140Q16×C59#4112yes—
141SD16×C59—42no—
142SD16×C59—42no—
143D8×C59—21no—
144D8×C59—21no—
145D8×C59—21no—
146D8×C59—21no—
147D8×C59—81no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59C2×D411yescharacteristic
153C2×C8×C59C2×D411yescharacteristic
154C2×C8×C59C2×D411yescharacteristic
155C2×C8×C59C2411yescharacteristic
156C4⋊C4×C59—21no—
157C4⋊C4×C59—41no—
158C4⋊C4×C59D811yescharacteristic
159C22⋊C4×C59—41no—
160C22⋊C4×C59—41no—
161C22⋊C4×C59D811yescharacteristic
162C22⋊C4×C59D811yescharacteristic
163C42×C59—41no—
164C42×C59C2×D411yescharacteristic
165C22⋊Dic59—41no—
166C22⋊Dic59—41no—
167C22⋊Dic59—41no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—41no—
172Dic59⋊C4—41no—
173Dic59⋊C4—41no—
174Dic59⋊C4—41no—
175Dic59⋊C4—41no—
176Dic59⋊C4—41no—
177C4×Dic59—41no—
178C4×Dic59—41no—
179Dic236—21no—
180Dic236—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59—41no—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×D811yescharacteristic
189Q8×C59—22no—
190Q8×C59—41no—
191D4×C59—41no—
192D4×C59—41no—
193D4×C59—41no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—82no—
197D4×C59—82no—
198D4×C59—82no—
199D4×C59—161no—
200D4×C59—161no—
201D4×C59—161no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—82no—
209C2×C4×C59—82no—
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C2×D811yescharacteristic
214C2×C4×C59C22×D411yescharacteristic
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219C2×Dic59—82no—
220Dic118—41no—
221Dic118—41no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227#33884—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—81no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59—81no—
240C22×C59C22×D811yescharacteristic
241C4×C59—21no—
242C4×C59—21no—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—42no—
246C4×C59—81no—
247C4×C59#54611yescharacteristic
248C4×C59#57711yescharacteristic
249Dic59—81no—
250Dic59—81no—
251Dic59—81no—
252Dic59—81no—
253#3336—591no—
254#3287D5911yescharacteristic
255#2244—591no—
256#2234—591no—
257#2233—591no—
258#2229—591no—
259#2220—591no—
260#2217—591no—
261#2207—591no—
262#2189—591no—
263#2189—591no—
264#2176—591no—
265#2168—591no—
266#2139—591no—
267#2138—591no—
268C2×C59—41no—
269C2×C59—41no—
270C2×C59—161no—
271C2×C59—161no—
272C2×C59#329411yescharacteristic
273C2×C59#341411yescharacteristic
274C2×C59#341611yescharacteristic
275C2×#135C2×D5911yescharacteristic
276C2×#135C2×D5911yescharacteristic
277#552—1181no—
278#550—21no—
279C2×Q32—591no—
280C2×SD32—591no—
281C2×SD32—591no—
282C2×M32—591no—
283C22×C16—591no—
284#499—591no—
285#493C2×D5911yescharacteristic
286#485—1181no—
287#483—1181no—
288#479—1181no—
289#467—591no—
290#467—591no—
291#466C2×D5911yescharacteristic
292#466C2×D5911yescharacteristic
293#460—21no—
294#449—21no—
295#449—21no—
296C4×Q16—591no—
297C4×Q16C2×D5911yescharacteristic
298C8×D4C2×D5911yescharacteristic
299#427—591no—
300C2×#107—591no—
301#367—591no—
302#367—591no—
303#366—591no—
304C4⋊C16—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#358—591no—
309#357—591no—
310#357—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315C22⋊C16—591no—
316C22⋊C16—591no—
317#346—591no—
318C59#3388411yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C2×#41—21no—
321#135—22no—
322#135—22no—
323#135—41no—
324#135—41no—
325C2×Q16—591no—
326C2×Q16C22×D5911yescharacteristic
327C2×SD16—21no—
328C2×SD16—21no—
329C2×D8—21no—
330C2×D8C22×D5911yescharacteristic
331C2×D8C22×D5911yescharacteristic
332C22×C8C22×D5911yescharacteristic
333C22×C8C22×D5911yescharacteristic
334#127—21no—
335#127—41no—
336#126—2361no—
337#125—1181no—
338#125—2361no—
339#123—2361no—
340C22⋊Q8—1181no—
341C22⋊Q8—1181no—
342#121—21no—
343#121—21no—
344#121—41no—
345#121—41no—
346C4×Q8—21no—
347C4×Q8—1181no—
348C4×D4—41no—
349C4×D4C22⋊D5911yescharacteristic
350#117—1181no—
351#117—2361no—
352C2×C4⋊C4—1181no—
353Q32—1182no—
354SD32—2361no—
355SD32—2361no—
356M32—2361no—
357C2×C16—591no—
358C2×C16—591no—
359C2×C16—591no—
360C2×C16—591no—
361C2×C16—2361no—
362#107—591no—
363#107—591no—
364#107—591no—
365#107—1181no—
366#107C22×D5911yescharacteristic
367#106—1181no—
368#106—1181no—
369C4⋊C8C22⋊D5911yescharacteristic
370#103—21no—
371#103—1181no—
372#103—1181no—
373#103—1181no—
374#102—21no—
375#102—21no—
376C22⋊C8C22⋊D5911yescharacteristic
377C22⋊C8C22⋊D5911yescharacteristic
378C4×C8C22×D5911yescharacteristic
379#41—42no—
380#41—42no—
381#41—81no—
382#41—81no—
383C2×Q8—21no—
384C2×Q8—1181no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—41no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4#1174566911yescharacteristic
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—2361no—
397C22×C4C2×C22⋊D5911yescharacteristic
398C22×C4C2×C22⋊D5911yescharacteristic
399Q16—21no—
400Q16—1181no—
401Q16—1181no—
402Q16#1174568712yes—
403SD16—42no—
404SD16—42no—
405D8—21no—
406D8—21no—
407D8—21no—
408D8—21no—
409D8—81no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—21no—
414C2×C8C2×C22⋊D5911yescharacteristic
415C2×C8C2×C22⋊D5911yescharacteristic
416C2×C8C2×C22⋊D5911yescharacteristic
417C2×C8C23×D5911yescharacteristic
418C4⋊C4—21no—
419C4⋊C4—41no—
420C4⋊C4—1181no—
421C4⋊C4—1181no—
422C4⋊C4—1181no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4—2361no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C4⋊C4#1174566911yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—2361no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4#1174566911yescharacteristic
436C22⋊C4#1174566911yescharacteristic
437C42—41no—
438C42—2361no—
439C42—2361no—
440C42C2×C22⋊D5911yescharacteristic
441C16—1181no—
442C16—1181no—
443C16—1181no—
444C16—1181no—
445C23—41no—
446C23—41no—
447C23C2×#1174566911yescharacteristic
448C23C2×#1174566911yescharacteristic
449Q8—22no—
450Q8—41no—
451Q8—2361no—
452Q8—2361no—
453D4—41no—
454D4—41no—
455D4—41no—
456D4—41no—
457D4—41no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—161no—
462D4—161no—
463D4—161no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—4722no—
473C2×C4—2361no—
474C2×C4—2361no—
475C2×C4—2361no—
476C2×C4—2361no—
477C2×C4C2×#1174566911yescharacteristic
478C2×C4C2×#1174566911yescharacteristic
479C2×C4C2×#1174566911yescharacteristic
480C2×C4C2×#1174566911yescharacteristic
481C2×C4C22×C22⋊D5911yescharacteristic
482C8—21no—
483C8—21no—
484C8—21no—
485C8C2×#1174568711yescharacteristic
486C8#1427377211yescharacteristic
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22C22×#1174566911yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—42no—
505C4—81no—
506C4—4721no—
507C4—4721no—
508C4—4721no—
509C4—4721no—
510C4#1683804611yescharacteristic
511C4#1683821311yescharacteristic
512C2—41no—
513C2—41no—
514C2—161no—
515C2—161no—
516C2#2143590911yescharacteristic
517C2#2143723011yescharacteristic
518C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
Fitting, prime radicalC59×#3287C2
socleC22×C59C22×D8
hypercenter#3287D59
hypocenterC59#33884
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3287

Complements

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

NH ≅ G/Ntype
1. #21436465C2semidirect
2. #21426337C2semidirect
3. #21426284C2semidirect
4. #21426279C2semidirect
5. #21426260C2semidirect
6. #21426207C2semidirect
7. #21426190C2semidirect
8. #21426133C2semidirect
9. #21426036C2semidirect
10. #21426036C2semidirect
11. #21425965C2semidirect
12. #21425922C2semidirect
13. #21425808C2semidirect
14. #21425803C2semidirect
318. C59#33884semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263372
#214262842
#214262792
#214262602
#214262072
#214261902
#214261332
#214260362
#214259652
#214259222
#214258082
#214258032
C59×#32872
#3388459