Finite Group Databasebeta

← #28745675

Subgroups — #28745675

Order 15104.

554 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#21436499C211yesmaximal, characteristic, complemented
2#21426341C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426291C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426138C211yesmaximal, characteristic, complemented
9#21426019C211yesmaximal, characteristic, complemented
10#21426019C211yesmaximal, characteristic, complemented
11#21425970C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14#21425803C211yesmaximal, characteristic, complemented
15C59×#3283C211yesmaximal, characteristic, complemented
16#16838098—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837804C2211yescharacteristic
23#16837727—21no—
24#16837721—21no—
25#16837703—21no—
26#16837638C2211yescharacteristic
27#16837638C2211yescharacteristic
28C4×D472C2211yescharacteristic
29#16837440C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837206C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C2×#135×C59C2211yescharacteristic
50C59×#547—21no—
51C59×#494C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#467C2211yescharacteristic
54C59×#461—21no—
55C59×#452—21no—
56C59×#452—21no—
57C4×D8×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×D472C2311yescharacteristic
60#14273722—41no—
61#14273719—21no—
62#14273709—41no—
63#14273701—21no—
64#14273701—21no—
65#14273699—41no—
66C4×D236—21no—
67(C22⋊D59)⋊C4—41no—
68#14273686—21no—
69C2×C4⋊Dic59—21no—
70#14273670—41no—
71#14273670—41no—
72#14273668—22no—
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#14273649—21no—
86#14273649—21no—
87#14273649—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×C59—21no—
95C2×Q16×C59C2311yescharacteristic
96C2×Q16×C59C2311yescharacteristic
97C2×SD16×C59—21no—
98C2×SD16×C59—21no—
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102C4⋊Q8×C59—21no—
103#123×C59—41no—
104#123×C59—41no—
105C22⋊Q8×C59—21no—
106C22⋊Q8×C59—21no—
107#121×C59—41no—
108C4×D4×C59—21no—
109C4×D4×C59D411yescharacteristic
110C2×C4⋊C4×C59—41no—
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×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×Q8×C59—21no—
125C2×D4×C59—21no—
126C2×D4×C59—41no—
127C2×D4×C59—41no—
128C2×D4×C59D811yescharacteristic
129C22×C4×C59—41no—
130C22×C4×C59—41no—
131C22×C4×C59—41no—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C2×C22⋊D59—41no—
135C2×C22⋊D59—41no—
136C22×Dic59—41no—
137C2×D236—21no—
138C2×C4×D59—41no—
139Q16×C59—21no—
140Q16×C59—21no—
141Q16×C59—21no—
142Q16×C59—21no—
143Q16×C59—81no—
144SD16×C59—42no—
145SD16×C59—42no—
146D8×C59—21no—
147D8×C59#4112yes—
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×C59—41no—
159C4⋊C4×C59—41no—
160C4⋊C4×C59—41no—
161C4⋊C4×C59D811yescharacteristic
162C22⋊C4×C59—41no—
163C22⋊C4×C59—41no—
164C22⋊C4×C59—41no—
165C22⋊C4×C59D811yescharacteristic
166C22⋊C4×C59D811yescharacteristic
167C42×C59C2×D411yescharacteristic
168C22⋊Dic59—41no—
169(C2×D59)⋊C4—41no—
170(C2×D59)⋊C4—41no—
171(C2×D59)⋊C4—41no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—41no—
176Dic59⋊C4—41no—
177Dic59⋊C4—41no—
178C4×Dic59—41no—
179D472—21no—
180D472—21no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59—41no—
186C23×C59C2×D811yescharacteristic
187C23×C59C2×D811yescharacteristic
188C22×D59—41no—
189Q8×C59—41no—
190Q8×C59—41no—
191Q8×C59—41no—
192Q8×C59—41no—
193D4×C59—22no—
194D4×C59—41no—
195D4×C59—41no—
196D4×C59—82no—
197D4×C59—82no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—41no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—82no—
206C2×C4×C59—82no—
207C2×C4×C59—82no—
208C2×C4×C59C2×D811yescharacteristic
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C22×D411yescharacteristic
213C22⋊D59—82no—
214C22⋊D59—161no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—161no—
219D236—41no—
220D236—41no—
221C4×D59—161no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59—21no—
225C8×C59C2×#4111yescharacteristic
226C8×C59H(5,𝔽2)11yescharacteristic
227#33874—591nomaximal, Sylow
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—42no—
235C22×C59—81no—
236C22×C59—81no—
237C22×C59C22×D811yescharacteristic
238C2×D59—81no—
239C2×D59—81no—
240C2×D59—81no—
241C4×C59—21no—
242C4×C59—21no—
243C4×C59—21no—
244C4×C59—21no—
245C4×C59—81no—
246C4×C59—81no—
247C4×C59—81no—
248C4×C59#54611yescharacteristic
249C4×C59#57711yescharacteristic
250Dic59—81no—
251Dic59—81no—
252Dic59—81no—
253#3339—591no—
254#3283D5911yescharacteristic, complemented
255#2245—591no—
256#2237—591no—
257#2235—591no—
258#2228—591no—
259#2220—591no—
260#2217—591no—
261#2208—591no—
262#2186—591no—
263#2186—591no—
264#2177—591no—
265#2168—591no—
266#2139—591no—
267#2138—591no—
268C2×C59—41no—
269C2×C59—41no—
270C2×C59—82no—
271C2×C59#329411yescharacteristic
272C2×C59#341411yescharacteristic
273C2×C59#341611yescharacteristic
274D59—161no—
275C2×#135C2×D5911yescharacteristic
276C2×#135C2×D5911yescharacteristic
277#553—1181no—
278#547—21no—
279C2×SD32—591no—
280C2×SD32—591no—
281C2×D16—591no—
282C2×M32—591no—
283C22×C16—591no—
284#499—591no—
285#494C2×D5911yescharacteristic
286#482—1181no—
287#481—1181no—
288#478—1181no—
289#467C2×D5911yescharacteristic
290#467C2×D5911yescharacteristic
291#466—591no—
292#466—591no—
293#461—21no—
294#452—21no—
295#452—21no—
296C4×D8—591no—
297C4×D8C2×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#358—591no—
310#358—591no—
311#357—591no—
312#357—591no—
313#357—591no—
314#357—591no—
315C22⋊C16—591no—
316C22⋊C16—591no—
317#346—591no—
318C59#3387411yescharacteristic, complemented, Sylow
319C2×#41—21no—
320C2×#41—21no—
321#135—22no—
322#135—22no—
323#135—41no—
324#135—41no—
325C2×Q16—21no—
326C2×Q16C22×D5911yescharacteristic
327C2×Q16C22×D5911yescharacteristic
328C2×SD16—21no—
329C2×SD16—21no—
330C2×D8—591no—
331C2×D8C22×D5911yescharacteristic
332C22×C8C22×D5911yescharacteristic
333C22×C8C22×D5911yescharacteristic
334C4⋊Q8—21no—
335#126—2361no—
336#125—1181no—
337#123—41no—
338#123—41no—
339#123—2361no—
340C22⋊Q8—21no—
341C22⋊Q8—21no—
342#121—41no—
343#121—1181no—
344#121—1181no—
345#121—2361no—
346C4×D4—21no—
347C4×D4—1181no—
348C4×D4—2361no—
349C4×D4C22⋊D5911yescharacteristic
350#117—1181no—
351C2×C4⋊C4—41no—
352C2×C4⋊C4—1181no—
353SD32—2361no—
354SD32—2361no—
355D16—1182no—
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—21no—
372#102—21no—
373#102—1181no—
374#102—1181no—
375#102—1181no—
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—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—1181no—
389C2×D4—2361no—
390C2×D4—2361no—
391C2×D4#1174566911yescharacteristic
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—2361no—
396C22×C4—2361no—
397C22×C4C2×C22⋊D5911yescharacteristic
398C22×C4C2×C22⋊D5911yescharacteristic
399Q16—21no—
400Q16—21no—
401Q16—21no—
402Q16—21no—
403Q16—81no—
404SD16—42no—
405SD16—42no—
406D8—21no—
407D8—1181no—
408D8—1181no—
409D8#1174568712yes—
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—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—1181no—
424C4⋊C4—1181no—
425C4⋊C4—1181no—
426C4⋊C4—2361no—
427C4⋊C4—2361no—
428C4⋊C4—2361no—
429C4⋊C4#1174566911yescharacteristic
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—2361no—
434C22⋊C4—2361no—
435C22⋊C4—2361no—
436C22⋊C4—2361no—
437C22⋊C4#1174566911yescharacteristic
438C22⋊C4#1174566911yescharacteristic
439C42—2361no—
440C42C2×C22⋊D5911yescharacteristic
441C16—1181no—
442C16—1181no—
443C16—1181no—
444C16—1181no—
445C23—41no—
446C23—2361no—
447C23C2×#1174566911yescharacteristic
448C23C2×#1174566911yescharacteristic
449Q8—41no—
450Q8—41no—
451Q8—41no—
452Q8—41no—
453D4—22no—
454D4—41no—
455D4—41no—
456D4—82no—
457D4—82no—
458D4—9441no—
459D4—4722no—
460D4—2361no—
461D4—2361no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—9441no—
473C2×C4—9441no—
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—42no—
494C22—81no—
495C22—81no—
496C22—4721no—
497C22—4721no—
498C22—4721no—
499C22C22×#1174566911yescharacteristic
500C4—21no—
501C4—21no—
502C4—21no—
503C4—21no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—4721no—
508C4—4721no—
509C4—4721no—
510C4#1683804611yescharacteristic
511C4#1683821311yescharacteristic
512C2—41no—
513C2—41no—
514C2—82no—
515C2—9441no—
516C2#2143590911yescharacteristic
517C2#2143723011yescharacteristic
518C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3283C2
socleC22×C59C22×D8
hypercenter#3283D59
hypocenterC59#33874
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #21436499C2semidirect
2. #21426341C2semidirect
3. #21426301C2semidirect
4. #21426291C2semidirect
5. #21426253C2semidirect
6. #21426211C2semidirect
7. #21426194C2semidirect
8. #21426138C2semidirect
9. #21426019C2semidirect
10. #21426019C2semidirect
11. #21425970C2semidirect
12. #21425922C2semidirect
13. #21425808C2semidirect
14. #21425803C2semidirect
15. C59×#3283C2semidirect
254. #3283D59semidirect
318. C59#33874semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214263412
#214263012
#214262912
#214262532
#214262112
#214261942
#214261382
#214260192
#214259702
#214259222
#214258082
#214258032
C59×#32832
#3387459