Finite Group Databasebeta

← #28745690

Subgroups — #28745690

Order 15104.

580 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436444C211yesmaximal, characteristic, complemented
2#21426332C211yesmaximal, characteristic, complemented
3#21426296C211yesmaximal, characteristic, complemented
4#21426274C211yesmaximal, characteristic, complemented
5#21426248C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426180C211yesmaximal, characteristic, complemented
8#21426148C211yesmaximal, characteristic, complemented
9#21426019C211yesmaximal, characteristic, complemented
10#21426013C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425808C211yesmaximal, characteristic, complemented
14C2×#16837206C211yesmaximal, characteristic, complemented
15C59×#3285C211yesmaximal, characteristic, complemented
16#16838056—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20C2×#14273667C2211yescharacteristic
21C22×C59⋊C16C2211yescharacteristic
22#16837807C2211yescharacteristic
23#16837716—21no—
24#16837716—21no—
25#16837683—21no—
26#16837638C2211yescharacteristic
27#16837638C2211yescharacteristic
28C4×D472C2211yescharacteristic
29C2×#14273661C2211yescharacteristic
30C2×#14273661C2211yescharacteristic
31#16837227C2211yescharacteristic
32#16837227C2211yescharacteristic
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
49C22×Q16×C59C2211yescharacteristic
50C59×#548—21no—
51C59×#495C2211yescharacteristic
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#464—21no—
55C59×#452—21no—
56C59×#448—21no—
57C4×Q16×C59C2211yescharacteristic
58C8×D4×C59C2211yescharacteristic
59C2×#11745687—41no—
60C2×D472C2311yescharacteristic
61#14273727—21no—
62#14273711—41no—
63#14273711—41no—
64#14273707—41no—
65#14273701—21no—
66#14273701—21no—
67C4×D236—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70#14273670—22no—
71#14273670—41no—
72#14273668—41no—
73#14273667—41no—
74C2×C59⋊C16—41no—
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79#14273661—21no—
80#14273661—21no—
81#14273661—21no—
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273649—21no—
86#14273649—21no—
87#14273649—21no—
88C2×#41×C59—21no—
89C22×Q8×C59—21no—
90#135×C59—22no—
91#135×C59—41no—
92C2×Q16×C59—22no—
93C2×Q16×C59—21no—
94C2×Q16×C59—41no—
95C2×Q16×C59C2311yescharacteristic
96C2×Q16×C59C2311yescharacteristic
97C2×SD16×C59—21no—
98C2×SD16×C59—21no—
99C2×D8×C59C2311yescharacteristic
100C22×C8×C59C2311yescharacteristic
101C22×C8×C59C2311yescharacteristic
102#124×C59—21no—
103#124×C59—41no—
104C22⋊Q8×C59—21no—
105C22⋊Q8×C59—41no—
106#121×C59—21no—
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#103×C59—21no—
115#102×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119#41×C59—42no—
120#41×C59—81no—
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123C2×Q8×C59—42no—
124C2×Q8×C59—41no—
125C2×Q8×C59—81no—
126C2×D4×C59—21no—
127C2×D4×C59—41no—
128C2×D4×C59—41no—
129C2×D4×C59D811yescharacteristic
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—
137C22×Dic59—41no—
138#11745687—84no—
139C2×D236—21no—
140C2×C4×D59—41no—
141C2×Dic118—41no—
142Q16×C59—21no—
143Q16×C59—21no—
144Q16×C59—21no—
145Q16×C59—42no—
146Q16×C59#4112yescomplemented
147SD16×C59—42no—
148SD16×C59—81no—
149D8×C59—21no—
150D8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59C2×D411yescharacteristic
156C2×C8×C59C2×D411yescharacteristic
157C2×C8×C59C2×D411yescharacteristic
158C2×C8×C59C2411yescharacteristic
159C4⋊C4×C59—21no—
160C4⋊C4×C59—41no—
161C4⋊C4×C59—41no—
162C4⋊C4×C59D811yescharacteristic
163C22⋊C4×C59—41no—
164C22⋊C4×C59—41no—
165C22⋊C4×C59—41no—
166C22⋊C4×C59—41no—
167C22⋊C4×C59D811yescharacteristic
168C22⋊C4×C59D811yescharacteristic
169C42×C59—41no—
170C42×C59C2×D411yescharacteristic
171(C2×D59)⋊C4—41no—
172(C2×D59)⋊C4—41no—
173(C2×D59)⋊C4—41no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—41no—
178C4⋊Dic59—41no—
179C4⋊Dic59—41no—
180D472—21no—
181D472—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C59⋊C16—21no—
186C23×C59—41no—
187C23×C59C2×D811yescharacteristic
188C23×C59C2×D811yescharacteristic
189C22×D59—41no—
190Q8×C59—22no—
191Q8×C59—41no—
192Q8×C59—41no—
193Q8×C59—41no—
194Q8×C59—82no—
195D4×C59—22no—
196D4×C59—41no—
197D4×C59—41no—
198D4×C59—82no—
199D4×C59—82no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—82no—
208C2×C4×C59—82no—
209C2×C4×C59C2×D811yescharacteristic
210C2×C4×C59C2×D811yescharacteristic
211C2×C4×C59C2×D811yescharacteristic
212C2×C4×C59C2×D811yescharacteristic
213C2×C4×C59C22×D411yescharacteristic
214C22⋊D59—82no—
215C22⋊D59—82no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219C2×Dic59—82no—
220C2×Dic59—82no—
221D236—41no—
222D236—41no—
223C4×D59—82no—
224Dic118—82no—
225C8×C59—21no—
226C8×C59—21no—
227C8×C59—21no—
228C8×C59C2×#4111yescharacteristic
229C8×C59H(5,𝔽2)11yescharacteristic
230#33875—591nomaximal, Sylow
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—81no—
237C22×C59—81no—
238C22×C59—81no—
239C22×C59#50912yes—
240C22×C59C22×D811yescharacteristic
241C2×D59—81no—
242C2×D59—81no—
243C2×D59—81no—
244C4×C59—21no—
245C4×C59—21no—
246C4×C59—21no—
247C4×C59—21no—
248C4×C59—42no—
249C4×C59—81no—
250C4×C59—81no—
251C4×C59#54611yescharacteristic
252C4×C59#57711yescharacteristic
253Dic59—81no—
254Dic59—81no—
255Dic59—81no—
256#3334—591no—
257#3285D5911yescharacteristic, complemented
258#2243—591no—
259#2236—591no—
260#2232—591no—
261#2227—591no—
262#2220—591no—
263#2215—591no—
264#2210—591no—
265#2186—591no—
266#2185—591no—
267#2175—591no—
268#2168—591no—
269#2139—591no—
270C2×#358—591no—
271C2×C59—22no—
272C2×C59—41no—
273C2×C59—161no—
274C2×C59#329411yescharacteristic
275C2×C59C2×#50911yescharacteristic
276C2×C59#341711yescharacteristic
277D59—161no—
278C2×#135C2×D5911yescharacteristic
279C22×Q16C2×D5911yescharacteristic
280#548—21no—
281#547—1181no—
282C2×SD32—591no—
283C2×SD32—591no—
284C2×D16—591no—
285C2×M32—591no—
286C22×C16—591no—
287#500—591no—
288#495C2×D5911yescharacteristic
289#480—1181no—
290#480—1181no—
291#474—1181no—
292#467C2×D5911yescharacteristic
293#466—591no—
294#466—591no—
295#466C2×D5911yescharacteristic
296#464—21no—
297#452—21no—
298#448—21no—
299C4×Q16C2×D5911yescharacteristic
300C4×D8—591no—
301C8×D4C2×D5911yescharacteristic
302C2×#107—591no—
303C2×#107—591no—
304#366—591no—
305#366—591no—
306#366—591no—
307C4⋊C16—591no—
308#358—591no—
309#358—591no—
310#358—591no—
311#358—591no—
312#358—591no—
313#358—591no—
314#357—591no—
315#357—591no—
316#357—591no—
317#357—591no—
318C22⋊C16—591no—
319C22⋊C16—591no—
320#346—591no—
321C59#3387511yescharacteristic, complemented, Sylow
322C2×#41—21no—
323C2×#41—2361no—
324C22×Q8—21no—
325#135—22no—
326#135—41no—
327C2×Q16—22no—
328C2×Q16—21no—
329C2×Q16—41no—
330C2×Q16C22×D5911yescharacteristic
331C2×Q16C22×D5911yescharacteristic
332C2×SD16—21no—
333C2×SD16—21no—
334C2×D8—591no—
335C2×D8C22×D5911yescharacteristic
336C22×C8C22×D5911yescharacteristic
337C22×C8C22×D5911yescharacteristic
338C4⋊Q8—1181no—
339#124—21no—
340#124—41no—
341#123—2361no—
342#123—2361no—
343C22⋊Q8—21no—
344C22⋊Q8—41no—
345C22⋊Q8—2361no—
346#121—21no—
347#121—41no—
348#121—1181no—
349#121—1181no—
350C4×Q8—21no—
351C4×D4—41no—
352C4×D4—1181no—
353C4×D4C22⋊D5911yescharacteristic
354C2×C4⋊C4—1181no—
355C2×C4⋊C4—1181no—
356SD32—1182no—
357SD32—2361no—
358D16—2361no—
359M32—2361no—
360C2×C16—591no—
361C2×C16—591no—
362C2×C16—591no—
363C2×C16—591no—
364C2×C16—2361no—
365#107—591no—
366#107—591no—
367#107—591no—
368#107—1181no—
369#107—1181no—
370#107—1181no—
371#107C22×D5911yescharacteristic
372C4⋊C8C22⋊D5911yescharacteristic
373#103—21no—
374#103—21no—
375#102—21no—
376#102—1181no—
377#102—1181no—
378#102—1181no—
379C22⋊C8C22⋊D5911yescharacteristic
380C22⋊C8C22⋊D5911yescharacteristic
381C4×C8C22×D5911yescharacteristic
382#41—42no—
383#41—81no—
384#41—4724no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—42no—
388C2×Q8—41no—
389C2×Q8—81no—
390C2×Q8—2361no—
391C2×D4—21no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—1181no—
395C2×D4—2361no—
396C2×D4—2361no—
397C2×D4#1174566911yescharacteristic
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—2361no—
401C22×C4—2361no—
402C22×C4—2361no—
403C22×C4C2×C22⋊D5911yescharacteristic
404C22×C4C2×C22⋊D5911yescharacteristic
405Q16—21no—
406Q16—21no—
407Q16—21no—
408Q16—42no—
409Q16#1174568712yescomplemented
410SD16—42no—
411SD16—81no—
412D8—21no—
413D8—21no—
414D8—1181no—
415D8—1181no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8C2×C22⋊D5911yescharacteristic
421C2×C8C2×C22⋊D5911yescharacteristic
422C2×C8C2×C22⋊D5911yescharacteristic
423C2×C8C23×D5911yescharacteristic
424C4⋊C4—21no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—1181no—
428C4⋊C4—1181no—
429C4⋊C4—1181no—
430C4⋊C4—2361no—
431C4⋊C4—2361no—
432C4⋊C4—2361no—
433C4⋊C4#1174566911yescharacteristic
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—2361no—
439C22⋊C4—2361no—
440C22⋊C4—2361no—
441C22⋊C4#1174566911yescharacteristic
442C22⋊C4#1174566911yescharacteristic
443C42—41no—
444C42C2×C22⋊D5911yescharacteristic
445C16—1181no—
446C16—1181no—
447C16—1181no—
448C16—1181no—
449C23—41no—
450C23—2361no—
451C23C2×#1174566911yescharacteristic
452C23C2×#1174566911yescharacteristic
453Q8—22no—
454Q8—41no—
455Q8—41no—
456Q8—41no—
457Q8—82no—
458Q8—4722no—
459D4—22no—
460D4—41no—
461D4—41no—
462D4—82no—
463D4—82no—
464D4—4722no—
465D4—4722no—
466D4—2361no—
467D4—2361no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—4722no—
478C2×C4—4722no—
479C2×C4—4722no—
480C2×C4—2361no—
481C2×C4—2361no—
482C2×C4—2361no—
483C2×C4C2×#1174566911yescharacteristic
484C2×C4C2×#1174566911yescharacteristic
485C2×C4C2×#1174566911yescharacteristic
486C2×C4C2×#1174566911yescharacteristic
487C2×C4C22×C22⋊D5911yescharacteristic
488C8—21no—
489C8—21no—
490C8—21no—
491C8C2×#1174568711yescharacteristic
492C8#1427377211yescharacteristic
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—4721no—
502C22—4721no—
503C22—4721no—
504C22#1683783712yes—
505C22C22×#1174566911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—4721no—
514C4—4721no—
515C4—4721no—
516C4#1683804611yescharacteristic
517C4#1683821311yescharacteristic
518C2—22no—
519C2—41no—
520C2—161no—
521C2—9441no—
522C2#2143590911yescharacteristic
523C2C2×#1683783711yescharacteristic
524C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3285C2
socleC22×C59C22×D8
hypercenter#3285D59
hypocenterC59#33875
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. #21436444C2semidirect
2. #21426332C2semidirect
3. #21426296C2semidirect
4. #21426274C2semidirect
5. #21426248C2semidirect
6. #21426211C2semidirect
7. #21426180C2semidirect
8. #21426148C2semidirect
9. #21426019C2semidirect
10. #21426013C2semidirect
11. #21425958C2semidirect
12. #21425922C2semidirect
13. #21425808C2semidirect
14. C2×#16837206C2semidirect
15. C59×#3285C2semidirect
146. Q16×C59#41semidirect
257. #3285D59semidirect
321. C59#33875semidirect
409. Q16#11745687semidirect

Maximal subgroups

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

subgroupindex
#214364442
#214263322
#214262962
#214262742
#214262482
#214262112
#214261802
#214261482
#214260192
#214260132
#214259582
#214259222
#214258082
C2×#168372062
C59×#32852
#3387559