Finite Group Databasebeta

← #28745078

Subgroups — #28745078

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434746C211yesmaximal, characteristic, complemented
2#21426260C211yesmaximal, characteristic, complemented
3#21426253C211yesmaximal, characteristic, complemented
4#21426240C211yesmaximal, characteristic, complemented
5#21426230C211yesmaximal, characteristic, complemented
6#21426211C211yesmaximal, characteristic, complemented
7#21426197C211yesmaximal, characteristic, complemented
8#21426194C211yesmaximal, characteristic, complemented
9#21426180C211yesmaximal, characteristic, complemented
10#21426148C211yesmaximal, characteristic, complemented
11#21426148C211yesmaximal, characteristic, complemented
12#21426136C211yesmaximal, characteristic, complemented
13#21426131C211yesmaximal, characteristic, complemented
14#21425855C211yesmaximal, characteristic, complemented
15C59×#3175C211yesmaximal, characteristic, complemented
16#16838007—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21C2×#14273667C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23#16837804C2211yescharacteristic
24#16837779C2211yescharacteristic
25#16837767—21no—
26#16837655—21no—
27#16837638C2211yescharacteristic
28#16837638C2211yescharacteristic
29#16837515—21no—
30C4×D472C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36C4⋊(C59⋊C16)C2211yescharacteristic
37C4⋊(C59⋊C16)C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39C4⋊(C59⋊C16)C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837203C2211yescharacteristic
43#16837203C2211yescharacteristic
44#16837202C2211yescharacteristic
45#16837202C2211yescharacteristic
46#16837202C2211yescharacteristic
47#16837202C2211yescharacteristic
48C59×#542—21no—
49C59×#500C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#489—21no—
52C59×#468—21no—
53C59×#467C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C59×#440—21no—
56C4×Q16×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C59×#405C2211yescharacteristic
59C2×D472—21no—
60C2×D472C2311yescharacteristic
61C2×D472C2311yescharacteristic
62#14273724—21no—
63#14273719—21no—
64#14273712—41no—
65#14273712—41no—
66#14273701—21no—
67#14273701—21no—
68#14273700—41no—
69#14273700—41no—
70C4×D236—21no—
71C4×D236—21no—
72#14273670—41no—
73#14273670—41no—
74#14273668—41no—
75#14273668—41no—
76#14273667—41no—
77C2×C59⋊C16—41no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273659—21no—
85#14273649—21no—
86#14273649—21no—
87#14273649—21no—
88#14273649—21no—
89C2×Q16×C59C2311yescharacteristic
90C2×SD16×C59—21no—
91C2×D8×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C4⋊Q8×C59—21no—
95#126×C59—41no—
96#124×C59—21no—
97#124×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—41no—
101#121×C59—21no—
102C4×Q8×C59—21no—
103C4×D4×C59—21no—
104#117×C59D411yescharacteristic
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C2×Q8×C59—21no—
117C2×Q8×C59—41no—
118C2×D4×C59—21no—
119C2×D4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59C2×D411yescharacteristic
122C2×C22⋊D59—41no—
123C2×C22⋊D59—41no—
124C2×D236—21no—
125C2×D236—21no—
126C2×D236—41no—
127C2×D236—41no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130Q16×C59—21no—
131Q16×C59—21no—
132SD16×C59—42no—
133D8×C59—21no—
134D8×C59—21no—
135M16×C59—41no—
136C2×C8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59C2×D411yescharacteristic
139C2×C8×C59C2×D411yescharacteristic
140C2×C8×C59C2×D411yescharacteristic
141C2×C8×C59C2411yescharacteristic
142C4⋊C4×C59—21no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—41no—
145C4⋊C4×C59—41no—
146C4⋊C4×C59—41no—
147C4⋊C4×C59D811yescharacteristic
148C4⋊C4×C59D811yescharacteristic
149C22⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C22⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C22⋊C4×C59D811yescharacteristic
155C22⋊C4×C59D811yescharacteristic
156C42×C59—41no—
157C42×C59C2×D411yescharacteristic
158C42×C59C2×D411yescharacteristic
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163C4⋊Dic59—21no—
164C4⋊Dic59—21no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167D472—21no—
168D472—21no—
169D472—21no—
170D472—21no—
171D472—42no—
172C59⋊C16—21no—
173C59⋊C16—21no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C23×C59—41no—
177C23×C59C2×D811yescharacteristic
178C22×D59—41no—
179C22×D59—41no—
180Q8×C59—41no—
181Q8×C59—41no—
182Q8×C59—161no—
183D4×C59—41no—
184D4×C59—41no—
185D4×C59—161no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—41no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—161no—
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C2×D811yescharacteristic
194C2×C4×C59C2×D811yescharacteristic
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C22×D411yescharacteristic
198C22⋊D59—161no—
199C22⋊D59—161no—
200C2×Dic59—41no—
201C2×Dic59—41no—
202D236—41no—
203D236—41no—
204D236—41no—
205D236—41no—
206D236—161no—
207D236—161no—
208C4×D59—161no—
209C4×D59—161no—
210C8×C59—21no—
211C8×C59—21no—
212C8×C59—21no—
213C8×C59H(5,𝔽2)11yescharacteristic
214C8×C592−1+411yescharacteristic
215#33823—591nomaximal, Sylow
216C22×C59—21no—
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—81no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59C22×D811yescharacteristic
223C2×D59—81no—
224C2×D59—81no—
225C2×D59—81no—
226C2×D59—81no—
227C2×D59—81no—
228C2×D59—81no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—21no—
232C4×C59—21no—
233C4×C59—21no—
234C4×C59—81no—
235C4×C59—81no—
236C4×C59—81no—
237C4×C59#53711yescharacteristic
238C4×C59#57511yescharacteristic
239Dic59—81no—
240Dic59—81no—
241#3176—591no—
242#3175D5911yescharacteristic, complemented
243#2229—591no—
244#2228—591no—
245#2226—591no—
246#2224—591no—
247#2220—591no—
248#2218—591no—
249#2217—591no—
250#2215—591no—
251#2210—591no—
252#2210—591no—
253#2207—591no—
254#2206—591no—
255#2151—591no—
256C2×C59—41no—
257C2×C59—161no—
258C2×C59#324511yescharacteristic
259C2×C59#341511yescharacteristic
260C2×C59#341611yescharacteristic
261D59—161no—
262D59—161no—
263#542—21no—
264#540—1181no—
265C2×SD32—591no—
266C2×SD32—591no—
267C2×D16—591no—
268C2×D16—591no—
269C2×M32—591no—
270C22×C16—591no—
271#500C2×D5911yescharacteristic
272#499—591no—
273#495C2×D5911yescharacteristic
274#493—591no—
275#490—1181no—
276#489—21no—
277#469—1181no—
278#468—21no—
279#467C2×D5911yescharacteristic
280#466—591no—
281#466—591no—
282#466C2×D5911yescharacteristic
283#442—1181no—
284#440—21no—
285C4×Q16C2×D5911yescharacteristic
286C4×D8—591no—
287C4×D8—591no—
288C4×D8C2×D5911yescharacteristic
289#405C2×D5911yescharacteristic
290#367—591no—
291#367—591no—
292#366—591no—
293#366—591no—
294C4⋊C16—591no—
295C4⋊C16—591no—
296C4⋊C16—591no—
297C4⋊C16—591no—
298#358—591no—
299#358—591no—
300#357—591no—
301#357—591no—
302#357—591no—
303#357—591no—
304#357—591no—
305#357—591no—
306C59#3382311yescharacteristic, complemented, Sylow
307C2×Q16C22×D5911yescharacteristic
308C2×SD16—21no—
309C2×D8—591no—
310C2×D8—591no—
311C2×D8—1181no—
312C2×D8C22×D5911yescharacteristic
313C2×M16C22⋊D5911yescharacteristic
314C22×C8C22×D5911yescharacteristic
315C4⋊Q8—21no—
316#127—1181no—
317#126—41no—
318#125—1181no—
319#124—21no—
320#124—41no—
321#123—41no—
322#123—2361no—
323#123—2361no—
324C22⋊Q8—21no—
325C22⋊Q8—41no—
326#121—21no—
327#121—1181no—
328#121—1181no—
329#121—2361no—
330#121—2361no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—1181no—
334C4×D4—1181no—
335#117C22⋊D5911yescharacteristic
336SD32—2361no—
337SD32—2361no—
338D16—2361no—
339D16—2361no—
340M32—2361no—
341C2×C16—591no—
342C2×C16—591no—
343C2×C16—591no—
344C2×C16—591no—
345C2×C16—2361no—
346#107—21no—
347#107—591no—
348#107—591no—
349#107C22×D5911yescharacteristic
350#107C22×D5911yescharacteristic
351#106—1181no—
352#103—21no—
353#103—21no—
354#102—21no—
355#102—21no—
356#102—1181no—
357#102—1181no—
358#102—1181no—
359#102—1181no—
360#97C22⋊D5911yescharacteristic
361#97C22⋊D5911yescharacteristic
362C4×C8C22×D5911yescharacteristic
363C4×C8C22×D5911yescharacteristic
364C2×Q8—21no—
365C2×Q8—41no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1181no—
369C2×D4—1181no—
370C2×D4—2361no—
371C2×D4—2361no—
372C2×D4—2361no—
373C2×D4—2361no—
374C22×C4—41no—
375C22×C4—2361no—
376C22×C4—2361no—
377C22×C4C2×C22⋊D5911yescharacteristic
378Q16—21no—
379Q16—21no—
380SD16—42no—
381D8—21no—
382D8—21no—
383D8—1181no—
384D8—1181no—
385D8—1181no—
386D8—1181no—
387D8—2362no—
388M16—41no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8C2×C22⋊D5911yescharacteristic
392C2×C8C2×C22⋊D5911yescharacteristic
393C2×C8C2×C22⋊D5911yescharacteristic
394C2×C8C23×D5911yescharacteristic
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—1181no—
401C4⋊C4—1181no—
402C4⋊C4—2361no—
403C4⋊C4—2361no—
404C4⋊C4#1174566911yescharacteristic
405C4⋊C4#1174566911yescharacteristic
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—2361no—
412C22⋊C4—2361no—
413C22⋊C4—2361no—
414C22⋊C4—2361no—
415C22⋊C4#1174566911yescharacteristic
416C22⋊C4#1174566911yescharacteristic
417C42—41no—
418C42C2×C22⋊D5911yescharacteristic
419C42C2×C22⋊D5911yescharacteristic
420C16—1181no—
421C16—1181no—
422C16—1181no—
423C16—1181no—
424C23—41no—
425C23—2361no—
426C23—2361no—
427C23C2×#1174566911yescharacteristic
428Q8—41no—
429Q8—41no—
430Q8—161no—
431D4—41no—
432D4—41no—
433D4—161no—
434D4—9441no—
435D4—9441no—
436D4—9441no—
437D4—9441no—
438D4—2361no—
439D4—2361no—
440D4—2361no—
441D4—2361no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—161no—
448C2×C4—9441no—
449C2×C4—9441no—
450C2×C4—2361no—
451C2×C4—2361no—
452C2×C4C2×#1174566911yescharacteristic
453C2×C4C2×#1174566911yescharacteristic
454C2×C4C2×#1174566911yescharacteristic
455C2×C4C2×#1174566911yescharacteristic
456C2×C4C2×#1174566911yescharacteristic
457C2×C4C22×C22⋊D5911yescharacteristic
458C8—21no—
459C8—21no—
460C8—21no—
461C8#1427377311yescharacteristic
462C8#1427377411yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—4721no—
470C22—4721no—
471C22—4721no—
472C22—4721no—
473C22—4721no—
474C22—4721no—
475C22C22×#1174566911yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—81no—
482C4—81no—
483C4—81no—
484C4—4721no—
485C4—4721no—
486C4#1683798011yescharacteristic
487C4#1683820011yescharacteristic
488C2—41no—
489C2—161no—
490C2—9441no—
491C2—9441no—
492C2#2143540311yescharacteristic
493C2#2143723511yescharacteristic
494C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3175C2
socleC22×C59C22×D8
hypercenter#3175D59
hypocenterC59#33823
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #21434746C2semidirect
2. #21426260C2semidirect
3. #21426253C2semidirect
4. #21426240C2semidirect
5. #21426230C2semidirect
6. #21426211C2semidirect
7. #21426197C2semidirect
8. #21426194C2semidirect
9. #21426180C2semidirect
10. #21426148C2semidirect
11. #21426148C2semidirect
12. #21426136C2semidirect
13. #21426131C2semidirect
14. #21425855C2semidirect
15. C59×#3175C2semidirect
242. #3175D59semidirect
306. C59#33823semidirect

Maximal subgroups

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

subgroupindex
#214347462
#214262602
#214262532
#214262402
#214262302
#214262112
#214261972
#214261942
#214261802
#214261482
#214261362
#214261312
#214258552
C59×#31752
#3382359