Finite Group Databasebeta

← #28735582

Subgroups — #28735582

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425679C211yesmaximal, characteristic, complemented
2#21425678C211yesmaximal, characteristic, complemented
3#21425673C211yesmaximal, characteristic, complemented
4#21425550C211yesmaximal, characteristic, complemented
5#21425546C211yesmaximal, characteristic, complemented
6#21425380C211yesmaximal, characteristic
7#21424810C211yesmaximal, characteristic, complemented
8#21424809C211yesmaximal, characteristic, complemented
9#21424460C211yesmaximal, characteristic
10#21424336C211yesmaximal, characteristic, complemented
11#21424334C211yesmaximal, characteristic, complemented
12#21424316C211yesmaximal, characteristic, complemented
13#21424315C211yesmaximal, characteristic, complemented
14#21424235C211yesmaximal, characteristic, complemented
15C59×#1881C211yesmaximal, characteristic
16#16838169C2211yescharacteristic
17#16837450C2211yescharacteristic
18#16837450C2211yescharacteristic
19#16837450C2211yescharacteristic
20#16837429C2211yescharacteristic
21#16837428C2211yescharacteristic
22#16837425C2211yescharacteristic
23#16837425C2211yescharacteristic
24#16837419C2211yescharacteristic
25C2×#14273654C2211yescharacteristic
26C2×#14273654C2211yescharacteristic
27#16837398C2211yescharacteristic
28#16837386C2211yescharacteristic
29#16837386C2211yescharacteristic
30#16837369C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32#16837174C2211yescharacteristic
33#16837173C2211yescharacteristic
34#16837172C2211yescharacteristic
35#16837167C2211yescharacteristic
36#16837167C2211yescharacteristic
37#16837166C2211yescharacteristic
38#16837166C2211yescharacteristic
39#16837166C2211yescharacteristic
40#16837163C2211yescharacteristic
41#16837161C2211yescharacteristic
42C59⋊#338C2211yescharacteristic
43#16837158C2211yescharacteristic
44C59×#575C2211yescharacteristic
45C59×#568C2211yescharacteristic
46C59×#421C2211yescharacteristic
47C59×#416C2211yescharacteristic
48C59×#408C2211yescharacteristic
49C59×#344C2211yescharacteristic
50C59×#341C2211yescharacteristic
51#14273734—41no—
52#14273734—41no—
53C2×#11745664—41no—
54C2×#11745664C2311yescharacteristic
55C2×#11745664C2311yescharacteristic
56C2×#11745664C2311yescharacteristic
57C2×#11745664C2311yescharacteristic
58C2×#11745664C2311yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60#14273686C2311yescharacteristic
61C2×C4×Dic59C2311yescharacteristic
62#14273662—41no—
63#14273662—41no—
64#14273662—41no—
65C4⋊(C59⋊C8)—41no—
66Dic59⋊C8—41no—
67Dic59⋊C8—41no—
68Dic59⋊C8—41no—
69#14273654—22no—
70#14273654—22no—
71#14273654—41no—
72#14273651—21no—
73#14273648—21no—
74#14273646—21no—
75#14273646—21no—
76#14273644—21no—
77#14273644—21no—
78C22⋊(C59⋊C8)—41no—
79#14273637—22no—
80C2×#41×C59C2311yescharacteristic
81C2×#41×C59C2311yescharacteristic
82#137×C59—21no—
83#137×C59—21no—
84AGL(1,ℤ/8)×C59—21no—
85AGL(1,ℤ/8)×C59—21no—
86#135×C59—41no—
87#131×C59—41no—
88#131×C59—41no—
89C2×M16×C59—41no—
90C2×M16×C59C2311yescharacteristic
91C2×M16×C59C2311yescharacteristic
92C2×M16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94#117×C59C2311yescharacteristic
95#104×C59—41no—
96#103×C59—21no—
97#102×C59—21no—
98C22⋊C8×C59—41no—
99#41×C59—21no—
100#41×C59—21no—
101#41×C59—21no—
102#41×C59—21no—
103#41×C59—21no—
104#41×C59—21no—
105#41×C59—21no—
106#41×C59—21no—
107C2×Q8×C59#4111yescharacteristic
108C2×Q8×C59#4111yescharacteristic
109C2×D4×C59—21no—
110C2×D4×C59—21no—
111C2×D4×C59#4111yescharacteristic
112C2×D4×C59#4111yescharacteristic
113C22×C4×C59—21no—
114C22×C4×C59—21no—
115C22×C4×C59C2411yescharacteristic
116C22×Dic59—21no—
117Q16×C59—41no—
118SD16×C59—41no—
119SD16×C59—41no—
120D8×C59—41no—
121M16×C59—21no—
122M16×C59—21no—
123M16×C59—21no—
124M16×C59—21no—
125M16×C59—21no—
126M16×C59—21no—
127M16×C59—42no—
128C2×C8×C59—21no—
129C2×C8×C59—21no—
130C2×C8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59—21no—
133C2×C8×C59—21no—
134C2×C8×C59—42no—
135C4⋊C4×C59—21no—
136C22⋊C4×C59—21no—
137C42×C59—21no—
138C22⋊Dic59—21no—
139C4⋊Dic59—21no—
140C4×Dic59—21no—
141C4×Dic59—21no—
142C4×Dic59#4112yes—
143#11745664—21no—
144#11745664—21no—
145#11745664—21no—
146#11745664—21no—
147#11745664—21no—
148#11745664—21no—
149#11745664—21no—
150#11745664—21no—
151#11745664—21no—
152#11745664—21no—
153#11745664—42no—
154C2×C59⋊C8—21no—
155C2×C59⋊C8—21no—
156C2×C59⋊C8—21no—
157C2×C59⋊C8—21no—
158C2×C59⋊C8—21no—
159C2×C59⋊C8—21no—
160C2×C59⋊C8—21no—
161C2×C59⋊C8—21no—
162C2×C59⋊C8—42no—
163C23×C59—21no—
164C23×C59—21no—
165C23×C592−1+411yescharacteristic
166Q8×C59—21no—
167Q8×C59—21no—
168Q8×C59—21no—
169Q8×C59—21no—
170D4×C59—21no—
171D4×C59—21no—
172D4×C59—21no—
173D4×C59—21no—
174D4×C59—41no—
175D4×C59—41no—
176D4×C59—41no—
177D4×C59—41no—
178C2×C4×C59—21no—
179C2×C4×C59—21no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59C2×#4111yescharacteristic
186C2×C4×C59C2×#4111yescharacteristic
187C2×C4×C59C2×#4111yescharacteristic
188C2×C4×C59H(5,𝔽2)11yescharacteristic
189C2×C4×C59H(5,𝔽2)11yescharacteristic
190C2×C4×C59H(5,𝔽2)11yescharacteristic
191C2×Dic59—22no—
192C2×Dic59—41no—
193C2×Dic59—81no—
194C8×C59—41no—
195C8×C59—41no—
196C8×C59—41no—
197C8×C59—41no—
198C59⋊C8—41no—
199C59⋊C8—41no—
200C59⋊C8—41no—
201C59⋊C8—41no—
202C59⋊C8—41no—
203C59⋊C8—41no—
204#32703—591nomaximal, Sylow
205C22×C59—41no—
206C22×C59—41no—
207C22×C59—41no—
208C22×C59—81no—
209C22×C59—81no—
210C22×C59#53911yescharacteristic
211C22×C59#53911yescharacteristic
212C22×C59#53911yescharacteristic
213C4×C59—41no—
214C4×C59—41no—
215C4×C59—81no—
216C4×C59#55111yescharacteristic
217C4×C59#55311yescharacteristic
218C4×C59#55311yescharacteristic
219C4×C59H(3,𝔽4)11yescharacteristic
220Dic59—42no—
221Dic59—81no—
222#2103—591no—
223#2103—591no—
224#2102—591no—
225#2080—591no—
226#2080—591no—
227#2050—591no—
228#1938—591no—
229#1938—591no—
230#1881—591no—
231#1881D5911yescharacteristic
232#1861—591no—
233#1861—591no—
234#1858—591no—
235#1858—591no—
236#1838—591no—
237C2×C59—21no—
238C2×C59—21no—
239C2×C59—21no—
240C2×C59—81no—
241C2×C59—81no—
242C2×C59#278211yescharacteristic
243#575C2×D5911yescharacteristic
244#568—591no—
245#568C2×D5911yescharacteristic
246#430—591no—
247#430—591no—
248#430—591no—
249#424—591no—
250#424—591no—
251#423—591no—
252#423—591no—
253#421—591no—
254#421C2×D5911yescharacteristic
255C2×#104—591no—
256C2×#104—591no—
257#416—591no—
258#416C2×D5911yescharacteristic
259#413—591no—
260#413—591no—
261#408—591no—
262#408C2×D5911yescharacteristic
263C2×#97—591no—
264#344—591no—
265#344—591no—
266#344—591no—
267#344C2×D5911yescharacteristic
268#341—591no—
269#341—591no—
270#341—591no—
271#341—591no—
272#341—591no—
273#341C2×D5911yescharacteristic
274#339—591no—
275#339—591no—
276#338—591no—
277H(3,ℤ/4)—591no—
278C59#3270311yescharacteristic, complemented, Sylow
279C2×#41C22×D5911yescharacteristic
280C2×#41C22×D5911yescharacteristic
281#137—21no—
282#137—21no—
283AGL(1,ℤ/8)—21no—
284AGL(1,ℤ/8)—21no—
285#135—41no—
286#131—41no—
287#131—41no—
288#131—2361no—
289#131—2361no—
290C2×M16—41no—
291C2×M16—591no—
292C2×M16—591no—
293C2×M16—591no—
294C2×M16—591no—
295C2×M16—591no—
296C2×M16—2361no—
297C2×M16C22×D5911yescharacteristic
298C2×M16C22×D5911yescharacteristic
299C2×M16C22×D5911yescharacteristic
300C22×C8—591no—
301C22×C8C22×D5911yescharacteristic
302#117—591no—
303#117C22×D5911yescharacteristic
304C2×C42—591no—
305#108—2361no—
306#108—2361no—
307#108—2361no—
308C4⋊C8—2361no—
309C4⋊C8—2361no—
310C4⋊C8—2361no—
311C4⋊C8—2361no—
312#104—41no—
313#104—1182no—
314#104—1182no—
315#104—2361no—
316#103—21no—
317#103—1181no—
318#102—21no—
319#102—1181no—
320#101—1181no—
321#101—1181no—
322#100—1181no—
323#100—1181no—
324C22⋊C8—41no—
325C22⋊C8—2361no—
326#97—1182no—
327#41—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—21no—
332#41—21no—
333#41—21no—
334#41—21no—
335C2×Q8#1174568511yescharacteristic
336C2×Q8#1174568711yescharacteristic
337C2×D4—21no—
338C2×D4—21no—
339C2×D4#1174568511yescharacteristic
340C2×D4#1174568711yescharacteristic
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—1181no—
344C22×C4C23×D5911yescharacteristic
345Q16—41no—
346SD16—41no—
347SD16—41no—
348D8—41no—
349M16—21no—
350M16—21no—
351M16—21no—
352M16—21no—
353M16—21no—
354M16—21no—
355M16—42no—
356M16—1181no—
357M16—1181no—
358M16—1181no—
359M16—1181no—
360M16—1181no—
361M16—1181no—
362M16—1181no—
363M16—1181no—
364M16—1181no—
365M16—1181no—
366M16—2362no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—42no—
374C2×C8—1181no—
375C2×C8—1181no—
376C2×C8—1181no—
377C2×C8—1181no—
378C2×C8—1181no—
379C2×C8—1181no—
380C2×C8—1181no—
381C2×C8—1181no—
382C2×C8—2362no—
383C4⋊C4—21no—
384C4⋊C4—1181no—
385C22⋊C4—21no—
386C22⋊C4—1181no—
387C42—21no—
388C42—592no—
389C42—1181no—
390C42—1181no—
391C23—21no—
392C23—21no—
393C23#1427377411yescharacteristic
394Q8—21no—
395Q8—21no—
396Q8—21no—
397Q8—21no—
398D4—21no—
399D4—21no—
400D4—21no—
401D4—21no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—1182no—
414C2×C4—4721no—
415C2×C4—2361no—
416C2×C4#41×D5911yescharacteristic
417C2×C4C2×#1174568711yescharacteristic
418C2×C4C2×#1174568511yescharacteristic
419C2×C4#1427377211yescharacteristic
420C2×C4#1427377211yescharacteristic
421C2×C4#1427377211yescharacteristic
422C8—41no—
423C8—41no—
424C8—41no—
425C8—41no—
426C8—2361no—
427C8—2361no—
428C8—2361no—
429C8—2361no—
430C8—2361no—
431C8—2361no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—81no—
436C22—81no—
437C22#1683799411yescharacteristic
438C22#1683799511yescharacteristic
439C22#1683799711yescharacteristic
440C4—41no—
441C4—41no—
442C4—81no—
443C4—4721no—
444C4—2362no—
445C4#1683807811yescharacteristic
446C4#1683809511yescharacteristic
447C4#1683810111yescharacteristic
448C4#1683814511yescharacteristic
449C2—21no—
450C2—21no—
451C2—21no—
452C2—81no—
453C2—81no—
454C2#2143176911yescharacteristic
subgroupG/H
centerC4#16838145
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1881C2
socleC2×C59#2782
prime radicalC59×#575C22
hypercenter#1881D59
hypocenterC59#32703
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1881

Complements

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

NH ≅ G/Ntype
1. #21425679C2semidirect
2. #21425678C2semidirect
3. #21425673C2semidirect
4. #21425550C2semidirect
5. #21425546C2semidirect
7. #21424810C2semidirect
8. #21424809C2semidirect
10. #21424336C2semidirect
11. #21424334C2semidirect
12. #21424316C2semidirect
13. #21424315C2semidirect
14. #21424235C2semidirect
278. C59#32703semidirect

Maximal subgroups

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

subgroupindex
#214256792
#214256782
#214256732
#214255502
#214255462
#214253802
#214248102
#214248092
#214244602
#214243362
#214243342
#214243162
#214243152
#214242352
C59×#18812
#3270359