Finite Group Databasebeta

← #28743333

Subgroups — #28743333

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434608C211yesmaximal, characteristic, complemented
2#21432867C211yesmaximal, characteristic
3#21426263C211yesmaximal, characteristic, complemented
4#21426259C211yesmaximal, characteristic, complemented
5#21426256C211yesmaximal, characteristic, complemented
6#21426252C211yesmaximal, characteristic, complemented
7#21426247C211yesmaximal, characteristic, complemented
8#21426242C211yesmaximal, characteristic, complemented
9#21426237C211yesmaximal, characteristic, complemented
10#21426232C211yesmaximal, characteristic, complemented
11#21425878C211yesmaximal, characteristic, complemented
12C2×Dic59⋊C16C211yesmaximal, characteristic, complemented
13#21425804C211yesmaximal, characteristic
14#21425801C211yesmaximal, characteristic
15C59×#2158C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17#16837955—21no—
18#16837870—21no—
19C22×C59⋊C16C2211yescharacteristic
20#16837806C2211yescharacteristic
21#16837806C2211yescharacteristic
22#16837803C2211yescharacteristic
23#16837803C2211yescharacteristic
24#16837802—21no—
25#16837799—21no—
26#16837496C2211yescharacteristic
27#16837496C2211yescharacteristic
28#16837489—21no—
29#16837489—21no—
30#16837484C2211yescharacteristic
31#16837484C2211yescharacteristic
32#16837440C2211yescharacteristic
33#16837438—21no—
34#16837396—21no—
35C2×C8×Dic59C2211yescharacteristic
36#16837229C2211yescharacteristic
37#16837229C2211yescharacteristic
38#16837227C2211yescharacteristic
39#16837227C2211yescharacteristic
40Dic59⋊C16C2211yescharacteristic
41Dic59⋊C16C2211yescharacteristic
42Dic59⋊C16C2211yescharacteristic
43Dic59⋊C16C2211yescharacteristic
44#16837207C2211yescharacteristic
45#16837207C2211yescharacteristic
46#16837205C2211yescharacteristic
47#16837205C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837204C2211yescharacteristic
50#16837202C2211yescharacteristic
51#16837202C2211yescharacteristic
52C22×C16×C59C2211yescharacteristic
53C59×#427C2211yescharacteristic
54C59×#427C2211yescharacteristic
55C59×#367C2211yescharacteristic
56C59×#367C2211yescharacteristic
57C59×#366C2211yescharacteristic
58C59×#366C2211yescharacteristic
59C2×#11745685—21no—
60#14273749—22no—
61#14273749Q814yescomplemented
62C2×Dic236C2311yescharacteristic
63C2×#11745660—21no—
64C2×D472C2311yescharacteristic
65C22×C59⋊C8D411yescharacteristic
66#14273726—21no—
67#14273726—21no—
68#14273718—21no—
69#14273718—21no—
70#14273704—41no—
71#14273704—41no—
72C4×Dic118—41no—
73C4×Dic118—41no—
74#14273692—21no—
75#14273692—21no—
76#14273689—21no—
77#14273689—21no—
78(C22⋊D59)⋊C4—41no—
79(C22⋊D59)⋊C4—41no—
80#14273686—21no—
81#14273684—41no—
82C2×C4×Dic59D411yescharacteristic
83C42×D59—41no—
84C2×C59⋊C16—22no—
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87#14273661C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273660—21no—
90#14273660—21no—
91#14273659—21no—
92#14273658—21no—
93#14273658—21no—
94#14273650—21no—
95#14273650—21no—
96#14273647—21no—
97#14273647—21no—
98C8×Dic59D411yescharacteristic
99C8×Dic59D411yescharacteristic
100C8×Dic59C2311yescharacteristic
101C8×Dic59C2311yescharacteristic
102C22×C8×C59C2311yescharacteristic
103#117×C59—21no—
104#117×C59—21no—
105C2×C16×C59—22no—
106C2×C16×C59C2311yescharacteristic
107C2×C16×C59C2311yescharacteristic
108#107×C59C2311yescharacteristic
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#106×C59—21no—
113#106×C59—21no—
114C22×C4×C59C2×D411yescharacteristic
115C2×C22⋊D59—41no—
116C22×Dic59D811yescharacteristic
117C22×Dic59D811yescharacteristic
118#11745685—24no—
119#11745685—42no—
120C2×D236—21no—
121C2×C4×D59—41no—
122C2×C4×D59—41no—
123C2×C4×D59—41no—
124C2×Dic118—21no—
125C2×Dic118—41no—
126C2×Dic118—41no—
127C2×C8×C59C2×D411yescharacteristic
128C2×C8×C59C2×Q812yes—
129C2×C8×C59#4112yes—
130C2×C8×C59C2411yescharacteristic
131C4⋊C4×C59—21no—
132C4⋊C4×C59—21no—
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C22⋊C4×C59—41no—
136C22⋊C4×C59—41no—
137C42×C59—41no—
138C42×C59—41no—
139C22⋊Dic59—41no—
140(C2×D59)⋊C4—41no—
141(C2×D59)⋊C4—41no—
142C4⋊Dic59—21no—
143C4⋊Dic59—21no—
144C4⋊Dic59—41no—
145C4⋊Dic59—41no—
146Dic59⋊C4—41no—
147Dic59⋊C4—41no—
148Dic59⋊C4—41no—
149Dic59⋊C4—41no—
150Dic59⋊C4—41no—
151Dic59⋊C4—41no—
152C4×Dic59—41no—
153C4×Dic59—41no—
154C4×Dic59—41no—
155C4×Dic59D811yescharacteristic
156C4×Dic59D811yescharacteristic
157C4×Dic59C2×D411yescharacteristic
158C4×Dic59C2×D411yescharacteristic
159C2×C59⋊C8—22no—
160C2×C59⋊C8C2×D411yescharacteristic
161C2×C59⋊C8C2×D411yescharacteristic
162Dic236—21no—
163Dic236C2×Q812yes—
164D472—21no—
165D472C2×Q812yes—
166#11745660—24no—
167C16×C59—21no—
168C16×C59—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C23×C59C2×D811yescharacteristic
172C22×D59—41no—
173C2×C4×C59—41no—
174C2×C4×C59—41no—
175C2×C4×C59—41no—
176C2×C4×C59—41no—
177C2×C4×C59C22⋊Q812yes—
178C2×C4×C59C2×D811yescharacteristic
179C2×C4×C59#13712yes—
180C2×C4×C59C22×D411yescharacteristic
181C22⋊D59—44no—
182C2×Dic59—22no—
183C2×Dic59—41no—
184C2×Dic59—41no—
185C2×Dic59—41no—
186C2×Dic59—41no—
187C2×Dic59C2×D811yescharacteristic
188C2×Dic59C2×D811yescharacteristic
189C2×Dic59C2×D811yescharacteristic
190C2×Dic59C2×D811yescharacteristic
191D236—22no—
192D236—41no—
193C4×D59—44no—
194C4×D59—82no—
195C4×D59—82no—
196Dic118—22no—
197Dic118—41no—
198Dic118—82no—
199Dic118—82no—
200C8×C59C22⋊Q812yes—
201C8×C59C22×Q811yescharacteristic
202C8×C59C2×#4111yescharacteristic
203C59⋊C8—21no—
204C59⋊C8—21no—
205#33626—591nomaximal, Sylow
206C22×C59—21no—
207C22×C59—21no—
208C22×C59#47412yes—
209C22×C59C22×D811yescharacteristic
210C2×D59—42no—
211C2×D59—81no—
212C2×D59—81no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59—81no—
217C4×C59#47412yes—
218C4×C59C2×C22⋊Q811yescharacteristic
219C4×C59C2×#13711yescharacteristic
220Dic59—21no—
221Dic59—21no—
222Dic59—21no—
223Dic59—42no—
224Dic59—81no—
225Dic59—81no—
226Dic59#50812yes—
227#3160—591no—
228C4×#135—591no—
229#2229—591no—
230#2229—591no—
231#2228—591no—
232#2228—591no—
233#2227—591no—
234#2226—591no—
235#2225—591no—
236#2224—591no—
237#2158—591no—
238#2158D5911yescharacteristic, complemented
239C2×C4⋊C16—591no—
240#2138—591no—
241#2138—591no—
242C2×C59—22no—
243C2×C59C2×#47411yescharacteristic
244C2×C59C2×#50811yescharacteristic
245C2×C59#341411yescharacteristic
246D59—82no—
247C2×#135—591no—
248#533—1181no—
249C4×#41—1181no—
250C22×C16—591no—
251C22×C16C2×D5911yescharacteristic
252#500—591no—
253#500—591no—
254#499—591no—
255#499—591no—
256#499—1181no—
257#498—1181no—
258C4×Q16—591no—
259C4×Q16—591no—
260C4×SD16—1181no—
261C4×SD16—1181no—
262C4×D8—591no—
263C4×D8—591no—
264#427—591no—
265#427—1181no—
266#427C2×D5911yescharacteristic
267#427C2×D5911yescharacteristic
268#416—1181no—
269C2×C4×C8—591no—
270#367—591no—
271#367—591no—
272#367C2×D5911yescharacteristic
273#367C2×D5911yescharacteristic
274#366—591no—
275#366—591no—
276#366C2×D5911yescharacteristic
277#366C2×D5911yescharacteristic
278C4⋊C16—591no—
279C4⋊C16—591no—
280C4⋊C16—591no—
281C4⋊C16—591no—
282#358—591no—
283#358—591no—
284#358—591no—
285#358—591no—
286#357—591no—
287#357—591no—
288#357—591no—
289#357—591no—
290C59#3362611yescharacteristic, complemented, Sylow
291C2×#41—1181no—
292#135—594no—
293#135—1182no—
294C2×Q16—591no—
295C2×SD16—1181no—
296C2×D8—591no—
297C22×C8—591no—
298C22×C8C22×D5911yescharacteristic
299C4⋊Q8—1181no—
300C4⋊Q8—1181no—
301#125—1181no—
302#125—1181no—
303C22⋊Q8—2361no—
304C22⋊Q8—2361no—
305C4×Q8—1181no—
306C4×Q8—1181no—
307C4×Q8—2361no—
308C4×Q8—2361no—
309C4×D4—1181no—
310C4×D4—1181no—
311C4×D4—2361no—
312C4×D4—2361no—
313#117—21no—
314#117—21no—
315#117—1181no—
316#117—2361no—
317C2×C42—591no—
318C2×C42—2361no—
319C2×C16—22no—
320C2×C16—591no—
321C2×C16—591no—
322C2×C16—1182no—
323C2×C16C22×D5911yescharacteristic
324C2×C16C22×D5911yescharacteristic
325#107—591no—
326#107—591no—
327#107—1181no—
328#107—1181no—
329#107C22×D5911yescharacteristic
330#107C22×D5911yescharacteristic
331#107C22×D5911yescharacteristic
332#107C22×D5911yescharacteristic
333#106—21no—
334#106—21no—
335#106—1181no—
336#106—1181no—
337#106—1181no—
338#103—1181no—
339#103—1181no—
340#102—1181no—
341#102—1181no—
342C4×C8—591no—
343C4×C8—591no—
344C4×C8—591no—
345C4×C8—591no—
346#41—1184no—
347#41—2362no—
348C2×Q8—1181no—
349C2×Q8—2361no—
350C2×Q8—2361no—
351C2×D4—1181no—
352C2×D4—2361no—
353C22×C4—591no—
354C22×C4—591no—
355C22×C4—2361no—
356C22×C4—2361no—
357C22×C4—2361no—
358C22×C4D4×D5911yescharacteristic
359Q16—592no—
360Q16—1181no—
361SD16—1184no—
362D8—592no—
363D8—1181no—
364C2×C8—591no—
365C2×C8—591no—
366C2×C8—1182no—
367C2×C8#1174568512yes—
368C2×C8D4×D5911yescharacteristic
369C2×C8Q8×D5912yes—
370C2×C8C23×D5911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—1181no—
376C4⋊C4—1181no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4—2361no—
381C4⋊C4—2361no—
382C4⋊C4—2361no—
383C4⋊C4—2361no—
384C4⋊C4—2361no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—2361no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C42—41no—
391C42—41no—
392C42—591no—
393C42—591no—
394C42—591no—
395C42—591no—
396C42—2361no—
397C42—2361no—
398C42—2361no—
399C16—21no—
400C16—21no—
401C16—1181no—
402C16—1181no—
403C23—2361no—
404C23D8×D5911yescharacteristic
405Q8—1182no—
406Q8—4722no—
407Q8—4722no—
408Q8—2361no—
409D4—1182no—
410D4—2364no—
411D4—2361no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—591no—
417C2×C4—591no—
418C2×C4—591no—
419C2×C4—591no—
420C2×C4—1182no—
421C2×C4—4722no—
422C2×C4—4722no—
423C2×C4—2361no—
424C2×C4—2361no—
425C2×C4—2361no—
426C2×C4—2361no—
427C2×C4—2364no—
428C2×C4#1427370512yes—
429C2×C4D8×D5911yescharacteristic
430C2×C4#1427375912yes—
431C2×C4C2×D4×D5911yescharacteristic
432C8—1181no—
433C8—1181no—
434C8#1427370512yes—
435C8C2×Q8×D5911yescharacteristic
436C8C2×#1174568511yescharacteristic
437C22—21no—
438C22—21no—
439C22—4721no—
440C22—4721no—
441C22—2362no—
442C22#1683768112yes—
443C22C2×D8×D5911yescharacteristic
444C4—81no—
445C4—81no—
446C4—81no—
447C4—81no—
448C4—592no—
449C4—1181no—
450C4—1181no—
451C4—1181no—
452C4—4721no—
453C4—4721no—
454C4—2362no—
455C4#1683768112yes—
456C4C2×#1427370511yescharacteristic
457C4C2×#1427375911yescharacteristic
458C2—22no—
459C2—4722no—
460C2C2×#1683768111yescharacteristic
461C2#508×D5911yescharacteristic
462C2#2143722911yescharacteristic
subgroupG/H
centerC2×C4D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2158C2
socleC22×C59C22×D8
prime radicalC2×#14273749C22
hypercenter#2158D59
hypocenterC59#33626
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2158

Complements

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

NH ≅ G/Ntype
1. #21434608C2semidirect
3. #21426263C2semidirect
4. #21426259C2semidirect
5. #21426256C2semidirect
6. #21426252C2semidirect
7. #21426247C2semidirect
8. #21426242C2semidirect
9. #21426237C2semidirect
10. #21426232C2semidirect
11. #21425878C2semidirect
12. C2×Dic59⋊C16C2semidirect
15. C59×#2158C2semidirect
61. #14273749Q8semidirect
238. #2158D59semidirect
290. C59#33626semidirect

Maximal subgroups

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

subgroupindex
#214346082
#214328672
#214262632
#214262592
#214262562
#214262522
#214262472
#214262422
#214262372
#214262322
#214258782
C2×Dic59⋊C162
#214258042
#214258012
C59×#21582
#3362659