Finite Group Databasebeta

← #28743331

Subgroups — #28743331

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#135×Dic59C211yesmaximal, characteristic
3#21426262C211yesmaximal, characteristic, complemented
4#21426258C211yesmaximal, characteristic, complemented
5#21426255C211yesmaximal, characteristic, complemented
6#21426251C211yesmaximal, characteristic, complemented
7#21426246C211yesmaximal, characteristic, complemented
8#21426241C211yesmaximal, characteristic, complemented
9#21426236C211yesmaximal, characteristic, complemented
10#21426231C211yesmaximal, characteristic, complemented
11#21425880C211yesmaximal, characteristic, complemented
12#21425878C211yesmaximal, characteristic, complemented
13C2×Dic59⋊C16C211yesmaximal, characteristic, complemented
14#21425803C211yesmaximal, characteristic
15C59×#2138C211yesmaximal, characteristic
16#16837955—21no—
17#41×Dic59—21no—
18C22×C59⋊C16C2211yescharacteristic
19#16837806C2211yescharacteristic
20#16837806C2211yescharacteristic
21#16837803C2211yescharacteristic
22#16837803C2211yescharacteristic
23#16837802—21no—
24#16837799—21no—
25Q16×Dic59C2211yescharacteristic
26Q16×Dic59C2211yescharacteristic
27SD16×Dic59—21no—
28SD16×Dic59—21no—
29D8×Dic59C2211yescharacteristic
30D8×Dic59C2211yescharacteristic
31#16837440C2211yescharacteristic
32#16837440C2211yescharacteristic
33#16837438—21no—
34#16837398—21no—
35C2×C8×Dic59C2211yescharacteristic
36#16837230C2211yescharacteristic
37#16837230C2211yescharacteristic
38#16837229C2211yescharacteristic
39#16837229C2211yescharacteristic
40#16837228C2211yescharacteristic
41#16837228C2211yescharacteristic
42#16837227C2211yescharacteristic
43#16837227C2211yescharacteristic
44Dic59⋊C16C2211yescharacteristic
45Dic59⋊C16C2211yescharacteristic
46Dic59⋊C16C2211yescharacteristic
47Dic59⋊C16C2211yescharacteristic
48#16837206C2211yescharacteristic
49#16837206C2211yescharacteristic
50#16837203C2211yescharacteristic
51#16837203C2211yescharacteristic
52C2×#135×C59C2211yescharacteristic
53C22×C16×C59C2211yescharacteristic
54C59×#427C2211yescharacteristic
55C59×#358C2211yescharacteristic
56C59×#358C2211yescharacteristic
57C59×#357C2211yescharacteristic
58C59×#357C2211yescharacteristic
59C22×C59⋊C8D411yescharacteristic
60#14273726—21no—
61#14273726—21no—
62#14273718—21no—
63#14273718—21no—
64#14273704—41no—
65#14273704—41no—
66Q8×Dic59—21no—
67Q8×Dic59—21no—
68C4×Dic118—41no—
69C4×Dic118—41no—
70D4×Dic59—21no—
71D4×Dic59—21no—
72D4×Dic59—41no—
73D4×Dic59—41no—
74#14273686—21no—
75#14273686—21no—
76#14273686—41no—
77C2×C4×Dic59—41no—
78C2×C4×Dic59D411yescharacteristic
79C2×C59⋊C16—22no—
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273661C2311yescharacteristic
86#14273660—21no—
87#14273660—21no—
88#14273659—21no—
89#14273659—21no—
90#14273658—21no—
91#14273658—21no—
92#14273651—21no—
93#14273651—21no—
94#14273648—21no—
95#14273648—21no—
96C8×Dic59D411yescharacteristic
97C8×Dic59D411yescharacteristic
98C8×Dic59C2311yescharacteristic
99C8×Dic59C2311yescharacteristic
100C2×#41×C59—21no—
101#135×C59—22no—
102#135×C59Q814yescomplemented
103C2×Q16×C59C2311yescharacteristic
104C2×SD16×C59—21no—
105C2×D8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107#117×C59—21no—
108C2×C16×C59—22no—
109C2×C16×C59C2311yescharacteristic
110C2×C16×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112#107×C59C2311yescharacteristic
113#106×C59—21no—
114#41×C59—24no—
115#41×C59—42no—
116C2×Q8×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59C2×D411yescharacteristic
121C22×Dic59—41no—
122C22×Dic59—41no—
123C22×Dic59D811yescharacteristic
124C22×Dic59D811yescharacteristic
125C2×Dic118—41no—
126C2×Dic118—41no—
127Q16×C59—21no—
128Q16×C59C2×Q812yes—
129SD16×C59—24no—
130D8×C59—21no—
131D8×C59C2×Q812yes—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2×Q812yes—
134C2×C8×C59#4112yes—
135C2×C8×C59C2411yescharacteristic
136C4⋊C4×C59—21no—
137C4⋊C4×C59—21no—
138C22⋊C4×C59—41no—
139C42×C59—41no—
140C22⋊Dic59—41no—
141C22⋊Dic59—41no—
142C22⋊Dic59—41no—
143C22⋊Dic59—41no—
144C4⋊Dic59—21no—
145C4⋊Dic59—21no—
146C4⋊Dic59—21no—
147C4⋊Dic59—21no—
148C4⋊Dic59—41no—
149C4⋊Dic59—41no—
150C4⋊Dic59—41no—
151C4⋊Dic59—41no—
152Dic59⋊C4—41no—
153Dic59⋊C4—41no—
154Dic59⋊C4—41no—
155Dic59⋊C4—41no—
156C4×Dic59—41no—
157C4×Dic59—41no—
158C4×Dic59—41no—
159C4×Dic59—41no—
160C4×Dic59D811yescharacteristic
161C4×Dic59D811yescharacteristic
162C4×Dic59C2×D411yescharacteristic
163C4×Dic59C2×D411yescharacteristic
164C2×C59⋊C8—22no—
165C2×C59⋊C8C2×D411yescharacteristic
166C2×C59⋊C8C2×D411yescharacteristic
167C16×C59—21no—
168C16×C59—21no—
169C59⋊C16—21no—
170C59⋊C16—21no—
171C23×C59—41no—
172C23×C59C2×D811yescharacteristic
173Q8×C59—22no—
174Q8×C59—41no—
175D4×C59—22no—
176D4×C59—44no—
177D4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—44no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59C22⋊Q812yes—
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59#13712yes—
185C2×C4×C59C22×D411yescharacteristic
186C2×Dic59—22no—
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—82no—
193C2×Dic59—82no—
194C2×Dic59C2×D811yescharacteristic
195C2×Dic59C2×D811yescharacteristic
196C2×Dic59C2×D811yescharacteristic
197C2×Dic59C2×D811yescharacteristic
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—42no—
209C22×C59—81no—
210C22×C59—81no—
211C22×C59#47412yes—
212C22×C59C22×D811yescharacteristic
213C4×C59—42no—
214C4×C59—81no—
215C4×C59—81no—
216C4×C59#47412yes—
217C4×C59C2×C22⋊Q811yescharacteristic
218C4×C59C2×#13711yescharacteristic
219Dic59—21no—
220Dic59—21no—
221Dic59—21no—
222Dic59—81no—
223Dic59—81no—
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#2158—591no—
239C2×C4⋊C16—591no—
240#2138—591no—
241#2138D5911yescharacteristic
242C2×C59—22no—
243C2×C59—82no—
244C2×C59C2×#47411yescharacteristic
245C2×C59C2×#50811yescharacteristic
246C2×C59#341411yescharacteristic
247C2×#135C2×D5911yescharacteristic
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—591no—
266#427—1181no—
267#427C2×D5911yescharacteristic
268#416—1181no—
269C2×C4×C8—591no—
270#367—591no—
271#367—591no—
272#367—591no—
273#367—591no—
274#366—591no—
275#366—591no—
276#366—591no—
277#366—591no—
278C4⋊C16—591no—
279C4⋊C16—591no—
280C4⋊C16—591no—
281C4⋊C16—591no—
282#358—591no—
283#358—591no—
284#358C2×D5911yescharacteristic
285#358C2×D5911yescharacteristic
286#357—591no—
287#357—591no—
288#357C2×D5911yescharacteristic
289#357C2×D5911yescharacteristic
290C59#3362611yescharacteristic, complemented, Sylow
291C2×#41—21no—
292#135—22no—
293#135Dic11814yescomplemented
294C2×Q16C22×D5911yescharacteristic
295C2×SD16—21no—
296C2×D8C22×D5911yescharacteristic
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—1181no—
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—591no—
328#107—591no—
329#107—1181no—
330#107—1181no—
331#107C22×D5911yescharacteristic
332#107C22×D5911yescharacteristic
333#106—21no—
334#106—1181no—
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—24no—
347#41—42no—
348C2×Q8—21no—
349C2×Q8—2361no—
350C2×Q8—2361no—
351C2×D4—21no—
352C2×D4—41no—
353C22×C4—41no—
354C22×C4—591no—
355C22×C4—591no—
356C22×C4—2361no—
357C22×C4—2361no—
358C22×C4D4×D5911yescharacteristic
359Q16—21no—
360Q16C2×Dic11812yes—
361SD16—24no—
362D8—21no—
363D8C2×Dic11812yes—
364C2×C8—591no—
365C2×C8—591no—
366C2×C8—1182no—
367C2×C8C2×Dic11812yes—
368C2×C8D4×D5911yescharacteristic
369C2×C8#1174568712yes—
370C2×C8C23×D5911yescharacteristic
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—1181no—
374C4⋊C4—1181no—
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—2361no—
387C22⋊C4—2361no—
388C22⋊C4—2361no—
389C22⋊C4—2361no—
390C42—41no—
391C42—591no—
392C42—591no—
393C42—591no—
394C42—591no—
395C42—2361no—
396C42—2361no—
397C42—2361no—
398C42—2361no—
399C16—21no—
400C16—21no—
401C16—1181no—
402C16—1181no—
403C23—41no—
404C23D8×D5911yescharacteristic
405Q8—22no—
406Q8—41no—
407Q8—4722no—
408Q8—4722no—
409D4—22no—
410D4—44no—
411D4—41no—
412C2×C4—41no—
413C2×C4—44no—
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—2361no—
428C2×C4#1427370412yes—
429C2×C4D8×D5911yescharacteristic
430C2×C4#1427375612yes—
431C2×C4C2×D4×D5911yescharacteristic
432C8—1181no—
433C8—1181no—
434C8#1427370412yes—
435C8C22×Dic11811yescharacteristic
436C8C2×#1174568711yescharacteristic
437C22—21no—
438C22—21no—
439C22—42no—
440C22—81no—
441C22—81no—
442C22#1683768012yes—
443C22C2×D8×D5911yescharacteristic
444C4—42no—
445C4—81no—
446C4—81no—
447C4—592no—
448C4—1181no—
449C4—1181no—
450C4—1181no—
451C4—4721no—
452C4—4721no—
453C4—4721no—
454C4—4721no—
455C4#1683768012yes—
456C4C2×#1427370411yescharacteristic
457C4C2×#1427375611yescharacteristic
458C2—22no—
459C2—82no—
460C2C2×#1683768011yescharacteristic
461C2#508×D5911yescharacteristic
462C2#2143722911yescharacteristic
subgroupG/H
centerC2×C4D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2138C2
socleC22×C59C22×D8
prime radicalC2×#135×C59C22
hypercenter#2138D59
hypocenterC59#33626
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. #21434608C2semidirect
3. #21426262C2semidirect
4. #21426258C2semidirect
5. #21426255C2semidirect
6. #21426251C2semidirect
7. #21426246C2semidirect
8. #21426241C2semidirect
9. #21426236C2semidirect
10. #21426231C2semidirect
11. #21425880C2semidirect
12. #21425878C2semidirect
13. C2×Dic59⋊C16C2semidirect
102. #135×C59Q8semidirect
290. C59#33626semidirect
293. #135Dic118semidirect

Maximal subgroups

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

subgroupindex
#214346082
#135×Dic592
#214262622
#214262582
#214262552
#214262512
#214262462
#214262412
#214262362
#214262312
#214258802
#214258782
C2×Dic59⋊C162
#214258032
C59×#21382
#3362659