Finite Group Databasebeta

← #28743625

Subgroups — #28743625

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273749C211yesmaximal, characteristic, complemented
2#21426301C211yesmaximal, characteristic, complemented
3#21426284C211yesmaximal, characteristic, complemented
4#21426257C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426155C211yesmaximal, characteristic, complemented
7#21426148C211yesmaximal, characteristic, complemented
8#21426034C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic
10C4×#14273673C211yesmaximal, characteristic, complemented
11C4×#14273670C211yesmaximal, characteristic, complemented
12C4×#14273671C211yesmaximal, characteristic, complemented
13C4×#14273668C211yesmaximal, characteristic, complemented
14#21425895C211yesmaximal, characteristic, complemented
15C59×#3168C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C4×#11745685—21no—
18C2×#14273673C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273670C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22C4×Dic236C2211yescharacteristic
23C4×Dic236C2211yescharacteristic
24C4×#11745660—21no—
25C4×#11745660—21no—
26C4×D472C2211yescharacteristic
27C4×D472C2211yescharacteristic
28#16837440C2211yescharacteristic
29#16837399—21no—
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35C4⋊(C59⋊C16)C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44C22⋊(C59⋊C16)C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C4×C59⋊C16C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#529—21no—
49C59×#499C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#488—21no—
52C59×#467C2211yescharacteristic
53C59×#466C2211yescharacteristic
54C59×#465—21no—
55C4×Q16×C59C2211yescharacteristic
56C4×SD16×C59—21no—
57C4×D8×C59C2211yescharacteristic
58C2×C4×C8×C59C2211yescharacteristic
59C2×#11745685—21no—
60#14273749—22no—
61#14273749—22no—
62C2×Dic236C2311yescharacteristic
63C2×#11745660—21no—
64C2×D472C2311yescharacteristic
65C4×Dic118—21no—
66C4×Dic118—21no—
67C4×D236—21no—
68C4×D236—21no—
69C4×C22⋊D59—41no—
70C4×C22⋊D59—41no—
71#14273686—21no—
72#14273684—41no—
73C42×D59—41no—
74#14273673—22no—
75#14273671—22no—
76#14273670—22no—
77#14273668—22no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661C2311yescharacteristic
83#14273661C2311yescharacteristic
84#14273659—21no—
85#14273652—21no—
86#14273652—21no—
87#14273649—21no—
88#14273649—21no—
89C2×Q16×C59C2311yescharacteristic
90C2×SD16×C59—21no—
91C2×D8×C59C2311yescharacteristic
92C22×C8×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94#126×C59—41no—
95#125×C59—21no—
96#124×C59—21no—
97#123×C59—41no—
98C22⋊Q8×C59—21no—
99#121×C59—21no—
100C4×Q8×C59—21no—
101C4×D4×C59—21no—
102C4×D4×C59—41no—
103#117×C59—41no—
104C2×C42×C59D411yescharacteristic
105#107×C59C2311yescharacteristic
106#107×C59C2311yescharacteristic
107#106×C59—21no—
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112C4×C8×C59D411yescharacteristic
113C4×C8×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C2×Q8×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—41no—
119C22×C4×C59—41no—
120C22×C4×C59D811yescharacteristic
121C22×C4×C59D811yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C2×C22⋊D59—41no—
124#11745685—24no—
125#11745685—42no—
126C2×D236—21no—
127C2×C4×D59—41no—
128C2×C4×D59—41no—
129C2×C4×D59—41no—
130C2×Dic118—21no—
131Q16×C59—21no—
132Q16×C59—21no—
133SD16×C59—42no—
134D8×C59—21no—
135D8×C59—21no—
136C2×C8×C59—21no—
137C2×C8×C59—21no—
138C2×C8×C59—41no—
139C2×C8×C59C2×D411yescharacteristic
140C2×C8×C59C2×D411yescharacteristic
141C2×C8×C59C2×D411yescharacteristic
142C2×C8×C59C2411yescharacteristic
143C4⋊C4×C59—21no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59—41no—
146C4⋊C4×C59—41no—
147C4⋊C4×C59—41no—
148C4⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C22⋊C4×C59—41no—
151C22⋊C4×C59—41no—
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C42×C59—41no—
155C42×C59D811yescharacteristic
156C42×C59D811yescharacteristic
157C42×C59C2×D411yescharacteristic
158C42×C59C2×D411yescharacteristic
159C22⋊Dic59—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166C4×Dic59—41no—
167C4×Dic59—41no—
168C4×Dic59—41no—
169Dic236—21no—
170Dic236#4112yes—
171D472—21no—
172D472#4112yes—
173#11745660—24no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—41no—
179C23×C59C2×D811yescharacteristic
180C22×D59—41no—
181Q8×C59—41no—
182Q8×C59—41no—
183D4×C59—41no—
184D4×C59—41no—
185D4×C59—82no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—22no—
189C2×C4×C59—41no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—82no—
194C2×C4×C59C2×D811yescharacteristic
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C22×D411yescharacteristic
200C22⋊D59—44no—
201C2×Dic59—41no—
202C2×Dic59—41no—
203C2×Dic59—41no—
204D236—22no—
205D236—41no—
206C4×D59—44no—
207C4×D59—82no—
208C4×D59—82no—
209Dic118—22no—
210Dic118—41no—
211C8×C59—21no—
212C8×C59—21no—
213C8×C59—21no—
214C8×C59C2×#4111yescharacteristic
215C8×C59C2×#4111yescharacteristic
216#33662—591nomaximal, Sylow
217C22×C59—21no—
218C22×C59—21no—
219C22×C59—21no—
220C22×C59—81no—
221C22×C59—81no—
222C22×C59—81no—
223C22×C59C22×D811yescharacteristic
224C2×D59—42no—
225C2×D59—81no—
226C2×D59—81no—
227C4×C59—21no—
228C4×C59—21no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—81no—
232C4×C59—81no—
233C4×C59—81no—
234C4×C59#50812yes—
235C4×C59#52511yescharacteristic
236C4×C59#57511yescharacteristic
237Dic59—42no—
238Dic59—81no—
239Dic59—81no—
240#3168D5911yescharacteristic, complemented
241C4×#135—591no—
242#2237—591no—
243#2234—591no—
244#2229—591no—
245#2228—591no—
246#2211—591no—
247#2210—591no—
248#2189—591no—
249#2186—591no—
250C4×Q32—591no—
251C4×SD32—591no—
252C4×SD32—591no—
253C4×D16—591no—
254#2162—591no—
255C2×C59—41no—
256C2×C59—161no—
257C2×C59#310011yescharacteristic
258C2×C59C2×#50811yescharacteristic
259C2×C59#341411yescharacteristic
260D59—82no—
261C2×#135—591no—
262#529—21no—
263C4×#41—1181no—
264C2×Q32—591no—
265C2×SD32—591no—
266C2×SD32—591no—
267C2×D16—591no—
268#499C2×D5911yescharacteristic
269#495C2×D5911yescharacteristic
270#488—21no—
271#467C2×D5911yescharacteristic
272#466C2×D5911yescharacteristic
273#465—21no—
274C4×Q16—591no—
275C4×Q16—591no—
276C4×Q16C2×D5911yescharacteristic
277C4×SD16—21no—
278C4×SD16—1181no—
279C4×SD16—1181no—
280C4×D8—591no—
281C4×D8—591no—
282C4×D8C2×D5911yescharacteristic
283#427—591no—
284#416—1181no—
285C2×C4×C8C2×D5911yescharacteristic
286#367—591no—
287#367—591no—
288#366—591no—
289#366—591no—
290C4⋊C16—591no—
291C4⋊C16—591no—
292#358—591no—
293#358—591no—
294#358—591no—
295#358—591no—
296#357—591no—
297#357—591no—
298#357—591no—
299#357—591no—
300C22⋊C16—591no—
301C22⋊C16—591no—
302C4×C16—591no—
303C4×C16—591no—
304C59#3366211yescharacteristic, complemented, Sylow
305C2×#41—1181no—
306#135—1182no—
307#135—1182no—
308C2×Q16—591no—
309C2×Q16C22×D5911yescharacteristic
310C2×SD16—21no—
311C2×SD16—1181no—
312C2×D8—591no—
313C2×D8C22×D5911yescharacteristic
314C22×C8C22⋊D5911yescharacteristic
315C22×C8C22×D5911yescharacteristic
316#126—41no—
317#125—21no—
318#124—21no—
319#123—41no—
320C22⋊Q8—21no—
321#121—21no—
322C4×Q8—21no—
323C4×Q8—1181no—
324C4×Q8—1181no—
325C4×D4—21no—
326C4×D4—41no—
327C4×D4—1181no—
328C4×D4—1181no—
329C4×D4—2361no—
330C4×D4—2361no—
331#117—41no—
332#117—1181no—
333#117—2361no—
334C2×C42—2361no—
335C2×C42C22⋊D5911yescharacteristic
336Q32—1182no—
337SD32—1182no—
338SD32—1182no—
339D16—1182no—
340C2×C16—591no—
341C2×C16—591no—
342C2×C16—591no—
343C2×C16—591no—
344#107—591no—
345#107—591no—
346#107C22×D5911yescharacteristic
347#107C22×D5911yescharacteristic
348#106—21no—
349#106—1181no—
350#103—21no—
351#103—21no—
352#103—1181no—
353#103—1181no—
354#102—21no—
355#102—21no—
356#102—1181no—
357#102—1181no—
358C4×C8C22⋊D5911yescharacteristic
359C4×C8C22⋊D5911yescharacteristic
360C4×C8C22×D5911yescharacteristic
361C4×C8C22×D5911yescharacteristic
362#41—1184no—
363#41—2362no—
364C2×Q8—21no—
365C2×Q8—1181no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1181no—
369C2×D4—2361no—
370C22×C4—41no—
371C22×C4—2361no—
372C22×C4—2361no—
373C22×C4—2361no—
374C22×C4#1174566911yescharacteristic
375C22×C4#1174566911yescharacteristic
376C22×C4C2×C22⋊D5911yescharacteristic
377Q16—21no—
378Q16—21no—
379Q16—592no—
380Q16—1181no—
381SD16—42no—
382SD16—1184no—
383D8—21no—
384D8—21no—
385D8—592no—
386D8—1181no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—41no—
390C2×C8C2×C22⋊D5911yescharacteristic
391C2×C8C2×C22⋊D5911yescharacteristic
392C2×C8C2×C22⋊D5911yescharacteristic
393C2×C8C23×D5911yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—1181no—
401C4⋊C4—1181no—
402C4⋊C4—2361no—
403C4⋊C4—2361no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—2361no—
410C22⋊C4—2361no—
411C22⋊C4—2361no—
412C42—41no—
413C42—2361no—
414C42—2361no—
415C42—2361no—
416C42#1174566911yescharacteristic
417C42#1174566911yescharacteristic
418C42C2×C22⋊D5911yescharacteristic
419C42C2×C22⋊D5911yescharacteristic
420C16—1181no—
421C16—1181no—
422C16—1181no—
423C16—1181no—
424C23—41no—
425C23—2361no—
426C23C2×#1174566911yescharacteristic
427Q8—41no—
428Q8—41no—
429Q8—1182no—
430Q8—2361no—
431D4—41no—
432D4—41no—
433D4—82no—
434D4—1182no—
435D4—2364no—
436D4—2361no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—82no—
445C2×C4—4722no—
446C2×C4—4722no—
447C2×C4—2364no—
448C2×C4—2361no—
449C2×C4—2361no—
450C2×C4—2361no—
451C2×C4C2×#1174566911yescharacteristic
452C2×C4C2×#1174566911yescharacteristic
453C2×C4C2×#1174566911yescharacteristic
454C2×C4C2×#1174566911yescharacteristic
455C2×C4C2×#1174566911yescharacteristic
456C2×C4C22×C22⋊D5911yescharacteristic
457C8—21no—
458C8—21no—
459C8—21no—
460C8#41×D5911yescharacteristic
461C8#41×D5911yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4721no—
469C22—4721no—
470C22—2362no—
471C22C22×#1174566911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—4721no—
480C4—4721no—
481C4—2362no—
482C4#1683783212yes—
483C4#1683790911yescharacteristic
484C4#1683820011yescharacteristic
485C2—41no—
486C2—161no—
487C2—4722no—
488C2#2143409211yescharacteristic
489C2C2×#1683783211yescharacteristic
490C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3168C2
socleC22×C59C22×D8
prime radical#21426017C2
hypercenter#3168D59
hypocenterC59#33662
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. C4×#14273749C2semidirect
2. #21426301C2semidirect
3. #21426284C2semidirect
4. #21426257C2semidirect
5. #21426250C2semidirect
6. #21426155C2semidirect
7. #21426148C2semidirect
8. #21426034C2semidirect
10. C4×#14273673C2semidirect
11. C4×#14273670C2semidirect
12. C4×#14273671C2semidirect
13. C4×#14273668C2semidirect
14. #21425895C2semidirect
15. C59×#3168C2semidirect
240. #3168D59semidirect
304. C59#33662semidirect

Maximal subgroups

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

subgroupindex
C4×#142737492
#214263012
#214262842
#214262572
#214262502
#214261552
#214261482
#214260342
#214260172
C4×#142736732
C4×#142736702
C4×#142736712
C4×#142736682
#214258952
C59×#31682
#3366259