Finite Group Databasebeta

← #28743631

Subgroups — #28743631

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
1#21434666C211yesmaximal, characteristic, complemented
2#21426304C211yesmaximal, characteristic, complemented
3#21426287C211yesmaximal, characteristic, complemented
4#21426260C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426158C211yesmaximal, characteristic, complemented
7#21426151C211yesmaximal, characteristic, complemented
8#21426036C211yesmaximal, characteristic, complemented
9#21426019C211yesmaximal, characteristic
10C4×#14273673C211yesmaximal, characteristic, complemented
11C4×#14273670C211yesmaximal, characteristic, complemented
12C4×#14273671C211yesmaximal, characteristic, complemented
13C4×#14273668C211yesmaximal, characteristic, complemented
14#21425895C211yesmaximal, characteristic, complemented
15C4×#135×C59C211yesmaximal, characteristic, complemented
16#16837936—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273671C2211yescharacteristic
19C2×#14273670C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21#16837804C2211yescharacteristic
22#16837786C2211yescharacteristic
23#16837757—21no—
24#16837643C2211yescharacteristic
25#16837638C2211yescharacteristic
26#16837631—21no—
27C4×Dic236C2211yescharacteristic
28C4×#11745660—21no—
29C4×D472C2211yescharacteristic
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
48C2×#135×C59C2211yescharacteristic
49C4×#41×C59—21no—
50C4×Q16×C59C2211yescharacteristic
51C4×Q16×C59C2211yescharacteristic
52C4×SD16×C59—21no—
53C4×SD16×C59—21no—
54C4×D8×C59C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C59×#427C2211yescharacteristic
57C59×#416—21no—
58C2×C4×C8×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60C2×#11745660—21no—
61C2×D472C2311yescharacteristic
62#14273721—41no—
63#14273719—21no—
64#14273715—21no—
65#14273710—41no—
66#14273706—21no—
67#14273701—21no—
68C4×Dic118—21no—
69C4×D236—21no—
70C4×C22⋊D59—41no—
71#14273684—41no—
72#14273673—22no—
73#14273671—22no—
74#14273670—22no—
75#14273668—22no—
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273659—21no—
83#14273652—21no—
84#14273652—21no—
85#14273649—21no—
86#14273649—21no—
87C2×#41×C59—21no—
88#135×C59—22no—
89#135×C59—22no—
90C2×Q16×C59C2311yescharacteristic
91C2×SD16×C59—21no—
92C2×D8×C59C2311yescharacteristic
93C22×C8×C59D411yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C4×Q8×C59—21no—
96C4×Q8×C59—21no—
97C4×D4×C59—21no—
98C4×D4×C59—21no—
99C4×D4×C59—41no—
100C4×D4×C59—41no—
101#117×C59—21no—
102#117×C59—41no—
103C2×C42×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
116#41×C59—24no—
117#41×C59—42no—
118C2×Q8×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—41no—
121C22×C4×C59—41no—
122C22×C4×C59—41no—
123C22×C4×C59—41no—
124C22×C4×C59D811yescharacteristic
125C22×C4×C59D811yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C2×C22⋊D59—41no—
128C2×D236—21no—
129C2×C4×D59—41no—
130C2×Dic118—21no—
131Q16×C59—21no—
132Q16×C59#4112yes—
133SD16×C59—24no—
134D8×C59—21no—
135D8×C59#4112yes—
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—
147C22⋊C4×C59—41no—
148C22⋊C4×C59—41no—
149C22⋊C4×C59—41no—
150C42×C59—41no—
151C42×C59—41no—
152C42×C59—41no—
153C42×C59D811yescharacteristic
154C42×C59D811yescharacteristic
155C42×C59C2×D411yescharacteristic
156C42×C59C2×D411yescharacteristic
157C22⋊Dic59—41no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164Dic59⋊C4—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—41no—
167Dic59⋊C4—41no—
168C4×Dic59—41no—
169Dic236—21no—
170Dic236—21no—
171D472—21no—
172D472—21no—
173#11745660—42no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—41no—
179C23×C59C2×D811yescharacteristic
180C22×D59—41no—
181Q8×C59—22no—
182Q8×C59—41no—
183D4×C59—22no—
184D4×C59—44no—
185D4×C59—41no—
186C2×C4×C59—21no—
187C2×C4×C59—21no—
188C2×C4×C59—22no—
189C2×C4×C59—41no—
190C2×C4×C59—44no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—82no—
195C2×C4×C59—82no—
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C2×D811yescharacteristic
198C2×C4×C59C2×D811yescharacteristic
199C2×C4×C59C2×D811yescharacteristic
200C2×C4×C59C2×D811yescharacteristic
201C2×C4×C59C22×D411yescharacteristic
202C22⋊D59—82no—
203C2×Dic59—41no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206D236—41no—
207D236—41no—
208C4×D59—82no—
209Dic118—41no—
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—42no—
221C22×C59—81no—
222C22×C59—81no—
223C22×C59C22×D811yescharacteristic
224C2×D59—81no—
225C2×D59—81no—
226C2×D59—81no—
227C4×C59—21no—
228C4×C59—21no—
229C4×C59—21no—
230C4×C59—21no—
231C4×C59—42no—
232C4×C59—81no—
233C4×C59—81no—
234C4×C59#50812yes—
235C4×C59#52511yescharacteristic
236C4×C59#57511yescharacteristic
237Dic59—81no—
238Dic59—81no—
239Dic59—81no—
240#3168—591no—
241C4×#135D5911yescharacteristic, complemented
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—82no—
257C2×C59#310011yescharacteristic
258C2×C59C2×#50811yescharacteristic
259C2×C59#341411yescharacteristic
260D59—161no—
261C2×#135C2×D5911yescharacteristic
262#529—1181no—
263C4×#41—21no—
264C2×Q32—591no—
265C2×SD32—591no—
266C2×SD32—591no—
267C2×D16—591no—
268#499—591no—
269#495—591no—
270#488—1181no—
271#467—591no—
272#466—591no—
273#465—1181no—
274C4×Q16—591no—
275C4×Q16C2×D5911yescharacteristic
276C4×Q16C2×D5911yescharacteristic
277C4×SD16—21no—
278C4×SD16—21no—
279C4×SD16—1181no—
280C4×D8—591no—
281C4×D8C2×D5911yescharacteristic
282C4×D8C2×D5911yescharacteristic
283#427C2×D5911yescharacteristic
284#416—21no—
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—21no—
306#135—22no—
307#135—22no—
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—2361no—
317#125—1181no—
318#124—1181no—
319#123—2361no—
320C22⋊Q8—1181no—
321#121—1181no—
322C4×Q8—21no—
323C4×Q8—21no—
324C4×Q8—1181no—
325C4×D4—21no—
326C4×D4—21no—
327C4×D4—41no—
328C4×D4—41no—
329C4×D4—1181no—
330C4×D4—2361no—
331#117—21no—
332#117—41no—
333#117—2361no—
334C2×C42—41no—
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—24no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—1181no—
366C2×D4—21no—
367C2×D4—41no—
368C2×D4—1181no—
369C2×D4—2361no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—2361no—
374C22×C4#1174566911yescharacteristic
375C22×C4#1174566911yescharacteristic
376C22×C4C2×C22⋊D5911yescharacteristic
377Q16—21no—
378Q16—1181no—
379Q16—1181no—
380Q16#1174568512yes—
381SD16—24no—
382SD16—2362no—
383D8—21no—
384D8—1181no—
385D8—1181no—
386D8#1174568512yes—
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—1181no—
399C4⋊C4—1181no—
400C4⋊C4—2361no—
401C4⋊C4—2361no—
402C4⋊C4—2361no—
403C4⋊C4—2361no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—2361no—
408C22⋊C4—2361no—
409C22⋊C4—2361no—
410C22⋊C4—2361no—
411C22⋊C4—2361no—
412C42—41no—
413C42—41no—
414C42—41no—
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—22no—
428Q8—41no—
429Q8—2361no—
430Q8—2361no—
431D4—22no—
432D4—44no—
433D4—41no—
434D4—4722no—
435D4—2361no—
436D4—2361no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—4722no—
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—
460C8C2×#1174568511yescharacteristic
461C8C2×#1174568511yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—4721no—
469C22—4721no—
470C22—4721no—
471C22C22×#1174566911yescharacteristic
472C4—21no—
473C4—21no—
474C4—21no—
475C4—21no—
476C4—42no—
477C4—81no—
478C4—81no—
479C4—4721no—
480C4—4721no—
481C4—4721no—
482C4#1683783212yes—
483C4#1683791311yescharacteristic
484C4#1683820011yescharacteristic
485C2—41no—
486C2—82no—
487C2—9441no—
488C2#2143409611yescharacteristic
489C2C2×#1683783211yescharacteristic
490C2#2143723011yescharacteristic
subgroupG/H
centerC2×C4C2×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC4×#135×C59C2
socleC22×C59C22×D8
prime radical#21426019C2
hypercenterC4×#135D59
hypocenterC59#33662
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. #21434666C2semidirect
2. #21426304C2semidirect
3. #21426287C2semidirect
4. #21426260C2semidirect
5. #21426253C2semidirect
6. #21426158C2semidirect
7. #21426151C2semidirect
8. #21426036C2semidirect
10. C4×#14273673C2semidirect
11. C4×#14273670C2semidirect
12. C4×#14273671C2semidirect
13. C4×#14273668C2semidirect
14. #21425895C2semidirect
15. C4×#135×C59C2semidirect
241. C4×#135D59semidirect
304. C59#33662semidirect

Maximal subgroups

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

subgroupindex
#214346662
#214263042
#214262872
#214262602
#214262532
#214261582
#214261512
#214260362
#214260192
C4×#142736732
C4×#142736702
C4×#142736712
C4×#142736682
#214258952
C4×#135×C592
#3366259