Finite Group Databasebeta

← #28743806

Subgroups — #28743806

Order 15104.

558 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432973C211yesmaximal, characteristic, complemented
2#21426279C211yesmaximal, characteristic, complemented
3#21426274C211yesmaximal, characteristic, complemented
4#21426257C211yesmaximal, characteristic, complemented
5#21426250C211yesmaximal, characteristic, complemented
6#21426133C211yesmaximal, characteristic, complemented
7#21426128C211yesmaximal, characteristic, complemented
8#21426017C211yesmaximal, characteristic, complemented
9#21426017C211yesmaximal, characteristic, complemented
10#21425970C211yesmaximal, characteristic, complemented
11#21425960C211yesmaximal, characteristic, complemented
12C4×#14273671C211yesmaximal, characteristic, complemented
13C4×#14273668C211yesmaximal, characteristic, complemented
14#21425903C211yesmaximal, characteristic, complemented
15C59×#3176C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17#16837882—21no—
18C2×#14273671C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21C2×#14273668C2211yescharacteristic
22#16837504—21no—
23#16837504—21no—
24C4×Dic236C2211yescharacteristic
25C4×Dic236C2211yescharacteristic
26C4×D472C2211yescharacteristic
27C4×D472C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837404—21no—
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35C4⋊(C59⋊C16)C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837203C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837203C2211yescharacteristic
41#16837203C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44C22⋊(C59⋊C16)C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#540—21no—
49C59×#499C2211yescharacteristic
50C59×#493C2211yescharacteristic
51C59×#490—21no—
52C59×#469—21no—
53C59×#466C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C59×#442—21no—
56C4×D8×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C59×#405C2211yescharacteristic
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—
69(C22⋊D59)⋊C4—41no—
70(C22⋊D59)⋊C4—41no—
71#14273685—41no—
72C2×C4⋊Dic59—21no—
73C4⋊C4×D59—41no—
74#14273671—22no—
75#14273671—22no—
76#14273668—22no—
77#14273668—22no—
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80C2×C59⋊C16C2311yescharacteristic
81C2×C59⋊C16C2311yescharacteristic
82#14273661—21no—
83#14273661C2311yescharacteristic
84#14273661C2311yescharacteristic
85#14273652—21no—
86#14273652—21no—
87#14273649—21no—
88#14273649—21no—
89C2×D8×C59—21no—
90C2×D8×C59C2311yescharacteristic
91C2×D8×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94#127×C59—21no—
95#125×C59—21no—
96#123×C59—41no—
97#123×C59—41no—
98#121×C59—21no—
99#121×C59—21no—
100#121×C59—41no—
101#121×C59—41no—
102C4×D4×C59—21no—
103C4×D4×C59—21no—
104#117×C59D411yescharacteristic
105#107×C59C2311yescharacteristic
106#107×C59C2311yescharacteristic
107#106×C59—21no—
108#102×C59—21no—
109#102×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×C59D411yescharacteristic
114C4×C8×C59C2311yescharacteristic
115C4×C8×C59C2311yescharacteristic
116C2×D4×C59—21no—
117C2×D4×C59—21no—
118C2×D4×C59—41no—
119C2×D4×C59—41no—
120C2×D4×C59—41no—
121C2×D4×C59—41no—
122C22×C4×C59—41no—
123C22×C4×C59—41no—
124C22×C4×C59C2×D411yescharacteristic
125C2×C22⋊D59—41no—
126C22×Dic59—41no—
127#11745685—24no—
128#11745685—42no—
129C2×D236—21no—
130C2×C4×D59—41no—
131C2×C4×D59—41no—
132C2×C4×D59—41no—
133C2×Dic118—21no—
134D8×C59—21no—
135D8×C59—21no—
136D8×C59—21no—
137D8×C59—21no—
138D8×C59—42no—
139M16×C59—41no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59C2×D411yescharacteristic
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2411yescharacteristic
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—41no—
149C4⋊C4×C59—41no—
150C4⋊C4×C59D811yescharacteristic
151C4⋊C4×C59D811yescharacteristic
152C22⋊C4×C59—41no—
153C22⋊C4×C59—41no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59—41no—
156C22⋊C4×C59D811yescharacteristic
157C22⋊C4×C59D811yescharacteristic
158C42×C59C2×D411yescharacteristic
159C42×C59C2×D411yescharacteristic
160(C2×D59)⋊C4—41no—
161(C2×D59)⋊C4—41no—
162C4⋊Dic59—21no—
163C4⋊Dic59—21no—
164C4⋊Dic59—41no—
165Dic59⋊C4—41no—
166Dic59⋊C4—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×C59—41no—
180C23×C59C2×D811yescharacteristic
181C22×D59—41no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—41no—
185D4×C59—41no—
186D4×C59—161no—
187D4×C59—161no—
188D4×C59—161no—
189D4×C59—161no—
190C2×C4×C59—21no—
191C2×C4×C59—21no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—161no—
195C2×C4×C59—161no—
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—44no—
203C2×Dic59—41no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206C2×Dic59—161no—
207D236—22no—
208D236—41no—
209C4×D59—44no—
210C4×D59—161no—
211C4×D59—161no—
212Dic118—22no—
213Dic118—41no—
214C8×C59—21no—
215C8×C59—21no—
216C8×C59—21no—
217C8×C59C2×#4111yescharacteristic
218C8×C59C2×#4111yescharacteristic
219#33679—591nomaximal, Sylow
220C22×C59—21no—
221C22×C59—21no—
222C22×C59—21no—
223C22×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59—81no—
227C22×C59—81no—
228C22×C59—81no—
229C22×C59C22×D811yescharacteristic
230C2×D59—42no—
231C2×D59—81no—
232C2×D59—81no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—21no—
238C4×C59—81no—
239C4×C59—81no—
240C4×C59#52511yescharacteristic
241C4×C59#57511yescharacteristic
242Dic59—42no—
243Dic59—81no—
244Dic59—81no—
245#3176D5911yescharacteristic, complemented
246#2951—591no—
247#2233—591no—
248#2232—591no—
249#2229—591no—
250#2228—591no—
251#2207—591no—
252#2206—591no—
253#2186—591no—
254#2186—591no—
255#2177—591no—
256#2175—591no—
257C4×SD32—591no—
258C4×D16—591no—
259#2164—591no—
260C2×C59—41no—
261C2×C59—161no—
262C2×C59—161no—
263C2×C59#310011yescharacteristic
264C2×C59#341511yescharacteristic
265C2×C59#341611yescharacteristic
266D59—82no—
267C2×#135—591no—
268#540—21no—
269#520—1181no—
270C2×SD32—591no—
271C2×SD32—591no—
272C2×D16—591no—
273C2×D16—591no—
274#499C2×D5911yescharacteristic
275#493C2×D5911yescharacteristic
276#490—21no—
277#469—21no—
278#466C2×D5911yescharacteristic
279#466C2×D5911yescharacteristic
280#442—21no—
281#440—1181no—
282#440—1181no—
283C4×Q16—591no—
284C4×Q16—591no—
285C4×D8—591no—
286C4×D8—591no—
287C4×D8C2×D5911yescharacteristic
288C4×D8C2×D5911yescharacteristic
289C2×#107—591no—
290#417—1181no—
291#405C2×D5911yescharacteristic
292#367—591no—
293#367—591no—
294#366—591no—
295#366—591no—
296C4⋊C16—591no—
297C4⋊C16—591no—
298#358—591no—
299#358—591no—
300#357—591no—
301#357—591no—
302#357—591no—
303#357—591no—
304#357—591no—
305#357—591no—
306C22⋊C16—591no—
307C22⋊C16—591no—
308#346—591no—
309C4×C16—591no—
310C59#3367911yescharacteristic, complemented, Sylow
311C2×#41—1181no—
312#135—1182no—
313#135—1182no—
314C2×Q16—591no—
315C2×SD16—1181no—
316C2×D8—21no—
317C2×D8—591no—
318C2×D8C22×D5911yescharacteristic
319C2×D8C22×D5911yescharacteristic
320C2×M16C22⋊D5911yescharacteristic
321C22×C8C22×D5911yescharacteristic
322#127—21no—
323#125—21no—
324#123—41no—
325#123—41no—
326#121—21no—
327#121—21no—
328#121—41no—
329#121—41no—
330C4×Q8—1181no—
331C4×Q8—1181no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—1181no—
335C4×D4—1181no—
336C4×D4—2361no—
337C4×D4—2361no—
338#117—2361no—
339#117C22⋊D5911yescharacteristic
340C2×C4⋊C4—1181no—
341C2×C4⋊C4—2361no—
342SD32—1182no—
343SD32—1182no—
344D16—1182no—
345D16—1182no—
346C2×C16—591no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16—591no—
350#107—591no—
351#107—591no—
352#107—1181no—
353#107C22×D5911yescharacteristic
354#107C22×D5911yescharacteristic
355#106—21no—
356#103—1181no—
357#103—1181no—
358#102—21no—
359#102—21no—
360#102—21no—
361#102—21no—
362#102—1181no—
363#102—1181no—
364#97C22⋊D5911yescharacteristic
365#97C22⋊D5911yescharacteristic
366C4×C8C22×D5911yescharacteristic
367C4×C8C22×D5911yescharacteristic
368#41—1184no—
369#41—2362no—
370C2×Q8—1181no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—1181no—
378C2×D4—2361no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—2361no—
382C22×C4—2361no—
383C22×C4—2361no—
384C22×C4—2361no—
385C22×C4C2×C22⋊D5911yescharacteristic
386Q16—592no—
387Q16—1181no—
388SD16—1184no—
389D8—21no—
390D8—21no—
391D8—21no—
392D8—21no—
393D8—42no—
394D8—592no—
395D8—1181no—
396M16—41no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8C2×C22⋊D5911yescharacteristic
400C2×C8C2×C22⋊D5911yescharacteristic
401C2×C8C2×C22⋊D5911yescharacteristic
402C2×C8C23×D5911yescharacteristic
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—1181no—
408C4⋊C4—1181no—
409C4⋊C4—2361no—
410C4⋊C4—2361no—
411C4⋊C4—2361no—
412C4⋊C4#1174566911yescharacteristic
413C4⋊C4#1174566911yescharacteristic
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4#1174566911yescharacteristic
421C22⋊C4#1174566911yescharacteristic
422C42—2361no—
423C42—2361no—
424C42C2×C22⋊D5911yescharacteristic
425C42C2×C22⋊D5911yescharacteristic
426C16—1181no—
427C16—1181no—
428C16—1181no—
429C16—1181no—
430C23—41no—
431C23—41no—
432C23—2361no—
433C23C2×#1174566911yescharacteristic
434Q8—1182no—
435Q8—2361no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—161no—
441D4—161no—
442D4—161no—
443D4—161no—
444D4—1182no—
445D4—2364no—
446D4—2361no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—161no—
452C2×C4—161no—
453C2×C4—9441no—
454C2×C4—9441no—
455C2×C4—9441no—
456C2×C4—2364no—
457C2×C4—2361no—
458C2×C4—2361no—
459C2×C4—2361no—
460C2×C4C2×#1174566911yescharacteristic
461C2×C4C2×#1174566911yescharacteristic
462C2×C4C2×#1174566911yescharacteristic
463C2×C4C2×#1174566911yescharacteristic
464C2×C4C2×#1174566911yescharacteristic
465C2×C4C22×C22⋊D5911yescharacteristic
466C8—21no—
467C8—21no—
468C8—21no—
469C8#41×D5911yescharacteristic
470C8#41×D5911yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—4721no—
481C22—4721no—
482C22—2362no—
483C22C22×#1174566911yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—81no—
490C4—81no—
491C4—4721no—
492C4—4721no—
493C4—2362no—
494C4#1683790911yescharacteristic
495C4#1683820011yescharacteristic
496C2—41no—
497C2—161no—
498C2—161no—
499C2—4722no—
500C2#2143409211yescharacteristic
501C2#2143723511yescharacteristic
502C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3176C2
socleC22×C59C22×D8
hypercenter#3176D59
hypocenterC59#33679
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #21432973C2semidirect
2. #21426279C2semidirect
3. #21426274C2semidirect
4. #21426257C2semidirect
5. #21426250C2semidirect
6. #21426133C2semidirect
7. #21426128C2semidirect
8. #21426017C2semidirect
9. #21426017C2semidirect
10. #21425970C2semidirect
11. #21425960C2semidirect
12. C4×#14273671C2semidirect
13. C4×#14273668C2semidirect
14. #21425903C2semidirect
15. C59×#3176C2semidirect
245. #3176D59semidirect
310. C59#33679semidirect

Maximal subgroups

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

subgroupindex
#214329732
#214262792
#214262742
#214262572
#214262502
#214261332
#214261282
#214260172
#214259702
#214259602
C4×#142736712
C4×#142736682
#214259032
C59×#31762
#3367959