Finite Group Databasebeta

← #28743812

Subgroups — #28743812

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#21434746C211yesmaximal, characteristic, complemented
2#21426281C211yesmaximal, characteristic, complemented
3#21426276C211yesmaximal, characteristic, complemented
4#21426260C211yesmaximal, characteristic, complemented
5#21426253C211yesmaximal, characteristic, complemented
6#21426136C211yesmaximal, characteristic, complemented
7#21426131C211yesmaximal, characteristic, complemented
8#21426019C211yesmaximal, characteristic, complemented
9#21426019C211yesmaximal, characteristic, complemented
10#21425970C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12C4×#14273670C211yesmaximal, characteristic, complemented
13C4×#14273668C211yesmaximal, characteristic, complemented
14#21425903C211yesmaximal, characteristic, complemented
15C59×#2951C211yesmaximal, characteristic, complemented
16#16838007—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273670C2211yescharacteristic
19C2×#14273668C2211yescharacteristic
20C2×#14273668C2211yescharacteristic
21#16837804C2211yescharacteristic
22#16837779C2211yescharacteristic
23#16837767—21no—
24#16837655—21no—
25#16837638C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837515—21no—
28C4×D472C2211yescharacteristic
29C4×D472C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837227C2211yescharacteristic
33#16837227C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35C4⋊(C59⋊C16)C2211yescharacteristic
36#16837206C2211yescharacteristic
37#16837206C2211yescharacteristic
38#16837203C2211yescharacteristic
39#16837203C2211yescharacteristic
40#16837202C2211yescharacteristic
41#16837202C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44C22⋊(C59⋊C16)C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46#16837178C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C2×#135×C59C2211yescharacteristic
49C59×#520—21no—
50C59×#440—21no—
51C59×#440—21no—
52C4×Q16×C59C2211yescharacteristic
53C4×Q16×C59C2211yescharacteristic
54C4×D8×C59C2211yescharacteristic
55C4×D8×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C59×#417—21no—
58C59×#405C2211yescharacteristic
59C2×D472—21no—
60C2×D472C2311yescharacteristic
61C2×D472C2311yescharacteristic
62#14273724—21no—
63#14273719—21no—
64#14273712—41no—
65#14273712—41no—
66#14273701—21no—
67#14273701—21no—
68#14273700—41no—
69#14273700—41no—
70C4×D236—21no—
71C4×D236—21no—
72#14273670—22no—
73#14273670—22no—
74#14273668—22no—
75#14273668—22no—
76C2×C59⋊C16C2311yescharacteristic
77C2×C59⋊C16C2311yescharacteristic
78C2×C59⋊C16C2311yescharacteristic
79C2×C59⋊C16C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273661C2311yescharacteristic
82#14273659—21no—
83#14273649—21no—
84#14273649—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
93C2×M16×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—41no—
102#117×C59D411yescharacteristic
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59—41no—
105#107×C59—21no—
106#107×C59C2311yescharacteristic
107#107×C59C2311yescharacteristic
108#103×C59—21no—
109#103×C59—21no—
110#102×C59—21no—
111#102×C59—21no—
112#97×C59D411yescharacteristic
113#97×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×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C2×D236—21no—
129C2×D236—21no—
130C2×D236—41no—
131C2×D236—41no—
132C2×C4×D59—41no—
133C2×C4×D59—41no—
134Q16×C59—21no—
135Q16×C59#4112yes—
136SD16×C59—24no—
137D8×C59—21no—
138D8×C59#4112yes—
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×C59—41no—
151C4⋊C4×C59D811yescharacteristic
152C4⋊C4×C59D811yescharacteristic
153C22⋊C4×C59—41no—
154C22⋊C4×C59—41no—
155C22⋊C4×C59D811yescharacteristic
156C22⋊C4×C59D811yescharacteristic
157C42×C59—41no—
158C42×C59—41no—
159C42×C59C2×D411yescharacteristic
160C42×C59C2×D411yescharacteristic
161(C2×D59)⋊C4—41no—
162(C2×D59)⋊C4—41no—
163(C2×D59)⋊C4—41no—
164(C2×D59)⋊C4—41no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167Dic59⋊C4—41no—
168Dic59⋊C4—41no—
169D472—21no—
170D472—21no—
171D472—21no—
172D472—21no—
173D472—42no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C59⋊C16—21no—
177C59⋊C16—21no—
178C23×C59—41no—
179C23×C59C2×D811yescharacteristic
180C22×D59—41no—
181C22×D59—41no—
182Q8×C59—22no—
183Q8×C59—41no—
184D4×C59—22no—
185D4×C59—44no—
186D4×C59—41no—
187C2×C4×C59—21no—
188C2×C4×C59—21no—
189C2×C4×C59—44no—
190C2×C4×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—161no—
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—161no—
203C22⋊D59—161no—
204C2×Dic59—41no—
205C2×Dic59—41no—
206D236—41no—
207D236—41no—
208D236—41no—
209D236—41no—
210D236—161no—
211D236—161no—
212C4×D59—161no—
213C4×D59—161no—
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—42no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59C22×D811yescharacteristic
227C2×D59—81no—
228C2×D59—81no—
229C2×D59—81no—
230C2×D59—81no—
231C2×D59—81no—
232C2×D59—81no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—21no—
236C4×C59—21no—
237C4×C59—21no—
238C4×C59—42no—
239C4×C59—81no—
240C4×C59—81no—
241C4×C59#52511yescharacteristic
242C4×C59#57511yescharacteristic
243Dic59—81no—
244Dic59—81no—
245#3176—591no—
246#2951D5911yescharacteristic, complemented
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—82no—
262C2×C59#310011yescharacteristic
263C2×C59#341511yescharacteristic
264C2×C59#341611yescharacteristic
265D59—161no—
266D59—161no—
267C2×#135C2×D5911yescharacteristic
268#540—1181no—
269#520—21no—
270C2×SD32—591no—
271C2×SD32—591no—
272C2×D16—591no—
273C2×D16—591no—
274#499—591no—
275#493—591no—
276#490—1181no—
277#469—1181no—
278#466—591no—
279#466—591no—
280#442—1181no—
281#440—21no—
282#440—21no—
283C4×Q16C2×D5911yescharacteristic
284C4×Q16C2×D5911yescharacteristic
285C4×D8—591no—
286C4×D8—591no—
287C4×D8C2×D5911yescharacteristic
288C4×D8C2×D5911yescharacteristic
289C2×#107C2×D5911yescharacteristic
290#417—21no—
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—21no—
312#135—22no—
313#135—22no—
314C2×Q16C22×D5911yescharacteristic
315C2×SD16—21no—
316C2×D8—591no—
317C2×D8—591no—
318C2×D8—1181no—
319C2×D8C22×D5911yescharacteristic
320C2×M16C22⋊D5911yescharacteristic
321C22×C8C22×D5911yescharacteristic
322#127—1181no—
323#125—1181no—
324#123—2361no—
325#123—2361no—
326#121—1181no—
327#121—1181no—
328#121—2361no—
329#121—2361no—
330C4×Q8—21no—
331C4×Q8—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—41no—
335C4×D4—41no—
336C4×D4—1181no—
337C4×D4—1181no—
338#117—41no—
339#117C22⋊D5911yescharacteristic
340C2×C4⋊C4—21no—
341C2×C4⋊C4—41no—
342SD32—1182no—
343SD32—1182no—
344D16—1182no—
345D16—1182no—
346C2×C16—591no—
347C2×C16—591no—
348C2×C16—591no—
349C2×C16—591no—
350#107—21no—
351#107—591no—
352#107—591no—
353#107C22×D5911yescharacteristic
354#107C22×D5911yescharacteristic
355#106—1181no—
356#103—21no—
357#103—21no—
358#102—21no—
359#102—21no—
360#102—1181no—
361#102—1181no—
362#102—1181no—
363#102—1181no—
364#97C22⋊D5911yescharacteristic
365#97C22⋊D5911yescharacteristic
366C4×C8C22×D5911yescharacteristic
367C4×C8C22×D5911yescharacteristic
368#41—24no—
369#41—42no—
370C2×Q8—21no—
371C2×D4—21no—
372C2×D4—41no—
373C2×D4—1181no—
374C2×D4—1181no—
375C2×D4—2361no—
376C2×D4—2361no—
377C2×D4—2361no—
378C2×D4—2361no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—2361no—
384C22×C4—2361no—
385C22×C4C2×C22⋊D5911yescharacteristic
386Q16—21no—
387Q16#1174568512yes—
388SD16—24no—
389D8—21no—
390D8—1181no—
391D8—1181no—
392D8—1181no—
393D8—1181no—
394D8—2362no—
395D8#1174568512yes—
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—41no—
408C4⋊C4—1181no—
409C4⋊C4—1181no—
410C4⋊C4—2361no—
411C4⋊C4—2361no—
412C4⋊C4#1174566911yescharacteristic
413C4⋊C4#1174566911yescharacteristic
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—2361no—
417C22⋊C4—2361no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4#1174566911yescharacteristic
421C22⋊C4#1174566911yescharacteristic
422C42—41no—
423C42—41no—
424C42C2×C22⋊D5911yescharacteristic
425C42C2×C22⋊D5911yescharacteristic
426C16—1181no—
427C16—1181no—
428C16—1181no—
429C16—1181no—
430C23—41no—
431C23—2361no—
432C23—2361no—
433C23C2×#1174566911yescharacteristic
434Q8—22no—
435Q8—41no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—9441no—
440D4—9441no—
441D4—9441no—
442D4—9441no—
443D4—2361no—
444D4—2361no—
445D4—2361no—
446D4—2361no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—161no—
454C2×C4—161no—
455C2×C4—161no—
456C2×C4—9441no—
457C2×C4—9441no—
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—
469C8C2×#1174568511yescharacteristic
470C8C2×#1174568511yescharacteristic
471C22—21no—
472C22—21no—
473C22—21no—
474C22—42no—
475C22—81no—
476C22—81no—
477C22—4721no—
478C22—4721no—
479C22—4721no—
480C22—4721no—
481C22—4721no—
482C22—4721no—
483C22C22×#1174566911yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—81no—
491C4—81no—
492C4—4721no—
493C4—4721no—
494C4#1683791311yescharacteristic
495C4#1683820011yescharacteristic
496C2—41no—
497C2—82no—
498C2—9441no—
499C2—9441no—
500C2#2143409611yescharacteristic
501C2#2143723511yescharacteristic
502C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2951C2
socleC22×C59C22×D8
hypercenter#2951D59
hypocenterC59#33679
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #2951

Complements

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

NH ≅ G/Ntype
1. #21434746C2semidirect
2. #21426281C2semidirect
3. #21426276C2semidirect
4. #21426260C2semidirect
5. #21426253C2semidirect
6. #21426136C2semidirect
7. #21426131C2semidirect
8. #21426019C2semidirect
9. #21426019C2semidirect
10. #21425970C2semidirect
11. #21425958C2semidirect
12. C4×#14273670C2semidirect
13. C4×#14273668C2semidirect
14. #21425903C2semidirect
15. C59×#2951C2semidirect
246. #2951D59semidirect
310. C59#33679semidirect

Maximal subgroups

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

subgroupindex
#214347462
#214262812
#214262762
#214262602
#214262532
#214261362
#214261312
#214260192
#214259702
#214259582
C4×#142736702
C4×#142736682
#214259032
C59×#29512
#3367959