Finite Group Databasebeta

← #28745093

Subgroups — #28745093

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434755C211yesmaximal, characteristic, complemented
2#21426260C211yesmaximal, characteristic, complemented
3#21426253C211yesmaximal, characteristic, complemented
4#21426245C211yesmaximal, characteristic, complemented
5#21426235C211yesmaximal, characteristic, complemented
6#21426207C211yesmaximal, characteristic
7#21426202C211yesmaximal, characteristic, complemented
8#21426190C211yesmaximal, characteristic
9#21426185C211yesmaximal, characteristic, complemented
10#21426155C211yesmaximal, characteristic, complemented
11#21426155C211yesmaximal, characteristic, complemented
12#21426146C211yesmaximal, characteristic, complemented
13#21426141C211yesmaximal, characteristic, complemented
14#21425855C211yesmaximal, characteristic, complemented
15C59×#3175C211yesmaximal, characteristic
16#16838014—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273673C2211yescharacteristic
19C2×#14273671C2211yescharacteristic
20C2×#14273671C2211yescharacteristic
21C2×#14273667C2211yescharacteristic
22C22×C59⋊C16C2211yescharacteristic
23#16837804C2211yescharacteristic
24#16837782C2211yescharacteristic
25#16837771—21no—
26#16837660—21no—
27#16837643C2211yescharacteristic
28#16837643C2211yescharacteristic
29#16837510—21no—
30C4×Dic236C2211yescharacteristic
31C4×Dic236C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36C4⋊(C59⋊C16)C2211yescharacteristic
37C4⋊(C59⋊C16)C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39C4⋊(C59⋊C16)C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837205C2211yescharacteristic
45#16837205C2211yescharacteristic
46#16837203C2211yescharacteristic
47#16837203C2211yescharacteristic
48C59×#542—21no—
49C59×#500C2211yescharacteristic
50C59×#495C2211yescharacteristic
51C59×#489—21no—
52C59×#468—21no—
53C59×#467C2211yescharacteristic
54C59×#466C2211yescharacteristic
55C59×#440—21no—
56C4×Q16×C59C2211yescharacteristic
57C4×D8×C59C2211yescharacteristic
58C59×#405C2211yescharacteristic
59C2×Dic236—21no—
60C2×Dic236C2311yescharacteristic
61C2×Dic236C2311yescharacteristic
62#14273727—21no—
63#14273720—41no—
64#14273720—41no—
65#14273719—21no—
66#14273717—41no—
67#14273717—41no—
68#14273706—21no—
69#14273706—21no—
70C4×Dic118—21no—
71C4×Dic118—21no—
72#14273673—41no—
73#14273673—41no—
74#14273671—41no—
75#14273671—41no—
76#14273667—41no—
77C2×C59⋊C16—41no—
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#14273652—21no—
88#14273652—21no—
89C2×Q16×C59C2311yescharacteristic
90C2×SD16×C59—21no—
91C2×D8×C59C2311yescharacteristic
92C2×M16×C59D411yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C4⋊Q8×C59—21no—
95#126×C59—41no—
96#124×C59—21no—
97#124×C59—41no—
98#123×C59—41no—
99C22⋊Q8×C59—21no—
100C22⋊Q8×C59—41no—
101#121×C59—21no—
102C4×Q8×C59—21no—
103C4×D4×C59—21no—
104#117×C59D411yescharacteristic
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
116C2×Q8×C59—21no—
117C2×Q8×C59—41no—
118C2×D4×C59—21no—
119C2×D4×C59—41no—
120C22×C4×C59—41no—
121C22×C4×C59C2×D411yescharacteristic
122C2×Dic118—21no—
123C2×Dic118—21no—
124Q16×C59—21no—
125Q16×C59—21no—
126SD16×C59—42no—
127D8×C59—21no—
128D8×C59—21no—
129M16×C59—41no—
130C2×C8×C59—21no—
131C2×C8×C59—21no—
132C2×C8×C59C2×D411yescharacteristic
133C2×C8×C59C2×D411yescharacteristic
134C2×C8×C59C2×D411yescharacteristic
135C2×C8×C59C2411yescharacteristic
136C4⋊C4×C59—21no—
137C4⋊C4×C59—21no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59—41no—
141C4⋊C4×C59D811yescharacteristic
142C4⋊C4×C59D811yescharacteristic
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59—41no—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C22⋊C4×C59D811yescharacteristic
149C22⋊C4×C59D811yescharacteristic
150C42×C59—41no—
151C42×C59C2×D411yescharacteristic
152C42×C59C2×D411yescharacteristic
153C22⋊Dic59—41no—
154C22⋊Dic59—41no—
155C4⋊Dic59—21no—
156C4⋊Dic59—21no—
157C4⋊Dic59—41no—
158C4⋊Dic59—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164Dic59⋊C4—41no—
165C4×Dic59—41no—
166C4×Dic59—41no—
167Dic236—21no—
168Dic236—21no—
169Dic236—21no—
170Dic236—21no—
171Dic236—42no—
172C59⋊C16—21no—
173C59⋊C16—21no—
174C59⋊C16—21no—
175C59⋊C16—21no—
176C23×C59—41no—
177C23×C59C2×D811yescharacteristic
178Q8×C59—41no—
179Q8×C59—41no—
180Q8×C59—161no—
181D4×C59—41no—
182D4×C59—41no—
183D4×C59—161no—
184C2×C4×C59—21no—
185C2×C4×C59—21no—
186C2×C4×C59—41no—
187C2×C4×C59—41no—
188C2×C4×C59—41no—
189C2×C4×C59—161no—
190C2×C4×C59C2×D811yescharacteristic
191C2×C4×C59C2×D811yescharacteristic
192C2×C4×C59C2×D811yescharacteristic
193C2×C4×C59C2×D811yescharacteristic
194C2×C4×C59C2×D811yescharacteristic
195C2×C4×C59C22×D411yescharacteristic
196C2×Dic59—41no—
197C2×Dic59—41no—
198C2×Dic59—41no—
199C2×Dic59—41no—
200Dic118—41no—
201Dic118—41no—
202Dic118—41no—
203Dic118—41no—
204C8×C59—21no—
205C8×C59—21no—
206C8×C59—21no—
207C8×C59H(5,𝔽2)11yescharacteristic
208C8×C592−1+411yescharacteristic
209#33824—591nomaximal, Sylow
210C22×C59—21no—
211C22×C59—21no—
212C22×C59—21no—
213C22×C59—81no—
214C22×C59—81no—
215C22×C59—81no—
216C22×C59C22×D811yescharacteristic
217C4×C59—21no—
218C4×C59—21no—
219C4×C59—21no—
220C4×C59—21no—
221C4×C59—21no—
222C4×C59—81no—
223C4×C59—81no—
224C4×C59—81no—
225C4×C59#53711yescharacteristic
226C4×C59#57511yescharacteristic
227Dic59—81no—
228Dic59—81no—
229Dic59—81no—
230Dic59—81no—
231#3177—591no—
232#3175D5911yescharacteristic
233#2229—591no—
234#2228—591no—
235#2227—591no—
236#2225—591no—
237#2220—591no—
238#2219—591no—
239#2217—591no—
240#2216—591no—
241#2211—591no—
242#2211—591no—
243#2209—591no—
244#2208—591no—
245#2151—591no—
246C2×C59—41no—
247C2×C59—161no—
248C2×C59#324511yescharacteristic
249C2×C59#341611yescharacteristic
250C2×C59#341711yescharacteristic
251#542—21no—
252#541—1181no—
253C2×Q32—591no—
254C2×Q32—591no—
255C2×SD32—591no—
256C2×SD32—591no—
257C2×M32—591no—
258C22×C16—591no—
259#500C2×D5911yescharacteristic
260#499—591no—
261#495C2×D5911yescharacteristic
262#494—591no—
263#491—1181no—
264#489—21no—
265#470—1181no—
266#468—21no—
267#467—591no—
268#467—591no—
269#467C2×D5911yescharacteristic
270#466C2×D5911yescharacteristic
271#441—1181no—
272#440—21no—
273C4×Q16—591no—
274C4×Q16—591no—
275C4×Q16C2×D5911yescharacteristic
276C4×D8C2×D5911yescharacteristic
277#405C2×D5911yescharacteristic
278#367—591no—
279#367—591no—
280#366—591no—
281#366—591no—
282C4⋊C16—591no—
283C4⋊C16—591no—
284C4⋊C16—591no—
285C4⋊C16—591no—
286#358—591no—
287#358—591no—
288#358—591no—
289#358—591no—
290#358—591no—
291#358—591no—
292#357—591no—
293#357—591no—
294C59#3382411yescharacteristic, complemented, Sylow
295C2×Q16—591no—
296C2×Q16—591no—
297C2×Q16—1181no—
298C2×Q16C22×D5911yescharacteristic
299C2×SD16—21no—
300C2×D8C22×D5911yescharacteristic
301C2×M16C22⋊D5911yescharacteristic
302C22×C8C22×D5911yescharacteristic
303C4⋊Q8—21no—
304C4⋊Q8—1181no—
305#126—41no—
306#126—2361no—
307#126—2361no—
308#125—1181no—
309#125—2361no—
310#125—2361no—
311#124—21no—
312#124—41no—
313#123—41no—
314C22⋊Q8—21no—
315C22⋊Q8—41no—
316C22⋊Q8—1181no—
317C22⋊Q8—1181no—
318#121—21no—
319C4×Q8—21no—
320C4×Q8—1181no—
321C4×Q8—1181no—
322C4×D4—21no—
323#117C22⋊D5911yescharacteristic
324Q32—2361no—
325Q32—2361no—
326SD32—2361no—
327SD32—2361no—
328M32—2361no—
329C2×C16—591no—
330C2×C16—591no—
331C2×C16—591no—
332C2×C16—591no—
333C2×C16—2361no—
334#107—21no—
335#107—591no—
336#107—591no—
337#107C22×D5911yescharacteristic
338#107C22×D5911yescharacteristic
339#106—1181no—
340#103—21no—
341#103—21no—
342#103—1181no—
343#103—1181no—
344#103—1181no—
345#103—1181no—
346#102—21no—
347#102—21no—
348#97C22⋊D5911yescharacteristic
349#97C22⋊D5911yescharacteristic
350C4×C8C22×D5911yescharacteristic
351C4×C8C22×D5911yescharacteristic
352C2×Q8—21no—
353C2×Q8—41no—
354C2×Q8—1181no—
355C2×Q8—1181no—
356C2×D4—21no—
357C2×D4—41no—
358C22×C4—41no—
359C22×C4C2×C22⋊D5911yescharacteristic
360Q16—21no—
361Q16—21no—
362Q16—1181no—
363Q16—1181no—
364Q16—1181no—
365Q16—1181no—
366Q16—2362no—
367SD16—42no—
368D8—21no—
369D8—21no—
370M16—41no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8C2×C22⋊D5911yescharacteristic
374C2×C8C2×C22⋊D5911yescharacteristic
375C2×C8C2×C22⋊D5911yescharacteristic
376C2×C8C23×D5911yescharacteristic
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—1181no—
383C4⋊C4—1181no—
384C4⋊C4—2361no—
385C4⋊C4—2361no—
386C4⋊C4—2361no—
387C4⋊C4—2361no—
388C4⋊C4—2361no—
389C4⋊C4—2361no—
390C4⋊C4—2361no—
391C4⋊C4—2361no—
392C4⋊C4#1174566911yescharacteristic
393C4⋊C4#1174566911yescharacteristic
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—2361no—
400C22⋊C4—2361no—
401C22⋊C4#1174566911yescharacteristic
402C22⋊C4#1174566911yescharacteristic
403C42—41no—
404C42—2361no—
405C42—2361no—
406C42C2×C22⋊D5911yescharacteristic
407C42C2×C22⋊D5911yescharacteristic
408C16—1181no—
409C16—1181no—
410C16—1181no—
411C16—1181no—
412C23—41no—
413C23C2×#1174566911yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—161no—
417Q8—2361no—
418Q8—2361no—
419Q8—2361no—
420Q8—2361no—
421D4—41no—
422D4—41no—
423D4—161no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—161no—
430C2×C4—2361no—
431C2×C4—2361no—
432C2×C4—2361no—
433C2×C4—2361no—
434C2×C4C2×#1174566911yescharacteristic
435C2×C4C2×#1174566911yescharacteristic
436C2×C4C2×#1174566911yescharacteristic
437C2×C4C2×#1174566911yescharacteristic
438C2×C4C2×#1174566911yescharacteristic
439C2×C4C22×C22⋊D5911yescharacteristic
440C8—21no—
441C8—21no—
442C8—21no—
443C8#1427377311yescharacteristic
444C8#1427377411yescharacteristic
445C22—21no—
446C22—21no—
447C22—21no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22C22×#1174566911yescharacteristic
452C4—21no—
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—81no—
458C4—81no—
459C4—81no—
460C4—4721no—
461C4—4721no—
462C4—4721no—
463C4—4721no—
464C4#1683798011yescharacteristic
465C4#1683820011yescharacteristic
466C2—41no—
467C2—161no—
468C2#2143540311yescharacteristic
469C2#2143724311yescharacteristic
470C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3175C2
socleC22×C59C22×D8
prime radicalC59×#466C22
hypercenter#3175D59
hypocenterC59#33824
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3175

Complements

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

NH ≅ G/Ntype
1. #21434755C2semidirect
2. #21426260C2semidirect
3. #21426253C2semidirect
4. #21426245C2semidirect
5. #21426235C2semidirect
7. #21426202C2semidirect
9. #21426185C2semidirect
10. #21426155C2semidirect
11. #21426155C2semidirect
12. #21426146C2semidirect
13. #21426141C2semidirect
14. #21425855C2semidirect
294. C59#33824semidirect

Maximal subgroups

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

subgroupindex
#214347552
#214262602
#214262532
#214262452
#214262352
#214262072
#214262022
#214261902
#214261852
#214261552
#214261462
#214261412
#214258552
C59×#31752
#3382459