Finite Group Databasebeta

← #28746244

Subgroups — #28746244

Order 15104.

476 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436499C211yesmaximal, characteristic
2#21426379C211yesmaximal, characteristic, complemented
3#21426301C211yesmaximal, characteristic, complemented
4#21426291C211yesmaximal, characteristic, complemented
5#21426287C211yesmaximal, characteristic, complemented
6#21426287C211yesmaximal, characteristic, complemented
7#21426260C211yesmaximal, characteristic, complemented
8#21426240C211yesmaximal, characteristic, complemented
9#21426211C211yesmaximal, characteristic, complemented
10#21426194C211yesmaximal, characteristic, complemented
11#21425958C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425885C211yesmaximal, characteristic, complemented
14#21425878C211yesmaximal, characteristic, complemented
15C59×#3342C211yesmaximal, characteristic, complemented
16#16838098—21no—
17C2×#14273670C2211yescharacteristic
18C2×#14273667C2211yescharacteristic
19C22×C59⋊C16C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837727—21no—
22#16837721—21no—
23#16837703—21no—
24#16837638C2211yescharacteristic
25#16837638C2211yescharacteristic
26C4×D472C2211yescharacteristic
27#16837440C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36#16837227C2211yescharacteristic
37#16837227C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#556—21no—
49C59×#500C2211yescharacteristic
50C59×#485—21no—
51C59×#485—21no—
52C59×#475—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C59×#427C2211yescharacteristic
59C2×D472C2311yescharacteristic
60#14273722—41no—
61#14273719—21no—
62#14273709—41no—
63#14273701—21no—
64#14273701—21no—
65#14273699—41no—
66C4×D236—21no—
67(C22⋊D59)⋊C4—41no—
68#14273686—21no—
69C2×C4⋊Dic59—21no—
70#14273670—22no—
71#14273667—22no—
72C2×C59⋊C16—22no—
73C2×C59⋊C16C2311yescharacteristic
74C2×C59⋊C16C2311yescharacteristic
75C2×C59⋊C16C2311yescharacteristic
76C2×C59⋊C16C2311yescharacteristic
77#14273661—21no—
78#14273661C2311yescharacteristic
79#14273661C2311yescharacteristic
80#14273661C2311yescharacteristic
81#14273659—21no—
82#14273659—21no—
83#14273649—21no—
84#14273649—21no—
85#14273649—21no—
86C2×Q16×C59C2311yescharacteristic
87C22×C8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89C4⋊Q8×C59—21no—
90#126×C59—41no—
91#126×C59—41no—
92#124×C59—41no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C4×Q8×C59—21no—
96C4×Q8×C59—41no—
97C4×D4×C59D411yescharacteristic
98#117×C59—21no—
99#117×C59—21no—
100#107×C59—21no—
101#107×C59C2311yescharacteristic
102#107×C59C2311yescharacteristic
103#107×C59C2311yescharacteristic
104#106×C59—21no—
105#106×C59—21no—
106C4⋊C8×C59D411yescharacteristic
107#103×C59—21no—
108#103×C59—21no—
109#103×C59—21no—
110C22⋊C8×C59D411yescharacteristic
111C22⋊C8×C59D411yescharacteristic
112C4×C8×C59C2311yescharacteristic
113C2×Q8×C59—21no—
114C2×Q8×C59—41no—
115C2×D4×C59D811yescharacteristic
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D411yescharacteristic
118C2×C22⋊D59—41no—
119C2×C22⋊D59—41no—
120C22×Dic59—41no—
121C2×D236—21no—
122C2×C4×D59—41no—
123Q16×C59—21no—
124Q16×C59—21no—
125C2×C8×C59—21no—
126C2×C8×C59—21no—
127C2×C8×C59—21no—
128C2×C8×C59—21no—
129C2×C8×C59C2×D411yescharacteristic
130C2×C8×C59C2×D411yescharacteristic
131C2×C8×C59C2×D411yescharacteristic
132C2×C8×C59C2411yescharacteristic
133C4⋊C4×C59—21no—
134C4⋊C4×C59—21no—
135C4⋊C4×C59—21no—
136C4⋊C4×C59—41no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59—41no—
140C4⋊C4×C59D811yescharacteristic
141C22⋊C4×C59—41no—
142C22⋊C4×C59—41no—
143C22⋊C4×C59—41no—
144C22⋊C4×C59—41no—
145C22⋊C4×C59D811yescharacteristic
146C22⋊C4×C59D811yescharacteristic
147C42×C59—41no—
148C42×C59—41no—
149C42×C59—41no—
150C42×C59C2×D411yescharacteristic
151C22⋊Dic59—41no—
152(C2×D59)⋊C4—41no—
153(C2×D59)⋊C4—41no—
154(C2×D59)⋊C4—41no—
155C4⋊Dic59—21no—
156C4⋊Dic59—21no—
157C4⋊Dic59—21no—
158C4⋊Dic59—41no—
159Dic59⋊C4—41no—
160Dic59⋊C4—41no—
161C4×Dic59—41no—
162D472—21no—
163D472—21no—
164C59⋊C16—21no—
165C59⋊C16—21no—
166C59⋊C16—21no—
167C59⋊C16#4112yes—
168C23×C59C2×D811yescharacteristic
169C23×C59C2×D811yescharacteristic
170C22×D59—41no—
171Q8×C59—41no—
172Q8×C59—41no—
173Q8×C59—82no—
174D4×C59—41no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—21no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C2×D811yescharacteristic
187C2×C4×C59C22×D411yescharacteristic
188C22⋊D59—82no—
189C22⋊D59—161no—
190C2×Dic59—41no—
191C2×Dic59—41no—
192C2×Dic59—41no—
193C2×Dic59—161no—
194D236—41no—
195D236—41no—
196C4×D59—161no—
197C8×C59—21no—
198C8×C59—21no—
199C8×C59—21no—
200C8×C59C2×#4111yescharacteristic
201C8×C592−1+411yescharacteristic
202#33917—591nomaximal, Sylow
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59—21no—
207C22×C59—21no—
208C22×C59—21no—
209C22×C59C22×D811yescharacteristic
210C2×D59—81no—
211C2×D59—81no—
212C2×D59—81no—
213C4×C59—21no—
214C4×C59—21no—
215C4×C59—21no—
216C4×C59—21no—
217C4×C59—81no—
218C4×C59—81no—
219C4×C59—81no—
220C4×C59—81no—
221C4×C59#54711yescharacteristic
222C4×C59#57811yescharacteristic
223Dic59—81no—
224Dic59—81no—
225Dic59—81no—
226#3342D5911yescharacteristic, complemented
227#3339—591no—
228#2255—591no—
229#2237—591no—
230#2235—591no—
231#2234—591no—
232#2234—591no—
233#2229—591no—
234#2226—591no—
235#2220—591no—
236#2217—591no—
237#2175—591no—
238#2168—591no—
239#2159—591no—
240#2158—591no—
241C2×C59—41no—
242C2×C59—41no—
243C2×C59#333411yescharacteristic
244C2×C59#341411yescharacteristic
245C2×C59#341611yescharacteristic
246D59—161no—
247#556—21no—
248#553—1181no—
249C2×SD32—591no—
250C2×M32—591no—
251C22×C16—591no—
252#500C2×D5911yescharacteristic
253#499—591no—
254#485—21no—
255#485—21no—
256#482—1181no—
257#481—1181no—
258#478—1181no—
259#475—21no—
260#467C2×D5911yescharacteristic
261#467C2×D5911yescharacteristic
262#466—591no—
263#466—591no—
264C4×Q16C2×D5911yescharacteristic
265C4×D8—591no—
266C8×D4C2×D5911yescharacteristic
267#427—591no—
268#427C2×D5911yescharacteristic
269#427C2×D5911yescharacteristic
270C2×#107—591no—
271#367—591no—
272#367—591no—
273#367—591no—
274#367—591no—
275#367—591no—
276#366—591no—
277#366—591no—
278#366—591no—
279#366—591no—
280C4⋊C16—591no—
281#358—591no—
282#358—591no—
283#358—591no—
284#357—591no—
285#357—591no—
286#357—591no—
287C22⋊C16—591no—
288C22⋊C16—591no—
289#346—591no—
290C59#3391711yescharacteristic, complemented, Sylow
291C2×Q16C22×D5911yescharacteristic
292C2×D8—591no—
293C22×C8C22×D5911yescharacteristic
294C22×C8C22×D5911yescharacteristic
295C4⋊Q8—21no—
296#126—41no—
297#126—41no—
298#126—2361no—
299#125—1181no—
300#124—41no—
301#123—2361no—
302C22⋊Q8—21no—
303C22⋊Q8—21no—
304#121—1181no—
305#121—1181no—
306#121—2361no—
307C4×Q8—21no—
308C4×Q8—41no—
309C4×D4—1181no—
310C4×D4—2361no—
311C4×D4C22⋊D5911yescharacteristic
312#117—21no—
313#117—21no—
314#117—1181no—
315C2×C4⋊C4—1181no—
316SD32—1182no—
317M32—1182no—
318C2×C16—591no—
319C2×C16—591no—
320C2×C16—591no—
321C2×C16—591no—
322C2×C16—1182no—
323#107—21no—
324#107—591no—
325#107—591no—
326#107—591no—
327#107—1181no—
328#107C22×D5911yescharacteristic
329#107C22×D5911yescharacteristic
330#107C22×D5911yescharacteristic
331#106—21no—
332#106—21no—
333#106—1181no—
334#106—1181no—
335C4⋊C8C22⋊D5911yescharacteristic
336#103—21no—
337#103—21no—
338#103—21no—
339#102—1181no—
340#102—1181no—
341#102—1181no—
342C22⋊C8C22⋊D5911yescharacteristic
343C22⋊C8C22⋊D5911yescharacteristic
344C4×C8C22×D5911yescharacteristic
345C2×Q8—21no—
346C2×Q8—41no—
347C2×D4—1181no—
348C2×D4—2361no—
349C2×D4—2361no—
350C2×D4#1174566911yescharacteristic
351C22×C4—2361no—
352C22×C4—2361no—
353C22×C4C2×C22⋊D5911yescharacteristic
354C22×C4C2×C22⋊D5911yescharacteristic
355Q16—21no—
356Q16—21no—
357D8—1181no—
358D8—1181no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8C2×C22⋊D5911yescharacteristic
364C2×C8C2×C22⋊D5911yescharacteristic
365C2×C8C2×C22⋊D5911yescharacteristic
366C2×C8C23×D5911yescharacteristic
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—1181no—
375C4⋊C4—1181no—
376C4⋊C4—1181no—
377C4⋊C4—2361no—
378C4⋊C4—2361no—
379C4⋊C4—2361no—
380C4⋊C4#1174566911yescharacteristic
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—2361no—
386C22⋊C4—2361no—
387C22⋊C4—2361no—
388C22⋊C4—2361no—
389C22⋊C4#1174566911yescharacteristic
390C22⋊C4#1174566911yescharacteristic
391C42—41no—
392C42—41no—
393C42—41no—
394C42—2361no—
395C42C2×C22⋊D5911yescharacteristic
396C16—592no—
397C16—1181no—
398C16—1181no—
399C16—1181no—
400C23—2361no—
401C23C2×#1174566911yescharacteristic
402C23C2×#1174566911yescharacteristic
403Q8—41no—
404Q8—41no—
405Q8—82no—
406D4—41no—
407D4—9441no—
408D4—4722no—
409D4—2361no—
410D4—2361no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—9441no—
420C2×C4—9441no—
421C2×C4—2361no—
422C2×C4—2361no—
423C2×C4—2361no—
424C2×C4C2×#1174566911yescharacteristic
425C2×C4C2×#1174566911yescharacteristic
426C2×C4C2×#1174566911yescharacteristic
427C2×C4C2×#1174566911yescharacteristic
428C2×C4C22×C22⋊D5911yescharacteristic
429C8—21no—
430C8—21no—
431C8—21no—
432C8#41×D5911yescharacteristic
433C8#1427377411yescharacteristic
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—4721no—
441C22—4721no—
442C22—4721no—
443C22C22×#1174566911yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—81no—
452C4—4721no—
453C4—4721no—
454C4—4721no—
455C4#1683805111yescharacteristic
456C4#1683821711yescharacteristic
457C2—41no—
458C2—41no—
459C2—9441no—
460C2#2143643911yescharacteristic
461C2#2143723011yescharacteristic
462C2#2143724011yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3342C2
socleC22×C59C22×D8
prime radical#21436499C2
hypercenter#3342D59
hypocenterC59#33917
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3342

Complements

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

NH ≅ G/Ntype
2. #21426379C2semidirect
3. #21426301C2semidirect
4. #21426291C2semidirect
5. #21426287C2semidirect
6. #21426287C2semidirect
7. #21426260C2semidirect
8. #21426240C2semidirect
9. #21426211C2semidirect
10. #21426194C2semidirect
11. #21425958C2semidirect
12. #21425922C2semidirect
13. #21425885C2semidirect
14. #21425878C2semidirect
15. C59×#3342C2semidirect
226. #3342D59semidirect
290. C59#33917semidirect

Maximal subgroups

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

subgroupindex
#214364992
#214263792
#214263012
#214262912
#214262872
#214262602
#214262402
#214262112
#214261942
#214259582
#214259222
#214258852
#214258782
C59×#33422
#3391759