Finite Group Databasebeta

← #28746268

Subgroups — #28746268

Order 15104.

464 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21436465C211yesmaximal, characteristic
2#21426372C211yesmaximal, characteristic, complemented
3#21426304C211yesmaximal, characteristic, complemented
4#21426301C211yesmaximal, characteristic, complemented
5#21426298C211yesmaximal, characteristic, complemented
6#21426296C211yesmaximal, characteristic, complemented
7#21426260C211yesmaximal, characteristic, complemented
8#21426257C211yesmaximal, characteristic, complemented
9#21426202C211yesmaximal, characteristic, complemented
10#21426202C211yesmaximal, characteristic, complemented
11C4×#14273673C211yesmaximal, characteristic, complemented
12#21425928C211yesmaximal, characteristic
13#21425885C211yesmaximal, characteristic, complemented
14#21425882C211yesmaximal, characteristic, complemented
15C59×#3336C211yesmaximal, characteristic
16#16838092—21no—
17C2×#14273673C2211yescharacteristic
18C2×#14273667C2211yescharacteristic
19C2×#14273667C2211yescharacteristic
20#16837804C2211yescharacteristic
21#16837744—21no—
22#16837733—21no—
23#16837710—21no—
24#16837643C2211yescharacteristic
25#16837643C2211yescharacteristic
26C4×Dic236C2211yescharacteristic
27#16837440C2211yescharacteristic
28C2×#14273661C2211yescharacteristic
29#16837229C2211yescharacteristic
30#16837229C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837227C2211yescharacteristic
34#16837227C2211yescharacteristic
35#16837227C2211yescharacteristic
36#16837227C2211yescharacteristic
37#16837227C2211yescharacteristic
38C4⋊(C59⋊C16)C2211yescharacteristic
39#16837206C2211yescharacteristic
40#16837206C2211yescharacteristic
41#16837206C2211yescharacteristic
42#16837205C2211yescharacteristic
43#16837205C2211yescharacteristic
44#16837205C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47C4×C59⋊C16C2211yescharacteristic
48C59×#552—21no—
49C59×#499C2211yescharacteristic
50C59×#485—21no—
51C59×#483—21no—
52C59×#479—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C59×#427C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×Dic236C2311yescharacteristic
60#14273720—41no—
61#14273719—21no—
62#14273717—41no—
63#14273708—41no—
64#14273706—21no—
65#14273706—21no—
66C4×Dic118—21no—
67#14273686—21no—
68#14273683—41no—
69C2×C4⋊Dic59—21no—
70#14273673—22no—
71#14273667—22no—
72#14273667—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#14273652—21no—
84#14273652—21no—
85#14273652—21no—
86C2×Q16×C59C2311yescharacteristic
87C22×C8×C59C2311yescharacteristic
88C22×C8×C59C2311yescharacteristic
89#126×C59—41no—
90#125×C59—21no—
91#125×C59—41no—
92#123×C59—41no—
93C22⋊Q8×C59—21no—
94C22⋊Q8×C59—21no—
95C4×Q8×C59—21no—
96C4×D4×C59D411yescharacteristic
97#117×C59—21no—
98#117×C59—41no—
99C2×C4⋊C4×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×D4×C59D811yescharacteristic
115C22×C4×C59—41no—
116C22×C4×C59C2×D411yescharacteristic
117C22×C4×C59C2×D411yescharacteristic
118C22×Dic59—41no—
119C2×Dic118—21no—
120Q16×C59—21no—
121Q16×C59—21no—
122C2×C8×C59—21no—
123C2×C8×C59—21no—
124C2×C8×C59—21no—
125C2×C8×C59—21no—
126C2×C8×C59C2×D411yescharacteristic
127C2×C8×C59C2×D411yescharacteristic
128C2×C8×C59C2×D411yescharacteristic
129C2×C8×C59C2411yescharacteristic
130C4⋊C4×C59—21no—
131C4⋊C4×C59—21no—
132C4⋊C4×C59—21no—
133C4⋊C4×C59—41no—
134C4⋊C4×C59—41no—
135C4⋊C4×C59—41no—
136C4⋊C4×C59—41no—
137C4⋊C4×C59—41no—
138C4⋊C4×C59—41no—
139C4⋊C4×C59D811yescharacteristic
140C22⋊C4×C59—41no—
141C22⋊C4×C59—41no—
142C22⋊C4×C59—41no—
143C22⋊C4×C59D811yescharacteristic
144C22⋊C4×C59D811yescharacteristic
145C42×C59—41no—
146C42×C59—41no—
147C42×C59C2×D411yescharacteristic
148C22⋊Dic59—41no—
149C22⋊Dic59—41no—
150C22⋊Dic59—41no—
151C4⋊Dic59—21no—
152C4⋊Dic59—21no—
153C4⋊Dic59—21no—
154C4⋊Dic59—41no—
155Dic59⋊C4—41no—
156Dic59⋊C4—41no—
157Dic59⋊C4—41no—
158Dic59⋊C4—41no—
159Dic59⋊C4—41no—
160C4×Dic59—41no—
161C4×Dic59—41no—
162Dic236—21no—
163Dic236—21no—
164C59⋊C16—21no—
165C59⋊C16—21no—
166C59⋊C16—21no—
167C59⋊C16#4112yes—
168C23×C59C2×D811yescharacteristic
169C23×C59C2×D811yescharacteristic
170Q8×C59—41no—
171Q8×C59—41no—
172D4×C59—41no—
173C2×C4×C59—21no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—41no—
178C2×C4×C59—41no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—82no—
182C2×C4×C59C2×D811yescharacteristic
183C2×C4×C59C2×D811yescharacteristic
184C2×C4×C59C2×D811yescharacteristic
185C2×C4×C59C2×D811yescharacteristic
186C2×C4×C59C22×D411yescharacteristic
187C2×Dic59—41no—
188C2×Dic59—41no—
189C2×Dic59—41no—
190C2×Dic59—41no—
191C2×Dic59—82no—
192Dic118—41no—
193Dic118—41no—
194C8×C59—21no—
195C8×C59—21no—
196C8×C59—21no—
197C8×C59C2×#4111yescharacteristic
198C8×C592−1+411yescharacteristic
199#33919—591nomaximal, Sylow
200C22×C59—21no—
201C22×C59—21no—
202C22×C59—21no—
203C22×C59—21no—
204C22×C59—21no—
205C22×C59—21no—
206C22×C59C22×D811yescharacteristic
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—81no—
212C4×C59—81no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59#54711yescharacteristic
216C4×C59#57811yescharacteristic
217Dic59—81no—
218Dic59—81no—
219Dic59—81no—
220Dic59—81no—
221#3336—591no—
222#3336D5911yescharacteristic
223#2253—591no—
224#2237—591no—
225#2237—591no—
226#2236—591no—
227#2236—591no—
228#2229—591no—
229#2229—591no—
230#2219—591no—
231#2219—591no—
232C4×Q32—591no—
233#2169—591no—
234#2159—591no—
235#2159—591no—
236C2×C59—41no—
237C2×C59—41no—
238C2×C59#333411yescharacteristic
239C2×C59#341411yescharacteristic
240C2×C59#341711yescharacteristic
241#552—21no—
242#552—1181no—
243C2×Q32—591no—
244C2×M32—591no—
245C2×M32—591no—
246#499—591no—
247#499C2×D5911yescharacteristic
248#485—21no—
249#485—1181no—
250#483—21no—
251#483—1181no—
252#479—21no—
253#479—1181no—
254#467—591no—
255#467—591no—
256#467C2×D5911yescharacteristic
257#467C2×D5911yescharacteristic
258C4×Q16—591no—
259C4×Q16C2×D5911yescharacteristic
260C8×D4C2×D5911yescharacteristic
261#427—591no—
262#427C2×D5911yescharacteristic
263C2×#107—591no—
264C2×#107C2×D5911yescharacteristic
265#367—591no—
266#367—591no—
267#367—591no—
268#367—591no—
269#366—591no—
270#366—591no—
271#366—591no—
272#366—591no—
273#366—591no—
274C4⋊C16—591no—
275#358—591no—
276#358—591no—
277#358—591no—
278#358—591no—
279#358—591no—
280#358—591no—
281C22⋊C16—591no—
282C22⋊C16—591no—
283C4×C16—591no—
284C59#3391911yescharacteristic, complemented, Sylow
285C2×Q16—591no—
286C2×Q16C22×D5911yescharacteristic
287C22×C8C22×D5911yescharacteristic
288C22×C8C22×D5911yescharacteristic
289#126—41no—
290#126—2361no—
291#125—21no—
292#125—41no—
293#125—1181no—
294#125—2361no—
295#123—41no—
296#123—2361no—
297C22⋊Q8—21no—
298C22⋊Q8—21no—
299C22⋊Q8—1181no—
300C22⋊Q8—1181no—
301C4×Q8—21no—
302C4×Q8—1181no—
303C4×D4C22⋊D5911yescharacteristic
304#117—21no—
305#117—41no—
306#117—1181no—
307#117—2361no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—1181no—
310Q32—1182no—
311M32—1182no—
312M32—1182no—
313C2×C16—591no—
314C2×C16—591no—
315C2×C16—591no—
316C2×C16—591no—
317#107—21no—
318#107—591no—
319#107—591no—
320#107—591no—
321#107—1181no—
322#107C22×D5911yescharacteristic
323#107C22×D5911yescharacteristic
324#107C22×D5911yescharacteristic
325#106—21no—
326#106—21no—
327#106—1181no—
328#106—1181no—
329C4⋊C8C22⋊D5911yescharacteristic
330#103—21no—
331#103—21no—
332#103—21no—
333#103—1181no—
334#103—1181no—
335#103—1181no—
336C22⋊C8C22⋊D5911yescharacteristic
337C22⋊C8C22⋊D5911yescharacteristic
338C4×C8C22×D5911yescharacteristic
339C2×Q8—21no—
340C2×Q8—1181no—
341C2×D4#1174566911yescharacteristic
342C22×C4—41no—
343C22×C4—2361no—
344C22×C4C2×C22⋊D5911yescharacteristic
345C22×C4C2×C22⋊D5911yescharacteristic
346Q16—21no—
347Q16—21no—
348Q16—1181no—
349Q16—1181no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8C2×C22⋊D5911yescharacteristic
355C2×C8C2×C22⋊D5911yescharacteristic
356C2×C8C2×C22⋊D5911yescharacteristic
357C2×C8C23×D5911yescharacteristic
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—1181no—
368C4⋊C4—1181no—
369C4⋊C4—1181no—
370C4⋊C4—2361no—
371C4⋊C4—2361no—
372C4⋊C4—2361no—
373C4⋊C4—2361no—
374C4⋊C4—2361no—
375C4⋊C4—2361no—
376C4⋊C4#1174566911yescharacteristic
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—2361no—
381C22⋊C4—2361no—
382C22⋊C4—2361no—
383C22⋊C4#1174566911yescharacteristic
384C22⋊C4#1174566911yescharacteristic
385C42—41no—
386C42—41no—
387C42—2361no—
388C42—2361no—
389C42C2×C22⋊D5911yescharacteristic
390C16—592no—
391C16—1181no—
392C16—1181no—
393C16—1181no—
394C23C2×#1174566911yescharacteristic
395C23C2×#1174566911yescharacteristic
396Q8—41no—
397Q8—41no—
398Q8—2361no—
399Q8—2361no—
400D4—41no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4—4722no—
411C2×C4—2361no—
412C2×C4—2361no—
413C2×C4—2361no—
414C2×C4—2361no—
415C2×C4C2×#1174566911yescharacteristic
416C2×C4C2×#1174566911yescharacteristic
417C2×C4C2×#1174566911yescharacteristic
418C2×C4C2×#1174566911yescharacteristic
419C2×C4C22×C22⋊D5911yescharacteristic
420C8—21no—
421C8—21no—
422C8—21no—
423C8#41×D5911yescharacteristic
424C8#1427377411yescharacteristic
425C22—21no—
426C22—21no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22C22×#1174566911yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—81no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—4721no—
441C4—4721no—
442C4—4721no—
443C4—4721no—
444C4#1683805111yescharacteristic
445C4#1683821711yescharacteristic
446C2—41no—
447C2—41no—
448C2#2143643911yescharacteristic
449C2#2143723011yescharacteristic
450C2#2143724711yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3336C2
socleC22×C59C22×D8
prime radicalC2×D4×C59D8
hypercenter#3336D59
hypocenterC59#33919
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #21426372C2semidirect
3. #21426304C2semidirect
4. #21426301C2semidirect
5. #21426298C2semidirect
6. #21426296C2semidirect
7. #21426260C2semidirect
8. #21426257C2semidirect
9. #21426202C2semidirect
10. #21426202C2semidirect
11. C4×#14273673C2semidirect
13. #21425885C2semidirect
14. #21425882C2semidirect
284. C59#33919semidirect

Maximal subgroups

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

subgroupindex
#214364652
#214263722
#214263042
#214263012
#214262982
#214262962
#214262602
#214262572
#214262022
C4×#142736732
#214259282
#214258852
#214258822
C59×#33362
#3391959