Finite Group Databasebeta

← #28743496

Subgroups — #28743496

Order 15104.

628 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433673C211yesmaximal, characteristic
2C2×#107×D59C211yesmaximal, characteristic, complemented
3#21426297C212yesmaximal, complemented
4#21426292C212yesmaximal, complemented
5#21426280C212yesmaximal, complemented
6#21426275C212yesmaximal, complemented
7#21425884C211yesmaximal, characteristic, complemented
8#21425882C211yesmaximal, characteristic, complemented
9#21425808C211yesmaximal, characteristic, complemented
10#21425706C211yesmaximal, characteristic, complemented
11C59×#2139C211yesmaximal, characteristic, complemented
12C22×C8×D59C2211yescharacteristic
13#16837909—21no—
14C2×C4⋊C4×D59—21no—
15C2×#14273667C2211yescharacteristic
16#16837640C2211yescharacteristic
17#16837640C2211yescharacteristic
18#16837635C2211yescharacteristic
19#16837635C2211yescharacteristic
20#16837628—21no—
21#16837628—21no—
22#16837440C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24C2×#14273660—21no—
25#107×D59—21no—
26#107×D59—21no—
27#107×D59C2212yes—
28#107×D59C2212yes—
29#16837398—21no—
30#16837230C2212yescomplemented
31#16837229C2212yescomplemented
32#16837228C2212yescomplemented
33#16837227C2212yescomplemented
34#16837206C2212yescomplemented
35#16837203C2212yescomplemented
36(C2×D59)⋊C16C2212yes—
37(C2×D59)⋊C16C2212yes—
38C2×#135×C59C2211yescharacteristic
39C2×M32×C59C2211yescharacteristic
40C2×#107×C59C2211yescharacteristic
41C59×#358C2212yescomplemented
42C59×#357C2212yescomplemented
43C22×C4×D59—41no—
44C22×C4×D59D411yescharacteristic
45C22×C59⋊C8D411yescharacteristic
46C2×C8×D59—22no—
47C2×C8×D59—22no—
48C2×C8×D59D411yescharacteristic
49C2×C8×D59D411yescharacteristic
50C2×C8×D59C2311yescharacteristic
51C2×C8×D59C2311yescharacteristic
52#14273708—41no—
53#14273703—21no—
54#14273703—21no—
55#14273698—21no—
56#14273698—21no—
57#14273695—41no—
58C4×C22⋊D59—41no—
59C4×C22⋊D59—41no—
60#14273686—21no—
61C2×C4⋊Dic59—21no—
62C2×Dic59⋊C4—41no—
63C4⋊C4×D59—22no—
64C4⋊C4×D59—22no—
65C4⋊C4×D59—41no—
66C4⋊C4×D59—41no—
67#14273667—22no—
68C2×C59⋊C16C2312yes—
69#14273661—21no—
70#14273661C2311yescharacteristic
71#14273661C2311yescharacteristic
72#14273661C2312yes—
73#14273660—22no—
74#14273660—22no—
75#14273659—21no—
76#14273651—21no—
77#14273651—21no—
78#14273648—21no—
79#14273648—21no—
80C2×#41×C59—21no—
81#135×C59—22no—
82#135×C59—22no—
83C2×Q16×C59C2311yescharacteristic
84C2×SD16×C59—21no—
85C2×D8×C59C2311yescharacteristic
86C22×C8×C59C2311yescharacteristic
87C2×C4⋊C4×C59—21no—
88M32×C59—22no—
89C2×C16×C59C2312yes—
90#107×C59—21no—
91#107×C59C2312yes—
92C23×D59D811yescharacteristic
93#41×C59—42no—
94#41×C59—42no—
95C2×Q8×C59—21no—
96C2×D4×C59—21no—
97C2×D4×C59—41no—
98C22×C4×C59—41no—
99C22×C4×C59—41no—
100C22×C4×C59C2×D411yescharacteristic
101C2×C22⋊D59—41no—
102C2×C22⋊D59—41no—
103C22×Dic59—41no—
104C22×Dic59D811yescharacteristic
105C2×C4×D59—22no—
106C2×C4×D59—22no—
107C2×C4×D59—42no—
108C2×C4×D59—42no—
109C2×C4×D59—84no—
110C2×C4×D59D811yescharacteristic
111C2×C4×D59D811yescharacteristic
112C2×C4×D59C2×D411yescharacteristic
113C2×C4×D59C2×D411yescharacteristic
114Q16×C59—21no—
115Q16×C59—21no—
116SD16×C59—42no—
117D8×C59—21no—
118D8×C59—21no—
119C2×C8×C59C2×D411yescharacteristic
120C2×C8×C59#4112yes—
121C2×C8×C59#4112yes—
122C2×C8×C59C2411yescharacteristic
123C4⋊C4×C59—22no—
124C4⋊C4×C59—41no—
125C22⋊Dic59—41no—
126C22⋊Dic59—41no—
127C22⋊Dic59—41no—
128(C2×D59)⋊C4—41no—
129(C2×D59)⋊C4—41no—
130C4⋊Dic59—21no—
131C4⋊Dic59—21no—
132C4⋊Dic59—22no—
133C4⋊Dic59—41no—
134Dic59⋊C4—42no—
135Dic59⋊C4—42no—
136Dic59⋊C4—41no—
137Dic59⋊C4—41no—
138C4×Dic59—41no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8C2×D411yescharacteristic
141C2×C59⋊C8C2×D411yescharacteristic
142C8×D59—22no—
143C8×D59—22no—
144C8×D59—21no—
145C8×D59—21no—
146C8×D59—21no—
147C8×D59—21no—
148C16×C59—22no—
149C59⋊C16—22no—
150C23×C59—41no—
151C23×C59C2×D811yescharacteristic
152C22×D59—22no—
153C22×D59—22no—
154C22×D59—22no—
155C22×D59C2×D811yescharacteristic
156C22×D59C2×D811yescharacteristic
157Q8×C59—41no—
158Q8×C59—41no—
159D4×C59—41no—
160D4×C59—41no—
161D4×C59—82no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—82no—
165C2×C4×C59—82no—
166C2×C4×C59#12312yes—
167C2×C4×C59C2×D811yescharacteristic
168C2×C4×C59#13712yes—
169C2×C4×C59C22×D411yescharacteristic
170C22⋊D59—82no—
171C22⋊D59—82no—
172C2×Dic59—22no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—42no—
176C2×Dic59—82no—
177C2×Dic59C2×D811yescharacteristic
178C2×Dic59C2×D811yescharacteristic
179C4×D59—21no—
180C4×D59—21no—
181C4×D59—21no—
182C4×D59—21no—
183C4×D59—22no—
184C4×D59—22no—
185C4×D59—84no—
186C4×D59—84no—
187C8×C59#12312yes—
188C8×C59C2×#4111yescharacteristic
189C8×C59C2×#4111yescharacteristic
190C59⋊C8—21no—
191C59⋊C8—21no—
192#33646—591nomaximal, Sylow
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59#48012yes—
199C22×C59C22×D811yescharacteristic
200C2×D59—28no—
201C2×D59—21no—
202C2×D59—21no—
203C2×D59—21no—
204C2×D59—21no—
205C2×D59#50912yes—
206C2×D59#51012yes—
207C4×C59—81no—
208C4×C59—82no—
209C4×C59#48012yes—
210C4×C59C2×#12311yescharacteristic
211C4×C59C2×#13711yescharacteristic
212Dic59—21no—
213Dic59—21no—
214Dic59—82no—
215Dic59—81no—
216Dic59—81no—
217#3050—591no—
218C22×#107—591no—
219#2236—592no—
220#2235—592no—
221#2233—592no—
222#2232—592no—
223#2159—591no—
224#2159—591no—
225#2139—591no—
226#2139D5911yescharacteristic, complemented
227#2112—591no—
228C2×C59—22no—
229C2×C59—161no—
230C2×C59C2×#48011yescharacteristic
231C2×C59C2×#50911yescharacteristic
232C2×C59C2×#51011yescharacteristic
233D59—22no—
234D59—22no—
235C2×#135C2×D5911yescharacteristic
236C23×C8—591no—
237#525—1181no—
238C22×C4⋊C4—1181no—
239C2×M32—591no—
240C2×M32C2×D5911yescharacteristic
241#467—591no—
242#467—591no—
243#466—591no—
244#466—591no—
245#465—1181no—
246#465—1181no—
247#427—591no—
248C2×#107—591no—
249C2×#107—592no—
250C2×#107—592no—
251C2×#107—1181no—
252C2×#107—1181no—
253C2×#107—1181no—
254C2×#107C2×D5911yescharacteristic
255#416—1181no—
256#367—592no—
257#367—592no—
258#366—592no—
259#366—592no—
260#358—592no—
261#358C2×D5912yescomplemented
262#357—592no—
263#357C2×D5912yescomplemented
264C22⋊C16—592no—
265C22⋊C16—592no—
266C59#3364611yescharacteristic, complemented, Sylow
267C2×#41—21no—
268C23×C4—591no—
269C23×C4—2361no—
270#135—22no—
271#135—22no—
272C2×Q16C22×D5911yescharacteristic
273C2×SD16—21no—
274C2×D8C22×D5911yescharacteristic
275C22×C8—591no—
276C22×C8—591no—
277C22×C8—591no—
278C22×C8—591no—
279C22×C8—591no—
280C22×C8—1182no—
281C22×C8—1182no—
282C22×C8C22×D5911yescharacteristic
283#123—2361no—
284C22⋊Q8—1181no—
285C22⋊Q8—1181no—
286#121—1181no—
287#121—1181no—
288#120—2361no—
289C4×D4—2361no—
290C4×D4—2361no—
291#117—1181no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—1182no—
294C2×C4⋊C4—1182no—
295C2×C4⋊C4—1181no—
296C2×C4⋊C4—2361no—
297C2×C4⋊C4—2361no—
298C2×C4⋊C4—2361no—
299M32—22no—
300M32—1182no—
301C2×C16—592no—
302C2×C16C22×D5912yes—
303#107—21no—
304#107—592no—
305#107—591no—
306#107—591no—
307#107—1182no—
308#107—1182no—
309#107—1181no—
310#107C22×D5912yes—
311#106—1181no—
312#103—1181no—
313#103—1181no—
314#102—1181no—
315#102—1181no—
316C24—591no—
317#41—42no—
318#41—42no—
319C2×Q8—21no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4—2361no—
323C2×D4—2361no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—591no—
327C22×C4—591no—
328C22×C4—591no—
329C22×C4—591no—
330C22×C4—591no—
331C22×C4—1182no—
332C22×C4—1182no—
333C22×C4—4724no—
334C22×C4—2362no—
335C22×C4—2362no—
336C22×C4—2361no—
337C22×C4D4×D5911yescharacteristic
338Q16—21no—
339Q16—21no—
340SD16—42no—
341D8—21no—
342D8—21no—
343C2×C8—591no—
344C2×C8—591no—
345C2×C8—1182no—
346C2×C8—1182no—
347C2×C8—1182no—
348C2×C8—1181no—
349C2×C8—1181no—
350C2×C8—1181no—
351C2×C8—1181no—
352C2×C8#1174568512yes—
353C2×C8D4×D5911yescharacteristic
354C2×C8#1174568712yes—
355C2×C8C23×D5911yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—41no—
358C4⋊C4—1181no—
359C4⋊C4—1181no—
360C4⋊C4—1182no—
361C4⋊C4—2362no—
362C4⋊C4—2362no—
363C4⋊C4—2361no—
364C4⋊C4—2361no—
365C4⋊C4—2361no—
366C22⋊C4—2361no—
367C22⋊C4—2361no—
368C22⋊C4—2361no—
369C22⋊C4—2361no—
370C22⋊C4—2361no—
371C42—2361no—
372C16—22no—
373C16—1182no—
374C23—41no—
375C23—591no—
376C23—591no—
377C23—1182no—
378C23—1182no—
379C23—1182no—
380C23D8×D5911yescharacteristic
381Q8—41no—
382Q8—41no—
383D4—41no—
384D4—41no—
385D4—82no—
386D4—4722no—
387D4—4722no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—591no—
393C2×C4—591no—
394C2×C4—1182no—
395C2×C4—1181no—
396C2×C4—1181no—
397C2×C4—1181no—
398C2×C4—1181no—
399C2×C4—1182no—
400C2×C4—1182no—
401C2×C4—4724no—
402C2×C4—4724no—
403C2×C4—4722no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2362no—
407C2×C4#1427370912yes—
408C2×C4D8×D5911yescharacteristic
409C2×C4#1427375612yes—
410C2×C4C2×D4×D5911yescharacteristic
411C8—1181no—
412C8—1181no—
413C8#1427370912yes—
414C8C2×#1174568711yescharacteristic
415C8C2×#1174568511yescharacteristic
416C22—21no—
417C22—21no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—592no—
422C22—592no—
423C22—1188no—
424C22—1181no—
425C22—1181no—
426C22—1181no—
427C22—1181no—
428C22#1683771412yes—
429C22C2×D8×D5911yescharacteristic
430C4—82no—
431C4—81no—
432C4—1181no—
433C4—1181no—
434C4—4722no—
435C4—4721no—
436C4—4721no—
437C4#1683771412yes—
438C4C2×#1427370911yescharacteristic
439C4C2×#1427375611yescharacteristic
440C2—22no—
441C2—161no—
442C2—1182no—
443C2—1182no—
444C2C2×#1683771411yescharacteristic
445C2#509×D5911yescharacteristic
446C2#510×D5911yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#2139C2
socleC22×C59C22×D8
prime radical#21433673C2
hypercenter#2139D59
hypocenterC59#33646
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2139

Complements

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

NH ≅ G/Ntype
2. C2×#107×D59C2semidirect
3. #21426297C2semidirect
4. #21426292C2semidirect
5. #21426280C2semidirect
6. #21426275C2semidirect
7. #21425884C2semidirect
8. #21425882C2semidirect
9. #21425808C2semidirect
10. #21425706C2semidirect
11. C59×#2139C2semidirect
30. #16837230C22semidirect
31. #16837229C22semidirect
32. #16837228C22semidirect
33. #16837227C22semidirect
34. #16837206C22semidirect
35. #16837203C22semidirect
41. C59×#358C22semidirect
42. C59×#357C22semidirect
226. #2139D59semidirect
261. #358C2×D59semidirect
263. #357C2×D59semidirect
266. C59#33646semidirect

Maximal subgroups

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

subgroupindex
#214336732
C2×#107×D592
#214262972
#214262922
#214262802
#214262752
#214258842
#214258822
#214258082
#214257062
C59×#21392
#3364659