Finite Group Databasebeta

← #28745804

Subgroups — #28745804

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21435839C211yesmaximal, characteristic
2#21426335C211yesmaximal, characteristic, complemented
3#21426293C211yesmaximal, characteristic, complemented
4#21426281C211yesmaximal, characteristic, complemented
5#21426235C211yesmaximal, characteristic, complemented
6#21426202C211yesmaximal, characteristic, complemented
7#21426194C211yesmaximal, characteristic, complemented
8#21426158C211yesmaximal, characteristic, complemented
9#21426034C211yesmaximal, characteristic, complemented
10#21426029C211yesmaximal, characteristic, complemented
11#21425965C211yesmaximal, characteristic, complemented
12#21425922C211yesmaximal, characteristic, complemented
13#21425806C211yesmaximal, characteristic, complemented
14C2×#16837205C211yesmaximal, characteristic, complemented
15C59×#3338C211yesmaximal, characteristic, complemented
16C2×#14273749C2211yescharacteristic
17C22×Dic236C2211yescharacteristic
18#16838061—21no—
19C2×#14273673C2211yescharacteristic
20C2×#14273673C2211yescharacteristic
21C2×#14273670C2211yescharacteristic
22C2×#14273667C2211yescharacteristic
23C22×C59⋊C16C2211yescharacteristic
24#16837786C2211yescharacteristic
25#16837643C2211yescharacteristic
26#16837638C2211yescharacteristic
27#16837625—21no—
28#16837566—21no—
29#16837543—21no—
30C4×Dic236C2211yescharacteristic
31#16837229C2211yescharacteristic
32#16837229C2211yescharacteristic
33#16837229C2211yescharacteristic
34C4⋊(C59⋊C16)C2211yescharacteristic
35#16837206C2211yescharacteristic
36#16837205C2211yescharacteristic
37#16837205C2211yescharacteristic
38#16837205C2211yescharacteristic
39#16837205C2211yescharacteristic
40#16837205C2211yescharacteristic
41#16837205C2211yescharacteristic
42#16837202C2211yescharacteristic
43#16837202C2211yescharacteristic
44#16837202C2211yescharacteristic
45C22⋊(C59⋊C16)C2211yescharacteristic
46C22⋊(C59⋊C16)C2211yescharacteristic
47#16837178C2211yescharacteristic
48C59×#554—21no—
49C59×#500C2211yescharacteristic
50C59×#484—21no—
51C59×#484—21no—
52C59×#477—21no—
53C59×#467C2211yescharacteristic
54C59×#467C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C8×D4×C59C2211yescharacteristic
57C2×#107×C59C2211yescharacteristic
58C2×#107×C59C2211yescharacteristic
59C2×#11745685—21no—
60C22×Dic118—21no—
61#14273749—22no—
62#14273749—41no—
63C2×Dic236—22no—
64C2×Dic236—21no—
65C2×Dic236—41no—
66C2×Dic236C2311yescharacteristic
67C2×Dic236C2311yescharacteristic
68C2×#11745660—21no—
69C2×#11745660—21no—
70C2×D472C2311yescharacteristic
71#14273715—21no—
72#14273713—41no—
73#14273706—21no—
74#14273705—41no—
75#14273701—21no—
76#14273699—41no—
77C4×Dic118—21no—
78(C22⋊D59)⋊C4—41no—
79#14273673—22no—
80#14273673—41no—
81#14273670—41no—
82#14273667—41no—
83C2×C59⋊C16—41no—
84C2×C59⋊C16C2311yescharacteristic
85C2×C59⋊C16C2311yescharacteristic
86C2×C59⋊C16C2311yescharacteristic
87C2×C59⋊C16C2311yescharacteristic
88#14273661C2311yescharacteristic
89#14273652—21no—
90#14273652—21no—
91#14273649—21no—
92C2×Q16×C59C2311yescharacteristic
93C22×C8×C59C2311yescharacteristic
94C22×C8×C59C2311yescharacteristic
95C4⋊Q8×C59—21no—
96#125×C59—41no—
97C22⋊Q8×C59—21no—
98C22⋊Q8×C59—21no—
99C22⋊Q8×C59—41no—
100C22⋊Q8×C59—41no—
101C4×Q8×C59—21no—
102C4×D4×C59—41no—
103C4×D4×C59D411yescharacteristic
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106#107×C59—21no—
107#107×C59—21no—
108#107×C59—21no—
109#107×C59C2311yescharacteristic
110#107×C59C2311yescharacteristic
111#107×C59C2311yescharacteristic
112C4⋊C8×C59D411yescharacteristic
113#103×C59—21no—
114#103×C59—21no—
115#103×C59—21no—
116C22⋊C8×C59D411yescharacteristic
117C22⋊C8×C59D411yescharacteristic
118C4×C8×C59C2311yescharacteristic
119C2×Q8×C59—21no—
120C2×Q8×C59—41no—
121C2×D4×C59D811yescharacteristic
122C22×C4×C59—41no—
123C22×C4×C59—41no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C2×C22⋊D59—41no—
127C2×C22⋊D59—41no—
128C22×Dic59—41no—
129#11745685—42no—
130#11745685—81no—
131C2×D236—21no—
132C2×C4×D59—41no—
133C2×Dic118—21no—
134C2×Dic118—21no—
135C2×Dic118—42no—
136C2×Dic118—41no—
137C2×Dic118—81no—
138Q16×C59—21no—
139Q16×C59—21no—
140C2×C8×C59—21no—
141C2×C8×C59—21no—
142C2×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59C2×D411yescharacteristic
145C2×C8×C59C2×D411yescharacteristic
146C2×C8×C59C2×D411yescharacteristic
147C2×C8×C59C2411yescharacteristic
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—41no—
152C4⋊C4×C59—41no—
153C4⋊C4×C59—41no—
154C4⋊C4×C59—41no—
155C4⋊C4×C59—41no—
156C4⋊C4×C59—41no—
157C4⋊C4×C59D811yescharacteristic
158C22⋊C4×C59—41no—
159C22⋊C4×C59—41no—
160C22⋊C4×C59D811yescharacteristic
161C22⋊C4×C59D811yescharacteristic
162C42×C59—41no—
163C42×C59C2×D411yescharacteristic
164C22⋊Dic59—41no—
165(C2×D59)⋊C4—41no—
166(C2×D59)⋊C4—41no—
167(C2×D59)⋊C4—41no—
168C4⋊Dic59—21no—
169Dic59⋊C4—41no—
170Dic59⋊C4—41no—
171C4×Dic59—41no—
172Dic236—21no—
173Dic236—21no—
174Dic236—21no—
175Dic236—42no—
176Dic236#4112yes—
177D472—21no—
178D472—21no—
179#11745660—42no—
180#11745660—81no—
181C59⋊C16—21no—
182C59⋊C16—21no—
183C59⋊C16—21no—
184C59⋊C16—21no—
185C23×C59C2×D811yescharacteristic
186C23×C59C2×D811yescharacteristic
187C22×D59—41no—
188Q8×C59—41no—
189Q8×C59—41no—
190Q8×C59—161no—
191D4×C59—22no—
192C2×C4×C59—21no—
193C2×C4×C59—21no—
194C2×C4×C59—21no—
195C2×C4×C59—21no—
196C2×C4×C59—41no—
197C2×C4×C59—41no—
198C2×C4×C59—41no—
199C2×C4×C59—41no—
200C2×C4×C59—161no—
201C2×C4×C59—161no—
202C2×C4×C59C2×D811yescharacteristic
203C2×C4×C59C2×D811yescharacteristic
204C2×C4×C59C2×D811yescharacteristic
205C2×C4×C59C2×D811yescharacteristic
206C2×C4×C59C22×D411yescharacteristic
207C22⋊D59—82no—
208C22⋊D59—82no—
209C2×Dic59—41no—
210C2×Dic59—41no—
211C2×Dic59—41no—
212C2×Dic59—82no—
213D236—41no—
214D236—41no—
215C4×D59—82no—
216Dic118—22no—
217Dic118—41no—
218Dic118—41no—
219Dic118—41no—
220Dic118—82no—
221C8×C59—21no—
222C8×C59—21no—
223C8×C59—21no—
224C8×C59C2×#4111yescharacteristic
225C8×C59H(5,𝔽2)11yescharacteristic
226#33883—591nomaximal, Sylow
227C22×C59—21no—
228C22×C59—21no—
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59#51012yes—
233C22×C59C22×D811yescharacteristic
234C2×D59—81no—
235C2×D59—81no—
236C2×D59—81no—
237C4×C59—21no—
238C4×C59—21no—
239C4×C59—21no—
240C4×C59—21no—
241C4×C59—81no—
242C4×C59—81no—
243C4×C59—81no—
244C4×C59—81no—
245C4×C59#54611yescharacteristic
246C4×C59#57711yescharacteristic
247Dic59—42no—
248Dic59—81no—
249Dic59—81no—
250#3338D5911yescharacteristic, complemented
251#3285—591no—
252#2243—591no—
253#2235—591no—
254#2233—591no—
255#2225—591no—
256#2219—591no—
257#2217—591no—
258#2211—591no—
259#2189—591no—
260#2188—591no—
261#2176—591no—
262#2168—591no—
263#2139—591no—
264C2×#358—591no—
265C2×C59—22no—
266C2×C59—41no—
267C2×C59#329411yescharacteristic
268C2×C59C2×#51011yescharacteristic
269C2×C59#341611yescharacteristic
270D59—161no—
271C2×#135—591no—
272C22×Q16—591no—
273#554—21no—
274#548—1181no—
275C2×Q32—591no—
276C2×Q32—591no—
277C2×SD32—591no—
278C2×M32—591no—
279C22×C16—591no—
280#500C2×D5911yescharacteristic
281#495—591no—
282#484—21no—
283#484—21no—
284#477—21no—
285#467—591no—
286#467C2×D5911yescharacteristic
287#467C2×D5911yescharacteristic
288#466—591no—
289#464—1181no—
290#452—1181no—
291#448—1181no—
292C4×Q16—591no—
293C4×Q16C2×D5911yescharacteristic
294C8×D4C2×D5911yescharacteristic
295C2×#107C2×D5911yescharacteristic
296C2×#107C2×D5911yescharacteristic
297#367—591no—
298#367—591no—
299#367—591no—
300C4⋊C16—591no—
301#358—591no—
302#358—591no—
303#358—591no—
304#358—591no—
305#358—591no—
306#358—591no—
307#358—591no—
308#357—591no—
309#357—591no—
310#357—591no—
311C22⋊C16—591no—
312C22⋊C16—591no—
313#346—591no—
314C59#3388311yescharacteristic, complemented, Sylow
315C2×#41—1181no—
316C22×Q8—1181no—
317#135—1182no—
318#135—2361no—
319C2×Q16—591no—
320C2×Q16—591no—
321C2×Q16—1182no—
322C2×Q16—1181no—
323C2×Q16—2361no—
324C2×Q16C22×D5911yescharacteristic
325C2×SD16—1181no—
326C2×SD16—1181no—
327C2×D8—591no—
328C22×C8C22×D5911yescharacteristic
329C22×C8C22×D5911yescharacteristic
330C4⋊Q8—21no—
331#125—41no—
332#124—1181no—
333#124—2361no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C22⋊Q8—41no—
337C22⋊Q8—41no—
338C22⋊Q8—1181no—
339C22⋊Q8—2361no—
340#121—1181no—
341#121—2361no—
342C4×Q8—21no—
343C4×Q8—1181no—
344C4×D4—41no—
345C4×D4—2361no—
346C4×D4C22⋊D5911yescharacteristic
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349Q32—1182no—
350Q32—2361no—
351SD32—2361no—
352M32—2361no—
353C2×C16—591no—
354C2×C16—591no—
355C2×C16—591no—
356C2×C16—591no—
357C2×C16—2361no—
358#107—21no—
359#107—21no—
360#107—21no—
361#107—591no—
362#107C22×D5911yescharacteristic
363#107C22×D5911yescharacteristic
364#107C22×D5911yescharacteristic
365C4⋊C8C22⋊D5911yescharacteristic
366#103—21no—
367#103—21no—
368#103—21no—
369#103—1181no—
370#103—1181no—
371#102—1181no—
372C22⋊C8C22⋊D5911yescharacteristic
373C22⋊C8C22⋊D5911yescharacteristic
374C4×C8C22×D5911yescharacteristic
375#41—4721no—
376#41—2362no—
377C2×Q8—21no—
378C2×Q8—41no—
379C2×Q8—1181no—
380C2×Q8—1181no—
381C2×Q8—4721no—
382C2×Q8—2362no—
383C2×Q8—2361no—
384C2×D4—1181no—
385C2×D4—2361no—
386C2×D4—2361no—
387C2×D4#1174566911yescharacteristic
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—2361no—
391C22×C4—2361no—
392C22×C4C2×C22⋊D5911yescharacteristic
393C22×C4C2×C22⋊D5911yescharacteristic
394Q16—21no—
395Q16—21no—
396Q16—592no—
397Q16—1181no—
398Q16—1181no—
399Q16—1181no—
400Q16—2362no—
401SD16—4721no—
402SD16—2362no—
403D8—1181no—
404D8—1181no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8C2×C22⋊D5911yescharacteristic
410C2×C8C2×C22⋊D5911yescharacteristic
411C2×C8C2×C22⋊D5911yescharacteristic
412C2×C8C23×D5911yescharacteristic
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—1181no—
423C4⋊C4—2361no—
424C4⋊C4—2361no—
425C4⋊C4#1174566911yescharacteristic
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—2361no—
429C22⋊C4—2361no—
430C22⋊C4—2361no—
431C22⋊C4—2361no—
432C22⋊C4#1174566911yescharacteristic
433C22⋊C4#1174566911yescharacteristic
434C42—41no—
435C42—2361no—
436C42C2×C22⋊D5911yescharacteristic
437C16—1181no—
438C16—1181no—
439C16—1181no—
440C16—1181no—
441C23—2361no—
442C23C2×#1174566911yescharacteristic
443C23C2×#1174566911yescharacteristic
444Q8—41no—
445Q8—41no—
446Q8—161no—
447Q8—1182no—
448Q8—4722no—
449Q8—2361no—
450Q8—2361no—
451Q8—2361no—
452D4—22no—
453D4—4722no—
454D4—4722no—
455D4—2361no—
456D4—2361no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—161no—
466C2×C4—161no—
467C2×C4—4722no—
468C2×C4—4722no—
469C2×C4—2361no—
470C2×C4—2361no—
471C2×C4—2361no—
472C2×C4C2×#1174566911yescharacteristic
473C2×C4C2×#1174566911yescharacteristic
474C2×C4C2×#1174566911yescharacteristic
475C2×C4C2×#1174566911yescharacteristic
476C2×C4C22×C22⋊D5911yescharacteristic
477C8—21no—
478C8—21no—
479C8—21no—
480C8#41×D5911yescharacteristic
481C8#1427377311yescharacteristic
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—4721no—
488C22—4721no—
489C22—4721no—
490C22#1683784312yes—
491C22C22×#1174566911yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—4721no—
501C4—4721no—
502C4—2362no—
503C4#1683804011yescharacteristic
504C4#1683821011yescharacteristic
505C2—22no—
506C2—41no—
507C2—9441no—
508C2#2143590311yescharacteristic
509C2C2×#1683784311yescharacteristic
510C2#2143724311yescharacteristic
subgroupG/H
centerC22C22×#11745669
commutator, cosocleC2×C8×C59C24
FrattiniC2×C8C23×D59
FittingC59×#3338C2
socleC22×C59C22×D8
prime radical#21435839C2
hypercenter#3338D59
hypocenterC59#33883
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3338

Complements

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

NH ≅ G/Ntype
2. #21426335C2semidirect
3. #21426293C2semidirect
4. #21426281C2semidirect
5. #21426235C2semidirect
6. #21426202C2semidirect
7. #21426194C2semidirect
8. #21426158C2semidirect
9. #21426034C2semidirect
10. #21426029C2semidirect
11. #21425965C2semidirect
12. #21425922C2semidirect
13. #21425806C2semidirect
14. C2×#16837205C2semidirect
15. C59×#3338C2semidirect
250. #3338D59semidirect
314. C59#33883semidirect

Maximal subgroups

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

subgroupindex
#214358392
#214263352
#214262932
#214262812
#214262352
#214262022
#214261942
#214261582
#214260342
#214260292
#214259652
#214259222
#214258062
C2×#168372052
C59×#33382
#3388359