Finite Group Databasebeta

← #28742015

Subgroups — #28742015

Order 15104.

674 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837844C211yesmaximal, characteristic
2C4×#14273750C211yesmaximal, characteristic
3C4×Q16×D59C211yesmaximal, characteristic, complemented
4#21425965C211yesmaximal, characteristic, complemented
5#21425962C211yesmaximal, characteristic, complemented
6#21425958C211yesmaximal, characteristic, complemented
7Q32×Dic59C211yesmaximal, characteristic, complemented
8#21425952C211yesmaximal, characteristic, complemented
9#21425947C211yesmaximal, characteristic, complemented
10#21425945C211yesmaximal, characteristic, complemented
11#21425883C211yesmaximal, characteristic, complemented
12#21425814C211yesmaximal, characteristic, complemented
13#21425807C211yesmaximal, characteristic
14#21425689C211yesmaximal, characteristic, complemented
15C59×#2176C211yesmaximal, characteristic, complemented
16C2×#14273750C2211yescharacteristic
17C2×Q16×D59C2211yescharacteristic
18C4×#11745689—21no—
19C4×Q8×D59—21no—
20#16837844C418yescomplemented
21C2×#14273673C2211yescharacteristic
22C2×#14273672C2211yescharacteristic
23C2×#14273670C2211yescharacteristic
24C2×#14273666C2211yescharacteristic
25C2×#14273666C2211yescharacteristic
26C4×Dic236C2211yescharacteristic
27Q16×Dic59C2211yescharacteristic
28Q16×Dic59C2211yescharacteristic
29#16837496C2211yescharacteristic
30C4×#11745672—21no—
31#16837494—21no—
32C4×#11745671—21no—
33#16837487—21no—
34C4×D472C2211yescharacteristic
35#16837484C2211yescharacteristic
36#16837439C2211yescharacteristic
37#107×D59C2211yescharacteristic
38#16837397—21no—
39#103×D59—21no—
40C4×C8×D59C2211yescharacteristic
41#16837229C2211yescharacteristic
42#16837228C2211yescharacteristic
43#16837227C2211yescharacteristic
44#16837207C2211yescharacteristic
45#16837206C2211yescharacteristic
46#16837206C2211yescharacteristic
47#16837205C2211yescharacteristic
48#16837204C2211yescharacteristic
49#16837202C2211yescharacteristic
50#16837178C2211yescharacteristic
51C16×Dic59C2211yescharacteristic
52C16×Dic59C2211yescharacteristic
53C2×Q32×C59C2211yescharacteristic
54C4×Q16×C59C2211yescharacteristic
55C4×Q16×C59C2211yescharacteristic
56C59×#367C2211yescharacteristic
57C59×#358C2211yescharacteristic
58C59×#358C2211yescharacteristic
59C59×#346C2211yescharacteristic
60C2×#11745689—21no—
61C2×Q8×D59—21no—
62#14273750—22no—
63#14273750C2×C414yes—
64C2×Dic236C2311yescharacteristic
65C2×#11745672—21no—
66Q16×D59—22no—
67Q16×D59C2×C414yescomplemented
68C2×#11745671—21no—
69C2×D472C2311yescharacteristic
70C2×C8×D59D411yescharacteristic
71C2×C8×D59C2311yescharacteristic
72Q8×Dic59—21no—
73Q8×Dic59—21no—
74C4×Dic118—21no—
75C4×Dic118—41no—
76#14273692—21no—
77#14273692—41no—
78C4×D236—21no—
79C4×D236—41no—
80#14273689—21no—
81#14273689—41no—
82#14273685—21no—
83#14273685—41no—
84C4⋊C4×D59—21no—
85C4⋊C4×D59—41no—
86C42×D59—41no—
87C42×D59—41no—
88C42×D59D411yescharacteristic
89#14273673C2×C414yescomplemented
90#14273672C2×C414yescomplemented
91#14273670C2×C414yescomplemented
92#14273666—22no—
93#14273666C2×C414yescomplemented
94C2×C59⋊C16C2311yescharacteristic
95C2×C59⋊C16C2311yescharacteristic
96#14273661C2311yescharacteristic
97#14273661C2311yescharacteristic
98#14273660—21no—
99#14273658—21no—
100#14273652—21no—
101#14273651—21no—
102#14273651—21no—
103#14273650—21no—
104#14273649—21no—
105#14273647—21no—
106C4×C59⋊C8D411yescharacteristic
107C8×Dic59D411yescharacteristic
108C8×Dic59C2311yescharacteristic
109C2×Q16×C59C2311yescharacteristic
110C2×Q16×C59C2311yescharacteristic
111C4×Q8×C59—21no—
112C4×Q8×C59—21no—
113Q32×C59C2×C414yescomplemented
114C2×C16×C59C2311yescharacteristic
115C2×C16×C59C2311yescharacteristic
116#107×C59C2311yescharacteristic
117#107×C59C2311yescharacteristic
118#103×C59—21no—
119#103×C59—21no—
120C4×C8×C59C2311yescharacteristic
121C2×Q8×C59—21no—
122C2×Q8×C59—21no—
123#11745689—24no—
124#11745689—42no—
125Q8×D59—24no—
126Q8×D59—42no—
127C2×D236—21no—
128C2×D236—41no—
129C2×C4×D59—41no—
130C2×C4×D59—41no—
131C2×C4×D59—41no—
132C2×C4×D59—41no—
133C2×C4×D59—41no—
134C2×C4×D59D811yescharacteristic
135C2×C4×D59D811yescharacteristic
136C2×C4×D59C2×D411yescharacteristic
137C2×Dic118—21no—
138C2×Dic118—41no—
139Q16×C59—21no—
140Q16×C59—21no—
141Q16×C59C22×C412yes—
142Q16×C59C22×C412yes—
143C2×C8×C59C2×D411yescharacteristic
144C2×C8×C59C2411yescharacteristic
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—41no—
148C4⋊C4×C59—41no—
149C42×C59—41no—
150C42×C59—41no—
151C42×C59C2×D411yescharacteristic
152#11745672—24no—
153#11745671—24no—
154(C2×D59)⋊C4—41no—
155(C2×D59)⋊C4—41no—
156(C2×D59)⋊C4—41no—
157C4⋊Dic59—21no—
158C4⋊Dic59—21no—
159C4⋊Dic59—41no—
160C4⋊Dic59—41no—
161Dic59⋊C4—41no—
162Dic59⋊C4—41no—
163Dic59⋊C4—41no—
164C4×Dic59—41no—
165C4×Dic59—41no—
166C4×Dic59—41no—
167C4×Dic59—41no—
168C4×Dic59—41no—
169C4×Dic59D811yescharacteristic
170C4×Dic59D811yescharacteristic
171C4×Dic59C2×D411yescharacteristic
172C2×C59⋊C8C2×D411yescharacteristic
173C2×C59⋊C8C2×D411yescharacteristic
174Dic236—21no—
175Dic236C22×C412yes—
176D472—21no—
177D472C22×C412yes—
178C8×D59—22no—
179C8×D59C22×C412yes—
180C8×D59#4112yes—
181C16×C59—21no—
182C16×C59C22×C412yes—
183C59⋊C16—21no—
184C59⋊C16C22×C412yes—
185C22×D59—41no—
186C22×D59C2×D811yescharacteristic
187Q8×C59—22no—
188Q8×C59—22no—
189Q8×C59—41no—
190Q8×C59—41no—
191C2×C4×C59—41no—
192C2×C4×C59—41no—
193C2×C4×C59—41no—
194C2×C4×C59—41no—
195C2×C4×C59C2×D811yescharacteristic
196C2×C4×C59C2×D811yescharacteristic
197C2×C4×C59C22×D411yescharacteristic
198C2×Dic59—41no—
199C2×Dic59—41no—
200C2×Dic59—41no—
201C2×Dic59C2×D811yescharacteristic
202C2×Dic59C2×D811yescharacteristic
203C2×Dic59C2×D811yescharacteristic
204D236—22no—
205D236—44no—
206D236—41no—
207C4×D59—22no—
208C4×D59—41no—
209C4×D59—44no—
210C4×D59—44no—
211C4×D59—82no—
212C4×D59—82no—
213C4×D59—82no—
214C4×D59C4×D412yes—
215C4×D59#13512yes—
216Dic118—22no—
217Dic118—41no—
218Dic118—44no—
219C8×C59—21no—
220C8×C59C23×C411yescharacteristic
221C8×C59C2×#4111yescharacteristic
222C59⋊C8—21no—
223C59⋊C8C4×D412yes—
224C4×#510—591nomaximal, Sylow
225C22×C59C22×D811yescharacteristic
226C2×D59—21no—
227C2×D59—21no—
228C2×D59—42no—
229C2×D59—81no—
230C2×D59—81no—
231C2×D59C4×D812yes—
232C4×C59—21no—
233C4×C59—21no—
234C4×C59—42no—
235C4×C59—42no—
236C4×C59—81no—
237C4×C59—81no—
238C4×C59C2×C4×D411yescharacteristic
239C4×C59C2×#13511yescharacteristic
240Dic59—21no—
241Dic59—42no—
242Dic59—81no—
243Dic59—81no—
244Dic59C4×D812yes—
245Dic59#51012yescomplemented
246C2×#510—591no—
247C4×#135—591no—
248C2×C4×Q16—591no—
249#2176—591no—
250#2176D5911yescharacteristic, complemented
251#2175—591no—
252#2175—591no—
253C4×Q32—591no—
254C4×Q32—591no—
255C4×SD32—591no—
256C4×SD32—591no—
257#2159—591no—
258#2141—591no—
259#2139—591no—
260C4×M32—591no—
261C2×C59C2×C4×D811yescharacteristic
262C2×C59C2×#51011yescharacteristic
263C2×C59#341411yescharacteristic
264D59—22no—
265D59—82no—
266C2×#135—591no—
267C22×Q16—591no—
268C4×#41—1181no—
269C2×C4×Q8—1181no—
270#510—598no—
271C2×Q32—591no—
272C2×Q32C2×D5911yescharacteristic
273C2×SD32—591no—
274C2×SD32—591no—
275C2×M32—591no—
276C2×M32—591no—
277C4×Q16—591no—
278C4×Q16—591no—
279C4×Q16—591no—
280C4×Q16—591no—
281C4×Q16—1181no—
282C4×Q16—1181no—
283C4×Q16C2×D5911yescharacteristic
284C4×Q16C2×D5911yescharacteristic
285C4×SD16—1181no—
286C4×SD16—1181no—
287C4×D8—591no—
288C4×D8—591no—
289#427—591no—
290C2×#107—591no—
291#416—1181no—
292C2×#103—1181no—
293C2×C4×C8—591no—
294#367—591no—
295#367C2×D5911yescharacteristic
296#366—591no—
297#366—591no—
298#358—591no—
299#358—591no—
300#358—591no—
301#358—591no—
302#358C2×D5911yescharacteristic
303#358C2×D5911yescharacteristic
304#357—591no—
305#357—591no—
306#346—591no—
307#346C2×D5911yescharacteristic
308C4×C16—591no—
309C4×C16—591no—
310C59C4×#51011yescharacteristic, complemented, Sylow
311C2×#41—1181no—
312C22×Q8—1181no—
313#135—594no—
314#135—1182no—
315C2×Q16—594no—
316C2×Q16—591no—
317C2×Q16—1182no—
318C2×Q16—1181no—
319C2×Q16C22×D5911yescharacteristic
320C2×Q16C22×D5911yescharacteristic
321C2×SD16—1181no—
322C2×D8—591no—
323C22×C8—591no—
324C22×C8—591no—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—1181no—
328C4×Q8—1181no—
329C4×Q8—1181no—
330C4×Q8—1181no—
331C4×Q8—2361no—
332C4×Q8—2361no—
333C4×D4—1181no—
334C4×D4—1181no—
335C4×D4—2361no—
336C4×D4—2361no—
337#117—1181no—
338#117—2361no—
339C2×C4⋊C4—1181no—
340C2×C4⋊C4—2361no—
341C2×C42—591no—
342C2×C42—2361no—
343C2×C42—2361no—
344Q32—594no—
345Q32C4×D5914yescomplemented
346SD32—594no—
347SD32—594no—
348M32—594no—
349M32—1182no—
350C2×C16—591no—
351C2×C16—591no—
352C2×C16C22×D5911yescharacteristic
353C2×C16C22×D5911yescharacteristic
354#107—591no—
355#107—591no—
356#107—1181no—
357#107C22×D5911yescharacteristic
358#107C22×D5911yescharacteristic
359#106—1181no—
360#103—21no—
361#103—21no—
362#103—1181no—
363#103—1181no—
364#103—1181no—
365#103—1181no—
366#102—1181no—
367#102—1181no—
368C4×C8—591no—
369C4×C8—591no—
370C4×C8—591no—
371C4×C8C22×D5911yescharacteristic
372#41—1184no—
373#41—2362no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—1184no—
377C2×Q8—1181no—
378C2×Q8—2361no—
379C2×Q8—2362no—
380C2×D4—1181no—
381C2×D4—2361no—
382C22×C4—591no—
383C22×C4—591no—
384C22×C4—591no—
385C22×C4—2361no—
386C22×C4—2361no—
387C22×C4—2361no—
388C22×C4—2361no—
389C22×C4—2361no—
390Q16—21no—
391Q16—21no—
392Q16—592no—
393Q16—1184no—
394Q16—1181no—
395Q16C2×C4×D5912yes—
396Q16C2×C4×D5912yes—
397SD16—1184no—
398D8—592no—
399D8—1181no—
400C2×C8—591no—
401C2×C8—592no—
402C2×C8—592no—
403C2×C8—591no—
404C2×C8—1182no—
405C2×C8D4×D5911yescharacteristic
406C2×C8C23×D5911yescharacteristic
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—1181no—
412C4⋊C4—1181no—
413C4⋊C4—2361no—
414C4⋊C4—2361no—
415C4⋊C4—2361no—
416C4⋊C4—2361no—
417C4⋊C4—2361no—
418C22⋊C4—2361no—
419C22⋊C4—2361no—
420C22⋊C4—2361no—
421C42—41no—
422C42—41no—
423C42—591no—
424C42—591no—
425C42—591no—
426C42—2361no—
427C42—2361no—
428C42—2361no—
429C42—2361no—
430C42—2361no—
431C42D4×D5911yescharacteristic
432C16—21no—
433C16—592no—
434C16—1181no—
435C16C2×C4×D5912yes—
436C23—591no—
437C23—2361no—
438Q8—22no—
439Q8—22no—
440Q8—41no—
441Q8—41no—
442Q8—1182no—
443Q8—2364no—
444Q8—2361no—
445D4—1182no—
446D4—2364no—
447D4—2361no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—591no—
453C2×C4—591no—
454C2×C4—592no—
455C2×C4—592no—
456C2×C4—591no—
457C2×C4—1182no—
458C2×C4—4722no—
459C2×C4—4722no—
460C2×C4—4722no—
461C2×C4—2361no—
462C2×C4—2364no—
463C2×C4—2364no—
464C2×C4—2361no—
465C2×C4—2361no—
466C2×C4—2361no—
467C2×C4D8×D5911yescharacteristic
468C2×C4D8×D5911yescharacteristic
469C2×C4C2×D4×D5911yescharacteristic
470C8—21no—
471C8—592no—
472C8—1181no—
473C8C22×C4×D5911yescharacteristic
474C8#41×D5911yescharacteristic
475C22—592no—
476C22—1181no—
477C22—1181no—
478C22—4721no—
479C22—4721no—
480C22—2362no—
481C22C2×D8×D5911yescharacteristic
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—592no—
489C4—592no—
490C4—1181no—
491C4—4721no—
492C4—4721no—
493C4—2362no—
494C4C4×D4×D5911yescharacteristic
495C4#135×D5911yescharacteristic
496C2—1182no—
497C2—4722no—
498C2C4×D8×D5911yescharacteristic
499C2#510×D5911yescharacteristic
500C2#2143723111yescharacteristic
subgroupG/H
centerC22C2×D8×D59
commutatorC8×C59C23×C4
FrattiniC2×C8C23×D59
FittingC59×#2176C2
socleC22×C59C22×D8
prime radicalC2×#14273750C22
cosocleC2×C8×C59C24
hypercenter#2176D59
hypocenterC59C4×#510
LCSG ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2176

Complements

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

NH ≅ G/Ntype
3. C4×Q16×D59C2semidirect
4. #21425965C2semidirect
5. #21425962C2semidirect
6. #21425958C2semidirect
7. Q32×Dic59C2semidirect
8. #21425952C2semidirect
9. #21425947C2semidirect
10. #21425945C2semidirect
11. #21425883C2semidirect
12. #21425814C2semidirect
14. #21425689C2semidirect
15. C59×#2176C2semidirect
20. #16837844C4semidirect
67. Q16×D59C2×C4semidirect
89. #14273673C2×C4semidirect
90. #14273672C2×C4semidirect
91. #14273670C2×C4semidirect
93. #14273666C2×C4semidirect
113. Q32×C59C2×C4semidirect
245. Dic59#510semidirect
250. #2176D59semidirect
310. C59C4×#510semidirect
345. Q32C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168378442
C4×#142737502
C4×Q16×D592
#214259652
#214259622
#214259582
Q32×Dic592
#214259522
#214259472
#214259452
#214258832
#214258142
#214258072
#214256892
C59×#21762
C4×#51059