Finite Group Databasebeta

← #28572529

Subgroups — #28572529

Order 15104.

710 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434671C211yesmaximal, characteristic
2#21433568C211yesmaximal, characteristic, complemented
3C2×Q8×C59⋊C8C211yesmaximal, characteristic, complemented
4#21423542C212yesmaximal, complemented
5#21423538C212yesmaximal, complemented
6#21423488C212yesmaximal, complemented
7#21423454C212yesmaximal, complemented
8#21422854C211yesmaximal, characteristic, complemented
9#21422774C211yesmaximal, characteristic, complemented
10#21422773C211yesmaximal, characteristic, complemented
11C59×#3037C211yesmaximal, characteristic, complemented
12C2×#14273758—21no—
13#16837952C2211yescharacteristic
14#16837805C2211yescharacteristic
15#16837778C2211yescharacteristic
16#16837750C2211yescharacteristic
17#16837748C2211yescharacteristic
18#16837636—21no—
19#16837636—21no—
20#16837624—22no—
21#16837612C2212yes—
22#16837602C2212yes—
23Q8×C59⋊C8—21no—
24Q8×C59⋊C8—21no—
25Q8×C59⋊C8C2212yes—
26Q8×C59⋊C8C2212yes—
27C4×#11745669—21no—
28C4×#11745669—21no—
29#16837426C2211yescharacteristic
30C2×C4⋊(C59⋊C8)—21no—
31C2×C4⋊(C59⋊C8)C2211yescharacteristic
32#16837401—21no—
33C2×C4×C59⋊C8—21no—
34C2×C4×C59⋊C8C2211yescharacteristic
35#16837154C2212yes—
36#16837154C2212yes—
37#16837149C2212yes—
38#16837148C2212yes—
39#16837146C2212yes—
40#16837145C2212yes—
41C2×#137×C59—21no—
42C59×#532C2211yescharacteristic
43C2×C4×Q8×C59C2211yescharacteristic
44C59×#464—22no—
45C59×#462C2212yes—
46C59×#460C2212yes—
47C59×#423C2211yescharacteristic
48C59×#417—21no—
49C2×#11745685—21no—
50#14273758—44no—
51C2×#11745672—22no—
52C2×#11745671—22no—
53C2×#11745669—21no—
54C2×#11745669—21no—
55C2×#11745664—21no—
56C22×C59⋊C8—21no—
57C22×C59⋊C8D411yescharacteristic
58C22×C59⋊C8D411yescharacteristic
59#14273727C2311yescharacteristic
60#14273724C2311yescharacteristic
61#14273715—21no—
62#14273701—21no—
63#14273701—21no—
64C4×D236—21no—
65C4×D236—21no—
66#14273660—21no—
67#14273660—21no—
68C4⋊(C59⋊C8)—21no—
69C4⋊(C59⋊C8)—21no—
70C4⋊(C59⋊C8)—22no—
71C4⋊(C59⋊C8)—22no—
72C4⋊(C59⋊C8)C2312yes—
73C4⋊(C59⋊C8)C2312yes—
74#14273650—22no—
75#14273648—21no—
76#14273648—21no—
77#14273647—22no—
78#14273647—21no—
79#14273647—21no—
80C4×C59⋊C8—22no—
81C4×C59⋊C8—22no—
82C4×C59⋊C8D411yescharacteristic
83C4×C59⋊C8D411yescharacteristic
84C4×C59⋊C8C2311yescharacteristic
85C4×C59⋊C8C2311yescharacteristic
86C2×#41×C59—21no—
87C22×Q8×C59D411yescharacteristic
88#137×C59—44no—
89C2×Q16×C59—22no—
90C2×SD16×C59—22no—
91C2×M16×C59—21no—
92C4⋊Q8×C59C2311yescharacteristic
93#127×C59C2311yescharacteristic
94#124×C59—21no—
95#121×C59—21no—
96#121×C59—21no—
97C4×Q8×C59—21no—
98C4×Q8×C59—21no—
99C4×Q8×C59D412yes—
100C4×Q8×C59C2312yes—
101C4×D4×C59—21no—
102C4×D4×C59—21no—
103C2×C4⋊C4×C59—21no—
104C2×C4⋊C4×C59D411yescharacteristic
105C2×C42×C59—21no—
106C2×C42×C59C2311yescharacteristic
107C4⋊C8×C59—21no—
108C4⋊C8×C59C2312yes—
109#103×C59—22no—
110#102×C59—22no—
111#41×C59—42no—
112#41×C59—42no—
113C2×Q8×C59—21no—
114C2×Q8×C59—22no—
115C2×Q8×C59—21no—
116C2×Q8×C59D814yescomplemented
117C2×Q8×C59C2×D412yes—
118C2×D4×C59—21no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—41no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59#4111yescharacteristic
128C2×C22⋊D59—41no—
129#11745685—42no—
130#11745685—42no—
131C2×D236—21no—
132C2×D236—21no—
133C2×D236—21no—
134C2×C4×D59—41no—
135C2×Dic118—21no—
136Q16×C59—44no—
137SD16×C59—44no—
138M16×C59—42no—
139C2×C8×C59—22no—
140C4⋊C4×C59—22no—
141C4⋊C4×C59—21no—
142C4⋊C4×C59—22no—
143C4⋊C4×C59—21no—
144C4⋊C4×C59—21no—
145C4⋊C4×C59C2×D412yes—
146C22⋊C4×C59—41no—
147C22⋊C4×C59—41no—
148C42×C59—22no—
149C42×C59—22no—
150C42×C59C2×D411yescharacteristic
151C42×C59C2×D411yescharacteristic
152C42×C59#4111yescharacteristic
153C42×C59C2411yescharacteristic
154#11745672—44no—
155#11745671—44no—
156#11745669—42no—
157#11745669—42no—
158(C2×D59)⋊C4—41no—
159(C2×D59)⋊C4—41no—
160C4⋊Dic59—21no—
161C4⋊Dic59—21no—
162#11745664—42no—
163C2×C59⋊C8—22no—
164C2×C59⋊C8—22no—
165C2×C59⋊C8—22no—
166C2×C59⋊C8—41no—
167C2×C59⋊C8—42no—
168C2×C59⋊C8C2×D411yescharacteristic
169C2×C59⋊C8C2×D411yescharacteristic
170C2×C59⋊C8C2×D411yescharacteristic
171C2×C59⋊C8C2×D411yescharacteristic
172C23×C59—41no—
173C23×C59#12111yescharacteristic
174C22×D59—41no—
175Q8×C59—22no—
176Q8×C59—24no—
177Q8×C59—41no—
178Q8×C59—41no—
179Q8×C59C2×D814yes—
180D4×C59—41no—
181D4×C59—41no—
182D4×C59—41no—
183D4×C59—41no—
184D4×C59—81no—
185D4×C59—82no—
186C2×C4×C59—22no—
187C2×C4×C59—24no—
188C2×C4×C59—21no—
189C2×C4×C59—21no—
190C2×C4×C59—22no—
191C2×C4×C59—41no—
192C2×C4×C59—42no—
193C2×C4×C59—41no—
194C2×C4×C59—82no—
195C2×C4×C59#12111yescharacteristic
196C2×C4×C59#12111yescharacteristic
197C2×C4×C59#12111yescharacteristic
198C2×C4×C59C2×D812yes—
199C2×C4×C59#13512yes—
200C2×C4×C59C22×D411yescharacteristic
201C2×C4×C59C22×D411yescharacteristic
202C2×C4×C59C2×#4111yescharacteristic
203C22⋊D59—82no—
204C2×Dic59—41no—
205D236—41no—
206D236—41no—
207D236—41no—
208D236—41no—
209D236—81no—
210C4×D59—82no—
211Dic118—41no—
212Dic118—41no—
213C8×C59—42no—
214C59⋊C8—21no—
215C59⋊C8—21no—
216C59⋊C8—21no—
217C59⋊C8—42no—
218C59⋊C8—42no—
219C59⋊C8#13712yescomplemented
220#19261—591nomaximal, Sylow
221C22×C59—21no—
222C22×C59—21no—
223C22×C59—81no—
224C22×C59—81no—
225C22×C59—81no—
226C22×C59#46612yes—
227C22×C59C2×#12111yescharacteristic
228C2×D59—81no—
229C2×D59—81no—
230C2×D59—81no—
231C4×C59—21no—
232C4×C59—24no—
233C4×C59—21no—
234C4×C59—21no—
235C4×C59—42no—
236C4×C59—81no—
237C4×C59#46612yes—
238C4×C59C22×D811yescharacteristic
239C4×C59C2×#13511yescharacteristic
240C4×C59C2×#13711yescharacteristic
241C4×C59#57511yescharacteristic
242Dic59—81no—
243#3169—591no—
244#3037—591no—
245#3037D5911yescharacteristic, complemented
246C2×C8×Q8—591no—
247#1682—592no—
248#1682—592no—
249#1674—592no—
250#1668—592no—
251#1567—591no—
252#1547—591no—
253#1547—591no—
254C2×C59—22no—
255C2×C59—161no—
256C2×C59#303711yescharacteristic
257C2×C59C2×#46611yescharacteristic
258C2×C59#306711yescharacteristic
259D59—161no—
260C2×#137—21no—
261C2×#137—1181no—
262#532—591no—
263#532C2×D5911yescharacteristic
264C2×C4×Q8C2×D5911yescharacteristic
265#500—591no—
266#493—591no—
267#486—591no—
268#486—591no—
269#466—1181no—
270#466—1181no—
271#464—22no—
272#464—1182no—
273#462—592no—
274#462C2×D5912yes—
275#460—592no—
276#460C2×D5912yes—
277C8×Q8—592no—
278C8×Q8—592no—
279C8×Q8—1181no—
280C8×Q8—1181no—
281C4×D8—1181no—
282C4×D8—1181no—
283#423—591no—
284#423C2×D5911yescharacteristic
285C2×C4⋊C8—591no—
286C2×C4⋊C8—1181no—
287#417—21no—
288#417—1181no—
289C2×C4×C8—591no—
290C2×C4×C8—1181no—
291#335—592no—
292#335—592no—
293#332—592no—
294#332—592no—
295#331—592no—
296#331—592no—
297C59#1926111yescharacteristic, complemented, Sylow
298C2×#41—21no—
299C2×#41—1181no—
300C22×Q8C22⋊D5911yescharacteristic
301#137—44no—
302#137—2364no—
303C2×Q16—22no—
304C2×Q16—1182no—
305C2×SD16—22no—
306C2×SD16—1182no—
307C2×D8—1181no—
308C2×D8—1181no—
309C2×M16—21no—
310C2×M16—1181no—
311C22×C8—591no—
312C22×C8—591no—
313C22×C8—1181no—
314C4⋊Q8—591no—
315C4⋊Q8C22×D5911yescharacteristic
316#127—591no—
317#127C22×D5911yescharacteristic
318#124—21no—
319#124—1181no—
320#121—21no—
321#121—21no—
322#121—1181no—
323#121—1181no—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22⋊D5912yes—
327C4×Q8C22×D5912yes—
328C4×D4—21no—
329C4×D4—21no—
330C4×D4—1181no—
331C4×D4—1181no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4C22⋊D5911yescharacteristic
334C2×C42—21no—
335C2×C42C22×D5911yescharacteristic
336#107—1181no—
337#107—1181no—
338C4⋊C8—21no—
339C4⋊C8—592no—
340C4⋊C8—592no—
341C4⋊C8—1182no—
342C4⋊C8—1182no—
343C4⋊C8—1181no—
344C4⋊C8—1181no—
345C4⋊C8C22×D5912yes—
346#103—22no—
347#103—1182no—
348#102—22no—
349#102—1181no—
350#102—1181no—
351#102—1181no—
352#102—1181no—
353#102—1182no—
354C4×C8—591no—
355C4×C8—591no—
356C4×C8—591no—
357C4×C8—591no—
358C4×C8—1182no—
359C4×C8—1182no—
360#41—42no—
361#41—42no—
362#41—2362no—
363#41—2362no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—21no—
367C2×Q8—1181no—
368C2×Q8#1174566914yescomplemented
369C2×Q8C2×C22⋊D5912yes—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4—41no—
374C2×D4—1181no—
375C2×D4—1181no—
376C2×D4—1181no—
377C2×D4—2361no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—2361no—
382C22×C4D4×D5911yescharacteristic
383C22×C4#1174568711yescharacteristic
384C22×C4C2×C22⋊D5911yescharacteristic
385Q16—44no—
386Q16—2364no—
387SD16—44no—
388SD16—2364no—
389D8—2362no—
390D8—2362no—
391M16—42no—
392M16—2362no—
393C2×C8—22no—
394C2×C8—591no—
395C2×C8—591no—
396C2×C8—591no—
397C2×C8—591no—
398C2×C8—1182no—
399C2×C8—1182no—
400C2×C8—1182no—
401C2×C8—2361no—
402C2×C8—2362no—
403C4⋊C4—22no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—1181no—
409C4⋊C4—1181no—
410C4⋊C4C2×C22⋊D5912yes—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—2361no—
414C22⋊C4—2361no—
415C42—22no—
416C42—22no—
417C42D4×D5911yescharacteristic
418C42#1174568711yescharacteristic
419C42C2×C22⋊D5911yescharacteristic
420C42C23×D5911yescharacteristic
421C23—41no—
422C23—2361no—
423C23#1427369911yescharacteristic
424Q8—24no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428Q8—2361no—
429Q8—2361no—
430Q8C2×#1174566914yes—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—41no—
435D4—81no—
436D4—82no—
437D4—4721no—
438D4—4722no—
439D4—2361no—
440D4—2361no—
441D4—2361no—
442D4—2361no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—82no—
452C2×C4—4722no—
453C2×C4—2361no—
454C2×C4#1427369911yescharacteristic
455C2×C4#1427369911yescharacteristic
456C2×C4#1427369911yescharacteristic
457C2×C4C2×#1174566912yes—
458C2×C4#1427374712yes—
459C2×C4C2×D4×D5911yescharacteristic
460C2×C4C22×C22⋊D5911yescharacteristic
461C2×C4C2×#1174568711yescharacteristic
462C8—42no—
463C8—592no—
464C8—1181no—
465C8—1181no—
466C8—1181no—
467C8—2362no—
468C8—2362no—
469C22—21no—
470C22—21no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—4721no—
475C22—4721no—
476C22—4721no—
477C22#1683763612yes—
478C22C2×#1427369911yescharacteristic
479C4—24no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—42no—
484C4—81no—
485C4—4721no—
486C4#1683763612yes—
487C4C22×#1174566911yescharacteristic
488C4C2×#1427374711yescharacteristic
489C4#137×D5911yescharacteristic
490C4#1683820111yescharacteristic
491C2—22no—
492C2—161no—
493C2—9441no—
494C2#2143356511yescharacteristic
495C2C2×#1683763611yescharacteristic
496C2#2143381311yescharacteristic
subgroupG/H
centerC22C2×#14273699
commutator, cosocleC42×C59C24
FrattiniC42C23×D59
FittingC59×#3037C2
socleC22×C59C2×#121
prime radical#21434671C2
hypercenter#3037D59
hypocenterC59#19261
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
2. #21433568C2semidirect
3. C2×Q8×C59⋊C8C2semidirect
4. #21423542C2semidirect
5. #21423538C2semidirect
6. #21423488C2semidirect
7. #21423454C2semidirect
8. #21422854C2semidirect
9. #21422774C2semidirect
10. #21422773C2semidirect
11. C59×#3037C2semidirect
116. C2×Q8×C59D8semidirect
219. C59⋊C8#137semidirect
245. #3037D59semidirect
297. C59#19261semidirect
368. C2×Q8#11745669semidirect

Maximal subgroups

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

subgroupindex
#214346712
#214335682
C2×Q8×C59⋊C82
#214235422
#214235382
#214234882
#214234542
#214228542
#214227742
#214227732
C59×#30372
#1926159