Finite Group Databasebeta

← #28627433

Subgroups — #28627433

Order 15104.

1034 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837541C211yesmaximal, characteristic
2C2×#16837459C211yesmaximal, characteristic, complemented
3#21432427C211yesmaximal, characteristic
4#21425491C212yesmaximal, complemented
5#21425411C212yesmaximal, complemented
6#21424980C212yesmaximal, complemented
7#21424789C212yesmaximal, complemented
8#21424413C211yesmaximal, characteristic
9#21424284C211yesmaximal, characteristic, complemented
10#21424031C211yesmaximal, characteristic, complemented
11C2×C59×#391C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C2×#14273710—21no—
14C2×#14273701C2211yescharacteristic
15#16837541—24no—
16#16837459—24no—
17C2×#14273660C2212yes—
18C2×C4⋊(C59⋊C8)C2212yes—
19C2×#14273651C2212yes—
20C2×#14273651C2212yes—
21C2×#14273651C2212yes—
22C2×#14273647C2212yes—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273638C2212yes—
26#16837304C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837293—21no—
31#16837293—21no—
32#16837281—21no—
33#16837262C2211yescharacteristic
34#16837164C2212yes—
35#16837164C2212yes—
36C23×Q8×C59C2211yescharacteristic
37C22×C4⋊C4×C59C2211yescharacteristic
38C22×C42×C59C2211yescharacteristic
39C59×#391—21no—
40C59×#391—21no—
41C59×#391C2212yes—
42C59×#391C2212yes—
43C2×#95×C59—21no—
44C2×#95×C59—21no—
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—21no—
47C2×#11745671—24no—
48C2×#11745671—24no—
49C2×#11745671—42no—
50C22×C59⋊C8C2312yes—
51C22×C59⋊C8C2312yes—
52#14273710—44no—
53#14273701—22no—
54#14273701—22no—
55C2×C4⋊Dic59C2311yescharacteristic
56C2×C4⋊Dic59C2311yescharacteristic
57C2×C4⋊Dic59C2311yescharacteristic
58C2×Dic59⋊C4—21no—
59C2×Dic59⋊C4—21no—
60C2×C22⋊Dic59—21no—
61C2×(C2×D59)⋊C4—21no—
62C2×(C2×D59)⋊C4—21no—
63C2×(C2×D59)⋊C4—21no—
64C2×(C2×D59)⋊C4—21no—
65C2×(C2×D59)⋊C4—21no—
66#14273660—42no—
67C4⋊(C59⋊C8)—24no—
68#14273651—24no—
69#14273651—24no—
70#14273651—42no—
71#14273647—42no—
72C22⋊(C59⋊C8)—22no—
73C22⋊(C59⋊C8)D414yes—
74#14273638—24no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C59⋊#95—21no—
78C22×Q8×C59—24no—
79C22×Q8×C59—21no—
80C22×Q8×C59—22no—
81C22×Q8×C59—22no—
82C22×Q8×C59—42no—
83C22×Q8×C59C2312yes—
84C23×C4×C59—21no—
85C23×C4×C59—21no—
86C23×C4×C59D411yescharacteristic
87C23×C4×C59D411yescharacteristic
88C23×C4×C59C2311yescharacteristic
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—42no—
91C2×C4⋊C4×C59C2312yes—
92C2×C42×C59—22no—
93C2×C42×C59—22no—
94C2×C42×C59D411yescharacteristic
95C2×C42×C59D411yescharacteristic
96C2×C42×C59C2311yescharacteristic
97C2×C42×C59C2311yescharacteristic
98#95×C59—22no—
99#95×C59—22no—
100#95×C59—22no—
101#95×C59—22no—
102C24×C59C2×D411yescharacteristic
103C23×D59—21no—
104C2×Q8×C59—24no—
105C2×Q8×C59—22no—
106C2×Q8×C59—22no—
107C2×Q8×C59—22no—
108C2×Q8×C59—22no—
109C2×Q8×C59—22no—
110C2×Q8×C59—48no—
111C2×Q8×C59—48no—
112C2×Q8×C59—42no—
113C2×Q8×C59—42no—
114C2×Q8×C59#4114yes—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—24no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—41no—
126C22×C4×C59—42no—
127C22×C4×C59—42no—
128C22×C4×C59—42no—
129C22×C4×C59—42no—
130C22×C4×C59—42no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D412yes—
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59#4112yes—
138C22×C4×C59C2411yescharacteristic
139C2×C22⋊D59—42no—
140C2×C22⋊D59—42no—
141C2×C22⋊D59—42no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C2×D236—21no—
146C2×D236—21no—
147C2×D236—22no—
148C2×D236—22no—
149C4⋊C4×C59—22no—
150C4⋊C4×C59—22no—
151C4⋊C4×C59—42no—
152C42×C59—22no—
153C42×C59—22no—
154C42×C59—21no—
155C42×C59—21no—
156C42×C59—21no—
157C42×C59—21no—
158C22⋊Dic59—42no—
159#11745671—48no—
160(C2×D59)⋊C4—42no—
161(C2×D59)⋊C4—42no—
162(C2×D59)⋊C4—42no—
163(C2×D59)⋊C4—42no—
164(C2×D59)⋊C4—42no—
165C4⋊Dic59—21no—
166C4⋊Dic59—22no—
167C4⋊Dic59—21no—
168C4⋊Dic59—22no—
169Dic59⋊C4—42no—
170Dic59⋊C4—42no—
171C2×C59⋊C8—22no—
172C2×C59⋊C8—22no—
173C2×C59⋊C8—22no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8C2×D414yes—
177C23×C59—21no—
178C23×C59—21no—
179C23×C59—21no—
180C23×C59—21no—
181C23×C59#12112yes—
182C23×C59AGL(1,ℤ/8)12yes—
183C23×C59#13712yes—
184C23×C59C22×D411yescharacteristic
185C22×D59—41no—
186C22×D59—41no—
187C22×D59—41no—
188C22×D59—41no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192Q8×C59—24no—
193Q8×C59—24no—
194Q8×C59—48no—
195Q8×C59—42no—
196Q8×C59—42no—
197C2×C4×C59—21no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—24no—
202C2×C4×C59—24no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—24no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—44no—
214C2×C4×C59—48no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—44no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—84no—
222C2×C4×C59#12112yes—
223C2×C4×C59#13712yes—
224C2×C4×C59C22×D411yescharacteristic
225C2×C4×C59C2×#4111yescharacteristic
226C2×C4×C59C2×#4112yes—
227C2×C4×C59H(5,𝔽2)12yes—
228C22⋊D59—84no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238D236—42no—
239D236—42no—
240C59⋊C8—24no—
241C59⋊C8—42no—
242#24509—591nomaximal, Sylow
243C22×C59—22no—
244C22×C59—22no—
245C22×C59—22no—
246C22×C59—22no—
247C22×C59—22no—
248C22×C59—22no—
249C22×C59#46814yes—
250C22×C59C2×#12111yescharacteristic
251C22×C59#52511yescharacteristic
252C22×C59#53511yescharacteristic
253C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
254C22×C59C2×#13711yescharacteristic
255C22×C59C2×#13711yescharacteristic
256C22×C59#57511yescharacteristic
257C2×D59—81no—
258C2×D59—81no—
259C2×D59—81no—
260C2×D59—81no—
261C2×D59—81no—
262C2×D59—81no—
263C2×D59—81no—
264C4×C59—22no—
265C4×C59—22no—
266C4×C59—44no—
267C4×C59—41no—
268C4×C59—41no—
269C4×C59—41no—
270C4×C59—82no—
271C4×C59#54612yes—
272C4×C59#55112yes—
273Dic59—81no—
274Dic59—81no—
275Dic59—81no—
276C2×#448—591no—
277C2×#432—591no—
278#2854—591no—
279C2×#391D5911yescharacteristic, complemented
280#2068—592no—
281#2055—592no—
282#1973—592no—
283#1935—592no—
284#1874—591no—
285#1852—591no—
286#1787—591no—
287C2×C59—24no—
288C2×C59#241111yescharacteristic
289C2×C59C2×#46811yescharacteristic
290C2×C59#305611yescharacteristic
291C2×C59#311211yescharacteristic
292C2×C59#311711yescharacteristic
293C2×C59#321311yescharacteristic
294C2×C59#321811yescharacteristic
295D59—161no—
296C23×Q8C2×D5911yescharacteristic
297C22×SD16—592no—
298C2×#123—1181no—
299C2×#121—591no—
300C22×C4⋊C4C2×D5911yescharacteristic
301C22×C42C2×D5911yescharacteristic
302#448—1184no—
303#432—1184no—
304C2×#107—592no—
305C2×C4⋊C8—592no—
306C2×#103—592no—
307C2×#103—592no—
308C2×#103—592no—
309C2×#102—592no—
310C2×C22⋊C8—591no—
311C2×C22⋊C8—591no—
312C2×#97—592no—
313#391—21no—
314#391—21no—
315#391C2×D5912yes—
316#391C2×D5912yes—
317#390—591no—
318#390—591no—
319#389—591no—
320#389—591no—
321#388—1181no—
322#388—1181no—
323#386—1181no—
324#380—591no—
325C2×#95—21no—
326C2×#95—21no—
327#340—592no—
328#340—592no—
329C59#2450911yescharacteristic, complemented, Sylow
330C22×Q8—24no—
331C22×Q8—21no—
332C22×Q8—22no—
333C22×Q8—22no—
334C22×Q8—42no—
335C22×Q8C22×D5912yes—
336C22×D4—591no—
337C22×D4—1181no—
338C23×C4—21no—
339C23×C4—21no—
340C23×C4C22⋊D5911yescharacteristic
341C23×C4C22⋊D5911yescharacteristic
342C23×C4C22×D5911yescharacteristic
343C2×SD16—1184no—
344C2×SD16—1184no—
345C2×SD16—2362no—
346C22×C8—592no—
347C22×C8—592no—
348#123—2364no—
349#121—1182no—
350#121—1182no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—591no—
354C2×C4⋊C4—591no—
355C2×C4⋊C4—591no—
356C2×C4⋊C4—1181no—
357C2×C4⋊C4—1181no—
358C2×C4⋊C4C22×D5912yes—
359C2×C22⋊C4—1181no—
360C2×C22⋊C4—1181no—
361C2×C22⋊C4—1181no—
362C2×C22⋊C4—1181no—
363C2×C22⋊C4—1181no—
364C2×C22⋊C4—1181no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C22⋊D5911yescharacteristic
368C2×C42C22⋊D5911yescharacteristic
369C2×C42C22×D5911yescharacteristic
370C2×C42C22×D5911yescharacteristic
371#107—2362no—
372C4⋊C8—1184no—
373#103—1184no—
374#103—1184no—
375#103—2362no—
376#102—2362no—
377C22⋊C8—594no—
378C22⋊C8—1182no—
379#97—1184no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—1181no—
385#95—1181no—
386#95—1181no—
387C24—1181no—
388C24C2×C22⋊D5911yescharacteristic
389C2×Q8—24no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—42no—
398C2×Q8—42no—
399C2×Q8#1174568514yes—
400C2×D4—1181no—
401C2×D4—1181no—
402C2×D4—1182no—
403C2×D4—1182no—
404C2×D4—2362no—
405C2×D4—2362no—
406C2×D4—2362no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—41no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4D4×D5912yes—
427C22×C4#1174568712yes—
428C22×C4C2×C22⋊D5911yescharacteristic
429C22×C4C2×C22⋊D5911yescharacteristic
430C22×C4C2×C22⋊D5911yescharacteristic
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C23×D5911yescharacteristic
434SD16—2368no—
435C2×C8—594no—
436C2×C8—1182no—
437C2×C8—1182no—
438C2×C8—1182no—
439C2×C8—1182no—
440C2×C8—1182no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—1181no—
445C4⋊C4—1181no—
446C4⋊C4—1182no—
447C4⋊C4—1182no—
448C4⋊C4—2362no—
449C4⋊C4—2362no—
450C22⋊C4—2362no—
451C22⋊C4—2362no—
452C22⋊C4—2362no—
453C22⋊C4—2362no—
454C22⋊C4—2362no—
455C22⋊C4—2362no—
456C42—22no—
457C42—22no—
458C42—21no—
459C42—21no—
460C42—21no—
461C42—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—2361no—
467C23—2361no—
468C23—2361no—
469C23—2361no—
470C23—2361no—
471C23—2361no—
472C23—2361no—
473C23#1427369912yes—
474C23#1427375212yes—
475C23#1427375812yes—
476C23C22×C22⋊D5911yescharacteristic
477Q8—24no—
478Q8—24no—
479Q8—48no—
480Q8—42no—
481Q8—42no—
482D4—4724no—
483D4—2362no—
484D4—2362no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—84no—
510C2×C4—2361no—
511C2×C4—2361no—
512C2×C4—2361no—
513C2×C4—2361no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4#1427369912yes—
520C2×C4#1427375812yes—
521C2×C4C2×D4×D5911yescharacteristic
522C2×C4C2×#1174568711yescharacteristic
523C2×C4C2×#1174568512yes—
524C2×C4#1427377212yes—
525C8—1184no—
526C8—2362no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—4721no—
534C22—4721no—
535C22—4721no—
536C22—4721no—
537C22—4721no—
538C22—4721no—
539C22—4721no—
540C22#1683765014yes—
541C22C2×#1427369911yescharacteristic
542C22#1683791311yescharacteristic
543C22#1683796911yescharacteristic
544C22C2×#1427375211yescharacteristic
545C22C2×#1427375811yescharacteristic
546C22C2×#1427375811yescharacteristic
547C22#1683820011yescharacteristic
548C4—22no—
549C4—22no—
550C4—44no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—82no—
555C4—4721no—
556C4—4721no—
557C4—4721no—
558C4#1683804212yes—
559C4#1683807812yes—
560C2—24no—
561C2—9441no—
562C2#2142748111yescharacteristic
563C2C2×#1683765011yescharacteristic
564C2#2143372211yescharacteristic
565C2#2143421811yescharacteristic
566C2#2143426711yescharacteristic
567C2#2143508311yescharacteristic
568C2#2143512311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#391C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenterC2×#391D59
hypocenterC59#24509
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. C2×#16837459C2semidirect
4. #21425491C2semidirect
5. #21425411C2semidirect
6. #21424980C2semidirect
7. #21424789C2semidirect
9. #21424284C2semidirect
10. #21424031C2semidirect
11. C2×C59×#391C2semidirect
279. C2×#391D59semidirect
329. C59#24509semidirect

Maximal subgroups

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

subgroupindex
C2×#168375412
C2×#168374592
#214324272
#214254912
#214254112
#214249802
#214247892
#214244132
#214242842
#214240312
C2×C59×#3912
#2450959