Finite Group Databasebeta

← #28600893

Subgroups — #28600893

Order 15104.

740 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837693C211yesmaximal, characteristic
2#21426844C211yesmaximal, characteristic, complemented
3#21426740C211yesmaximal, characteristic
4#21425047C211yesmaximal, characteristic, complemented
5#21424866C211yesmaximal, characteristic, complemented
6#21424839C211yesmaximal, characteristic, complemented
7#21424824C211yesmaximal, characteristic, complemented
8#21424793C211yesmaximal, characteristic
9#21424779C211yesmaximal, characteristic
10#21424724C211yesmaximal, characteristic
11#21424717C211yesmaximal, characteristic
12#21424682C211yesmaximal, characteristic, complemented
13#21424675C211yesmaximal, characteristic, complemented
14#21424107C211yesmaximal, characteristic
15C59×#1768C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837693C418yescomplemented
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27#16837306—21no—
28#16837306—21no—
29#16837304C2211yescharacteristic
30#16837304C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837300C2211yescharacteristic
35#16837291—21no—
36#16837291—21no—
37#16837285—21no—
38#16837285—21no—
39#16837268C2211yescharacteristic
40#16837264—21no—
41#16837261—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44C4×C4⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×C4⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C59×#376C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×Dic118—21no—
61C22×D236C2311yescharacteristic
62C22×C4×D59—21no—
63#14273727—22no—
64#14273727C2×C414yes—
65C4×D236—22no—
66C4×D236C2×C414yes—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87#14273661—41no—
88#14273659—41no—
89#14273659C2×C414yes—
90#14273659C2×C414yes—
91#14273649—41no—
92#14273649—41no—
93#14273649C2×C414yes—
94#14273649C2×C414yes—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C4⋊C8×C59—22no—
112C4⋊C8×C59C2×C414yes—
113#97×C59—41no—
114C4×C8×C59—41no—
115#95×C59D411yescharacteristic
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118#95×C59D411yescharacteristic
119C23×D59—21no—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236C22×C412yes—
140C2×C4×D59—24no—
141C2×C4×D59—81no—
142C2×C4×D59—81no—
143C2×Dic118—24no—
144C2×Dic118—81no—
145C2×Dic118—81no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59—21no—
153C2×C8×C59—21no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59C22×C412yes—
157C2×C8×C59C22×C412yes—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59C22×C412yes—
163C42×C59#4112yes—
164(C2×D59)⋊C4—24no—
165(C2×D59)⋊C4—81no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168(C2×D59)⋊C4—81no—
169C4⋊Dic59—24no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C22×C412yes—
182C4⋊Dic59C22×C412yes—
183C4⋊Dic59C22×C412yes—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187Dic59⋊C4—81no—
188C4×Dic59—81no—
189C4×Dic59—81no—
190C4×Dic59—81no—
191C23×C59C22×D411yescharacteristic
192C22×D59—22no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
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—21no—
214C2×C4×C59—21no—
215C2×C4×C59C4×D412yes—
216C2×C4×C59C4×D412yes—
217C2×C4×C59C23×C411yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59H(5,𝔽2)11yescharacteristic
220C2×C4×C592−1+411yescharacteristic
221C2×C4×C592−1+411yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×Dic59—22no—
224C2×Dic59—22no—
225C2×Dic59—22no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
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—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244D236—22no—
245D236—22no—
246D236—41no—
247D236—41no—
248C4×D59—84no—
249Dic118—84no—
250C8×C59—22no—
251C8×C59—22no—
252C8×C59—41no—
253C8×C59—41no—
254#22123—591nomaximal, Sylow
255C22×C59C2×C4×D411yescharacteristic
256C22×C59#53611yescharacteristic
257C22×C59#53711yescharacteristic
258C22×C59#57611yescharacteristic
259C22×C59#57711yescharacteristic
260C22×C59#57711yescharacteristic
261C22×C59#57711yescharacteristic
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C4×C59—22no—
270C4×C59—22no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59#51911yescharacteristic
276C4×C59#52011yescharacteristic
277C4×C59#54111yescharacteristic
278C4×C59#54211yescharacteristic
279Dic59—42no—
280Dic59—42no—
281Dic59—42no—
282Dic59—81no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286C2×#476—591no—
287#2331—591no—
288#2319—591no—
289#1988—591no—
290#1949—591no—
291#1944—591no—
292#1941—591no—
293#1936—591no—
294#1934—591no—
295#1926—591no—
296#1925—591no—
297#1920—591no—
298#1919—591no—
299#1804—591no—
300#1768D5911yescharacteristic, complemented
301C2×C59#232311yescharacteristic
302C2×C59#294611yescharacteristic
303C2×C59#294911yescharacteristic
304C2×C59#324111yescharacteristic
305C2×C59#324411yescharacteristic
306C2×C59#325711yescharacteristic
307C2×C59#326311yescharacteristic
308D59—82no—
309C2×C4⋊Q8—591no—
310C2×C4×D4—591no—
311#476—598no—
312C2×#107—591no—
313C2×#106—591no—
314C2×#106—591no—
315C2×#106—591no—
316C2×C4⋊C8C2×D5911yescharacteristic
317C2×C4⋊C8C2×D5911yescharacteristic
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#97C2×D5911yescharacteristic
323C2×C4×C8C2×D5911yescharacteristic
324#391—1181no—
325#391—1181no—
326#390—591no—
327#390—591no—
328#389—591no—
329#389—591no—
330#389—591no—
331#389—591no—
332#388—1181no—
333#388—1181no—
334#387—1181no—
335#387—1181no—
336#382—591no—
337#381—1181no—
338#380—1181no—
339C4×C4⋊C4—591no—
340C4×C4⋊C4—591no—
341C4×C4⋊C4—591no—
342#376C2×D5911yescharacteristic
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#336C2×D5911yescharacteristic
352#336C2×D5911yescharacteristic
353C59#2212311yescharacteristic, complemented, Sylow
354C22×Q8—1181no—
355C22×D4—591no—
356C23×C4—1181no—
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360C22×C8C22×D5911yescharacteristic
361C4⋊Q8—594no—
362C4⋊Q8—1182no—
363C4×D4—594no—
364C4×D4—1182no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42—1181no—
385C2×C42C22×D5911yescharacteristic
386C2×C42C22×D5911yescharacteristic
387C2×C42C22×D5911yescharacteristic
388#107—2361no—
389#106—594no—
390#106—594no—
391#106—2361no—
392C4⋊C8—22no—
393C4⋊C8C4×D5914yes—
394#102—594no—
395#102—594no—
396#102—2361no—
397#102—2361no—
398#97—41no—
399C4×C8—41no—
400#95—1181no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95D23611yescharacteristic
410#95D23611yescharacteristic
411#95D23611yescharacteristic
412#95D23611yescharacteristic
413C24—1181no—
414C2×Q8—1184no—
415C2×Q8—4721no—
416C2×Q8—4721no—
417C2×D4—592no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C2×D4—1181no—
422C2×D4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—1181no—
426C22×C4—1184no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—4721no—
432C22×C4—4721no—
433C22×C4C2×D23611yescharacteristic
434C22×C4C2×D23611yescharacteristic
435C22×C4C2×D23611yescharacteristic
436C22×C4C2×D23611yescharacteristic
437C22×C4C2×D23611yescharacteristic
438C22×C4C2×D23611yescharacteristic
439C22×C4C23×D5911yescharacteristic
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8C2×C4×D5912yes—
451C2×C8C2×C4×D5912yes—
452C4⋊C4—592no—
453C4⋊C4—592no—
454C4⋊C4—592no—
455C4⋊C4—1184no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—4721no—
468C4⋊C4—4721no—
469C4⋊C4—4721no—
470C4⋊C4—4721no—
471C22⋊C4—1184no—
472C22⋊C4—4721no—
473C22⋊C4—4721no—
474C22⋊C4—4721no—
475C22⋊C4—4721no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—4721no—
481C42—4721no—
482C42—4721no—
483C42C2×C4×D5912yes—
484C42#1174568512yes—
485C23—1182no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23C22×D23611yescharacteristic
493Q8—4724no—
494D4—1182no—
495D4—1182no—
496D4—2361no—
497D4—2361no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—1182no—
515C2×C4—1182no—
516C2×C4—1182no—
517C2×C4—4724no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4C4×D23612yes—
537C2×C4C4×D23612yes—
538C2×C4C22×C4×D5911yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C2×C4#1427377211yescharacteristic
541C2×C4#1427377411yescharacteristic
542C2×C4#1427377411yescharacteristic
543C2×C4#1427377511yescharacteristic
544C8—22no—
545C8—22no—
546C8—41no—
547C8—41no—
548C22—4721no—
549C22—4721no—
550C22—4721no—
551C22—4721no—
552C22—2362no—
553C22—2362no—
554C22—2362no—
555C22C2×C4×D23611yescharacteristic
556C22#1683797811yescharacteristic
557C22#1683798311yescharacteristic
558C22#1683820711yescharacteristic
559C22#1683821411yescharacteristic
560C22#1683821411yescharacteristic
561C22#1683821411yescharacteristic
562C4—22no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—4721no—
569C4—4721no—
570C4—4721no—
571C4—4721no—
572C4—2362no—
573C4—2362no—
574C4—2362no—
575C4#1683787811yescharacteristic
576C4#1683788111yescharacteristic
577C4#1683801011yescharacteristic
578C4#1683801711yescharacteristic
579C2—4722no—
580C2#2142677911yescharacteristic
581C2#2143293011yescharacteristic
582C2#2143295711yescharacteristic
583C2#2143535111yescharacteristic
584C2#2143539611yescharacteristic
585C2#2143550911yescharacteristic
586C2#2143556711yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1768C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1768D59
hypocenterC59#22123
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
2. #21426844C2semidirect
4. #21425047C2semidirect
5. #21424866C2semidirect
6. #21424839C2semidirect
7. #21424824C2semidirect
12. #21424682C2semidirect
13. #21424675C2semidirect
15. C59×#1768C2semidirect
18. #16837693C4semidirect
300. #1768D59semidirect
353. C59#22123semidirect

Maximal subgroups

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

subgroupindex
C2×#168376932
#214268442
#214267402
#214250472
#214248662
#214248392
#214248242
#214247932
#214247792
#214247242
#214247172
#214246822
#214246752
#214241072
C59×#17682
#2212359