Finite Group Databasebeta

← #28495879

Subgroups — #28495879

Order 15104.

882 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837375C211yesmaximal, characteristic
2#21432540C211yesmaximal, characteristic
3C2×#16837368C211yesmaximal, characteristic
4C2×#16837173C211yesmaximal, characteristic, complemented
5C2×#16837172C211yesmaximal, characteristic, complemented
6#21423792C211yesmaximal, characteristic, complemented
7C2×C59×#424C211yesmaximal, characteristic, complemented
8C22×#11745685C2211yescharacteristic
9C22×#11745664C2211yescharacteristic
10C22×#11745664C2211yescharacteristic
11C2×#14273686C2211yescharacteristic
12#16837375—22no—
13#16837375—22no—
14C2×#14273641C414yes—
15#16837370—22no—
16#16837368—22no—
17#16837368—22no—
18#16837368—22no—
19#16837365—22no—
20C2×(C2×D59)⋊C8—22no—
21#16837173—22no—
22#16837172—22no—
23#16837156—22no—
24C22×M16×C59C2211yescharacteristic
25C23×C8×C59C2211yescharacteristic
26C2×#117×C59C2211yescharacteristic
27C59×#424—22no—
28C59×#424—22no—
29C2×C4⋊C8×C59—22no—
30C2×#11745685—21no—
31C2×#11745685—21no—
32C2×#11745685—21no—
33C2×#11745685—21no—
34C2×#11745685—21no—
35C2×#11745685—21no—
36C2×#11745685—21no—
37C2×#11745685—21no—
38C2×#11745685—21no—
39C2×#11745685—21no—
40C2×#11745685—21no—
41C2×#11745685—21no—
42C22×Dic118Q811yescharacteristic
43C22×D236Q811yescharacteristic
44C22×C22⋊D59C2×C412yes—
45C22×C4×D59D412yes—
46C2×#11745664—21no—
47C2×#11745664—21no—
48C2×#11745664—21no—
49C2×#11745664—21no—
50C2×#11745664—21no—
51C2×#11745664—21no—
52C2×#11745664—21no—
53C2×#11745664—21no—
54C2×#11745664—21no—
55C2×#11745664—21no—
56C2×#11745664—21no—
57C2×#11745664D412yes—
58C22×C59⋊C8—21no—
59C22×C59⋊C8D412yes—
60#14273686—21no—
61#14273686—21no—
62#14273686—21no—
63#14273686—21no—
64C2×C4⋊Dic59—21no—
65C2×C22⋊Dic59C2×C412yes—
66C2×C4×Dic59—21no—
67#14273641—28no—
68(C2×D59)⋊C8—28no—
69(C2×D59)⋊C8—28no—
70C23×C4×C59C2311yescharacteristic
71C2×M16×C59—21no—
72C2×M16×C59—21no—
73C2×M16×C59—21no—
74C2×M16×C59—21no—
75C2×M16×C59—21no—
76C2×M16×C59—21no—
77C22×C8×C59—21no—
78C22×C8×C59—21no—
79C22×C8×C59—21no—
80C22×C8×C59—21no—
81C22×C8×C59—21no—
82C22×C8×C59—21no—
83C22×C8×C59D412yes—
84C22×C8×C59D412yes—
85#117×C59—21no—
86#117×C59—21no—
87#117×C59D412yes—
88#117×C59D412yes—
89C2×C4⋊C4×C59—21no—
90C2×C22⋊C4×C59C2×C412yes—
91C2×C42×C59—21no—
92C4⋊C8×C59—28no—
93C24×C59C22×C411yescharacteristic
94C23×D59C4⋊C412yes—
95C22×C4×C59—22no—
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×D411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59C2×Q811yescharacteristic
104C22×C4×C59#4111yescharacteristic
105C22×C4×C59#4111yescharacteristic
106C22×C4×C59#4111yescharacteristic
107C22×C4×C59#4111yescharacteristic
108C22×C4×C59#4111yescharacteristic
109C22×C4×C59#4111yescharacteristic
110C2×C22⋊D59—22no—
111C2×C22⋊D59—22no—
112C2×C22⋊D59—22no—
113C2×C22⋊D59—22no—
114C2×C22⋊D59—22no—
115C2×C22⋊D59—22no—
116C22×Dic59—22no—
117C22×Dic59C4⋊C412yes—
118#11745685—41no—
119#11745685—41no—
120#11745685—41no—
121#11745685—41no—
122#11745685—41no—
123#11745685—41no—
124#11745685—41no—
125#11745685—41no—
126#11745685—41no—
127#11745685—41no—
128#11745685—41no—
129#11745685—41no—
130#11745685—41no—
131#11745685—41no—
132#11745685—41no—
133#11745685—41no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×C4×D59—22no—
141C2×C4×D59—22no—
142C2×C4×D59—22no—
143C2×C4×D59—22no—
144C2×C4×D59—22no—
145C2×C4×D59—22no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118—21no—
151C2×Dic118—21no—
152M16×C59—41no—
153M16×C59—41no—
154M16×C59—41no—
155M16×C59—41no—
156C2×C8×C59—22no—
157C2×C8×C59—22no—
158C2×C8×C59—22no—
159C2×C8×C59—22no—
160C2×C8×C59—22no—
161C2×C8×C59—22no—
162C2×C8×C59—41no—
163C2×C8×C59—41no—
164C2×C8×C59—41no—
165C2×C8×C59—41no—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—22no—
168C22⋊C4×C59—22no—
169C22⋊C4×C59C22⋊C414yes—
170C42×C59—22no—
171C42×C59—22no—
172C22⋊Dic59—22no—
173C22⋊Dic59—22no—
174C4⋊Dic59—41no—
175C4⋊Dic59—41no—
176C4×Dic59—41no—
177C4×Dic59—41no—
178#11745664—22no—
179#11745664—22no—
180#11745664—41no—
181#11745664—41no—
182#11745664—41no—
183#11745664—41no—
184#11745664—41no—
185#11745664—41no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—22no—
190C2×C59⋊C8—41no—
191C2×C59⋊C8—41no—
192C23×C59—42no—
193C23×C59C2×C22⋊C411yescharacteristic
194C23×C59C2×C4⋊C411yescharacteristic
195C23×C59#11711yescharacteristic
196C23×C59C4×D411yescharacteristic
197C23×C59C4×D411yescharacteristic
198C23×C59C4×D411yescharacteristic
199C23×C59C4×D411yescharacteristic
200C22×D59—22no—
201C22×D59—22no—
202C22×D59—22no—
203C22×D59—44no—
204C2×C4×C59—24no—
205C2×C4×C59—44no—
206C2×C4×C59#12011yescharacteristic
207C2×C4×C59#12011yescharacteristic
208C2×C4×C59#12011yescharacteristic
209C2×C4×C59#12111yescharacteristic
210C2×C4×C59#12111yescharacteristic
211C2×C4×C59#12111yescharacteristic
212C2×C4×C59#12111yescharacteristic
213C2×C4×C59#12111yescharacteristic
214C2×C4×C59#12111yescharacteristic
215C2×C4×C59#12111yescharacteristic
216C2×C4×C59#12111yescharacteristic
217C2×C4×C59C22⋊Q811yescharacteristic
218C2×C4×C59C22⋊Q811yescharacteristic
219C2×C4×C59C22⋊Q811yescharacteristic
220C2×C4×C59C22⋊Q811yescharacteristic
221C2×C4×C59C22⋊Q811yescharacteristic
222C2×C4×C59C22⋊Q811yescharacteristic
223C2×C4×C59#12311yescharacteristic
224C2×C4×C59#12311yescharacteristic
225C2×C4×C59#12311yescharacteristic
226C2×C4×C59#12311yescharacteristic
227C2×C4×C59#12311yescharacteristic
228C2×C4×C59#12411yescharacteristic
229C2×C4×C59#12411yescharacteristic
230C2×C4×C59#12411yescharacteristic
231C2×C4×C59#12611yescharacteristic
232C2×C4×C59#12611yescharacteristic
233C2×C4×C59#12711yescharacteristic
234C22⋊D59—42no—
235C22⋊D59—42no—
236C22⋊D59—42no—
237C22⋊D59—42no—
238C2×Dic59—22no—
239C2×Dic59—22no—
240C2×Dic59—22no—
241C2×Dic59—44no—
242C2×Dic59—42no—
243D236—41no—
244D236—41no—
245D236—41no—
246D236—41no—
247C4×D59—42no—
248C4×D59—42no—
249C4×D59—42no—
250C4×D59—42no—
251Dic118—41no—
252Dic118—41no—
253Dic118—41no—
254Dic118—41no—
255C8×C59—42no—
256C8×C59—42no—
257C59⋊C8—42no—
258C59⋊C8—42no—
259#11088—591nomaximal, Sylow
260C22×C59—22no—
261C22×C59—22no—
262C22×C59—22no—
263C22×C59—41no—
264C22×C59—41no—
265C22×C59—41no—
266C22×C59—41no—
267C22×C59#38011yescharacteristic
268C22×C59#38511yescharacteristic
269C22×C59#38511yescharacteristic
270C22×C59#38611yescharacteristic
271C22×C59#38811yescharacteristic
272C22×C59#38811yescharacteristic
273C22×C59#39011yescharacteristic
274C2×D59—44no—
275C2×D59—44no—
276C2×D59—42no—
277C2×D59—82no—
278C4×C59—44no—
279C4×C59#39211yescharacteristic
280C4×C59#39311yescharacteristic
281C4×C59#39411yescharacteristic
282C4×C59#39411yescharacteristic
283C4×C59#39611yescharacteristic
284C4×C59#39611yescharacteristic
285C4×C59#39711yescharacteristic
286C4×C59#39911yescharacteristic
287Dic59—42no—
288Dic59—82no—
289C2×#424D5911yescharacteristic, complemented
290C2×#410—591no—
291#2879—591no—
292C2×#408—591no—
293C2×#344—591no—
294C2×#344—591no—
295#1729—591no—
296C2×C59—21no—
297C2×C59—21no—
298C2×C59—21no—
299C2×C59—21no—
300C2×C59—21no—
301C2×C59—21no—
302C2×C59#143911yescharacteristic
303C2×C59#186812yes—
304D59—82no—
305C22×#41—591no—
306C22×M16—591no—
307C22×M16—591no—
308C22×M16C2×D5911yescharacteristic
309C23×C8C2×D5911yescharacteristic
310C2×#117—591no—
311C2×#117C2×D5911yescharacteristic
312#424—22no—
313#424—22no—
314C2×C4⋊C8—22no—
315#410—1182no—
316#410—1182no—
317C2×#99—594no—
318#408—1182no—
319#408—1182no—
320#408—1182no—
321#408—1182no—
322#407—1182no—
323C2×C22⋊C8—1182no—
324#344—1182no—
325#344—1182no—
326#336—1182no—
327C59#1108811yescharacteristic, complemented, Sylow
328C2×#41—1181no—
329C2×#41—1181no—
330C2×#41—1181no—
331C2×#41—1181no—
332C2×#41—1181no—
333C2×#41—1181no—
334C2×#41—1181no—
335C2×#41—1181no—
336C2×#41—1181no—
337C2×#41—1181no—
338C2×#41—1181no—
339C2×#41—1181no—
340C22×Q8—591no—
341C22×D4—592no—
342C22×D4—591no—
343C23×C4—592no—
344C23×C4C22×D5911yescharacteristic
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—21no—
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—592no—
352C2×M16—1181no—
353C2×M16—1181no—
354C2×M16—1181no—
355C2×M16—1181no—
356C2×M16—1181no—
357C2×M16—1181no—
358C2×M16—1181no—
359C2×M16—1181no—
360C2×M16—1181no—
361C2×M16—1181no—
362C2×M16—1181no—
363C22×C8—21no—
364C22×C8—21no—
365C22×C8—21no—
366C22×C8—21no—
367C22×C8—21no—
368C22×C8—21no—
369C22×C8—592no—
370C22×C8—1181no—
371C22×C8C22⋊D5912yes—
372C22×C8C22⋊D5912yes—
373#117—21no—
374#117—21no—
375#117—1181no—
376#117—1181no—
377#117—1181no—
378#117—1181no—
379#117D23612yes—
380#117C22⋊D5912yes—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—1181no—
383C2×C22⋊C4—592no—
384C2×C22⋊C4C4×D5912yes—
385C2×C42—21no—
386C2×C42—1181no—
387C4⋊C8—28no—
388#99—1188no—
389C22⋊C8—1188no—
390C22⋊C8—1188no—
391C24—592no—
392C24C2×C4×D5911yescharacteristic
393#41—2361no—
394#41—2361no—
395#41—2361no—
396#41—2361no—
397#41—2361no—
398#41—2361no—
399#41—2361no—
400#41—2361no—
401#41—2361no—
402#41—2361no—
403#41—2361no—
404#41—2361no—
405#41—2361no—
406#41—2361no—
407#41—2361no—
408#41—2361no—
409C2×Q8—1181no—
410C2×Q8—1181no—
411C2×Q8—1181no—
412C2×Q8—1181no—
413C2×Q8—1181no—
414C2×Q8—1181no—
415C2×D4—1182no—
416C2×D4—1182no—
417C2×D4—1182no—
418C2×D4—1182no—
419C2×D4—1182no—
420C2×D4—1182no—
421C2×D4—1181no—
422C2×D4—1181no—
423C2×D4—1181no—
424C2×D4—1181no—
425C2×D4—1181no—
426C2×D4—1181no—
427C22×C4—22no—
428C22×C4—592no—
429C22×C4—1182no—
430C22×C4—1182no—
431C22×C4—1182no—
432C22×C4—1182no—
433C22×C4—1182no—
434C22×C4—1182no—
435C22×C4—1182no—
436C22×C4C2×D23611yescharacteristic
437C22×C4#1174568511yescharacteristic
438C22×C4#1174568511yescharacteristic
439C22×C4D4×D5911yescharacteristic
440C22×C4D4×D5911yescharacteristic
441C22×C4D4×D5911yescharacteristic
442C22×C4#1174568711yescharacteristic
443C22×C4Q8×D5911yescharacteristic
444C22×C4#1174568911yescharacteristic
445C22×C4#1174568911yescharacteristic
446C22×C4#1174568911yescharacteristic
447C22×C4C2×C22⋊D5911yescharacteristic
448C22×C4C2×C22⋊D5911yescharacteristic
449C22×C4C2×C22⋊D5911yescharacteristic
450M16—41no—
451M16—41no—
452M16—41no—
453M16—41no—
454M16—1182no—
455M16—1182no—
456M16—2361no—
457M16—2361no—
458M16—2361no—
459M16—2361no—
460M16—2361no—
461M16—2361no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—22no—
465C2×C8—22no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—41no—
469C2×C8—41no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—1182no—
473C2×C8—1182no—
474C2×C8—1182no—
475C2×C8—1182no—
476C2×C8—2361no—
477C2×C8—2361no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—2361no—
481C4⋊C4—2361no—
482C22⋊C4—22no—
483C22⋊C4—1182no—
484C22⋊C4—1182no—
485C22⋊C4(C2×D59)⋊C414yes—
486C42—22no—
487C42—22no—
488C42—2361no—
489C42—2361no—
490C23—42no—
491C23—1182no—
492C23—1182no—
493C23—1182no—
494C23—2364no—
495C23C2×(C2×D59)⋊C411yescharacteristic
496C23C4⋊C4×D5911yescharacteristic
497C23#1427368511yescharacteristic
498C23C4×C22⋊D5911yescharacteristic
499C23C4×C22⋊D5911yescharacteristic
500C23#1427368911yescharacteristic
501C23#1427368911yescharacteristic
502Q8—2361no—
503Q8—2361no—
504Q8—2361no—
505Q8—2361no—
506D4—2362no—
507D4—2362no—
508D4—2362no—
509D4—2362no—
510D4—2361no—
511D4—2361no—
512D4—2361no—
513D4—2361no—
514C2×C4—24no—
515C2×C4—44no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—1182no—
519C2×C4—2364no—
520C2×C4—2362no—
521C2×C4—2362no—
522C2×C4—2362no—
523C2×C4—2362no—
524C2×C4—2362no—
525C2×C4#1427369511yescharacteristic
526C2×C4#1427369511yescharacteristic
527C2×C4#1427369611yescharacteristic
528C2×C4#1427369811yescharacteristic
529C2×C4#1427369911yescharacteristic
530C2×C4#1427369911yescharacteristic
531C2×C4#1427370111yescharacteristic
532C2×C4#1427370111yescharacteristic
533C2×C4#1427370211yescharacteristic
534C2×C4#1427370211yescharacteristic
535C2×C4#1427370211yescharacteristic
536C2×C4#1427370311yescharacteristic
537C2×C4#1427370311yescharacteristic
538C2×C4#1427370311yescharacteristic
539C2×C4#1427370511yescharacteristic
540C2×C4#1427370511yescharacteristic
541C2×C4#1427370711yescharacteristic
542C2×C4#1427371011yescharacteristic
543C2×C4#1427371211yescharacteristic
544C2×C4#1427371211yescharacteristic
545C2×C4#1427371211yescharacteristic
546C2×C4#1427371211yescharacteristic
547C2×C4#1427371611yescharacteristic
548C2×C4#1427371611yescharacteristic
549C2×C4#1427371611yescharacteristic
550C2×C4#1427372211yescharacteristic
551C2×C4#1427372211yescharacteristic
552C2×C4#1427372311yescharacteristic
553C8—42no—
554C8—42no—
555C8—2362no—
556C8—2362no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—4722no—
565C22—2364no—
566C22—2362no—
567C22—2364no—
568C22#1683725911yescharacteristic
569C22#1683727311yescharacteristic
570C22#1683727311yescharacteristic
571C22#1683728111yescharacteristic
572C22#1683729211yescharacteristic
573C22#1683729211yescharacteristic
574C22#1683730311yescharacteristic
575C4—44no—
576C4—4722no—
577C4—2362no—
578C4#1683731011yescharacteristic
579C4#1683731311yescharacteristic
580C4#1683731911yescharacteristic
581C4#1683731911yescharacteristic
582C4#1683732411yescharacteristic
583C4#1683732411yescharacteristic
584C4#1683732811yescharacteristic
585C4#1683733711yescharacteristic
586C2—21no—
587C2—21no—
588C2—21no—
589C2—21no—
590C2—21no—
591C2—21no—
592C2—4722no—
593C2#2142233811yescharacteristic
594C2#2142438012yes—
subgroupG/H
centerC2×C4#14273695
commutatorC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#424C2
socleC22×C59#385
prime radicalC22×#11745685C22
cosocleC23×C4×C59C23
hypercenterC2×#424D59
hypocenterC59#11088
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#424

Complements

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

NH ≅ G/Ntype
4. C2×#16837173C2semidirect
5. C2×#16837172C2semidirect
6. #21423792C2semidirect
7. C2×C59×#424C2semidirect
289. C2×#424D59semidirect
327. C59#11088semidirect

Maximal subgroups

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

subgroupindex
C2×#168373752
#214325402
C2×#168373682
C2×#168371732
C2×#168371722
#214237922
C2×C59×#4242
#1108859