Finite Group Databasebeta

← #28597271

Subgroups — #28597271

Order 15104.

1062 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837797C211yesmaximal, characteristic
2C2×#16837507C211yesmaximal, characteristic, complemented
3C2×#16837502C211yesmaximal, characteristic
4C2×C4×#11745672C211yesmaximal, characteristic, complemented
5C2×C4×#11745670C211yesmaximal, characteristic
6#21426815C211yesmaximal, characteristic, complemented
7#21424976C211yesmaximal, characteristic
8#21424966C211yesmaximal, characteristic, complemented
9#21424956C211yesmaximal, characteristic, complemented
10#21424949C211yesmaximal, characteristic
11#21424908C211yesmaximal, characteristic
12#21424897C211yesmaximal, characteristic, complemented
13#21424262C211yesmaximal, characteristic, complemented
14#21423783C211yesmaximal, characteristic, complemented
15C59×#2326C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745672C2211yescharacteristic
18C22×#11745670C2211yescharacteristic
19C22×#11745670C2211yescharacteristic
20C2×#14273727C2211yescharacteristic
21C2×C4×Dic118C2211yescharacteristic
22C2×C4×Dic118C2211yescharacteristic
23#16837797C418yescomplemented
24#16837507—24no—
25#16837502—24no—
26C4×#11745672—24no—
27C4×#11745670—24no—
28C2×#14273660C2211yescharacteristic
29C2×#14273660C2211yescharacteristic
30C2×#14273658C2211yescharacteristic
31C2×#14273658C2211yescharacteristic
32C2×#14273651C2211yescharacteristic
33C2×#14273651C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×#14273650C2211yescharacteristic
36C2×#14273650C2211yescharacteristic
37C2×#14273650C2211yescharacteristic
38C2×#14273648C2211yescharacteristic
39C2×#14273648C2211yescharacteristic
40C2×#14273638C2211yescharacteristic
41C2×#14273638C2211yescharacteristic
42C2×#14273638C2211yescharacteristic
43C2×C4×C59⋊C8C2211yescharacteristic
44C2×C4×C59⋊C8C2211yescharacteristic
45C2×C4×C59⋊C8C2211yescharacteristic
46#16837308C2211yescharacteristic
47#16837308C2211yescharacteristic
48#16837287—22no—
49#16837287—22no—
50#16837268C2211yescharacteristic
51C2×#124×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C2×C4×D4×C59C2211yescharacteristic
54C59×#391C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#388—22no—
57C59×#385—22no—
58C59×#384C2211yescharacteristic
59C59×#376C2211yescharacteristic
60C22×Dic118C2311yescharacteristic
61C22×Dic118C2311yescharacteristic
62C2×#11745672—22no—
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745672C2×C414yescomplemented
67C2×#11745672C2×C414yescomplemented
68C2×#11745670—22no—
69C2×#11745670—22no—
70C2×#11745670—22no—
71C2×#11745670—22no—
72C2×#11745670C2×C414yes—
73C2×#11745670C2×C414yes—
74C22×C59⋊C8C2311yescharacteristic
75C22×C59⋊C8C2311yescharacteristic
76C22×C59⋊C8C2311yescharacteristic
77C22×C59⋊C8C2311yescharacteristic
78#14273727—22no—
79#14273727C2×C414yescomplemented
80C4×Dic118—22no—
81C4×Dic118—22no—
82C4×Dic118—22no—
83C4×Dic118—22no—
84C2×C4⋊Dic59—22no—
85C2×C4⋊Dic59C2311yescharacteristic
86C2×C4⋊Dic59C2311yescharacteristic
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91#14273660—22no—
92#14273660—22no—
93#14273658—22no—
94#14273658—22no—
95#14273651—22no—
96#14273651—22no—
97#14273650—22no—
98#14273650—22no—
99#14273650—22no—
100#14273650—22no—
101#14273648—22no—
102#14273648—22no—
103#14273638—22no—
104#14273638C2×C414yescomplemented
105#14273638D414yes—
106C4×C59⋊C8—22no—
107C4×C59⋊C8—22no—
108C4×C59⋊C8D414yes—
109C59⋊#95—22no—
110C59⋊#95—22no—
111C59⋊#95—22no—
112C22×Q8×C59C2311yescharacteristic
113C22×D4×C59C2311yescharacteristic
114C23×C4×C59—21no—
115#124×C59—22no—
116#124×C59C2×C414yes—
117C4×Q8×C59—22no—
118C4×Q8×C59—22no—
119C4×D4×C59—22no—
120C4×D4×C59—22no—
121C2×C4⋊C4×C59—21no—
122C2×C4⋊C4×C59—22no—
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C22⋊C4×C59—22no—
126C2×C22⋊C4×C59—21no—
127C2×C22⋊C4×C59—22no—
128C2×C42×C59—21no—
129C2×C42×C59C2311yescharacteristic
130C2×C42×C59C2311yescharacteristic
131C2×C42×C59C2311yescharacteristic
132#95×C59—22no—
133#95×C59D412yes—
134#95×C59D412yes—
135C24×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59C22×C412yes—
142C2×D4×C59—21no—
143C2×D4×C59—21no—
144C2×D4×C59—21no—
145C2×D4×C59—21no—
146C2×D4×C59—21no—
147C2×D4×C59C22×C412yes—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—42no—
152C22×C4×C59—82no—
153C22×C4×C59C2×D412yes—
154C22×C4×C59C2×D412yes—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2411yescharacteristic
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118C22×C412yes—
173C2×Dic118C22×C412yes—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—42no—
179C4⋊C4×C59—82no—
180C22⋊C4×C59—28no—
181C22⋊C4×C59—42no—
182C22⋊C4×C59—82no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186C42×C59C22×C412yes—
187C42×C59C2×D412yes—
188C42×C59C2×D412yes—
189C42×C59#4112yes—
190#11745672—28no—
191#11745672—28no—
192#11745670—28no—
193#11745670—28no—
194C4⋊Dic59—28no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199Dic59⋊C4—42no—
200Dic59⋊C4—42no—
201C4×Dic59—42no—
202C4×Dic59—42no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8C22×C412yes—
208C2×C59⋊C8C22×C412yes—
209C2×C59⋊C8C2×D412yes—
210C2×C59⋊C8C2×D412yes—
211C2×C59⋊C8C2×D412yes—
212C2×C59⋊C8C2×D412yes—
213C2×C59⋊C8#4112yes—
214C2×C59⋊C8#4112yes—
215C23×C59—22no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—42no—
219C23×C59—42no—
220C23×C59C22×D411yescharacteristic
221Q8×C59—22no—
222Q8×C59—22no—
223Q8×C59—41no—
224Q8×C59—41no—
225D4×C59—22no—
226D4×C59—22no—
227D4×C59—41no—
228D4×C59—41no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—84no—
244C2×C4×C59C4×D414yes—
245C2×C4×C59#12712yes—
246C2×C4×C59#12712yes—
247C2×C4×C59C23×C411yescharacteristic
248C2×C4×C59C22×D411yescharacteristic
249C2×C4×C59C22×D411yescharacteristic
250C2×C4×C59C2×#4111yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×Dic59—22no—
254C2×Dic59—22no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—42no—
262C2×Dic59—42no—
263Dic118—22no—
264Dic118—22no—
265Dic118—22no—
266Dic118—22no—
267Dic118—41no—
268Dic118—41no—
269Dic118—41no—
270Dic118—41no—
271C59⋊C8—22no—
272C59⋊C8—22no—
273C59⋊C8C4×D414yes—
274C59⋊C8C4×D414yes—
275#21703—591nomaximal, Sylow
276C22×C59—42no—
277C22×C59—44no—
278C22×C59—81no—
279C22×C59—81no—
280C22×C59—82no—
281C22×C59C2×C4×D411yescharacteristic
282C22×C59C2×#12711yescharacteristic
283C22×C59#53211yescharacteristic
284C22×C59#57712yes—
285C22×C59#57812yes—
286C4×C59—22no—
287C4×C59—22no—
288C4×C59—24no—
289C4×C59—42no—
290C4×C59—42no—
291C4×C59—81no—
292C4×C59—81no—
293C4×C59C2×C4×D411yescharacteristic
294C4×C59C2×C4×D411yescharacteristic
295C4×C59#53211yescharacteristic
296C4×C59#53211yescharacteristic
297Dic59—42no—
298Dic59—42no—
299Dic59—81no—
300Dic59—81no—
301C2×#497—591no—
302C2×#441—591no—
303C2×#440—591no—
304C2×C4×Q16—591no—
305C2×C4×SD16—591no—
306#2327—591no—
307#2326D5911yescharacteristic
308#1973—591no—
309#1971—591no—
310#1969—591no—
311#1968—591no—
312#1958—591no—
313#1956—591no—
314#1847—591no—
315#1726—591no—
316C2×C59—82no—
317C2×C59C4×#12711yescharacteristic
318C2×C59#294712yes—
319C2×C59#315411yescharacteristic
320C2×C59#315611yescharacteristic
321C2×C59#317812yes—
322C22×Q16—591no—
323C22×Q16—591no—
324C22×SD16—591no—
325C22×SD16—591no—
326C2×C4⋊Q8—591no—
327C2×#124C2×D5911yescharacteristic
328C2×C4×Q8—591no—
329C2×C4×Q8—591no—
330C2×C4×Q8C2×D5911yescharacteristic
331C2×C4×D4C2×D5911yescharacteristic
332#497—598no—
333#441—1184no—
334#440—1184no—
335C4×Q16—1184no—
336C4×SD16—1184no—
337C2×#107—591no—
338C2×#107—591no—
339C2×#106—591no—
340C2×#106—591no—
341C2×#103—591no—
342C2×#103—591no—
343C2×#103—591no—
344C2×#103—591no—
345C2×#103—591no—
346C2×#103—591no—
347C2×#102—591no—
348C2×#102—591no—
349C2×#97—591no—
350C2×#97—591no—
351C2×#97—591no—
352C2×C4×C8—591no—
353C2×C4×C8—591no—
354C2×C4×C8—591no—
355#391—591no—
356#391—591no—
357#391C2×D5911yescharacteristic
358#390C2×D5911yescharacteristic
359#388—22no—
360#387—1182no—
361#387—1182no—
362#385—22no—
363#384C2×D5911yescharacteristic
364#382—591no—
365#376C2×D5911yescharacteristic
366C59#2170311yescharacteristic, complemented, Sylow
367C22×Q8—591no—
368C22×Q8—591no—
369C22×Q8C22×D5911yescharacteristic
370C22×D4C22×D5911yescharacteristic
371C23×C4—21no—
372C2×Q16—594no—
373C2×Q16—594no—
374C2×Q16—1182no—
375C2×Q16—1182no—
376C2×Q16—1182no—
377C2×Q16—1182no—
378C2×SD16—594no—
379C2×SD16—594no—
380C2×SD16—1182no—
381C2×SD16—1182no—
382C2×SD16—1182no—
383C2×SD16—1182no—
384C22×C8—591no—
385C22×C8—591no—
386C22×C8—591no—
387C22×C8—591no—
388C4⋊Q8—594no—
389C4⋊Q8—1182no—
390#124—22no—
391#124C4×D5914yes—
392C4×Q8—22no—
393C4×Q8—22no—
394C4×Q8—1182no—
395C4×Q8—1182no—
396C4×Q8—1182no—
397C4×Q8—1182no—
398C4×D4—22no—
399C4×D4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—591no—
403C2×C4⋊C4—591no—
404C2×C4⋊C4—1182no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4C22×D5911yescharacteristic
408C2×C4⋊C4C22×D5911yescharacteristic
409C2×C22⋊C4—22no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—22no—
412C2×C42—21no—
413C2×C42—1181no—
414C2×C42—1181no—
415C2×C42C22×D5911yescharacteristic
416C2×C42C22×D5911yescharacteristic
417C2×C42C22×D5911yescharacteristic
418#107—1182no—
419#107—1182no—
420#106—1182no—
421#106—1182no—
422#103—1182no—
423#103—1182no—
424#103—1182no—
425#103—1182no—
426#103—1182no—
427#103—1182no—
428#102—1182no—
429#102—1182no—
430#97—594no—
431#97—594no—
432#97—1182no—
433C4×C8—594no—
434C4×C8—1182no—
435C4×C8—1182no—
436#95—22no—
437#95—1182no—
438#95—1182no—
439#95—1182no—
440#95C22⋊D5912yes—
441#95C22⋊D5912yes—
442C24—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—592no—
449C2×Q8—592no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C2×Q8—1181no—
453C2×Q8—1181no—
454C2×Q8—1181no—
455C2×Q8—1181no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8C2×C4×D5912yes—
461C2×D4—21no—
462C2×D4—21no—
463C2×D4—21no—
464C2×D4—21no—
465C2×D4—21no—
466C2×D4C2×C4×D5912yes—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—42no—
471C22×C4—82no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—1181no—
476C22×C4C2×C22⋊D5912yes—
477C22×C4C2×C22⋊D5912yes—
478C22×C4C2×C22⋊D5911yescharacteristic
479C22×C4C2×C22⋊D5911yescharacteristic
480C22×C4C23×D5911yescharacteristic
481Q16—1188no—
482Q16—1188no—
483SD16—1188no—
484SD16—1188no—
485C2×C8—592no—
486C2×C8—592no—
487C2×C8—592no—
488C2×C8—592no—
489C2×C8—592no—
490C2×C8—592no—
491C2×C8—592no—
492C2×C8—592no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—42no—
502C4⋊C4—82no—
503C4⋊C4—1188no—
504C4⋊C4—1181no—
505C4⋊C4—1181no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C22⋊C4—28no—
511C22⋊C4—42no—
512C22⋊C4—82no—
513C42—21no—
514C42—21no—
515C42—42no—
516C42—2362no—
517C42—2362no—
518C42C2×C4×D5912yes—
519C42#1174568512yes—
520C42D4×D5912yes—
521C42D4×D5912yes—
522C23—22no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—42no—
527C23C22×C22⋊D5911yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532Q8—1182no—
533Q8—1182no—
534Q8—1182no—
535Q8—1182no—
536Q8—2361no—
537Q8—2361no—
538Q8—2361no—
539Q8—2361no—
540D4—22no—
541D4—22no—
542D4—41no—
543D4—41no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—84no—
559C2×C4—1182no—
560C2×C4—1182no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2362no—
564C2×C4—2362no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4C4×C22⋊D5914yes—
570C2×C4#1427372312yes—
571C2×C4#1427372312yes—
572C2×C4C22×C4×D5911yescharacteristic
573C2×C4C2×D4×D5911yescharacteristic
574C2×C4C2×D4×D5911yescharacteristic
575C2×C4#41×D5911yescharacteristic
576C2×C4#41×D5911yescharacteristic
577C2×C4C2×#1174568511yescharacteristic
578C8—594no—
579C8—594no—
580C8—1182no—
581C8—1182no—
582C22—44no—
583C22—42no—
584C22—82no—
585C22—81no—
586C22—81no—
587C22C2×C4×C22⋊D5911yescharacteristic
588C22C2×#1427372311yescharacteristic
589C22#1683794811yescharacteristic
590C22#1683821312yes—
591C22#1683821712yes—
592C4—24no—
593C4—22no—
594C4—22no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—4721no—
600C4—4721no—
601C4—2362no—
602C4—2362no—
603C4C4×D4×D5911yescharacteristic
604C4C4×D4×D5911yescharacteristic
605C4#1683794911yescharacteristic
606C4#1683794911yescharacteristic
607C2—82no—
608C2C4×#1427372311yescharacteristic
609C2#2143293512yes—
610C2#2143456811yescharacteristic
611C2#2143458711yescharacteristic
612C2#2143475812yes—
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#2326C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenter#2326D59
hypocenterC59#21703
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. C2×#16837507C2semidirect
4. C2×C4×#11745672C2semidirect
6. #21426815C2semidirect
8. #21424966C2semidirect
9. #21424956C2semidirect
12. #21424897C2semidirect
13. #21424262C2semidirect
14. #21423783C2semidirect
23. #16837797C4semidirect
66. C2×#11745672C2×C4semidirect
67. C2×#11745672C2×C4semidirect
79. #14273727C2×C4semidirect
104. #14273638C2×C4semidirect
366. C59#21703semidirect

Maximal subgroups

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

subgroupindex
C2×#168377972
C2×#168375072
C2×#168375022
C2×C4×#117456722
C2×C4×#117456702
#214268152
#214249762
#214249662
#214249562
#214249492
#214249082
#214248972
#214242622
#214237832
C59×#23262
#2170359