Finite Group Databasebeta

← #28624205

Subgroups — #28624205

Order 15104.

1456 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837604C211yesmaximal, characteristic, complemented
2C2×#16837541C212yesmaximal, complemented
3C2×#16837500C211yesmaximal, characteristic, complemented
4#21428997C211yesmaximal, characteristic
5#21425296C212yesmaximal, complemented
6#21425097C212yesmaximal, complemented
7#21424917C211yesmaximal, characteristic, complemented
8#21424895C211yesmaximal, characteristic, complemented
9#21424111C211yesmaximal, characteristic, complemented
10#21424031C212yesmaximal, complemented
11C2×D4×Q8×C59C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C22×#11745671C2211yescharacteristic
14C2×#14273701C2212yes—
15C2×C4×D236C2211yescharacteristic
16#16837604—24no—
17#16837541—28no—
18#16837500—24no—
19C2×#14273660C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2212yes—
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2212yes—
24C2×#14273651C2212yes—
25C2×#14273651C2212yes—
26C2×#14273647C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2212yes—
28C2×C22⋊(C59⋊C8)C2212yes—
29C2×#14273638C2211yescharacteristic
30#16837328—21no—
31#16837310—21no—
32#16837278—22no—
33#16837268C2211yescharacteristic
34#16837262C2212yes—
35#16837164C2212yes—
36C23×Q8×C59C2212yes—
37D4×Q8×C59—24no—
38D4×Q8×C59—24no—
39C2×C4⋊Q8×C59—21no—
40C2×C4⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59—22no—
42C2×C22⋊Q8×C59C2212yes—
43C2×C4×Q8×C59C2211yescharacteristic
44C2×C4×D4×C59—21no—
45C2×C4×D4×C59C2211yescharacteristic
46C22×D236C2311yescharacteristic
47C22×C22⋊D59—22no—
48C22×C4×D59—21no—
49C2×#11745671—22no—
50C2×#11745671—22no—
51C2×#11745671—24no—
52C2×#11745671—24no—
53C2×#11745671—42no—
54C2×#11745671D414yescomplemented
55C22×C59⋊C8C2311yescharacteristic
56C22×C59⋊C8C2312yes—
57C22×C59⋊C8C2311yescharacteristic
58#14273701—24no—
59#14273701—42no—
60C4×D236—22no—
61C4×D236—22no—
62C2×C4⋊Dic59C2312yes—
63C2×C4⋊Dic59C2311yescharacteristic
64C2×C22⋊Dic59—22no—
65C2×(C2×D59)⋊C4—22no—
66C2×(C2×D59)⋊C4—21no—
67#14273660—22no—
68C4⋊(C59⋊C8)—22no—
69#14273651—22no—
70#14273651—24no—
71#14273651—24no—
72#14273651—24no—
73#14273651—42no—
74#14273647—22no—
75C22⋊(C59⋊C8)—24no—
76C22⋊(C59⋊C8)—42no—
77#14273638—22no—
78C59⋊#95—22no—
79C59⋊#95—21no—
80C59⋊#95—21no—
81C22×Q8×C59—24no—
82C22×Q8×C59—24no—
83C22×Q8×C59—22no—
84C22×Q8×C59—24no—
85C22×Q8×C59—24no—
86C22×Q8×C59—42no—
87C22×Q8×C59—42no—
88C22×Q8×C59C2312yes—
89C22×Q8×C59C2311yescharacteristic
90C22×D4×C59D411yescharacteristic
91C23×C4×C59—22no—
92C23×C4×C59C2312yes—
93C4⋊Q8×C59—28no—
94C4⋊Q8×C59—22no—
95C4⋊Q8×C59D414yescomplemented
96C22⋊Q8×C59—216no—
97C22⋊Q8×C59—24no—
98C22⋊Q8×C59—24no—
99C4×Q8×C59—22no—
100C4×Q8×C59D414yescomplemented
101C4×D4×C59—28no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—22no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—22no—
111C2×C22⋊C4×C59D412yes—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C24×C59C2×D412yes—
115C23×D59—21no—
116C2×Q8×C59—28no—
117C2×Q8×C59—21no—
118C2×Q8×C59—22no—
119C2×Q8×C59—21no—
120C2×Q8×C59—21no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—22no—
125C2×Q8×C59—44no—
126C2×Q8×C59—44no—
127C2×Q8×C59—48no—
128C2×Q8×C59—48no—
129C2×Q8×C59—48no—
130C2×Q8×C59—48no—
131C2×Q8×C59C2×D414yes—
132C2×Q8×C59C2×D412yes—
133C2×Q8×C59C2×D412yes—
134C2×Q8×C59#4112yes—
135C2×D4×C59—24no—
136C2×D4×C59—22no—
137C22×C4×C59—21no—
138C22×C4×C59—28no—
139C22×C4×C59—21no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—22no—
147C22×C4×C59—44no—
148C22×C4×C59—44no—
149C22×C4×C59C2×D412yes—
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C2×C22⋊D59—44no—
154C2×C22⋊D59—44no—
155C2×C22⋊D59—82no—
156C22×Dic59—22no—
157C22×Dic59—21no—
158C2×D236—21no—
159C2×D236—21no—
160C2×D236—21no—
161C2×D236—21no—
162C2×D236—21no—
163C2×D236C2×D412yes—
164C2×C4×D59—42no—
165C2×C4×D59—82no—
166C4⋊C4×C59—28no—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—22no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59C2×D412yes—
173C22⋊C4×C59—28no—
174C22⋊C4×C59—24no—
175C42×C59—24no—
176C42×C59C2×D412yes—
177C42×C59C2×D412yes—
178C22⋊Dic59—82no—
179#11745671—28no—
180#11745671—48no—
181(C2×D59)⋊C4—44no—
182(C2×D59)⋊C4—42no—
183C4⋊Dic59—21no—
184C4⋊Dic59—22no—
185C4⋊Dic59—21no—
186C4⋊Dic59—22no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—22no—
192C2×C59⋊C8—21no—
193C2×C59⋊C8—22no—
194C2×C59⋊C8—21no—
195C2×C59⋊C8C2×D412yes—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—22no—
201C23×C59—22no—
202C23×C59#13714yes—
203C23×C59C22×D411yescharacteristic
204C22×D59—22no—
205C22×D59—41no—
206C22×D59—41no—
207C22×D59—42no—
208C22×D59—42no—
209Q8×C59—22no—
210Q8×C59—22no—
211Q8×C59—22no—
212Q8×C59—24no—
213Q8×C59—24no—
214Q8×C59—416no—
215Q8×C59—42no—
216Q8×C59—42no—
217Q8×C59#12114yes—
218D4×C59—28no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—24no—
226C2×C4×C59—22no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—416no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59—42no—
233C2×C4×C59—42no—
234C2×C4×C59—42no—
235C2×C4×C59—42no—
236C2×C4×C59—42no—
237C2×C4×C59#12012yes—
238C2×C4×C59#12012yes—
239C2×C4×C59C22×D411yescharacteristic
240C2×C4×C59C22×D411yescharacteristic
241C2×C4×C59C22×D411yescharacteristic
242C2×C4×C59C2×#4111yescharacteristic
243C2×C4×C59H(5,𝔽2)11yescharacteristic
244C2×C4×C59H(5,𝔽2)11yescharacteristic
245C22⋊D59—88no—
246C2×Dic59—41no—
247C2×Dic59—42no—
248C2×Dic59—42no—
249C2×Dic59—42no—
250C2×Dic59—42no—
251D236—22no—
252D236—22no—
253D236—41no—
254D236—41no—
255C4×D59—84no—
256C59⋊C8—22no—
257C59⋊C8—41no—
258C59⋊C8—42no—
259#24251—591nomaximal, Sylow
260C22×C59—24no—
261C22×C59—24no—
262C22×C59—24no—
263C22×C59—42no—
264C22×C59—42no—
265C22×C59—42no—
266C22×C59—42no—
267C22×C59C2×#12011yescharacteristic
268C22×C59D4211yescharacteristic
269C22×C59#54611yescharacteristic
270C22×C59C2×#13712yes—
271C22×C59#57712yes—
272C2×D59—42no—
273C2×D59—44no—
274C2×D59—81no—
275C2×D59—81no—
276C2×D59—82no—
277C4×C59—24no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—42no—
281C4×C59—44no—
282C4×C59—42no—
283C4×C59C2×#12111yescharacteristic
284C4×C59#53511yescharacteristic
285C4×C59D4211yescharacteristic
286C4×C59#54611yescharacteristic
287Dic59—81no—
288Dic59—82no—
289C2×D4×Q8D5911yescharacteristic, complemented
290C2×#461—591no—
291C2×#448—592no—
292C2×#440—591no—
293#2535—591no—
294#2036—592no—
295#2000—592no—
296#1960—591no—
297#1955—591no—
298#1805—591no—
299#1787—592no—
300C2×C59—44no—
301C2×C59#240811yescharacteristic
302C2×C59#300411yescharacteristic
303C2×C59#301011yescharacteristic
304C2×C59#326612yes—
305C2×C59#332012yes—
306D59—82no—
307C23×Q8C2×D5912yes—
308C22×SD16—591no—
309C22×SD16—592no—
310D4×Q8—24no—
311D4×Q8—24no—
312C2×C4⋊Q8—21no—
313C2×C4⋊Q8C2×D5911yescharacteristic
314C2×C22⋊Q8—22no—
315C2×C22⋊Q8C2×D5912yes—
316C2×#121—592no—
317C2×C4×Q8C2×D5911yescharacteristic
318C2×C4×D4—21no—
319C2×C4×D4—591no—
320C2×C4×D4C2×D5911yescharacteristic
321#461—1184no—
322#448—1188no—
323#440—1184no—
324C2×#107—591no—
325C2×C4⋊C8—591no—
326C2×#103—592no—
327C2×#103—591no—
328C2×#103—592no—
329C2×#103—592no—
330C2×#103—592no—
331C2×#102—591no—
332C2×C22⋊C8—592no—
333C2×C22⋊C8—592no—
334C2×#97—591no—
335#397—1181no—
336#392—1181no—
337#386—1182no—
338#382—591no—
339#380—592no—
340#340—592no—
341C59#2425111yescharacteristic, complemented, Sylow
342C22×Q8—24no—
343C22×Q8—24no—
344C22×Q8—22no—
345C22×Q8—24no—
346C22×Q8—24no—
347C22×Q8—42no—
348C22×Q8—42no—
349C22×Q8C22×D5912yes—
350C22×Q8C22×D5911yescharacteristic
351C22×D4—591no—
352C22×D4—1182no—
353C22×D4C22⋊D5911yescharacteristic
354C23×C4—22no—
355C23×C4—1181no—
356C23×C4C22×D5912yes—
357C2×SD16—594no—
358C2×SD16—1182no—
359C2×SD16—1182no—
360C2×SD16—1184no—
361C2×SD16—1184no—
362C2×SD16—2362no—
363C22×C8—591no—
364C22×C8—592no—
365C22×C8—591no—
366C4⋊Q8—28no—
367C4⋊Q8—22no—
368C4⋊Q8C22⋊D5914yescomplemented
369C22⋊Q8—216no—
370C22⋊Q8—24no—
371C22⋊Q8—24no—
372#121—1184no—
373#121—2362no—
374C4×Q8—22no—
375C4×Q8C22⋊D5914yescomplemented
376C4×D4—28no—
377C4×D4—22no—
378C4×D4—22no—
379C4×D4—1182no—
380C4×D4—1182no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—22no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—592no—
387C2×C4⋊C4C22⋊D5911yescharacteristic
388C2×C4⋊C4C22×D5911yescharacteristic
389C2×C22⋊C4—22no—
390C2×C22⋊C4—1182no—
391C2×C22⋊C4—1181no—
392C2×C22⋊C4—1182no—
393C2×C22⋊C4C22⋊D5912yes—
394C2×C42—21no—
395C2×C42C22×D5911yescharacteristic
396#107—1182no—
397C4⋊C8—1182no—
398#103—1184no—
399#103—1184no—
400#103—1182no—
401#103—1184no—
402#103—2362no—
403#102—1182no—
404C22⋊C8—1184no—
405C22⋊C8—2362no—
406#97—1182no—
407#95—1182no—
408#95—1181no—
409#95—1181no—
410C24—1181no—
411C24C2×C22⋊D5912yes—
412C2×Q8—28no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—21no—
416C2×Q8—21no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—44no—
422C2×Q8—44no—
423C2×Q8—48no—
424C2×Q8—48no—
425C2×Q8—48no—
426C2×Q8—48no—
427C2×Q8D4×D5912yes—
428C2×Q8#1174568712yes—
429C2×Q8C2×C22⋊D5914yes—
430C2×Q8C2×C22⋊D5912yes—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—592no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—1181no—
437C2×D4—1181no—
438C2×D4—1181no—
439C2×D4—4722no—
440C2×D4—2364no—
441C2×D4—2364no—
442C22×C4—21no—
443C22×C4—28no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—1182no—
455C22×C4—1181no—
456C22×C4—4722no—
457C22×C4—2362no—
458C22×C4C2×C22⋊D5912yes—
459C22×C4C2×C22⋊D5911yescharacteristic
460C22×C4C2×C22⋊D5911yescharacteristic
461C22×C4C23×D5911yescharacteristic
462SD16—1188no—
463SD16—2368no—
464C2×C8—592no—
465C2×C8—1181no—
466C2×C8—1181no—
467C2×C8—1182no—
468C2×C8—1181no—
469C2×C8—1182no—
470C2×C8—1181no—
471C2×C8—1182no—
472C2×C8—1181no—
473C4⋊C4—28no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—22no—
478C4⋊C4—21no—
479C4⋊C4—1182no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1182no—
483C4⋊C4C2×C22⋊D5912yes—
484C22⋊C4—28no—
485C22⋊C4—24no—
486C22⋊C4—4722no—
487C22⋊C4—2364no—
488C22⋊C4—2362no—
489C42—24no—
490C42C2×C22⋊D5912yes—
491C42C2×C22⋊D5912yes—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—1182no—
499C23—2361no—
500C23—2362no—
501C23—2362no—
502C23—2361no—
503C23#1427375814yes—
504C23C22×C22⋊D5911yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8—24no—
509Q8—24no—
510Q8—416no—
511Q8—42no—
512Q8—42no—
513Q8#1427369814yes—
514D4—28no—
515D4—1182no—
516D4—1182no—
517D4—4728no—
518D4—2361no—
519D4—2361no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—416no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—4724no—
539C2×C4—2361no—
540C2×C4—2362no—
541C2×C4—2362no—
542C2×C4—2362no—
543C2×C4—2362no—
544C2×C4#1427369612yes—
545C2×C4#1427369612yes—
546C2×C4C2×D4×D5911yescharacteristic
547C2×C4C22×C22⋊D5911yescharacteristic
548C2×C4C22×C22⋊D5911yescharacteristic
549C2×C4C2×#1174568711yescharacteristic
550C2×C4#1427377211yescharacteristic
551C2×C4#1427377211yescharacteristic
552C8—1182no—
553C8—2361no—
554C8—2362no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—42no—
562C22—4721no—
563C22—4722no—
564C22—4721no—
565C22—2364no—
566C22—2362no—
567C22C2×#1427369611yescharacteristic
568C22D4×C22⋊D5911yescharacteristic
569C22#1683804611yescharacteristic
570C22C2×#1427375812yes—
571C22#1683821012yes—
572C4—22no—
573C4—22no—
574C4—24no—
575C4—42no—
576C4—42no—
577C4—44no—
578C4—4721no—
579C4—4722no—
580C4C2×#1427369811yescharacteristic
581C4#1683796911yescharacteristic
582C4D4×C22⋊D5911yescharacteristic
583C4#1683804611yescharacteristic
584C2—44no—
585C2—4722no—
586C2#2142744611yescharacteristic
587C2#2143333011yescharacteristic
588C2#2143338211yescharacteristic
589C2#2143560712yes—
590C2#2143626712yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×D4×Q8×C59C2
socleC23×C59C22×D4
prime radical#21428997C2
hypercenterC2×D4×Q8D59
hypocenterC59#24251
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837604C2semidirect
2. C2×#16837541C2semidirect
3. C2×#16837500C2semidirect
5. #21425296C2semidirect
6. #21425097C2semidirect
7. #21424917C2semidirect
8. #21424895C2semidirect
9. #21424111C2semidirect
10. #21424031C2semidirect
11. C2×D4×Q8×C59C2semidirect
54. C2×#11745671D4semidirect
95. C4⋊Q8×C59D4semidirect
100. C4×Q8×C59D4semidirect
289. C2×D4×Q8D59semidirect
341. C59#24251semidirect
368. C4⋊Q8C22⋊D59semidirect
375. C4×Q8C22⋊D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168376042
C2×#168375412
C2×#168375002
#214289972
#214252962
#214250972
#214249172
#214248952
#214241112
#214240312
C2×D4×Q8×C592
#2425159