Finite Group Databasebeta

← #28504163

Subgroups — #28504163

Order 15104.

1228 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837192C212yesmaximal, complemented
2C2×#16837189C212yesmaximal, complemented
3#21424017C211yesmaximal, characteristic, complemented
4#21424011C211yesmaximal, characteristic, complemented
5C2×C59×#534C211yesmaximal, characteristic
6C2×#14273644—21no—
7C2×#14273644C2212yes—
8C2×#14273642—21no—
9C2×#14273642C2212yes—
10#16837366—21no—
11#16837258—21no—
12#16837192—24no—
13#16837189—24no—
14C23×D4×C59C411yescharacteristic, complemented
15C23×D4×C59C2211yescharacteristic
16C59×#534—24no—
17C59×#534—24no—
18C2×#123×C59—21no—
19C2×#123×C59C411yescharacteristic, complemented
20C2×#123×C59C2211yescharacteristic
21C2×#121×C59—21no—
22C2×#121×C59—21no—
23C2×#120×C59—21no—
24C2×#120×C59C411yescharacteristic, complemented
25C2×#120×C59C2211yescharacteristic
26C22×C22⋊C4×C59C411yescharacteristic, complemented
27C2×#11745664—22no—
28C2×C22⋊Dic59—22no—
29C2×C22⋊Dic59—42no—
30#14273644—22no—
31#14273644—22no—
32#14273644—42no—
33#14273642—22no—
34#14273642—22no—
35#14273642—42no—
36C22⋊(C59⋊C8)—42no—
37C25×C59C2×C411yescharacteristic
38C25×C59C2×C411yescharacteristic
39C25×C59D411yescharacteristic
40C22×D4×C59—21no—
41C22×D4×C59—21no—
42C22×D4×C59—22no—
43C22×D4×C59—21no—
44C22×D4×C59—21no—
45C22×D4×C59—22no—
46C22×D4×C59—22no—
47C22×D4×C59—22no—
48C22×D4×C59—42no—
49C22×D4×C59—42no—
50C22×D4×C59—42no—
51C22×D4×C59—42no—
52C22×D4×C59C2×C411yescharacteristic
53C22×D4×C59D411yescharacteristic
54C22×D4×C59D411yescharacteristic
55C22×D4×C59C2311yescharacteristic
56C23×C4×C59—21no—
57C23×C4×C59C2×C411yescharacteristic
58C23×C4×C59D411yescharacteristic
59#123×C59—28no—
60#123×C59—22no—
61#123×C59—22no—
62#123×C59—22no—
63#123×C59D414yes—
64#121×C59—28no—
65#121×C59—28no—
66#120×C59—28no—
67#120×C59—22no—
68#120×C59—22no—
69#120×C59—22no—
70#120×C59D414yes—
71C2×C4⋊C4×C59—21no—
72C2×C4⋊C4×C59—21no—
73C2×C22⋊C4×C59—21no—
74C2×C22⋊C4×C59—21no—
75C2×C22⋊C4×C59—21no—
76C2×C22⋊C4×C59—21no—
77C2×C22⋊C4×C59—21no—
78C2×C22⋊C4×C59—22no—
79C2×C22⋊C4×C59—22no—
80C2×C22⋊C4×C59C2×C411yescharacteristic
81C2×C22⋊C4×C59C2×C411yescharacteristic
82C24×C59—22no—
83C24×C59—22no—
84C24×C59—21no—
85C24×C59—21no—
86C24×C59—21no—
87C24×C59—22no—
88C24×C59—21no—
89C24×C59—22no—
90C24×C59—41no—
91C24×C59—41no—
92C24×C59—41no—
93C24×C59—82no—
94C24×C59—82no—
95C24×C59—82no—
96C24×C59C22⋊C411yescharacteristic
97C24×C59C22⋊C411yescharacteristic
98C24×C59C22⋊C411yescharacteristic
99C24×C59C22⋊C412yescomplemented
100C24×C59C22×C411yescharacteristic
101C24×C59C2×D411yescharacteristic
102C2×D4×C59—24no—
103C2×D4×C59—24no—
104C2×D4×C59—22no—
105C2×D4×C59—22no—
106C2×D4×C59—21no—
107C2×D4×C59—21no—
108C2×D4×C59—22no—
109C2×D4×C59—24no—
110C2×D4×C59—24no—
111C2×D4×C59—22no—
112C2×D4×C59—41no—
113C2×D4×C59—41no—
114C2×D4×C59—41no—
115C2×D4×C59—41no—
116C2×D4×C59—41no—
117C2×D4×C59—41no—
118C2×D4×C59—48no—
119C2×D4×C59—48no—
120C2×D4×C59—48no—
121C2×D4×C59—42no—
122C2×D4×C59—42no—
123C2×D4×C59—48no—
124C2×D4×C59—42no—
125C2×D4×C59—42no—
126C2×D4×C59—82no—
127C2×D4×C59—82no—
128C2×D4×C59C2×D412yes—
129C2×D4×C59C2×D412yes—
130C22×C4×C59—22no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—24no—
136C22×C4×C59—22no—
137C22×C4×C59—41no—
138C22×C4×C59—41no—
139C22×C4×C59—41no—
140C22×C4×C59—82no—
141C22×C4×C59C22⋊C411yescharacteristic
142C22×C4×C59C22⋊C412yescomplemented
143C22×C4×C59C2×D411yescharacteristic
144C22×Dic59—42no—
145C4⋊C4×C59—24no—
146C4⋊C4×C59—24no—
147C22⋊C4×C59—24no—
148C22⋊C4×C59—24no—
149C22⋊C4×C59—24no—
150C22⋊C4×C59—21no—
151C22⋊C4×C59—21no—
152C22⋊C4×C59—24no—
153C22⋊C4×C59—24no—
154C22⋊C4×C59C22⋊C412yescomplemented
155C22⋊C4×C59C22⋊C412yescomplemented
156C22⋊Dic59—42no—
157C22⋊Dic59—42no—
158C22⋊Dic59—84no—
159#11745664—42no—
160#11745664—42no—
161C2×C59⋊C8—42no—
162C23×C59—24no—
163C23×C59—24no—
164C23×C59—21no—
165C23×C59—21no—
166C23×C59—21no—
167C23×C59—21no—
168C23×C59—21no—
169C23×C59—21no—
170C23×C59—21no—
171C23×C59—21no—
172C23×C59—21no—
173C23×C59—22no—
174C23×C59—21no—
175C23×C59—21no—
176C23×C59—42no—
177C23×C59—44no—
178C23×C59—42no—
179C23×C59—42no—
180C23×C59—42no—
181C23×C59—42no—
182C23×C59—44no—
183C23×C59—42no—
184C23×C59—42no—
185C23×C59—42no—
186C23×C59—42no—
187C23×C59—41no—
188C23×C59—41no—
189C23×C59—82no—
190C23×C59—82no—
191C23×C59—82no—
192C23×C59—82no—
193C23×C59—84no—
194C23×C59—82no—
195C23×C59—82no—
196C23×C59—82no—
197C23×C59—84no—
198C23×C59—84no—
199C23×C59—84no—
200C23×C59—84no—
201C23×C59#9912yes—
202C23×C59#9912yes—
203C23×C59C2×C22⋊C411yescharacteristic
204C23×C59C2×C22⋊C411yescharacteristic
205C23×C59C2×C22⋊C411yescharacteristic
206C23×C59#12012yes—
207D4×C59—42no—
208D4×C59—42no—
209D4×C59—44no—
210D4×C59—44no—
211D4×C59—42no—
212D4×C59—42no—
213D4×C59—48no—
214D4×C59—48no—
215D4×C59—84no—
216D4×C59—84no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—21no—
222C2×C4×C59—22no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—44no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59—42no—
232C2×C4×C59—84no—
233C2×C4×C59#12012yes—
234C2×Dic59—82no—
235C2×Dic59—84no—
236C59⋊C8—82no—
237#12690—591nomaximal, Sylow
238C22×C59—22no—
239C22×C59—22no—
240C22×C59—22no—
241C22×C59—22no—
242C22×C59—22no—
243C22×C59—21no—
244C22×C59—21no—
245C22×C59—22no—
246C22×C59—42no—
247C22×C59—41no—
248C22×C59—41no—
249C22×C59—42no—
250C22×C59—42no—
251C22×C59—42no—
252C22×C59—42no—
253C22×C59—42no—
254C22×C59—42no—
255C22×C59—42no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—42no—
261C22×C59—41no—
262C22×C59—42no—
263C22×C59—81no—
264C22×C59—81no—
265C22×C59—82no—
266C22×C59—84no—
267C22×C59—84no—
268C22×C59—88no—
269C22×C59—84no—
270C22×C59—84no—
271C22×C59—82no—
272C22×C59#37911yescharacteristic
273C22×C59C2×#9911yescharacteristic
274C22×C59C2×#9911yescharacteristic
275C4×C59—22no—
276C4×C59—42no—
277C4×C59—42no—
278C4×C59—41no—
279C4×C59—42no—
280Dic59—162no—
281C2×#534D5911yescharacteristic
282C2×#352—592no—
283C2×AΣL(1,𝔽16)—592no—
284#1783—591no—
285#1781—591no—
286C2×C59—21no—
287C2×C59—21no—
288C2×C59—42no—
289C2×C59—42no—
290C2×C59—42no—
291C2×C59—41no—
292C2×C59—41no—
293C2×C59—81no—
294C2×C59—82no—
295C2×C59#178111yescharacteristic
296C2×C59#212112yes—
297C23×D4Dic5911yescharacteristic, complemented
298C23×D4C2×D5911yescharacteristic
299#534—24no—
300#534—24no—
301C2×#123—21no—
302C2×#123Dic5911yescharacteristic, complemented
303C2×#123C2×D5911yescharacteristic
304C2×#121—21no—
305C2×#121—21no—
306C2×#120—21no—
307C2×#120Dic5911yescharacteristic, complemented
308C2×#120C2×D5911yescharacteristic
309C22×C22⋊C4Dic5911yescharacteristic, complemented
310C2×#100—592no—
311C2×#100—1181no—
312C2×#99—592no—
313C2×#99—1181no—
314#407—1181no—
315#379—1181no—
316#352—1184no—
317AΣL(1,𝔽16)—1184no—
318C59#1269011yescharacteristic, complemented, Sylow
319C25C2×Dic5911yescharacteristic
320C25C2×Dic5911yescharacteristic
321C25C22⋊D5911yescharacteristic
322C22×D4—21no—
323C22×D4—21no—
324C22×D4—22no—
325C22×D4—21no—
326C22×D4—21no—
327C22×D4—22no—
328C22×D4—22no—
329C22×D4—22no—
330C22×D4—42no—
331C22×D4—42no—
332C22×D4—42no—
333C22×D4—42no—
334C22×D4C2×Dic5911yescharacteristic
335C22×D4C22⋊D5911yescharacteristic
336C22×D4C22⋊D5911yescharacteristic
337C22×D4C22×D5911yescharacteristic
338C23×C4—21no—
339C23×C4C2×Dic5911yescharacteristic
340C23×C4C22⋊D5911yescharacteristic
341C2×M16—1182no—
342#123—28no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123C22⋊D5914yes—
347#121—28no—
348#121—28no—
349#120—28no—
350#120—22no—
351#120—22no—
352#120—22no—
353#120C22⋊D5914yes—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—1182no—
364C2×C22⋊C4—2362no—
365C2×C22⋊C4C2×Dic5911yescharacteristic
366C2×C22⋊C4C2×Dic5911yescharacteristic
367#100—1182no—
368#100—1182no—
369#100—2362no—
370#99—1182no—
371#99—1182no—
372#99—2362no—
373C22⋊C8—2362no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—82no—
386C24—82no—
387C24—82no—
388C24C22⋊Dic5911yescharacteristic
389C24C22⋊Dic5911yescharacteristic
390C24C22⋊Dic5911yescharacteristic
391C24C22⋊Dic5912yescomplemented
392C24C22×Dic5911yescharacteristic
393C24C2×C22⋊D5911yescharacteristic
394C2×D4—24no—
395C2×D4—24no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—41no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—42no—
414C2×D4—42no—
415C2×D4—48no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4—82no—
419C2×D4—82no—
420C2×D4C2×C22⋊D5912yes—
421C2×D4C2×C22⋊D5912yes—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—82no—
433C22×C4—2362no—
434C22×C4C22⋊Dic5911yescharacteristic
435C22×C4C22⋊Dic5912yescomplemented
436C22×C4C2×C22⋊D5911yescharacteristic
437M16—2362no—
438M16—2362no—
439C2×C8—2362no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—24no—
448C22⋊C4—24no—
449C22⋊C4—4724no—
450C22⋊C4—2362no—
451C22⋊C4—2362no—
452C22⋊C4C22⋊Dic5912yescomplemented
453C22⋊C4C22⋊Dic5912yescomplemented
454C23—24no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—41no—
480C23—41no—
481C23—82no—
482C23—82no—
483C23—82no—
484C23—82no—
485C23—84no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23—84no—
490C23—84no—
491C23—84no—
492C23—84no—
493C23#1427364212yes—
494C23#1427364212yes—
495C23C2×C22⋊Dic5911yescharacteristic
496C23C2×C22⋊Dic5911yescharacteristic
497C23C2×C22⋊Dic5911yescharacteristic
498C23#1427369612yes—
499D4—42no—
500D4—42no—
501D4—44no—
502D4—44no—
503D4—42no—
504D4—42no—
505D4—48no—
506D4—48no—
507D4—84no—
508D4—84no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—84no—
525C2×C4—4722no—
526C2×C4—4724no—
527C2×C4#1427369612yes—
528C8—4722no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—41no—
539C22—41no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—41no—
553C22—42no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—84no—
558C22—84no—
559C22—88no—
560C22—84no—
561C22—84no—
562C22—82no—
563C22#1683725811yescharacteristic
564C22C2×#1427364211yescharacteristic
565C22C2×#1427364211yescharacteristic
566C4—22no—
567C4—42no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—9442no—
572C2—21no—
573C2—21no—
574C2—42no—
575C2—42no—
576C2—41no—
577C2—41no—
578C2—42no—
579C2—81no—
580C2—82no—
581C2#2142401111yescharacteristic
582C2#2142574212yes—
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
Fitting, prime radicalC2×C59×#534C2
socleC22×C59C2×#99
cosocleC22×D4×C59C23
hypercenterC2×#534D59
hypocenterC59#12690
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ C2×#534

Complements

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

NH ≅ G/Ntype
1. C2×#16837192C2semidirect
2. C2×#16837189C2semidirect
3. #21424017C2semidirect
4. #21424011C2semidirect
14. C23×D4×C59C4semidirect
19. C2×#123×C59C4semidirect
24. C2×#120×C59C4semidirect
26. C22×C22⋊C4×C59C4semidirect
99. C24×C59C22⋊C4semidirect
142. C22×C4×C59C22⋊C4semidirect
154. C22⋊C4×C59C22⋊C4semidirect
155. C22⋊C4×C59C22⋊C4semidirect
297. C23×D4Dic59semidirect
302. C2×#123Dic59semidirect
307. C2×#120Dic59semidirect
309. C22×C22⋊C4Dic59semidirect
318. C59#12690semidirect
391. C24C22⋊Dic59semidirect
435. C22×C4C22⋊Dic59semidirect
452. C22⋊C4C22⋊Dic59semidirect
453. C22⋊C4C22⋊Dic59semidirect

Maximal subgroups

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

subgroupindex
C2×#168371922
C2×#168371892
#214240172
#214240112
C2×C59×#5342
#1269059