Finite Group Databasebeta

← #28753881

Subgroups — #28753881

Order 15104.

2132 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16838109C211yesmaximal, characteristic
2#21427256C214yesmaximal, complemented
3#21427188C212yesmaximal, complemented
4#21427185C211yesmaximal, characteristic, complemented
5#21427123C212yesmaximal, complemented
6#21427044C212yesmaximal
7#21427008C211yesmaximal, characteristic, complemented
8#21426967C212yesmaximal, complemented
9#21426833C212yesmaximal, complemented
10#21426774C212yesmaximal, complemented
11#21426771C214yesmaximal, complemented
12#21426733C212yesmaximal
13#21426712C212yesmaximal, complemented
14#21426663C211yesmaximal, characteristic, complemented
15#21426570C211yesmaximal, characteristic, complemented
16#21426561C211yesmaximal, characteristic
17C59×#2354C211yesmaximal, characteristic
18#16838109C4116yescomplemented
19C2×#14273727C2211yescharacteristic
20C2×#14273718C2212yes—
21C2×#14273706C2212yes—
22C2×#14273704C2214yes—
23C2×C4×Dic118C2211yescharacteristic
24C2×#14273692C2212yes—
25C2×#14273692C2212yes—
26C2×D4×Dic59C2212yes—
27C2×#14273686C2212yes—
28C2×#14273683C2214yes—
29C22×C4⋊Dic59C2212yes—
30#16837307C2212yes—
31#16837307C2211yescharacteristic
32#16837307C2211yescharacteristic
33#16837306C2214yes—
34#16837305C2211yescharacteristic
35#16837305C2211yescharacteristic
36#16837301C2211yescharacteristic
37#16837301C2211yescharacteristic
38#16837300C2214yes—
39#16837300C2214yes—
40#16837300C2212yes—
41#16837299C2214yes—
42#16837298C2211yescharacteristic
43#16837298C2212yes—
44#16837298C2211yescharacteristic
45#16837298C2212yes—
46#16837298C2212yes—
47#16837298C2212yes—
48#16837297C2214yes—
49#16837296C2214yes—
50#16837296C2214yes—
51#16837295C2212yes—
52#16837295C2212yes—
53#16837290C2214yes—
54#16837290C2214yes—
55#16837289C2214yes—
56#16837286C2214yes—
57#16837285C2214yes—
58#16837285C2214yes—
59#16837284C2212yes—
60#16837284C2212yes—
61#16837271C2212yes—
62#16837263C2214yes—
63#16837260C2214yes—
64#16837260C2214yes—
65C4⋊C4×Dic59C2212yes—
66C4⋊C4×Dic59C2212yes—
67C4⋊C4×Dic59C2212yes—
68C4⋊C4×Dic59C2212yes—
69C4×Dic59⋊C4C2214yes—
70C4×Dic59⋊C4C2212yes—
71C4×C4⋊Dic59C2211yescharacteristic
72C4×C4⋊Dic59C2212yes—
73C4×C22⋊Dic59C2212yes—
74#16837248C2212yes—
75C42×Dic59C2211yescharacteristic
76C2×C4×D4×C59C2211yescharacteristic
77C2×#117×C59C2212yes—
78C59×#390C2212yes—
79C59×#388C2214yes—
80C59×#384C2211yescharacteristic
81C59×#380C2212yes—
82C4×C4⋊C4×C59C2211yescharacteristic
83C4×C4⋊C4×C59C2212yes—
84C22×Dic118C2311yescharacteristic
85C22×Dic118C2312yes—
86C23×Dic59C2314yes—
87#14273727C2×C418yes—
88#14273718C2×C4116yes—
89#14273706C2×C4116yes—
90#14273704C2×C4132yes—
91C4×Dic118C2×C418yes—
92#14273692—44no—
93#14273692—44no—
94D4×Dic59C2×C4116yes—
95#14273686—44no—
96#14273683—48no—
97C2×C4⋊Dic59C2×C4116yes—
98C2×C4⋊Dic59C2311yescharacteristic
99C2×C4⋊Dic59C2311yescharacteristic
100C2×C4⋊Dic59C2312yes—
101C2×C4⋊Dic59C2312yes—
102C2×C4⋊Dic59C2314yes—
103C2×C4⋊Dic59C2312yes—
104C2×C4⋊Dic59C2312yes—
105C2×C4⋊Dic59C2311yescharacteristic
106C2×Dic59⋊C4C2312yes—
107C2×Dic59⋊C4C2312yes—
108C2×Dic59⋊C4C2314yes—
109C2×Dic59⋊C4C2314yes—
110C2×Dic59⋊C4C2314yes—
111C2×Dic59⋊C4C2314yes—
112C2×Dic59⋊C4C2312yes—
113C2×Dic59⋊C4C2314yes—
114C2×C22⋊Dic59C2312yes—
115C2×C22⋊Dic59C2312yes—
116C2×C22⋊Dic59C2312yes—
117C2×C22⋊Dic59C2312yes—
118C2×C22⋊Dic59C2314yes—
119C2×C4×Dic59C2311yescharacteristic
120C2×C4×Dic59C2311yescharacteristic
121C2×C4×Dic59C2311yescharacteristic
122C2×C4×Dic59C2311yescharacteristic
123C2×C4×Dic59C2311yescharacteristic
124C2×C4×Dic59C2312yes—
125C2×C4×Dic59C2314yes—
126C2×C4×Dic59C2311yescharacteristic
127C2×C4×Dic59C2314yes—
128C2×C4×Dic59C2314yes—
129C2×C4×Dic59C2312yes—
130C59⋊#95C2312yes—
131C59⋊#95C2312yes—
132C59⋊#95C2312yes—
133C59⋊#95C2312yes—
134C59⋊#95C2312yes—
135C59⋊#95C2312yes—
136C59⋊#95C2314yes—
137C59⋊#95C2312yes—
138C59⋊#95C2312yes—
139C59⋊#95C2312yes—
140C59⋊#95C2314yes—
141C59⋊#95C2314yes—
142C59⋊#95C2314yes—
143C59⋊#95C2314yes—
144C22×D4×C59C2311yescharacteristic
145C23×C4×C59C2312yes—
146C4×D4×C59C2×C418yes—
147#117×C59—44no—
148C2×C4⋊C4×C59C2311yescharacteristic
149C2×C4⋊C4×C59C2312yes—
150C2×C4⋊C4×C59C2311yescharacteristic
151C2×C4⋊C4×C59C2312yes—
152C2×C4⋊C4×C59C2312yes—
153C2×C4⋊C4×C59C2311yescharacteristic
154C2×C22⋊C4×C59C2314yes—
155C2×C22⋊C4×C59C2314yes—
156C2×C22⋊C4×C59C2312yes—
157C2×C42×C59C2312yes—
158C2×C42×C59C2314yes—
159C2×C42×C59C2311yescharacteristic
160#95×C59C2314yes—
161#95×C59C2312yes—
162C24×C59C2412yes—
163C2×D4×C59—22no—
164C2×D4×C59—22no—
165C2×D4×C59C22×C414yes—
166C22×C4×C59—24no—
167C22×C4×C59—24no—
168C22×C4×C59C22×C418yes—
169C22×C4×C59C2412yes—
170C22×C4×C59C2412yes—
171C22×C4×C59C2414yes—
172C22×C4×C59C2412yes—
173C22×C4×C59C2411yescharacteristic
174C22×C4×C59C2411yescharacteristic
175C22×C4×C59C2411yescharacteristic
176C22×Dic59—44no—
177C22×Dic59—44no—
178C22×Dic59C22×C4116yes—
179C22×Dic59C2411yescharacteristic
180C22×Dic59C2412yes—
181C22×Dic59C2411yescharacteristic
182C22×Dic59C2411yescharacteristic
183C22×Dic59C2412yes—
184C22×Dic59C2412yes—
185C22×Dic59C2414yes—
186C22×Dic59C2412yes—
187C22×Dic59C2411yescharacteristic
188C2×Dic118—22no—
189C2×Dic118—22no—
190C2×Dic118—42no—
191C2×Dic118—42no—
192C2×Dic118C22×C414yes—
193C2×Dic118C22×C418yes—
194C4⋊C4×C59—22no—
195C4⋊C4×C59—24no—
196C4⋊C4×C59—24no—
197C4⋊C4×C59—22no—
198C4⋊C4×C59—42no—
199C4⋊C4×C59C22×C414yes—
200C22⋊C4×C59—44no—
201C22⋊C4×C59—44no—
202C22⋊C4×C59C22×C418yes—
203C42×C59—24no—
204C42×C59—44no—
205C42×C59C22×C414yes—
206C22⋊Dic59—24no—
207C22⋊Dic59—42no—
208C22⋊Dic59—42no—
209C22⋊Dic59—42no—
210C22⋊Dic59C22×C4116yes—
211C4⋊Dic59—22no—
212C4⋊Dic59—22no—
213C4⋊Dic59—42no—
214C4⋊Dic59—42no—
215C4⋊Dic59C22×C414yes—
216C4⋊Dic59C22×C4116yes—
217C4⋊Dic59C22×C418yes—
218C4⋊Dic59C22×C418yes—
219Dic59⋊C4—24no—
220Dic59⋊C4—44no—
221Dic59⋊C4—42no—
222Dic59⋊C4—44no—
223Dic59⋊C4—44no—
224Dic59⋊C4—44no—
225Dic59⋊C4C22×C418yes—
226Dic59⋊C4C22×C4116yes—
227C4×Dic59—22no—
228C4×Dic59—24no—
229C4×Dic59—22no—
230C4×Dic59—22no—
231C4×Dic59—22no—
232C4×Dic59—28no—
233C4×Dic59—22no—
234C4×Dic59—41no—
235C4×Dic59—44no—
236C4×Dic59—44no—
237C4×Dic59C22×C418yes—
238C23×C59—22no—
239C23×C59—22no—
240C23×C59—22no—
241C23×C59—22no—
242C23×C59—22no—
243C23×C59—22no—
244C23×C59C23×C414yes—
245C23×C59C2511yescharacteristic
246D4×C59—28no—
247C2×C4×C59—21no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—216no—
252C2×C4×C59—24no—
253C2×C4×C59—22no—
254C2×C4×C59—24no—
255C2×C4×C59—22no—
256C2×C4×C59—22no—
257C2×C4×C59—22no—
258C2×C4×C59—22no—
259C2×C4×C59—24no—
260C2×C4×C59—22no—
261C2×C4×C59C23×C414yes—
262C2×C4×C59C23×C412yes—
263C2×C4×C59C23×C412yes—
264C2×C4×C59C23×C412yes—
265C2×C4×C59H(5,𝔽2)12yes—
266C2×C4×C592−1+412yes—
267C2×C4×C592−1+412yes—
268C2×C4×C592−1+412yes—
269C2×Dic59—24no—
270C2×Dic59—22no—
271C2×Dic59—22no—
272C2×Dic59—21no—
273C2×Dic59—21no—
274C2×Dic59—22no—
275C2×Dic59—24no—
276C2×Dic59—22no—
277C2×Dic59—22no—
278C2×Dic59—21no—
279C2×Dic59—22no—
280C2×Dic59—21no—
281C2×Dic59—21no—
282C2×Dic59—22no—
283C2×Dic59—21no—
284C2×Dic59—21no—
285C2×Dic59—21no—
286C2×Dic59—21no—
287C2×Dic59—22no—
288C2×Dic59—22no—
289C2×Dic59—21no—
290C2×Dic59—21no—
291C2×Dic59—22no—
292C2×Dic59—416no—
293C2×Dic59C23×C414yes—
294C2×Dic59C23×C414yes—
295C2×Dic59C23×C418yes—
296C2×Dic59H(5,𝔽2)12yes—
297Dic118—28no—
298Dic118—48no—
299#34484—591nomaximal, Sylow
300C22×C59—24no—
301C22×C59—24no—
302C22×C59—24no—
303C22×C59—42no—
304C22×C59—42no—
305C22×C59—42no—
306C22×C59—42no—
307C22×C59C24×C411yescharacteristic
308C22×C59C2×H(5,𝔽2)11yescharacteristic
309C22×C59C2×H(5,𝔽2)11yescharacteristic
310C22×C59C2×2−1+411yescharacteristic
311C22×C59C2×2−1+411yescharacteristic
312C22×C59C2×2−1+411yescharacteristic
313C22×C59#58511yescharacteristic
314C4×C59—24no—
315C4×C59—22no—
316C4×C59—42no—
317C4×C59—42no—
318C4×C59—44no—
319C4×C59#51914yes—
320C4×C59#52014yes—
321Dic59—24no—
322Dic59—28no—
323Dic59—24no—
324Dic59—22no—
325Dic59—42no—
326Dic59—41no—
327Dic59—41no—
328Dic59—42no—
329Dic59—41no—
330C2×#554—591no—
331#2378—594no—
332#2370—592no—
333#2369—591no—
334#2362—592no—
335#2354—592no—
336#2354D5911yescharacteristic
337Q8×C4⋊C4—591no—
338#2344—592no—
339#2330—592no—
340#2323—594no—
341#2323—592no—
342#2319—592no—
343#2316—592no—
344C4×C4⋊Q8—591no—
345#2295—591no—
346#2293—591no—
347C2×C59—44no—
348C2×C59C2×#51911yescharacteristic
349C2×C59C2×#52011yescharacteristic
350C2×C59C4×H(5,𝔽2)11yescharacteristic
351C2×C59#354411yescharacteristic
352C2×C59#354711yescharacteristic
353C2×C59#354811yescharacteristic
354C2×C59#355511yescharacteristic
355#554—5916no—
356C2×C4⋊Q8—591no—
357C2×#125—592no—
358C2×C22⋊Q8—592no—
359C2×C22⋊Q8—594no—
360C2×C4×Q8—592no—
361C2×C4×Q8—592no—
362C2×C4×Q8—591no—
363C2×C4×D4—592no—
364C2×C4×D4C2×D5911yescharacteristic
365C2×#117—592no—
366C2×#117—594no—
367C2×#117C2×D5912yes—
368C22×C4⋊C4—592no—
369#391—592no—
370#391—591no—
371#391—591no—
372#391—594no—
373#390—591no—
374#390—591no—
375#390C2×D5912yes—
376#389—592no—
377#389—592no—
378#389—591no—
379#389—592no—
380#389—592no—
381#389—594no—
382#389—591no—
383#389—591no—
384#389—591no—
385#389—594no—
386#389—594no—
387#389—592no—
388#389—594no—
389#388—594no—
390#388—594no—
391#388—592no—
392#388—594no—
393#388—594no—
394#388—592no—
395#388C2×D5914yes—
396#387—594no—
397#387—592no—
398#387—594no—
399#387—594no—
400#387—594no—
401#387—592no—
402#384—592no—
403#384C2×D5911yescharacteristic
404#381—594no—
405#380—594no—
406#380—594no—
407#380C2×D5912yes—
408C4×C4⋊C4—592no—
409C4×C4⋊C4—592no—
410C4×C4⋊C4—592no—
411C4×C4⋊C4—592no—
412C4×C4⋊C4—594no—
413C4×C4⋊C4—592no—
414C4×C4⋊C4—591no—
415C4×C4⋊C4—592no—
416C4×C4⋊C4C2×D5912yes—
417C4×C4⋊C4C2×D5911yescharacteristic
418C4×C22⋊C4—592no—
419#376—592no—
420C43—591no—
421C59#3448411yescharacteristic, complemented, Sylow
422C22×Q8—591no—
423C22×Q8—592no—
424C22×D4C22×D5911yescharacteristic
425C23×C4—594no—
426C23×C4C22×D5912yes—
427C4⋊Q8—598no—
428#125—5916no—
429C22⋊Q8—5916no—
430C22⋊Q8—5932no—
431C4×Q8—598no—
432C4×Q8—2364no—
433C4×Q8—2364no—
434C4×D4—5916no—
435C4×D4C4×D5918yes—
436#117—44no—
437#117—2364no—
438#117—2368no—
439C2×C4⋊C4—5916no—
440C2×C4⋊C4—591no—
441C2×C4⋊C4—594no—
442C2×C4⋊C4—592no—
443C2×C4⋊C4—592no—
444C2×C4⋊C4—592no—
445C2×C4⋊C4—594no—
446C2×C4⋊C4—592no—
447C2×C4⋊C4—591no—
448C2×C4⋊C4—591no—
449C2×C4⋊C4—594no—
450C2×C4⋊C4—592no—
451C2×C4⋊C4—592no—
452C2×C4⋊C4—594no—
453C2×C4⋊C4—594no—
454C2×C4⋊C4—592no—
455C2×C4⋊C4—594no—
456C2×C4⋊C4C22×D5911yescharacteristic
457C2×C4⋊C4C22×D5912yes—
458C2×C4⋊C4C22×D5911yescharacteristic
459C2×C4⋊C4C22×D5912yes—
460C2×C4⋊C4C22×D5912yes—
461C2×C4⋊C4C22×D5911yescharacteristic
462C2×C22⋊C4—594no—
463C2×C22⋊C4—592no—
464C2×C22⋊C4—592no—
465C2×C22⋊C4—592no—
466C2×C22⋊C4—592no—
467C2×C22⋊C4C22×D5914yes—
468C2×C22⋊C4C22×D5914yes—
469C2×C22⋊C4C22×D5912yes—
470C2×C42—592no—
471C2×C42—591no—
472C2×C42—592no—
473C2×C42—594no—
474C2×C42—591no—
475C2×C42—591no—
476C2×C42—591no—
477C2×C42—591no—
478C2×C42—594no—
479C2×C42—594no—
480C2×C42—591no—
481C2×C42C22×D5912yes—
482C2×C42C22×D5914yes—
483C2×C42C22×D5911yescharacteristic
484#95—592no—
485#95—592no—
486#95—592no—
487#95—592no—
488#95—592no—
489#95—592no—
490#95—592no—
491#95—594no—
492#95—594no—
493#95—594no—
494#95—594no—
495#95—594no—
496#95—592no—
497#95—592no—
498#95C22×D5914yes—
499#95C22×D5912yes—
500C24C23×D5912yes—
501C2×Q8—594no—
502C2×Q8—598no—
503C2×Q8—1182no—
504C2×Q8—1182no—
505C2×Q8—2362no—
506C2×Q8—2362no—
507C2×D4—22no—
508C2×D4—22no—
509C2×D4C2×C4×D5914yes—
510C22×C4—24no—
511C22×C4—24no—
512C22×C4—591no—
513C22×C4—592no—
514C22×C4—594no—
515C22×C4—592no—
516C22×C4—5916no—
517C22×C4—591no—
518C22×C4—592no—
519C22×C4—591no—
520C22×C4—591no—
521C22×C4—592no—
522C22×C4—2364no—
523C22×C4—2364no—
524C22×C4C2×C4×D5918yes—
525C22×C4C23×D5912yes—
526C22×C4C23×D5912yes—
527C22×C4C23×D5914yes—
528C22×C4C23×D5911yescharacteristic
529C22×C4C23×D5912yes—
530C22×C4C23×D5911yescharacteristic
531C22×C4C23×D5911yescharacteristic
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—22no—
536C4⋊C4—42no—
537C4⋊C4—594no—
538C4⋊C4—5916no—
539C4⋊C4—598no—
540C4⋊C4—598no—
541C4⋊C4—598no—
542C4⋊C4—5916no—
543C4⋊C4—1184no—
544C4⋊C4—1182no—
545C4⋊C4—1182no—
546C4⋊C4—2364no—
547C4⋊C4—2362no—
548C4⋊C4—2362no—
549C4⋊C4—2364no—
550C4⋊C4—2364no—
551C4⋊C4—2362no—
552C4⋊C4—2364no—
553C4⋊C4C2×C4×D5914yes—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—5916no—
557C22⋊C4—1184no—
558C22⋊C4—2362no—
559C22⋊C4—2362no—
560C22⋊C4—2362no—
561C22⋊C4C2×C4×D5918yes—
562C42—24no—
563C42—44no—
564C42—598no—
565C42—1188no—
566C42—1182no—
567C42—1182no—
568C42—1184no—
569C42—1182no—
570C42—1182no—
571C42—1182no—
572C42—2364no—
573C42—2364no—
574C42—2361no—
575C42C2×C4×D5914yes—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C22×C4×D5914yes—
583C23C24×D5911yescharacteristic
584Q8—1188no—
585Q8—2368no—
586D4—28no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—216no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—594no—
602C2×C4—594no—
603C2×C4—598no—
604C2×C4—592no—
605C2×C4—1184no—
606C2×C4—1182no—
607C2×C4—1182no—
608C2×C4—1181no—
609C2×C4—1181no—
610C2×C4—1182no—
611C2×C4—1182no—
612C2×C4—1184no—
613C2×C4—1181no—
614C2×C4—1182no—
615C2×C4—1182no—
616C2×C4—1181no—
617C2×C4—1181no—
618C2×C4—1182no—
619C2×C4—1181no—
620C2×C4—1181no—
621C2×C4—1181no—
622C2×C4—1181no—
623C2×C4—1182no—
624C2×C4—1182no—
625C2×C4—1181no—
626C2×C4—1181no—
627C2×C4—1182no—
628C2×C4—23616no—
629C2×C4C22×C4×D5912yes—
630C2×C4C22×C4×D5914yes—
631C2×C4C22×C4×D5912yes—
632C2×C4C22×C4×D5912yes—
633C2×C4#1427377212yes—
634C2×C4#1427377412yes—
635C2×C4#1427377412yes—
636C2×C4#1427377412yes—
637C22—24no—
638C22—24no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C23×C4×D5911yescharacteristic
645C22H(5,𝔽2)×D5911yescharacteristic
646C22C2×#1427377211yescharacteristic
647C22C2×#1427377411yescharacteristic
648C22C2×#1427377411yescharacteristic
649C22C2×#1427377411yescharacteristic
650C22#1683823911yescharacteristic
651C4—24no—
652C4—22no—
653C4—42no—
654C4—44no—
655C4—42no—
656C4—1184no—
657C4—1184no—
658C4—1188no—
659C4—1182no—
660C4—2362no—
661C4—2361no—
662C4—2361no—
663C4—2362no—
664C4—2361no—
665C4#1683787914yes—
666C4#1683788114yes—
667C2—44no—
668C2C2×#1683787911yescharacteristic
669C2C2×#1683788111yescharacteristic
670C2#2143730211yescharacteristic
671C2#2143843111yescharacteristic
672C2#2143847311yescharacteristic
673C2#2143848911yescharacteristic
674C2#2143857211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D59
commutatorC22×C59C24×C4
FittingC59×#2354C2
socle, cosocleC23×C59C25
prime radicalC22×D4×C59C23
hypercenter#2354D59
hypocenterC59#34484
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #21427256C2semidirect
3. #21427188C2semidirect
4. #21427185C2semidirect
5. #21427123C2semidirect
7. #21427008C2semidirect
8. #21426967C2semidirect
9. #21426833C2semidirect
10. #21426774C2semidirect
11. #21426771C2semidirect
13. #21426712C2semidirect
14. #21426663C2semidirect
15. #21426570C2semidirect
18. #16838109C4semidirect
421. C59#34484semidirect

Maximal subgroups

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

subgroupindex
C2×#168381092
#214272562
#214271882
#214271852
#214271232
#214270442
#214270082
#214269672
#214268332
#214267742
#214267712
#214267332
#214267122
#214266632
#214265702
#214265612
C59×#23542
#3448459