Finite Group Databasebeta

← #28515232

Subgroups — #28515232

Order 15104.

1926 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837277C211yesmaximal, characteristic
2C2×#16837274C211yesmaximal, characteristic
3C2×#16837260C211yesmaximal, characteristic
4C2×#16837260C211yesmaximal, characteristic
5C22×C59⋊#95C211yesmaximal, characteristic
6#21422360C212yesmaximal, complemented
7#21422354C212yesmaximal, complemented
8#21422350C212yesmaximal, complemented
9#21422347C212yesmaximal, complemented
10#21422319C211yesmaximal, characteristic
11C59×C22⋊#95C211yesmaximal, characteristic
12C24×Dic59C2211yescharacteristic
13C24×Dic59C2211yescharacteristic
14C22×C4⋊Dic59C2212yes—
15C22×Dic59⋊C4C2212yes—
16C22×Dic59⋊C4C2211yescharacteristic
17C22×Dic59⋊C4C2211yescharacteristic
18C22×Dic59⋊C4C2212yes—
19C22×C22⋊Dic59C2211yescharacteristic
20C22×C22⋊Dic59C2211yescharacteristic
21#16837277—24no—
22#16837274—24no—
23#16837260—24no—
24#16837260C418yescomplemented
25C2×C59⋊#95—24no—
26C2×C59⋊#95C2211yescharacteristic
27C2×C59⋊#95C2212yes—
28C2×C59⋊#95C2212yes—
29C2×C59⋊#95C2211yescharacteristic
30C2×C59⋊#95C2212yes—
31C2×C59⋊#95C2212yes—
32C2×C59⋊#95C2212yes—
33C2×C59⋊#95C2212yes—
34C2×C59⋊#95C2212yes—
35C24×C4×C59C2211yescharacteristic
36C22×C22⋊C4×C59C2211yescharacteristic
37C22×C22⋊C4×C59C2211yescharacteristic
38C2×#95×C59C2212yes—
39C2×#95×C59C2212yes—
40C23×Dic59—22no—
41C23×Dic59—22no—
42C23×Dic59—42no—
43C23×Dic59—42no—
44C23×Dic59—42no—
45C23×Dic59—42no—
46C23×Dic59C2×C414yes—
47C23×Dic59D414yes—
48C23×Dic59D414yes—
49C23×Dic59Q814yes—
50C23×Dic59C2311yescharacteristic
51C23×Dic59C2311yescharacteristic
52C23×Dic59C2312yes—
53C23×Dic59C2312yes—
54C23×Dic59C2311yescharacteristic
55C23×Dic59C2311yescharacteristic
56C2×C4⋊Dic59—24no—
57C2×C4⋊Dic59—42no—
58C2×C4⋊Dic59—42no—
59C2×Dic59⋊C4—22no—
60C2×Dic59⋊C4—22no—
61C2×Dic59⋊C4—24no—
62C2×Dic59⋊C4—42no—
63C2×Dic59⋊C4—42no—
64C2×Dic59⋊C4—42no—
65C2×Dic59⋊C4—42no—
66C2×Dic59⋊C4—42no—
67C2×Dic59⋊C4—42no—
68C2×Dic59⋊C4C2×C418yescomplemented
69C2×C22⋊Dic59—22no—
70C2×C22⋊Dic59—22no—
71C2×C22⋊Dic59—22no—
72C2×C22⋊Dic59—22no—
73C2×C22⋊Dic59—22no—
74C2×C22⋊Dic59C2×C414yes—
75C59⋊#95—24no—
76C59⋊#95—24no—
77C59⋊#95—22no—
78C59⋊#95—22no—
79C59⋊#95—24no—
80C59⋊#95—24no—
81C59⋊#95—42no—
82C59⋊#95—42no—
83C59⋊#95C2×C418yescomplemented
84C25×C59C2311yescharacteristic
85C23×C4×C59—22no—
86C23×C4×C59—22no—
87C23×C4×C59—42no—
88C23×C4×C59—42no—
89C23×C4×C59Q814yes—
90C23×C4×C59C2311yescharacteristic
91C23×C4×C59C2311yescharacteristic
92C23×C4×C59C2312yes—
93C23×C4×C59C2312yes—
94C2×C22⋊C4×C59—22no—
95C2×C22⋊C4×C59—22no—
96C2×C22⋊C4×C59—22no—
97C2×C22⋊C4×C59—22no—
98C2×C22⋊C4×C59—22no—
99C2×C22⋊C4×C59C2×C414yes—
100#95×C59—42no—
101#95×C59—42no—
102C24×C59—22no—
103C24×C59—22no—
104C24×C59—22no—
105C24×C59—22no—
106C24×C59C22×C412yes—
107C24×C59C2×D412yes—
108C24×C59C2×D412yes—
109C24×C59C2×Q812yes—
110C24×C59C2×Q812yes—
111C24×C59#4112yes—
112C24×C59#4112yes—
113C24×C59C2411yescharacteristic
114C22×C4×C59—28no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—22no—
120C22×C4×C59—22no—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—22no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—22no—
137C22×C4×C59—48no—
138C22×C4×C59—44no—
139C22×C4×C59—44no—
140C22×C4×C59—48no—
141C22×C4×C59C22×C414yes—
142C22×C4×C59C2×Q814yes—
143C22×Dic59—22no—
144C22×Dic59—22no—
145C22×Dic59—22no—
146C22×Dic59—22no—
147C22×Dic59—28no—
148C22×Dic59—22no—
149C22×Dic59—22no—
150C22×Dic59—22no—
151C22×Dic59—22no—
152C22×Dic59—22no—
153C22×Dic59—22no—
154C22×Dic59—22no—
155C22×Dic59—22no—
156C22×Dic59—22no—
157C22×Dic59—22no—
158C22×Dic59—22no—
159C22×Dic59—22no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C22×Dic59—22no—
167C22×Dic59—22no—
168C22×Dic59—22no—
169C22×Dic59—22no—
170C22×Dic59—48no—
171C22×Dic59—44no—
172C22×Dic59—44no—
173C22×Dic59—48no—
174C22×Dic59—48no—
175C22×Dic59—44no—
176C22×Dic59—44no—
177C22×Dic59—48no—
178C22×Dic59C4⋊C4116yescomplemented
179C22×Dic59C22×C414yes—
180C22×Dic59C22×C412yes—
181C22×Dic59C22×C412yes—
182C22×Dic59C2×D414yes—
183C22×Dic59C2×D412yes—
184C22×Dic59C2×D412yes—
185C22×Dic59C2×Q812yes—
186C22×Dic59C2×Q812yes—
187C22⋊C4×C59—28no—
188C22⋊C4×C59—44no—
189C22⋊Dic59—28no—
190C22⋊Dic59—44no—
191C4⋊Dic59—48no—
192Dic59⋊C4—48no—
193Dic59⋊C4—44no—
194Dic59⋊C4—44no—
195Dic59⋊C4—48no—
196C23×C59—24no—
197C23×C59—24no—
198C23×C59—22no—
199C23×C59—22no—
200C23×C59—24no—
201C23×C59—22no—
202C23×C59—22no—
203C23×C59—24no—
204C23×C59—24no—
205C23×C59—24no—
206C23×C59—24no—
207C23×C59—24no—
208C23×C59—24no—
209C23×C59—24no—
210C23×C59—24no—
211C23×C59—24no—
212C23×C59C2×C4⋊C414yes—
213C23×C59C4×D414yes—
214C23×C59C4×Q814yes—
215C23×C59#12114yes—
216C23×C59C22⋊Q814yes—
217C23×C59#12514yes—
218C23×C59C4⋊Q814yes—
219C23×C59C23×C411yescharacteristic
220C23×C59C22×D411yescharacteristic
221C23×C59C22×D411yescharacteristic
222C23×C59C22×Q811yescharacteristic
223C23×C59C22×Q811yescharacteristic
224C23×C59C2×#4111yescharacteristic
225C23×C59C2×#4111yescharacteristic
226C23×C59H(5,𝔽2)12yes—
227C23×C59H(5,𝔽2)12yes—
228C23×C59H(5,𝔽2)12yes—
229C23×C592−1+412yes—
230C2×C4×C59—24no—
231C2×C4×C59—24no—
232C2×C4×C59—24no—
233C2×C4×C59—24no—
234C2×C4×C59—24no—
235C2×C4×C59—24no—
236C2×C4×C59—416no—
237C2×C4×C59—42no—
238C2×C4×C59—42no—
239C2×C4×C59—42no—
240C2×C4×C59—42no—
241C2×C4×C59—42no—
242C2×C4×C59—42no—
243C2×C4×C59—42no—
244C2×C4×C59—42no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—42no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—42no—
253C2×C4×C59—42no—
254C2×C4×C59—42no—
255C2×Dic59—24no—
256C2×Dic59—24no—
257C2×Dic59—22no—
258C2×Dic59—22no—
259C2×Dic59—22no—
260C2×Dic59—22no—
261C2×Dic59—24no—
262C2×Dic59—24no—
263C2×Dic59—24no—
264C2×Dic59—24no—
265C2×Dic59—24no—
266C2×Dic59—24no—
267C2×Dic59—24no—
268C2×Dic59—24no—
269C2×Dic59—416no—
270C2×Dic59—416no—
271C2×Dic59—42no—
272C2×Dic59—42no—
273C2×Dic59—42no—
274C2×Dic59—42no—
275C2×Dic59—42no—
276C2×Dic59—42no—
277C2×Dic59—42no—
278C2×Dic59—42no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—42no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—42no—
287C2×Dic59—42no—
288C2×Dic59—42no—
289C2×Dic59C2×C4⋊C414yes—
290C2×Dic59C2×C4⋊C414yes—
291#14275—591nomaximal, Sylow
292C22×C59—24no—
293C22×C59—24no—
294C22×C59—28no—
295C22×C59—28no—
296C22×C59—24no—
297C22×C59—24no—
298C22×C59—24no—
299C22×C59—24no—
300C22×C59—28no—
301C22×C59—28no—
302C22×C59#38918yes—
303C22×C59C22×C4⋊C411yescharacteristic
304C22×C59C2×C4×D411yescharacteristic
305C22×C59C2×C4×Q811yescharacteristic
306C22×C59#51812yes—
307C22×C59#52012yes—
308C22×C59C2×#12111yescharacteristic
309C22×C59C2×C22⋊Q811yescharacteristic
310C22×C59C2×#12511yescharacteristic
311C22×C59C2×C4⋊Q811yescharacteristic
312C22×C59#53412yes—
313C22×C59#53411yescharacteristic
314C22×C59#53411yescharacteristic
315C22×C59#53511yescharacteristic
316C22×C59#53611yescharacteristic
317C22×C59#53712yes—
318C22×C59#53812yes—
319C22×C59#53812yes—
320C22×C59#53912yes—
321C22×C59#53912yes—
322C22×C59#54312yes—
323C22×C59#54312yes—
324C22×C59#54412yes—
325C22×C59#54412yes—
326C4×C59—44no—
327C4×C59—44no—
328C4×C59—81no—
329C4×C59—81no—
330Dic59—24no—
331Dic59—24no—
332Dic59—44no—
333Dic59—44no—
334Dic59—81no—
335Dic59—81no—
336C2×#385—591no—
337C2×#385—591no—
338C2×#380—591no—
339C2×#380—591no—
340C22×#95—591no—
341#1444—592no—
342#1442—592no—
343#1441—592no—
344#1441—592no—
345C22⋊#95—591no—
346C22⋊#95D5911yescharacteristic
347C2×C59—28no—
348C2×C59C2×#38911yescharacteristic
349C2×C59#231511yescharacteristic
350C2×C59#231711yescharacteristic
351C2×C59#232512yes—
352C2×C59#232912yes—
353C2×C59#261711yescharacteristic
354C2×C59#261811yescharacteristic
355C2×C59#262611yescharacteristic
356C2×C59#262711yescharacteristic
357C2×C59#264612yes—
358C2×C59#265912yes—
359C24×C4—591no—
360C24×C4—591no—
361C24×C4C2×D5911yescharacteristic
362C22×C4⋊C4—592no—
363C22×C4⋊C4—592no—
364C22×C4⋊C4—592no—
365C22×C4⋊C4—591no—
366C22×C4⋊C4—591no—
367C22×C22⋊C4—591no—
368C22×C22⋊C4—591no—
369C22×C22⋊C4C2×D5911yescharacteristic
370C22×C22⋊C4C2×D5911yescharacteristic
371#385—1184no—
372#385—1184no—
373#380—598no—
374#380—1184no—
375C2×#95—592no—
376C2×#95—592no—
377C2×#95—591no—
378C2×#95—591no—
379C2×#95—592no—
380C2×#95—592no—
381C2×#95—592no—
382C2×#95—592no—
383C2×#95—592no—
384C2×#95—1184no—
385C2×#95C2×D5912yes—
386C2×#95C2×D5912yes—
387C59#1427511yescharacteristic, complemented, Sylow
388C25C22×D5911yescharacteristic
389C23×C4—22no—
390C23×C4—22no—
391C23×C4—42no—
392C23×C4—42no—
393C23×C4—591no—
394C23×C4—591no—
395C23×C4—594no—
396C23×C4—594no—
397C23×C4—594no—
398C23×C4—592no—
399C23×C4—592no—
400C23×C4—594no—
401C23×C4—591no—
402C23×C4—591no—
403C23×C4—1182no—
404C23×C4—1182no—
405C23×C4—2362no—
406C23×C4—2362no—
407C23×C4—2362no—
408C23×C4—2362no—
409C23×C4Dic11814yes—
410C23×C4C22×D5911yescharacteristic
411C23×C4C22×D5911yescharacteristic
412C23×C4C22×D5912yes—
413C23×C4C22×D5912yes—
414C2×C4⋊C4—598no—
415C2×C4⋊C4—1182no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4—1184no—
418C2×C4⋊C4—1184no—
419C2×C4⋊C4—2362no—
420C2×C4⋊C4—2362no—
421C2×C4⋊C4—2362no—
422C2×C4⋊C4—2362no—
423C2×C4⋊C4—2362no—
424C2×C4⋊C4—2362no—
425C2×C4⋊C4—2362no—
426C2×C4⋊C4—2362no—
427C2×C22⋊C4—22no—
428C2×C22⋊C4—22no—
429C2×C22⋊C4—22no—
430C2×C22⋊C4—22no—
431C2×C22⋊C4—22no—
432C2×C22⋊C4—594no—
433C2×C22⋊C4—1182no—
434C2×C22⋊C4—1182no—
435C2×C22⋊C4—1182no—
436C2×C22⋊C4—1182no—
437C2×C22⋊C4—1182no—
438C2×C22⋊C4C4×D5914yes—
439#95—42no—
440#95—42no—
441#95—598no—
442#95—1184no—
443#95—1184no—
444#95—1182no—
445#95—1182no—
446#95—1184no—
447#95—1184no—
448#95—2362no—
449#95—2362no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×Dic11812yes—
455C24C2×C4×D5912yes—
456C24#1174568512yes—
457C24D4×D5912yes—
458C24D4×D5912yes—
459C24#1174568712yes—
460C24Q8×D5912yes—
461C24C23×D5911yescharacteristic
462C22×C4—28no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—48no—
489C22×C4—592no—
490C22×C4—592no—
491C22×C4—592no—
492C22×C4—592no—
493C22×C4—592no—
494C22×C4—592no—
495C22×C4—5916no—
496C22×C4—594no—
497C22×C4—594no—
498C22×C4—1182no—
499C22×C4—1182no—
500C22×C4—1182no—
501C22×C4—1182no—
502C22×C4—1188no—
503C22×C4—1182no—
504C22×C4—1182no—
505C22×C4—1182no—
506C22×C4—1182no—
507C22×C4—1182no—
508C22×C4—1182no—
509C22×C4—1182no—
510C22×C4—1182no—
511C22×C4—1182no—
512C22×C4—1182no—
513C22×C4—1182no—
514C22×C4—1182no—
515C22×C4—1181no—
516C22×C4—1181no—
517C22×C4—1181no—
518C22×C4—1181no—
519C22×C4—1181no—
520C22×C4—1181no—
521C22×C4—1182no—
522C22×C4—1182no—
523C22×C4—1182no—
524C22×C4—1182no—
525C22×C4—2364no—
526C22×C4—2364no—
527C22×C4—2368no—
528C22×C4—2368no—
529C22×C4—2368no—
530C22×C4—2364no—
531C22×C4—2364no—
532C22×C4—2368no—
533C22×C4C2×Dic11814yes—
534C22×C4C2×C4×D5914yes—
535C4⋊C4—2368no—
536C4⋊C4—2368no—
537C4⋊C4—2364no—
538C4⋊C4—2364no—
539C4⋊C4—2368no—
540C22⋊C4—28no—
541C22⋊C4—44no—
542C22⋊C4—1188no—
543C22⋊C4—2364no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23C4⋊C4×D5914yes—
561C23(C22⋊D59)⋊C414yes—
562C23C4×Dic11814yes—
563C23#1427370014yes—
564C23#1427370414yes—
565C23#1427371714yes—
566C23#1427372614yes—
567C23C22×C4×D5911yescharacteristic
568C23C2×D4×D5911yescharacteristic
569C23C2×D4×D5911yescharacteristic
570C23C2×Q8×D5911yescharacteristic
571C23C22×Dic11811yescharacteristic
572C23C2×#1174568711yescharacteristic
573C23C2×#1174568511yescharacteristic
574C23#1427377212yes—
575C23#1427377212yes—
576C23#1427377312yes—
577C23#1427377412yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—594no—
604C2×C4—594no—
605C2×C4—1182no—
606C2×C4—1182no—
607C2×C4—1182no—
608C2×C4—1182no—
609C2×C4—1184no—
610C2×C4—1184no—
611C2×C4—1184no—
612C2×C4—1184no—
613C2×C4—1184no—
614C2×C4—1184no—
615C2×C4—1184no—
616C2×C4—1184no—
617C2×C4—1184no—
618C2×C4—1184no—
619C2×C4—23616no—
620C2×C4—2361no—
621C2×C4—2361no—
622C2×C4—2362no—
623C2×C4—2361no—
624C2×C4—2361no—
625C2×C4—2361no—
626C2×C4—2361no—
627C2×C4—2362no—
628C2×C4—2362no—
629C2×C4—2362no—
630C2×C4—23616no—
631C2×C4—2362no—
632C2×C4—2362no—
633C2×C4—2362no—
634C2×C4—2362no—
635C2×C4—2362no—
636C2×C4—2362no—
637C2×C4—2362no—
638C2×C4—2362no—
639C22—24no—
640C22—24no—
641C22—28no—
642C22—24no—
643C22—24no—
644C22—28no—
645C22—28no—
646C22—24no—
647C22—24no—
648C22—28no—
649C22#1683729918yes—
650C22C2×C4⋊C4×D5911yescharacteristic
651C22C2×(C22⋊D59)⋊C411yescharacteristic
652C22C2×C4×Dic11811yescharacteristic
653C22#1683787412yes—
654C22#1683788112yes—
655C22C2×#1427370011yescharacteristic
656C22C2×#1427370411yescharacteristic
657C22C2×#1427371711yescharacteristic
658C22C2×#1427372611yescharacteristic
659C22#1683796012yes—
660C22#1683796211yescharacteristic
661C22#1683796311yescharacteristic
662C22#1683796711yescharacteristic
663C22#1683797411yescharacteristic
664C22#1683797912yes—
665C22#1683798512yes—
666C22#1683798712yes—
667C22#1683799412yes—
668C22#1683799612yes—
669C22#1683802712yes—
670C22#1683802812yes—
671C22#1683803012yes—
672C22#1683803112yes—
673C4—44no—
674C4—44no—
675C4—81no—
676C4—81no—
677C4—1184no—
678C4—1184no—
679C4—4721no—
680C4—4721no—
681C4—2364no—
682C4—2364no—
683C2—28no—
684C2C2×#1683729911yescharacteristic
685C2#2142670711yescharacteristic
686C2#2142671511yescharacteristic
687C2#2142679712yes—
688C2#2142682412yes—
689C2#2142983611yescharacteristic
690C2#2142984411yescharacteristic
691C2#2142990511yescharacteristic
692C2#2142991311yescharacteristic
693C2#2143006812yes—
694C2#2143021312yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC59×C22⋊#95C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC22⋊#95D59
hypocenterC59#14275
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
6. #21422360C2semidirect
7. #21422354C2semidirect
8. #21422350C2semidirect
9. #21422347C2semidirect
24. #16837260C4semidirect
68. C2×Dic59⋊C4C2×C4semidirect
83. C59⋊#95C2×C4semidirect
178. C22×Dic59C4⋊C4semidirect
387. C59#14275semidirect

Maximal subgroups

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

subgroupindex
C2×#168372772
C2×#168372742
C2×#168372602
C22×C59⋊#952
#214223602
#214223542
#214223502
#214223472
#214223192
C59×C22⋊#952
#1427559