Finite Group Databasebeta

← #28610993

Subgroups — #28610993

Order 15104.

958 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837622C211yesmaximal, characteristic
2C2×#16837494C211yesmaximal, characteristic, complemented
3#21427169C211yesmaximal, characteristic
4#21427014C211yesmaximal, characteristic, complemented
5#21424906C211yesmaximal, characteristic
6#21424865C211yesmaximal, characteristic, complemented
7#21424776C211yesmaximal, characteristic
8#21424772C211yesmaximal, characteristic, complemented
9#21424453C211yesmaximal, characteristic
10#21424437C211yesmaximal, characteristic, complemented
11#21424361C211yesmaximal, characteristic, complemented
12#21424343C211yesmaximal, characteristic
13C8×Dic59⋊C4C211yesmaximal, characteristic, complemented
14#103×Dic59C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C59C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273714C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×Q8×Dic59C2211yescharacteristic
20C2×#14273692C2211yescharacteristic
21#16837622C418yescomplemented
22#16837494—24no—
23C2×#14273660C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×C8×Dic59C2211yescharacteristic
32C2×C8×Dic59C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837305C2211yescharacteristic
35#16837300—21no—
36#16837300C2211yescharacteristic
37#16837298—21no—
38#16837298C2211yescharacteristic
39#16837296—21no—
40#16837290—21no—
41#16837275—21no—
42#16837267—21no—
43#16837267C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4×Dic59⋊C4—21no—
46C4×Dic59⋊C4C2211yescharacteristic
47C4×C22⋊Dic59—21no—
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837157C2211yescharacteristic
53#16837156C2211yescharacteristic
54C22×SD16×C59C2211yescharacteristic
55C2×C4×Q8×C59C2211yescharacteristic
56C2×C4×D4×C59C2211yescharacteristic
57C4×SD16×C59—24no—
58C2×#106×C59C2211yescharacteristic
59C2×#103×C59C2211yescharacteristic
60C2×#102×C59C2211yescharacteristic
61C2×C4×C8×C59C2211yescharacteristic
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672C2×C414yescomplemented
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273714—22no—
69#14273714C2×C414yes—
70Q8×Dic59—22no—
71Q8×Dic59—22no—
72Q8×Dic59—41no—
73Q8×Dic59C2×C414yescomplemented
74#14273692—22no—
75#14273692—41no—
76C2×C4⋊Dic59—21no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4D411yescharacteristic
85C2×Dic59⋊C4D411yescharacteristic
86C2×C22⋊Dic59—21no—
87C2×C22⋊Dic59—21no—
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59C2311yescharacteristic
95C2×C4×Dic59C2311yescharacteristic
96#14273660—22no—
97Dic59⋊C8C2×C414yescomplemented
98#14273652—22no—
99#14273652C2×C414yescomplemented
100#14273651—22no—
101#14273650—41no—
102#14273648—41no—
103#14273648C2×C414yes—
104C8×Dic59—22no—
105C8×Dic59—22no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C59⋊#95D411yescharacteristic
114C59⋊#95D411yescharacteristic
115C22×Q8×C59C2311yescharacteristic
116C22×D4×C59C2311yescharacteristic
117C23×C4×C59—21no—
118C2×SD16×C59—22no—
119C2×SD16×C59—22no—
120C2×SD16×C59C2×C414yes—
121C22×C8×C59C2311yescharacteristic
122C22×C8×C59C2311yescharacteristic
123C4×Q8×C59—22no—
124C4×Q8×C59—41no—
125C4×D4×C59—22no—
126C4×D4×C59—41no—
127C2×C4⋊C4×C59—21no—
128C2×C4⋊C4×C59C2311yescharacteristic
129C2×C4⋊C4×C59C2311yescharacteristic
130C2×C22⋊C4×C59—21no—
131C2×C42×C59—21no—
132C2×C42×C59C2311yescharacteristic
133#106×C59—22no—
134#103×C59D414yes—
135#102×C59—22no—
136C4×C8×C59—22no—
137C24×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59C22×C412yes—
142C2×Q8×C59C2×D412yes—
143C2×Q8×C59#4112yes—
144C2×D4×C59—21no—
145C2×D4×C59—21no—
146C2×D4×C59—21no—
147C2×D4×C59—21no—
148C2×D4×C59—21no—
149C2×D4×C59C22×C412yes—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59—42no—
154C22×C4×C59—42no—
155C22×C4×C59—81no—
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2411yescharacteristic
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—21no—
162C22×Dic59—21no—
163C22×Dic59C2×D411yescharacteristic
164C22×Dic59C2×D411yescharacteristic
165C22×Dic59C2×D411yescharacteristic
166C22×Dic59C2×D411yescharacteristic
167C2×Dic118—21no—
168C2×Dic118—21no—
169C2×Dic118—21no—
170C2×Dic118—21no—
171C2×Dic118—21no—
172C2×Dic118C22×C412yes—
173SD16×C59—28no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—21no—
178C2×C8×C59C22×C412yes—
179C2×C8×C59C2×D412yes—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59C2×D412yes—
185C22⋊C4×C59—42no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189C22⋊Dic59—24no—
190C22⋊Dic59—24no—
191C22⋊Dic59—42no—
192C22⋊Dic59—81no—
193#11745672—28no—
194C4⋊Dic59—21no—
195C4⋊Dic59—24no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—42no—
201C4⋊Dic59C22×C412yes—
202Dic59⋊C4—22no—
203Dic59⋊C4—22no—
204Dic59⋊C4—42no—
205Dic59⋊C4—42no—
206Dic59⋊C4—42no—
207C4×Dic59—21no—
208C4×Dic59—21no—
209C4×Dic59—24no—
210C4×Dic59—42no—
211C4×Dic59—42no—
212C4×Dic59—81no—
213C4×Dic59C22×C412yes—
214C4×Dic59#4112yes—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8C22×C412yes—
221C23×C59—22no—
222C23×C59—41no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59C22×D411yescharacteristic
229Q8×C59—22no—
230Q8×C59—22no—
231Q8×C59—22no—
232Q8×C59C4×D414yes—
233D4×C59—22no—
234D4×C59—22no—
235D4×C59—41no—
236D4×C59—41no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—84no—
252C2×C4×C59#13512yes—
253C2×C4×C59C23×C411yescharacteristic
254C2×C4×C59C22×D411yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59C2×#4111yescharacteristic
257C2×C4×C59H(5,𝔽2)11yescharacteristic
258C2×C4×C592−1+411yescharacteristic
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—21no—
264C2×Dic59—21no—
265C2×Dic59—22no—
266C2×Dic59—22no—
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—21no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59C4×D412yes—
281C2×Dic59C4×D412yes—
282C2×Dic59#13512yes—
283Dic118—22no—
284Dic118—22no—
285Dic118—41no—
286Dic118—41no—
287C8×C59—22no—
288C8×C59—22no—
289C59⋊C8—22no—
290C59⋊C8—41no—
291#23000—591nomaximal, Sylow
292C22×C59—42no—
293C22×C59—42no—
294C22×C59—42no—
295C22×C59—81no—
296C22×C59—81no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59C2×C4×D411yescharacteristic
300C22×C59D4211yescharacteristic
301C22×C59#54811yescharacteristic
302C22×C59C2×#13511yescharacteristic
303C22×C59C2×#13511yescharacteristic
304C22×C59#57711yescharacteristic
305C22×C59#57811yescharacteristic
306C4×C59—24no—
307C4×C59—22no—
308C4×C59—42no—
309C4×C59—41no—
310C4×C59—81no—
311C4×C59C2×C4×D411yescharacteristic
312C4×C59#52011yescharacteristic
313C4×C59#54711yescharacteristic
314C4×C59#55311yescharacteristic
315Dic59—22no—
316Dic59—22no—
317Dic59—22no—
318Dic59—42no—
319Dic59—42no—
320Dic59—41no—
321Dic59—81no—
322Dic59—81no—
323C2×#464—591no—
324C2×C4×Q16—591no—
325C2×C4×SD16D5911yescharacteristic
326#2367—591no—
327#2351—591no—
328#1958—591no—
329#1949—591no—
330#1934—591no—
331#1933—591no—
332#1880—591no—
333#1878—591no—
334#1865—591no—
335#1862—591no—
336C8×C4⋊C4—591no—
337C4×#103—591no—
338C2×C59—82no—
339C2×C59#236111yescharacteristic
340C2×C59C4×#13511yescharacteristic
341C2×C59#294711yescharacteristic
342C2×C59#328211yescharacteristic
343C2×C59#328511yescharacteristic
344C2×C59#336411yescharacteristic
345C2×C59#336511yescharacteristic
346C22×Q16—591no—
347C22×SD16C2×D5911yescharacteristic
348C2×#124—591no—
349C2×C4×Q8—591no—
350C2×C4×Q8—591no—
351C2×C4×Q8—591no—
352C2×C4×Q8C2×D5911yescharacteristic
353C2×C4×D4C2×D5911yescharacteristic
354#464—598no—
355C4×Q16—1184no—
356C4×SD16—24no—
357C2×#107—591no—
358C2×#106C2×D5911yescharacteristic
359C2×C4⋊C8—591no—
360C2×#103—591no—
361C2×#103—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103C2×D5911yescharacteristic
365C2×#102—591no—
366C2×#102—591no—
367C2×#102C2×D5911yescharacteristic
368C2×C4×C8—591no—
369C2×C4×C8—591no—
370C2×C4×C8C2×D5911yescharacteristic
371#391—591no—
372#390—591no—
373#389—591no—
374#389—591no—
375#389—1181no—
376#389—1181no—
377#388—1181no—
378#388—1181no—
379#385—1181no—
380#382—591no—
381#382—1181no—
382C4×C4⋊C4—591no—
383C4×C4⋊C4—591no—
384C4×C4⋊C4—1181no—
385C4×C22⋊C4—1181no—
386#340—591no—
387#340—591no—
388#340—591no—
389#340—591no—
390#336—591no—
391#336—591no—
392C59#2300011yescharacteristic, complemented, Sylow
393C22×Q8—591no—
394C22×Q8C22×D5911yescharacteristic
395C22×D4C22×D5911yescharacteristic
396C23×C4—21no—
397C2×Q16—594no—
398C2×Q16—1182no—
399C2×Q16—1182no—
400C2×SD16—22no—
401C2×SD16—22no—
402C2×SD16C4×D5914yes—
403C22×C8—591no—
404C22×C8—591no—
405C22×C8C22×D5911yescharacteristic
406C22×C8C22×D5911yescharacteristic
407#124—594no—
408#124—1182no—
409C4×Q8—22no—
410C4×Q8—41no—
411C4×Q8—594no—
412C4×Q8—1182no—
413C4×Q8—1182no—
414C4×Q8—1182no—
415C4×Q8—2361no—
416C4×Q8—2361no—
417C4×D4—22no—
418C4×D4—41no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4C22×D5911yescharacteristic
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C22⋊C4—21no—
433C2×C22⋊C4—1181no—
434C2×C22⋊C4—1181no—
435C2×C22⋊C4—1181no—
436C2×C22⋊C4—1181no—
437C2×C42—21no—
438C2×C42—591no—
439C2×C42—591no—
440C2×C42—1181no—
441C2×C42—1181no—
442C2×C42—1181no—
443C2×C42—1181no—
444C2×C42C22×D5911yescharacteristic
445#107—1182no—
446#106—22no—
447C4⋊C8—594no—
448#103—594no—
449#103—1182no—
450#103—1182no—
451#103—2361no—
452#103C22⋊D5914yes—
453#102—22no—
454#102—594no—
455#102—2361no—
456C4×C8—22no—
457C4×C8—1182no—
458C4×C8—1182no—
459#95—591no—
460#95—591no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95—1181no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468C24—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—592no—
473C2×Q8—1181no—
474C2×Q8—1181no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8C2×C4×D5912yes—
479C2×Q8#1174568512yes—
480C2×Q8C2×C22⋊D5912yes—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4C2×C4×D5912yes—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—42no—
491C22×C4—42no—
492C22×C4—81no—
493C22×C4—591no—
494C22×C4—591no—
495C22×C4—591no—
496C22×C4—591no—
497C22×C4—1181no—
498C22×C4—1181no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4D4×D5911yescharacteristic
502C22×C4D4×D5911yescharacteristic
503C22×C4C23×D5911yescharacteristic
504Q16—1188no—
505SD16—28no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—592no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8C2×C4×D5912yes—
517C2×C8C2×C22⋊D5912yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—592no—
523C4⋊C4—1182no—
524C4⋊C4—1184no—
525C4⋊C4—1182no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4C2×C22⋊D5912yes—
536C22⋊C4—42no—
537C22⋊C4—1184no—
538C22⋊C4—1184no—
539C22⋊C4—4721no—
540C22⋊C4—2362no—
541C42—21no—
542C42—21no—
543C42—42no—
544C42—592no—
545C42—592no—
546C42—1184no—
547C42—1181no—
548C42—1181no—
549C42—4721no—
550C42—2362no—
551C42—2362no—
552C23—22no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D5911yescharacteristic
560Q8—22no—
561Q8—22no—
562Q8—22no—
563Q8—1182no—
564Q8—1182no—
565Q8—2361no—
566Q8—2361no—
567Q8C4×C22⋊D5914yes—
568D4—22no—
569D4—22no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—84no—
587C2×C4—592no—
588C2×C4—592no—
589C2×C4—592no—
590C2×C4—1181no—
591C2×C4—1181no—
592C2×C4—1181no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1181no—
597C2×C4—1181no—
598C2×C4—1181no—
599C2×C4—1182no—
600C2×C4—1182no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4#1427374812yes—
612C2×C4C22×C4×D5911yescharacteristic
613C2×C4C22×C22⋊D5911yescharacteristic
614C2×C4#41×D5911yescharacteristic
615C2×C4C2×#1174568511yescharacteristic
616C2×C4#1427377211yescharacteristic
617C2×C4#1427377411yescharacteristic
618C8—22no—
619C8—22no—
620C8—1182no—
621C8—2361no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—81no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22C4×D4×D5911yescharacteristic
630C22D4×C22⋊D5911yescharacteristic
631C22#1683806111yescharacteristic
632C22#135×D5911yescharacteristic
633C22C2×#1427374811yescharacteristic
634C22#1683820811yescharacteristic
635C22#1683821611yescharacteristic
636C4—22no—
637C4—24no—
638C4—42no—
639C4—41no—
640C4—81no—
641C4—1182no—
642C4—1182no—
643C4—1182no—
644C4—4721no—
645C4—4721no—
646C4—2362no—
647C4—2361no—
648C4—2362no—
649C4C2×C4×C22⋊D5911yescharacteristic
650C4#1683788111yescharacteristic
651C4#1683805111yescharacteristic
652C4#1683810111yescharacteristic
653C2—82no—
654C2#2142711311yescharacteristic
655C2C4×#1427374811yescharacteristic
656C2#2143293211yescharacteristic
657C2#2143580311yescharacteristic
658C2#2143583111yescharacteristic
659C2#2143677911yescharacteristic
660C2#2143679311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C4×SD16×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
cosocleC22×C4×C59C24
hypercenterC2×C4×SD16D59
hypocenterC59#23000
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×SD16

Complements

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

NH ≅ G/Ntype
2. C2×#16837494C2semidirect
4. #21427014C2semidirect
6. #21424865C2semidirect
8. #21424772C2semidirect
10. #21424437C2semidirect
11. #21424361C2semidirect
13. C8×Dic59⋊C4C2semidirect
14. #103×Dic59C2semidirect
21. #16837622C4semidirect
65. C2×#11745672C2×C4semidirect
73. Q8×Dic59C2×C4semidirect
97. Dic59⋊C8C2×C4semidirect
99. #14273652C2×C4semidirect
392. C59#23000semidirect

Maximal subgroups

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

subgroupindex
C2×#168376222
C2×#168374942
#214271692
#214270142
#214249062
#214248652
#214247762
#214247722
#214244532
#214244372
#214243612
#214243432
C8×Dic59⋊C42
#103×Dic592
C2×C4×SD16×C592
#2300059