Finite Group Databasebeta

← #28611168

Subgroups — #28611168

Order 15104.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837622C211yesmaximal, characteristic
2C2×#16837506C211yesmaximal, characteristic, complemented
3#21427169C211yesmaximal, characteristic
4Q8×Dic59⋊C4C211yesmaximal, characteristic, complemented
5#21424911C211yesmaximal, characteristic
6#21424856C211yesmaximal, characteristic, complemented
7#21424790C211yesmaximal, characteristic
8#21424772C211yesmaximal, characteristic, complemented
9#21424453C211yesmaximal, characteristic
10#21424437C211yesmaximal, characteristic, complemented
11#21424361C211yesmaximal, characteristic, complemented
12#21424357C211yesmaximal, characteristic
13#21423970C211yesmaximal, characteristic, complemented
14#21423928C211yesmaximal, characteristic, complemented
15C2×C4×SD16×C59C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C2×#14273714C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C2×Q8×Dic59C2211yescharacteristic
20C2×#14273692C2211yescharacteristic
21#16837622C418yescomplemented
22#16837506—24no—
23C2×#14273658C2211yescharacteristic
24C2×Dic59⋊C8C2211yescharacteristic
25C2×#14273652C2211yescharacteristic
26C2×#14273652C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273637C2211yescharacteristic
32C2×#14273637C2211yescharacteristic
33#16837308C2211yescharacteristic
34#16837305C2211yescharacteristic
35#16837300—21no—
36#16837300C2211yescharacteristic
37#16837298—21no—
38#16837298C2211yescharacteristic
39#16837296—21no—
40#16837290—21no—
41#16837275—21no—
42#16837271—21no—
43#16837271C2211yescharacteristic
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⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4D411yescharacteristic
87C2×Dic59⋊C4D411yescharacteristic
88C2×C22⋊Dic59—21no—
89C2×C22⋊Dic59—21no—
90C2×C22⋊Dic59—21no—
91C2×C22⋊Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96C2×C4×Dic59C2311yescharacteristic
97C2×C4×Dic59C2311yescharacteristic
98#14273658—22no—
99Dic59⋊C8C2×C414yescomplemented
100#14273652—22no—
101#14273652C2×C414yescomplemented
102#14273651—22no—
103#14273650—41no—
104#14273648—41no—
105#14273648C2×C414yes—
106#14273637—22no—
107#14273637—22no—
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—41no—
204Dic59⋊C4—42no—
205Dic59⋊C4—42no—
206Dic59⋊C4—81no—
207Dic59⋊C4—81no—
208Dic59⋊C4—81no—
209C4×Dic59—21no—
210C4×Dic59—21no—
211C4×Dic59—24no—
212C4×Dic59—42no—
213C4×Dic59—42no—
214C4×Dic59—81no—
215C4×Dic59C22×C412yes—
216C4×Dic59#4112yes—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8—21no—
222C2×C59⋊C8C22×C412yes—
223C23×C59—22no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59—41no—
230C23×C59C22×D411yescharacteristic
231Q8×C59—22no—
232Q8×C59—22no—
233Q8×C59—22no—
234Q8×C59C4×D414yes—
235D4×C59—22no—
236D4×C59—22no—
237D4×C59—41no—
238D4×C59—41no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—22no—
243C2×C4×C59—21no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—84no—
254C2×C4×C59#13712yes—
255C2×C4×C59C23×C411yescharacteristic
256C2×C4×C59C22×D411yescharacteristic
257C2×C4×C59C2×#4111yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59H(5,𝔽2)11yescharacteristic
260C2×C4×C592−1+411yescharacteristic
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—21no—
271C2×Dic59—21no—
272C2×Dic59—21no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59C4×D412yes—
284C2×Dic59C4×D412yes—
285Dic118—22no—
286Dic118—22no—
287Dic118—41no—
288Dic118—41no—
289C8×C59—22no—
290C8×C59—22no—
291C59⋊C8—22no—
292C59⋊C8—41no—
293#23013—591nomaximal, Sylow
294C22×C59—42no—
295C22×C59—42no—
296C22×C59—42no—
297C22×C59—81no—
298C22×C59—81no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59C2×C4×D411yescharacteristic
302C22×C59D4211yescharacteristic
303C22×C59#54811yescharacteristic
304C22×C59C2×#13711yescharacteristic
305C22×C59#57511yescharacteristic
306C22×C59#57711yescharacteristic
307C22×C59#57711yescharacteristic
308C4×C59—24no—
309C4×C59—22no—
310C4×C59—42no—
311C4×C59—41no—
312C4×C59—81no—
313C4×C59C2×C4×D411yescharacteristic
314C4×C59#52011yescharacteristic
315C4×C59#54711yescharacteristic
316C4×C59#55311yescharacteristic
317Dic59—22no—
318Dic59—22no—
319Dic59—42no—
320Dic59—42no—
321Dic59—41no—
322Dic59—41no—
323Dic59—81no—
324Dic59—81no—
325C2×#464—591no—
326C2×#441—591no—
327C2×C4×SD16D5911yescharacteristic
328#2367—591no—
329Q8×C4⋊C4—591no—
330#1959—591no—
331#1948—591no—
332#1936—591no—
333#1933—591no—
334#1880—591no—
335#1878—591no—
336#1865—591no—
337#1864—591no—
338#1771—591no—
339#1763—591no—
340C2×C59—82no—
341C2×C59#236111yescharacteristic
342C2×C59C4×#13711yescharacteristic
343C2×C59#295211yescharacteristic
344C2×C59#326611yescharacteristic
345C2×C59#327111yescharacteristic
346C2×C59#336911yescharacteristic
347C2×C59#337111yescharacteristic
348C22×Q16—591no—
349C22×SD16C2×D5911yescharacteristic
350C2×#124—591no—
351C2×C4×Q8—591no—
352C2×C4×Q8—591no—
353C2×C4×Q8—591no—
354C2×C4×Q8C2×D5911yescharacteristic
355C2×C4×D4C2×D5911yescharacteristic
356#464—598no—
357#441—1184no—
358C4×SD16—24no—
359C2×#106—591no—
360C2×#106C2×D5911yescharacteristic
361C2×C4⋊C8—591no—
362C2×#103—591no—
363C2×#103—591no—
364C2×#103—591no—
365C2×#103—591no—
366C2×#103C2×D5911yescharacteristic
367C2×#102—591no—
368C2×#102—591no—
369C2×#102C2×D5911yescharacteristic
370C2×#97—591no—
371C2×#97—591no—
372C2×C4×C8C2×D5911yescharacteristic
373#391—591no—
374#390—591no—
375#389—591no—
376#389—591no—
377#389—1181no—
378#389—1181no—
379#388—1181no—
380#388—1181no—
381#385—1181no—
382#384—591no—
383#384—1181no—
384C4×C4⋊C4—591no—
385C4×C4⋊C4—591no—
386C4×C4⋊C4—1181no—
387C4×C22⋊C4—1181no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#336—591no—
393#336—591no—
394C59#2301311yescharacteristic, complemented, Sylow
395C22×Q8—591no—
396C22×Q8C22×D5911yescharacteristic
397C22×D4C22×D5911yescharacteristic
398C23×C4—21no—
399C2×Q16—594no—
400C2×Q16—1182no—
401C2×Q16—1182no—
402C2×SD16—22no—
403C2×SD16—22no—
404C2×SD16C4×D5914yes—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8C22×D5911yescharacteristic
408C22×C8C22×D5911yescharacteristic
409#124—594no—
410#124—1182no—
411C4×Q8—22no—
412C4×Q8—41no—
413C4×Q8—594no—
414C4×Q8—1182no—
415C4×Q8—1182no—
416C4×Q8—1182no—
417C4×Q8—2361no—
418C4×Q8—2361no—
419C4×D4—22no—
420C4×D4—41no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—591no—
426C2×C4⋊C4—591no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4—1181no—
432C2×C4⋊C4—1181no—
433C2×C4⋊C4—1181no—
434C2×C4⋊C4C22×D5911yescharacteristic
435C2×C4⋊C4C22×D5911yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1181no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4—1181no—
441C2×C42—21no—
442C2×C42—591no—
443C2×C42—591no—
444C2×C42—1181no—
445C2×C42—1181no—
446C2×C42—1181no—
447C2×C42—1181no—
448C2×C42C22×D5911yescharacteristic
449#106—22no—
450#106—1182no—
451C4⋊C8—594no—
452#103—594no—
453#103—1182no—
454#103—1182no—
455#103—2361no—
456#103C22⋊D5914yes—
457#102—22no—
458#102—594no—
459#102—2361no—
460#97—1182no—
461#97—1182no—
462C4×C8—22no—
463#95—591no—
464#95—591no—
465#95—1181no—
466#95—1181no—
467#95—1181no—
468#95—1181no—
469#95—1181no—
470C24—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—592no—
475C2×Q8—1181no—
476C2×Q8—1181no—
477C2×Q8—1181no—
478C2×Q8—1181no—
479C2×Q8—1181no—
480C2×Q8C2×C4×D5912yes—
481C2×Q8#1174568512yes—
482C2×Q8C2×C22⋊D5912yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D5912yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—81no—
495C22×C4—591no—
496C22×C4—591no—
497C22×C4—591no—
498C22×C4—591no—
499C22×C4—1181no—
500C22×C4—1181no—
501C22×C4—1181no—
502C22×C4—1181no—
503C22×C4D4×D5911yescharacteristic
504C22×C4D4×D5911yescharacteristic
505C22×C4C23×D5911yescharacteristic
506Q16—1188no—
507SD16—28no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—592no—
513C2×C8—1181no—
514C2×C8—1181no—
515C2×C8—1181no—
516C2×C8—1181no—
517C2×C8—1181no—
518C2×C8C2×C4×D5912yes—
519C2×C8C2×C22⋊D5912yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—42no—
524C4⋊C4—592no—
525C4⋊C4—1184no—
526C4⋊C4—1182no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—1181no—
532C4⋊C4—4721no—
533C4⋊C4—4721no—
534C4⋊C4—4721no—
535C4⋊C4—2361no—
536C4⋊C4—2362no—
537C4⋊C4—2362no—
538C4⋊C4—2362no—
539C4⋊C4C2×C22⋊D5912yes—
540C22⋊C4—42no—
541C22⋊C4—1184no—
542C22⋊C4—1184no—
543C22⋊C4—4721no—
544C22⋊C4—2362no—
545C42—21no—
546C42—21no—
547C42—42no—
548C42—592no—
549C42—592no—
550C42—1184no—
551C42—1181no—
552C42—1181no—
553C42—4721no—
554C42—2362no—
555C42—2362no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D5911yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—1182no—
568Q8—1182no—
569Q8—2361no—
570Q8—2361no—
571Q8C4×C22⋊D5914yes—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—592no—
592C2×C4—592no—
593C2×C4—1181no—
594C2×C4—1181no—
595C2×C4—1181no—
596C2×C4—1181no—
597C2×C4—1181no—
598C2×C4—1181no—
599C2×C4—1181no—
600C2×C4—1181no—
601C2×C4—1181no—
602C2×C4—1181no—
603C2×C4—1182no—
604C2×C4—1182no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4—2361no—
612C2×C4—2361no—
613C2×C4—2361no—
614C2×C4—2361no—
615C2×C4#1427375612yes—
616C2×C4C22×C4×D5911yescharacteristic
617C2×C4C22×C22⋊D5911yescharacteristic
618C2×C4#41×D5911yescharacteristic
619C2×C4C2×#1174568511yescharacteristic
620C2×C4#1427377211yescharacteristic
621C2×C4#1427377411yescharacteristic
622C8—22no—
623C8—22no—
624C8—1182no—
625C8—2361no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C4×D4×D5911yescharacteristic
634C22D4×C22⋊D5911yescharacteristic
635C22#1683806111yescharacteristic
636C22C2×#1427375611yescharacteristic
637C22#1683819911yescharacteristic
638C22#1683820811yescharacteristic
639C22#1683820911yescharacteristic
640C4—24no—
641C4—22no—
642C4—41no—
643C4—42no—
644C4—81no—
645C4—1182no—
646C4—1182no—
647C4—4721no—
648C4—4721no—
649C4—2361no—
650C4—2362no—
651C4—2362no—
652C4—2361no—
653C4C2×C4×C22⋊D5911yescharacteristic
654C4#1683788111yescharacteristic
655C4#1683805111yescharacteristic
656C4#1683810111yescharacteristic
657C2—82no—
658C2#2142711311yescharacteristic
659C2C4×#1427375611yescharacteristic
660C2#2143297411yescharacteristic
661C2#2143560611yescharacteristic
662C2#2143568111yescharacteristic
663C2#2143683411yescharacteristic
664C2#2143686011yescharacteristic
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#23013
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×#16837506C2semidirect
4. Q8×Dic59⋊C4C2semidirect
6. #21424856C2semidirect
8. #21424772C2semidirect
10. #21424437C2semidirect
11. #21424361C2semidirect
13. #21423970C2semidirect
14. #21423928C2semidirect
21. #16837622C4semidirect
65. C2×#11745672C2×C4semidirect
73. Q8×Dic59C2×C4semidirect
99. Dic59⋊C8C2×C4semidirect
101. #14273652C2×C4semidirect
394. C59#23013semidirect

Maximal subgroups

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

subgroupindex
C2×#168376222
C2×#168375062
#214271692
Q8×Dic59⋊C42
#214249112
#214248562
#214247902
#214247722
#214244532
#214244372
#214243612
#214243572
#214239702
#214239282
C2×C4×SD16×C592
#2301359