Finite Group Databasebeta

← #28638139

Subgroups — #28638139

Order 15104.

792 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837772C211yesmaximal, characteristic, complemented
2C2×#16837699C211yesmaximal, characteristic
3#21429091C211yesmaximal, characteristic
4#21425488C211yesmaximal, characteristic, complemented
5#21425474C211yesmaximal, characteristic, complemented
6#21425337C211yesmaximal, characteristic
7#21425337C211yesmaximal, characteristic
8#21425028C211yesmaximal, characteristic
9#21425028C211yesmaximal, characteristic
10#21425019C211yesmaximal, characteristic, complemented
11#21424820C211yesmaximal, characteristic, complemented
12#21424737C211yesmaximal, characteristic, complemented
13#21424730C211yesmaximal, characteristic, complemented
14#21423924C211yesmaximal, characteristic
15C59×#2370C211yesmaximal, characteristic, complemented
16C2×#14273727C2211yescharacteristic
17C2×#14273715C2211yescharacteristic
18C2×#14273715C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837772—24no—
21#16837699—24no—
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×C4⋊(C59⋊C8)C2211yescharacteristic
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×#14273650C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273650C2211yescharacteristic
31C2×#14273650C2211yescharacteristic
32C2×#14273650C2211yescharacteristic
33C2×#14273650C2211yescharacteristic
34C2×#14273650C2211yescharacteristic
35C2×#14273647C2211yescharacteristic
36C2×#14273647C2211yescharacteristic
37C2×#14273647C2211yescharacteristic
38C2×#14273647C2211yescharacteristic
39C2×#14273638C2211yescharacteristic
40#16837329—21no—
41#16837329—21no—
42#16837308C2211yescharacteristic
43#16837308C2211yescharacteristic
44#16837259—21no—
45#16837259—21no—
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#125×C59C2211yescharacteristic
51C59×#389—21no—
52C59×#389C2211yescharacteristic
53C59×#389C2211yescharacteristic
54C59×#387—21no—
55C59×#387—21no—
56C4×C4⋊C4×C59—21no—
57C4×C4⋊C4×C59C2211yescharacteristic
58C4×C4⋊C4×C59C2211yescharacteristic
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C22×Dic118C2311yescharacteristic
63C22×D236C2311yescharacteristic
64C22×C4×D59—21no—
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68C22×C59⋊C8C2311yescharacteristic
69#14273727—22no—
70#14273727—22no—
71#14273715—22no—
72#14273715—22no—
73#14273715—41no—
74#14273715—41no—
75C4×D236—22no—
76C4×D236—22no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59—21no—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85#14273660—22no—
86#14273658—22no—
87C4⋊(C59⋊C8)—22no—
88C4⋊(C59⋊C8)—41no—
89C4⋊(C59⋊C8)—41no—
90#14273650—22no—
91#14273650—22no—
92#14273650—22no—
93#14273650—22no—
94#14273650—22no—
95#14273650—22no—
96#14273650—41no—
97#14273650—41no—
98#14273647—22no—
99#14273647—22no—
100#14273647—22no—
101#14273647—22no—
102#14273638—22no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107#125×C59—22no—
108#125×C59D414yes—
109C2×C4⋊C4×C59—21no—
110C2×C4⋊C4×C59—21no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C42×C59—21no—
120C2×C42×C59—21no—
121C2×C42×C59—21no—
122C2×C42×C59C2311yescharacteristic
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125#95×C59—21no—
126#95×C59—21no—
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C23×D59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59—21no—
136C22×C4×C59—21no—
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2411yescharacteristic
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C2×D236—21no—
148C2×D236—21no—
149C2×D236—21no—
150C2×D236—21no—
151C2×D236—21no—
152C2×D236#4112yes—
153C2×C4×D59—42no—
154C2×C4×D59—42no—
155C2×C4×D59—42no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—21no—
165C2×Dic118—21no—
166C2×Dic118—21no—
167C2×Dic118—21no—
168C4⋊C4×C59—24no—
169C4⋊C4×C59—24no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—41no—
176C4⋊C4×C59—41no—
177C4⋊C4×C59—42no—
178C4⋊C4×C59—81no—
179C4⋊C4×C59C2×D412yes—
180C4⋊C4×C59C2×D412yes—
181C4⋊C4×C59#4112yes—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—42no—
187C42×C59—81no—
188C42×C59—81no—
189C42×C59C2×D412yes—
190C42×C59#4112yes—
191(C2×D59)⋊C4—42no—
192(C2×D59)⋊C4—42no—
193(C2×D59)⋊C4—42no—
194(C2×D59)⋊C4—42no—
195(C2×D59)⋊C4—42no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—42no—
199C4⋊Dic59—42no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—21no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C23×C59C22×D411yescharacteristic
213C22×D59—22no—
214C22×D59—41no—
215C22×D59—41no—
216C22×D59—41no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—22no—
225C2×C4×C59—22no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59#12112yes—
249C2×C4×C59#12112yes—
250C2×C4×C59C22×D411yescharacteristic
251C2×C4×C59C2×#4111yescharacteristic
252C2×C4×C59C2×#4111yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59H(5,𝔽2)11yescharacteristic
255C2×C4×C592−1+411yescharacteristic
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265D236—22no—
266D236—22no—
267D236—41no—
268D236—41no—
269C4×D59—84no—
270Dic118—41no—
271Dic118—41no—
272Dic118—41no—
273Dic118—41no—
274Dic118—41no—
275Dic118—41no—
276Dic118—41no—
277Dic118—41no—
278C59⋊C8—41no—
279C59⋊C8—41no—
280C59⋊C8—41no—
281C59⋊C8—41no—
282#25289—591nomaximal, Sylow
283C22×C59C2×#12111yescharacteristic
284C22×C59#54611yescharacteristic
285C22×C59#54811yescharacteristic
286C22×C59#57511yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57711yescharacteristic
289C22×C59#57811yescharacteristic
290C2×D59—42no—
291C2×D59—42no—
292C2×D59—42no—
293C2×D59—81no—
294C2×D59—81no—
295C2×D59—81no—
296C2×D59—81no—
297C4×C59—22no—
298C4×C59—22no—
299C4×C59—42no—
300C4×C59—42no—
301C4×C59—42no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—81no—
307C4×C59#52911yescharacteristic
308C4×C59#54211yescharacteristic
309C4×C59#54811yescharacteristic
310C4×C59#55011yescharacteristic
311Dic59—81no—
312Dic59—81no—
313Dic59—81no—
314C2×#491—591no—
315C2×#478—591no—
316#2547—591no—
317#2370D5911yescharacteristic, complemented
318#2068—591no—
319#2066—591no—
320#2042—591no—
321#2042—591no—
322#1983—591no—
323#1983—591no—
324#1981—591no—
325#1940—591no—
326#1928—591no—
327#1927—591no—
328#1762—591no—
329C2×C59#242611yescharacteristic
330C2×C59#304011yescharacteristic
331C2×C59#304611yescharacteristic
332C2×C59#330711yescharacteristic
333C2×C59#331511yescharacteristic
334C2×C59#337611yescharacteristic
335C2×C59#337711yescharacteristic
336D59—82no—
337C2×C4⋊Q8—591no—
338C2×#125C2×D5911yescharacteristic
339C2×#124—591no—
340C2×#124—591no—
341C2×C4×D4—591no—
342#491—1184no—
343#478—1184no—
344C2×#107—591no—
345C2×#106—591no—
346C2×C4⋊C8—591no—
347C2×C4⋊C8—591no—
348C2×C4⋊C8—591no—
349C2×#103—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#103—591no—
356C2×#103—591no—
357C2×#102—591no—
358C2×#102—591no—
359C2×#102—591no—
360C2×#102—591no—
361C2×#97—591no—
362#397—1181no—
363#397—1181no—
364#391—591no—
365#391—591no—
366#389—21no—
367#389C2×D5911yescharacteristic
368#389C2×D5911yescharacteristic
369#387—21no—
370#387—21no—
371#380—1181no—
372#380—1181no—
373C4×C4⋊C4—21no—
374C4×C4⋊C4C2×D5911yescharacteristic
375C4×C4⋊C4C2×D5911yescharacteristic
376C4×C4⋊C4C2×D5911yescharacteristic
377C4×C4⋊C4C2×D5911yescharacteristic
378#340—591no—
379#340—591no—
380#336—591no—
381#336—591no—
382C59#2528911yescharacteristic, complemented, Sylow
383C22×Q8—591no—
384C22×Q8—591no—
385C22×D4—591no—
386C23×C4—1181no—
387C22×C8—591no—
388C22×C8—591no—
389C22×C8—591no—
390C22×C8—591no—
391C4⋊Q8—1182no—
392C4⋊Q8—1182no—
393#125—22no—
394#125D23614yes—
395#124—1182no—
396#124—1182no—
397#124—2361no—
398#124—2361no—
399C4×D4—1182no—
400C4×D4—1182no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—591no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4C22⋊D5911yescharacteristic
409C2×C4⋊C4C22⋊D5911yescharacteristic
410C2×C4⋊C4C22×D5911yescharacteristic
411C2×C4⋊C4C22×D5911yescharacteristic
412C2×C4⋊C4C22×D5911yescharacteristic
413C2×C4⋊C4C22×D5911yescharacteristic
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4—1181no—
417C2×C22⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C22×D5911yescharacteristic
423C2×C42C22×D5911yescharacteristic
424C2×C42C22×D5911yescharacteristic
425#107—1182no—
426#106—1182no—
427C4⋊C8—1182no—
428C4⋊C8—2361no—
429C4⋊C8—2361no—
430#103—1182no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—1182no—
435#103—1182no—
436#103—2361no—
437#103—2361no—
438#102—1182no—
439#102—1182no—
440#102—1182no—
441#102—1182no—
442#97—1182no—
443#95—21no—
444#95—21no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95C22⋊D5911yescharacteristic
453#95C22⋊D5911yescharacteristic
454C24—1181no—
455C2×Q8—1181no—
456C2×Q8—1181no—
457C2×Q8—1181no—
458C2×Q8—1181no—
459C2×Q8—1181no—
460C2×Q8—1181no—
461C2×Q8—1181no—
462C2×Q8—1181no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×D4—592no—
468C2×D4—1181no—
469C2×D4—1181no—
470C2×D4—1181no—
471C2×D4—1181no—
472C2×D4—1181no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—2362no—
481C22×C4—2362no—
482C22×C4—2362no—
483C22×C4C2×C22⋊D5911yescharacteristic
484C22×C4C2×C22⋊D5911yescharacteristic
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C23×D5911yescharacteristic
490C2×C8—1181no—
491C2×C8—1181no—
492C2×C8—1181no—
493C2×C8—1181no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—41no—
510C4⋊C4—41no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—2362no—
516C4⋊C4—2362no—
517C4⋊C4C2×D23612yes—
518C4⋊C4C2×D23612yes—
519C4⋊C4#1174568912yes—
520C22⋊C4—2362no—
521C22⋊C4—2362no—
522C22⋊C4—2362no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—42no—
530C42—81no—
531C42—81no—
532C42C2×D23612yes—
533C42#1174568512yes—
534C23—1182no—
535C23—2361no—
536C23—2361no—
537C23—2361no—
538C23—2361no—
539C23—2361no—
540C23—2361no—
541C23C22×C22⋊D5911yescharacteristic
542Q8—2361no—
543Q8—2361no—
544Q8—2361no—
545Q8—2361no—
546Q8—2361no—
547Q8—2361no—
548Q8—2361no—
549Q8—2361no—
550D4—1182no—
551D4—1182no—
552D4—2361no—
553D4—2361no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—4724no—
583C2×C4—2361no—
584C2×C4—2361no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4#1427370112yes—
593C2×C4#1427370112yes—
594C2×C4C22×D23611yescharacteristic
595C2×C4#41×D5911yescharacteristic
596C2×C4C2×#1174568511yescharacteristic
597C2×C4C2×#1174568911yescharacteristic
598C2×C4#1427377311yescharacteristic
599C2×C4#1427377511yescharacteristic
600C8—2361no—
601C8—2361no—
602C8—2361no—
603C8—2361no—
604C22—4721no—
605C22—4721no—
606C22—4721no—
607C22—4721no—
608C22—2362no—
609C22—2362no—
610C22—2362no—
611C22C2×#1427370111yescharacteristic
612C22#1683804011yescharacteristic
613C22#1683806311yescharacteristic
614C22#1683820011yescharacteristic
615C22#1683820011yescharacteristic
616C22#1683821311yescharacteristic
617C22#1683821711yescharacteristic
618C4—22no—
619C4—22no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—42no—
626C4—42no—
627C4—81no—
628C4—4721no—
629C4—4721no—
630C4—4721no—
631C4#1683793911yescharacteristic
632C4#1683802411yescharacteristic
633C4#1683806411yescharacteristic
634C4#1683807411yescharacteristic
635C2—4722no—
636C2#2142765611yescharacteristic
637C2#2143359011yescharacteristic
638C2#2143364611yescharacteristic
639C2#2143606611yescharacteristic
640C2#2143618611yescharacteristic
641C2#2143692311yescharacteristic
642C2#2143693811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2370C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#2370D59
hypocenterC59#25289
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2370

Complements

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

NH ≅ G/Ntype
1. C2×#16837772C2semidirect
4. #21425488C2semidirect
5. #21425474C2semidirect
10. #21425019C2semidirect
11. #21424820C2semidirect
12. #21424737C2semidirect
13. #21424730C2semidirect
15. C59×#2370C2semidirect
317. #2370D59semidirect
382. C59#25289semidirect

Maximal subgroups

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

subgroupindex
C2×#168377722
C2×#168376992
#214290912
#214254882
#214254742
#214253372
#214250282
#214250192
#214248202
#214247372
#214247302
#214239242
C59×#23702
#2528959