Finite Group Databasebeta

← #28607055

Subgroups — #28607055

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837500C211yesmaximal, characteristic
2C2×#16837500C211yesmaximal, characteristic
3C2×C4×#11745671C211yesmaximal, characteristic
4#21427007C211yesmaximal, characteristic
5#21424831C211yesmaximal, characteristic, complemented
6#21424826C211yesmaximal, characteristic, complemented
7#21424364C211yesmaximal, characteristic, complemented
8#21424356C211yesmaximal, characteristic
9#21424350C211yesmaximal, characteristic, complemented
10#21424342C211yesmaximal, characteristic
11#21424262C211yesmaximal, characteristic, complemented
12#21424111C211yesmaximal, characteristic, complemented
13#21424093C211yesmaximal, characteristic
14#21423971C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C59C211yesmaximal, characteristic, complemented
16C22×#11745671C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20#16837500—24no—
21#16837500C418yescomplemented
22C4×#11745671—24no—
23C2×#14273660C2211yescharacteristic
24C2×#14273660C2211yescharacteristic
25C2×#14273660C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×#14273658C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273651C2211yescharacteristic
30C2×#14273651C2211yescharacteristic
31C2×#14273651C2211yescharacteristic
32C2×#14273647C2211yescharacteristic
33C2×#14273647C2211yescharacteristic
34C2×#14273647C2211yescharacteristic
35C2×#14273638C2211yescharacteristic
36C2×#14273638C2211yescharacteristic
37C2×C4×C59⋊C8C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837300C2211yescharacteristic
40#16837273—21no—
41#16837273—21no—
42#16837268C2211yescharacteristic
43#16837261—21no—
44#95×D59—21no—
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837157C2211yescharacteristic
50#16837157C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C2×C4×Q8×C59C2211yescharacteristic
53C2×C4×Q8×C59C2211yescharacteristic
54C59×#389—21no—
55C59×#389—21no—
56C59×#389C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#384—21no—
59C59×#384C2211yescharacteristic
60C4×C4⋊C4×C59—21no—
61C4×C4⋊C4×C59C2211yescharacteristic
62C22×D236C2311yescharacteristic
63C22×C4×D59—21no—
64C22×C4×D59—21no—
65C22×C4×D59—21no—
66C2×#11745671C2×C414yes—
67C2×#11745671D414yes—
68C2×#11745671Q814yes—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73C4×D236—22no—
74C4×D236—22no—
75C4×D236—22no—
76C4×D236—41no—
77C4×D236—41no—
78C4×D236C2×C414yes—
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87#14273660—22no—
88#14273660—41no—
89#14273660C2×C414yescomplemented
90#14273658—22no—
91#14273658—41no—
92C4⋊(C59⋊C8)—22no—
93#14273651—22no—
94#14273651—22no—
95#14273651C2×C414yescomplemented
96#14273647—22no—
97#14273647—22no—
98#14273647C2×C414yes—
99#14273638—22no—
100#14273638C2×C414yescomplemented
101C4×C59⋊C8—22no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C22×Q8×C59C2311yescharacteristic
107C4×Q8×C59—22no—
108C4×Q8×C59—22no—
109C4×Q8×C59—22no—
110C4×Q8×C59—41no—
111C4×Q8×C59—41no—
112C4×Q8×C59C2×C414yescomplemented
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—21no—
115C2×C4⋊C4×C59—21no—
116C2×C4⋊C4×C59—21no—
117C2×C4⋊C4×C59—21no—
118C2×C4⋊C4×C59—21no—
119C2×C4⋊C4×C59—21no—
120C2×C4⋊C4×C59D411yescharacteristic
121C2×C4⋊C4×C59D411yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C4⋊C4×C59C2311yescharacteristic
125C2×C42×C59—21no—
126C2×C42×C59—21no—
127C2×C42×C59—21no—
128C2×C42×C59C2311yescharacteristic
129C2×C42×C59C2311yescharacteristic
130C2×C42×C59C2311yescharacteristic
131#95×C59—21no—
132#95×C59—21no—
133#95×C59D411yescharacteristic
134#95×C59D411yescharacteristic
135C23×D59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59C22×C412yes—
140C2×Q8×C59C2×D412yes—
141C2×Q8×C59C2×Q812yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2411yescharacteristic
153C22×Dic59—21no—
154C22×Dic59—21no—
155C22×Dic59—21no—
156C2×D236—21no—
157C2×D236—21no—
158C2×D236—21no—
159C2×D236C22×C412yes—
160C2×D236C2×D412yes—
161C2×D236C2×Q812yes—
162C2×C4×D59—24no—
163C2×C4×D59—42no—
164C2×C4×D59—42no—
165C2×C4×D59—42no—
166C2×C4×D59—42no—
167C2×C4×D59—81no—
168C2×C4×D59—81no—
169C2×C4×D59—81no—
170C2×C4×D59—81no—
171C4⋊C4×C59—24no—
172C4⋊C4×C59—22no—
173C4⋊C4×C59—21no—
174C4⋊C4×C59—21no—
175C4⋊C4×C59—21no—
176C4⋊C4×C59—21no—
177C4⋊C4×C59—21no—
178C4⋊C4×C59—41no—
179C4⋊C4×C59—42no—
180C4⋊C4×C59—42no—
181C4⋊C4×C59—42no—
182C4⋊C4×C59—81no—
183C4⋊C4×C59—81no—
184C4⋊C4×C59—81no—
185C4⋊C4×C59C22×C412yes—
186C42×C59—24no—
187C42×C59—21no—
188C42×C59—21no—
189C42×C59—21no—
190C42×C59—21no—
191C42×C59—42no—
192C42×C59—42no—
193C42×C59C22×C412yes—
194C42×C59#4112yes—
195#11745671C4⋊C4116yescomplemented
196(C2×D59)⋊C4—24no—
197(C2×D59)⋊C4—42no—
198(C2×D59)⋊C4—42no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—21no—
203C4⋊Dic59—21no—
204C4⋊Dic59C22×C412yes—
205Dic59⋊C4—81no—
206Dic59⋊C4—81no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8C22×C412yes—
216C2×C59⋊C8C22×C412yes—
217C2×C59⋊C8C2×D412yes—
218C2×C59⋊C8C2×Q812yes—
219C23×C59C22×D411yescharacteristic
220C22×D59—22no—
221C22×D59—22no—
222C22×D59—22no—
223C22×D59—41no—
224C22×D59—41no—
225C22×D59—41no—
226C22×D59—41no—
227Q8×C59—22no—
228Q8×C59—22no—
229Q8×C59—22no—
230Q8×C59C2×C4⋊C414yes—
231C2×C4×C59—22no—
232C2×C4×C59—21no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59C4×D412yes—
259C2×C4×C59C4×D412yes—
260C2×C4×C59#13712yes—
261C2×C4×C59C23×C411yescharacteristic
262C2×C4×C59C22×D411yescharacteristic
263C2×C4×C59C22×Q811yescharacteristic
264C2×C4×C59C2×#4111yescharacteristic
265C2×C4×C59H(5,𝔽2)11yescharacteristic
266C2×C4×C592−1+411yescharacteristic
267C2×Dic59—22no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276D236—22no—
277D236—22no—
278D236—22no—
279D236C2×C4⋊C414yes—
280C4×D59—84no—
281C4×D59—84no—
282C4×D59—84no—
283C59⋊C8—22no—
284C59⋊C8—41no—
285C59⋊C8—41no—
286C59⋊C8C2×C4⋊C414yes—
287#22678—591nomaximal, Sylow
288C22×C59C2×C4×D411yescharacteristic
289C22×C59D4211yescharacteristic
290C22×C59D4×Q811yescharacteristic
291C22×C59C2×#13711yescharacteristic
292C22×C59#57511yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57711yescharacteristic
295C2×D59—24no—
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—42no—
299C2×D59—42no—
300C2×D59—42no—
301C2×D59—42no—
302C4×C59—24no—
303C4×C59—22no—
304C4×C59—22no—
305C4×C59—22no—
306C4×C59—42no—
307C4×C59—41no—
308C4×C59—41no—
309C4×C59—41no—
310C4×C59—81no—
311C4×C59—81no—
312C4×C59C22×C4⋊C411yescharacteristic
313C4×C59#52011yescharacteristic
314C4×C59#54711yescharacteristic
315C4×C59#55411yescharacteristic
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319C2×#440—591no—
320C2×#440—591no—
321C2×C4×SD16—591no—
322Q8×C4⋊C4D5911yescharacteristic, complemented
323#2349—591no—
324#1943—591no—
325#1942—591no—
326#1865—591no—
327#1864—591no—
328#1863—591no—
329#1862—591no—
330#1847—591no—
331#1805—591no—
332#1802—591no—
333#1771—591no—
334C2×C59D4×C4⋊C411yescharacteristic
335C2×C59C4×#13711yescharacteristic
336C2×C59#294611yescharacteristic
337C2×C59#326611yescharacteristic
338C2×C59#326811yescharacteristic
339C2×C59#338511yescharacteristic
340C2×C59#338811yescharacteristic
341D59—44no—
342C22×SD16—591no—
343C2×C4×Q8C2×D5911yescharacteristic
344C2×C4×Q8C2×D5911yescharacteristic
345C2×C4×Q8C2×D5911yescharacteristic
346C2×C4×D4—591no—
347C2×C4×D4—591no—
348C2×C4×D4—591no—
349#440—598no—
350#440—1184no—
351C4×SD16—1184no—
352C2×#107—591no—
353C2×#107—591no—
354C2×#107—591no—
355C2×#106—591no—
356C2×#106—591no—
357C2×C4⋊C8—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#102—591no—
362C2×#102—591no—
363C2×#102—591no—
364C2×#97—591no—
365C2×#97—591no—
366C2×C4×C8—591no—
367#389—21no—
368#389—21no—
369#389—591no—
370#389—591no—
371#389C2×D5911yescharacteristic
372#389C2×D5911yescharacteristic
373#385—1181no—
374#385—1181no—
375#384—21no—
376#384C2×D5911yescharacteristic
377#382—591no—
378#380—1181no—
379C4×C4⋊C4—21no—
380C4×C4⋊C4C2×D5911yescharacteristic
381C2×#95—1181no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#336—591no—
387#336—591no—
388C59#2267811yescharacteristic, complemented, Sylow
389C22×Q8C22×D5911yescharacteristic
390C22×D4—591no—
391C23×C4—1181no—
392C23×C4—1181no—
393C23×C4—1181no—
394C2×SD16—594no—
395C2×SD16—594no—
396C2×SD16—594no—
397C22×C8—591no—
398C22×C8—591no—
399C22×C8—591no—
400C22×C8—591no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—41no—
406C4×Q8C4×D5914yescomplemented
407C4×D4—594no—
408C4×D4—1182no—
409C4×D4—1182no—
410C4×D4—1182no—
411C4×D4—2361no—
412C4×D4—2361no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—21no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—591no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4C22⋊D5911yescharacteristic
426C2×C4⋊C4C22⋊D5911yescharacteristic
427C2×C4⋊C4C22×D5911yescharacteristic
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C22⋊C4—1181no—
431C2×C22⋊C4—1181no—
432C2×C22⋊C4—1181no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D5911yescharacteristic
437C2×C42C22×D5911yescharacteristic
438C2×C42C22×D5911yescharacteristic
439#107—594no—
440#107—1182no—
441#107—2361no—
442#106—1182no—
443#106—2361no—
444C4⋊C8—1182no—
445#103—594no—
446#103—1182no—
447#103—1182no—
448#102—594no—
449#102—1182no—
450#102—1182no—
451#97—594no—
452#97—1182no—
453C4×C8—1182no—
454#95—21no—
455#95—21no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460#95C22⋊D5911yescharacteristic
461#95C22⋊D5911yescharacteristic
462C24—1181no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×C4×D5912yes—
467C2×Q8D4×D5912yes—
468C2×Q8Q8×D5912yes—
469C2×D4—592no—
470C2×D4—592no—
471C2×D4—592no—
472C2×D4—1181no—
473C2×D4—1181no—
474C2×D4—1181no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—1181no—
480C22×C4—1184no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—4721no—
484C22×C4—4721no—
485C22×C4—4721no—
486C22×C4—4721no—
487C22×C4—2362no—
488C22×C4—2362no—
489C22×C4—2362no—
490C22×C4—2362no—
491C22×C4C2×C22⋊D5911yescharacteristic
492C22×C4C2×C22⋊D5911yescharacteristic
493C22×C4C2×C22⋊D5911yescharacteristic
494C22×C4C2×C22⋊D5911yescharacteristic
495C22×C4C2×C22⋊D5911yescharacteristic
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C23×D5911yescharacteristic
498SD16—5916no—
499C2×C8—592no—
500C2×C8—592no—
501C2×C8—592no—
502C2×C8—592no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—41no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4—592no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—1181no—
530C4⋊C4—1181no—
531C4⋊C4—4721no—
532C4⋊C4—4721no—
533C4⋊C4C2×C4×D5912yes—
534C22⋊C4—1184no—
535C22⋊C4—2362no—
536C22⋊C4—2362no—
537C42—24no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—42no—
544C42C2×C4×D5912yes—
545C42#1174568512yes—
546C23—1182no—
547C23—1182no—
548C23—1182no—
549C23—2361no—
550C23—2361no—
551C23—2361no—
552C23—2361no—
553C23C22×C22⋊D5911yescharacteristic
554Q8—22no—
555Q8—22no—
556Q8—22no—
557Q8C4⋊C4×D5914yes—
558D4—594no—
559D4—1182no—
560D4—1182no—
561D4—1182no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
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—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—1182no—
590C2×C4—4724no—
591C2×C4—4724no—
592C2×C4—4724no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4C4×C22⋊D5912yes—
602C2×C4C4×C22⋊D5912yes—
603C2×C4#1427375812yes—
604C2×C4C22×C4×D5911yescharacteristic
605C2×C4C2×D4×D5911yescharacteristic
606C2×C4C2×Q8×D5911yescharacteristic
607C2×C4C2×#1174568511yescharacteristic
608C2×C4#1427377211yescharacteristic
609C2×C4#1427377511yescharacteristic
610C8—594no—
611C8—1182no—
612C8—2361no—
613C8—2361no—
614C22—1184no—
615C22—2362no—
616C22—2362no—
617C22—2362no—
618C22—2362no—
619C22—2362no—
620C22—2362no—
621C22C2×C4×C22⋊D5911yescharacteristic
622C22D4×C22⋊D5911yescharacteristic
623C22Q8×C22⋊D5911yescharacteristic
624C22C2×#1427375811yescharacteristic
625C22#1683820011yescharacteristic
626C22#1683820511yescharacteristic
627C22#1683821011yescharacteristic
628C4—24no—
629C4—22no—
630C4—22no—
631C4—22no—
632C4—42no—
633C4—41no—
634C4—41no—
635C4—41no—
636C4—81no—
637C4—81no—
638C4—4721no—
639C4—4721no—
640C4—2362no—
641C4C2×C4⋊C4×D5911yescharacteristic
642C4#1683788311yescharacteristic
643C4#1683805311yescharacteristic
644C4#1683810711yescharacteristic
645C2—2364no—
646C2C4⋊C4×C22⋊D5911yescharacteristic
647C2C4×#1427375811yescharacteristic
648C2#2143292411yescharacteristic
649C2#2143560711yescharacteristic
650C2#2143563711yescharacteristic
651C2#2143700211yescharacteristic
652C2#2143703511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingQ8×C4⋊C4×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenterQ8×C4⋊C4D59
hypocenterC59#22678
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
5. #21424831C2semidirect
6. #21424826C2semidirect
7. #21424364C2semidirect
9. #21424350C2semidirect
11. #21424262C2semidirect
12. #21424111C2semidirect
14. #21423971C2semidirect
15. Q8×C4⋊C4×C59C2semidirect
21. #16837500C4semidirect
89. #14273660C2×C4semidirect
95. #14273651C2×C4semidirect
100. #14273638C2×C4semidirect
112. C4×Q8×C59C2×C4semidirect
195. #11745671C4⋊C4semidirect
322. Q8×C4⋊C4D59semidirect
388. C59#22678semidirect
406. C4×Q8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168375002
C2×C4×#117456712
#214270072
#214248312
#214248262
#214243642
#214243562
#214243502
#214243422
#214242622
#214241112
#214240932
#214239712
Q8×C4⋊C4×C592
#2267859