Finite Group Databasebeta

← #28605937

Subgroups — #28605937

Order 15104.

982 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837403C211yesmaximal, characteristic
2#21426975C211yesmaximal, characteristic, complemented
3#21426937C211yesmaximal, characteristic
4#21424790C211yesmaximal, characteristic, complemented
5#21424776C211yesmaximal, characteristic, complemented
6#21424726C211yesmaximal, characteristic, complemented
7#21424719C211yesmaximal, characteristic, complemented
8#21424362C211yesmaximal, characteristic, complemented
9#21424348C211yesmaximal, characteristic, complemented
10#21424183C211yesmaximal, characteristic, complemented
11#21424127C211yesmaximal, characteristic, complemented
12#21423928C211yesmaximal, characteristic, complemented
13#103×Dic59C211yesmaximal, characteristic, complemented
14#21423882C211yesmaximal, characteristic, complemented
15C59×#1788C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×Q8×Dic59C2211yescharacteristic
18C2×Q8×Dic59C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×#14273658C2211yescharacteristic
22#16837403C418yescomplemented
23C2×#14273651C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273648C2211yescharacteristic
26C2×#14273648C2211yescharacteristic
27C2×#14273637C2211yescharacteristic
28C2×C8×Dic59C2211yescharacteristic
29#16837305—21no—
30#16837305—21no—
31#16837305C2211yescharacteristic
32#16837305C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837287—21no—
36#16837287—21no—
37#16837262—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C22⋊C4×Dic59—21no—
43C4×C22⋊Dic59—21no—
44C4×C22⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×#41×C59C2211yescharacteristic
54C2×#103×C59C2211yescharacteristic
55C2×#103×C59C2211yescharacteristic
56C2×#102×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×C22⋊C8×C59C2211yescharacteristic
59C59×#380C2211yescharacteristic
60C23×Dic59—21no—
61C2×#11745664—41no—
62C2×#11745664—41no—
63C2×#11745664C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66Q8×Dic59—22no—
67Q8×Dic59—22no—
68Q8×Dic59—41no—
69Q8×Dic59—41no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2×C414yes—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×C22⋊Dic59—21no—
86C2×C22⋊Dic59D411yescharacteristic
87C2×C22⋊Dic59D411yescharacteristic
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59C2311yescharacteristic
93C2×C4×Dic59C2311yescharacteristic
94#14273660—41no—
95#14273658—41no—
96#14273651C2×C414yes—
97#14273651C2×C414yes—
98#14273648C2×C414yes—
99#14273648C2×C414yes—
100#14273637—41no—
101C8×Dic59—41no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95D411yescharacteristic
108C59⋊#95D411yescharacteristic
109C2×#41×C59—22no—
110C2×#41×C59—22no—
111C2×#41×C59—22no—
112C2×#41×C59—41no—
113C2×#41×C59—41no—
114C2×#41×C59C2×C414yes—
115C22×Q8×C59C2311yescharacteristic
116C22×D4×C59—21no—
117C22×D4×C59C2311yescharacteristic
118C23×C4×C59—21no—
119C23×C4×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C22×C8×C59C2311yescharacteristic
122C2×C4⋊C4×C59C2311yescharacteristic
123C2×C4⋊C4×C59C2311yescharacteristic
124C2×C22⋊C4×C59—21no—
125#103×C59—22no—
126#103×C59—22no—
127#102×C59—41no—
128#102×C59—41no—
129C22⋊C8×C59—22no—
130#95×C59—21no—
131C24×C59—21no—
132C24×C59C2×D411yescharacteristic
133#41×C59—28no—
134#41×C59—44no—
135#41×C59—44no—
136#41×C59—44no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59C22×C412yes—
141C2×Q8×C59#4112yes—
142C2×Q8×C59#4112yes—
143C2×D4×C59—24no—
144C2×D4×C59—21no—
145C2×D4×C59—21no—
146C2×D4×C59—21no—
147C2×D4×C59—21no—
148C2×D4×C59—21no—
149C2×D4×C59—42no—
150C2×D4×C59—42no—
151C2×D4×C59C22×C412yes—
152C22×C4×C59—21no—
153C22×C4×C59—21no—
154C22×C4×C59—21no—
155C22×C4×C59—21no—
156C22×C4×C59—21no—
157C22×C4×C59—24no—
158C22×C4×C59—21no—
159C22×C4×C59—21no—
160C22×C4×C59—42no—
161C22×C4×C59—42no—
162C22×C4×C59C22×C412yes—
163C22×C4×C59C2×D411yescharacteristic
164C22×C4×C59#4112yes—
165C22×C4×C59C2411yescharacteristic
166C22×Dic59—21no—
167C22×Dic59—21no—
168C22×Dic59—24no—
169C22×Dic59—21no—
170C22×Dic59—21no—
171C22×Dic59—81no—
172C22×Dic59—81no—
173C22×Dic59C2×D411yescharacteristic
174C22×Dic59C2×D411yescharacteristic
175C22×Dic59C2×D411yescharacteristic
176C22×Dic59C2×D411yescharacteristic
177C2×C8×C59—21no—
178C2×C8×C59—21no—
179C2×C8×C59—21no—
180C2×C8×C59—21no—
181C2×C8×C59—21no—
182C2×C8×C59—21no—
183C4⋊C4×C59—21no—
184C4⋊C4×C59—21no—
185C4⋊C4×C59—21no—
186C4⋊C4×C59—21no—
187C22⋊C4×C59—42no—
188C22⋊Dic59—22no—
189C22⋊Dic59—41no—
190C22⋊Dic59—42no—
191C22⋊Dic59—42no—
192C22⋊Dic59—81no—
193C22⋊Dic59—81no—
194C22⋊Dic59—81no—
195C4⋊Dic59—24no—
196C4⋊Dic59—21no—
197C4⋊Dic59—21no—
198C4⋊Dic59—21no—
199C4⋊Dic59—21no—
200C4⋊Dic59—21no—
201C4⋊Dic59—21no—
202C4⋊Dic59—42no—
203C4⋊Dic59—42no—
204C4⋊Dic59C22×C412yes—
205C4⋊Dic59C22×C412yes—
206Dic59⋊C4—81no—
207Dic59⋊C4—81no—
208C4×Dic59—21no—
209C4×Dic59—21no—
210C4×Dic59—21no—
211C4×Dic59—21no—
212C4×Dic59—42no—
213C4×Dic59—42no—
214C4×Dic59—81no—
215C4×Dic59—81no—
216#11745664—44no—
217C2×C59⋊C8—21no—
218C2×C59⋊C8—21no—
219C2×C59⋊C8—21no—
220C2×C59⋊C8—21no—
221C2×C59⋊C8C22×C412yes—
222C2×C59⋊C8C22×C412yes—
223C23×C59—21no—
224C23×C59—21no—
225C23×C59—21no—
226C23×C59—21no—
227C23×C59—21no—
228C23×C59—21no—
229C23×C59—22no—
230C23×C59—41no—
231C23×C59—41no—
232C23×C59—41no—
233C23×C59—41no—
234C23×C59—41no—
235C23×C59—41no—
236C23×C59C4×D412yes—
237C23×C59C22×D411yescharacteristic
238Q8×C59—22no—
239Q8×C59—22no—
240Q8×C59—22no—
241Q8×C59#11714yes—
242D4×C59—22no—
243D4×C59—22no—
244D4×C59—48no—
245D4×C59—41no—
246D4×C59—41no—
247C2×C4×C59—22no—
248C2×C4×C59—22no—
249C2×C4×C59—22no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—22no—
253C2×C4×C59—22no—
254C2×C4×C59—22no—
255C2×C4×C59—22no—
256C2×C4×C59—48no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—41no—
263C2×C4×C59C4×D412yes—
264C2×C4×C59C23×C411yescharacteristic
265C2×C4×C59C2×#4111yescharacteristic
266C2×C4×C59C2×#4111yescharacteristic
267C2×C4×C59C2×#4111yescharacteristic
268C2×C4×C59H(5,𝔽2)11yescharacteristic
269C2×C4×C59H(5,𝔽2)11yescharacteristic
270C2×Dic59—21no—
271C2×Dic59—21no—
272C2×Dic59—22no—
273C2×Dic59—22no—
274C2×Dic59—21no—
275C2×Dic59—21no—
276C2×Dic59—21no—
277C2×Dic59—21no—
278C2×Dic59—21no—
279C2×Dic59—21no—
280C2×Dic59—21no—
281C2×Dic59—21no—
282C2×Dic59—21no—
283C2×Dic59—41no—
284C2×Dic59—41no—
285C2×Dic59—41no—
286C2×Dic59—41no—
287C2×Dic59—41no—
288C2×Dic59—41no—
289C2×Dic59—41no—
290C2×Dic59—41no—
291C2×Dic59—41no—
292C2×Dic59—41no—
293C2×Dic59—84no—
294C2×Dic59#13712yes—
295C8×C59—41no—
296C8×C59—41no—
297C59⋊C8—22no—
298C59⋊C8—22no—
299#22579—591nomaximal, Sylow
300C22×C59—22no—
301C22×C59—22no—
302C22×C59—22no—
303C22×C59—41no—
304C22×C59—41no—
305C22×C59—41no—
306C22×C59—41no—
307C22×C59—42no—
308C22×C59—42no—
309C22×C59—42no—
310C22×C59—81no—
311C22×C59—81no—
312C22×C59—81no—
313C22×C59—81no—
314C22×C59C2×C4×D411yescharacteristic
315C22×C59#54611yescharacteristic
316C22×C59#54611yescharacteristic
317C22×C59C2×#13711yescharacteristic
318C22×C59#57511yescharacteristic
319C22×C59#57611yescharacteristic
320C22×C59#57711yescharacteristic
321C4×C59—22no—
322C4×C59—24no—
323C4×C59—81no—
324C4×C59—81no—
325C4×C59C2×#11711yescharacteristic
326C4×C59#51811yescharacteristic
327C4×C59#55111yescharacteristic
328C4×C59#55111yescharacteristic
329Dic59—22no—
330Dic59—41no—
331Dic59—42no—
332Dic59—42no—
333Dic59—41no—
334Dic59—41no—
335Dic59—81no—
336Dic59—81no—
337C2×#417—591no—
338#2345—591no—
339#2341—591no—
340#1936—591no—
341#1934—591no—
342#1926—591no—
343#1925—591no—
344#1865—591no—
345#1863—591no—
346#1823—591no—
347#1809—591no—
348#1788D5911yescharacteristic
349#1763—591no—
350C4×#103—591no—
351#1753—591no—
352C2×C59—42no—
353C2×C59—82no—
354C2×C59#234211yescharacteristic
355C2×C59C4×#13711yescharacteristic
356C2×C59#294611yescharacteristic
357C2×C59#330911yescharacteristic
358C2×C59#331611yescharacteristic
359C2×C59#332611yescharacteristic
360C2×C59#333311yescharacteristic
361C22×#41C2×D5911yescharacteristic
362C22×M16—591no—
363C2×C4×Q8—591no—
364C2×C4×Q8—591no—
365C22×C4⋊C4—591no—
366C2×#107—591no—
367C2×#106—591no—
368#417—598no—
369C2×#103—591no—
370C2×#103—591no—
371C2×#103C2×D5911yescharacteristic
372C2×#103C2×D5911yescharacteristic
373C2×#102—591no—
374C2×#102—591no—
375C2×#102C2×D5911yescharacteristic
376C2×#102C2×D5911yescharacteristic
377C2×C22⋊C8C2×D5911yescharacteristic
378C2×#97—591no—
379C2×C4×C8—591no—
380#390—591no—
381#390—591no—
382#390—1181no—
383#390—1181no—
384#389—591no—
385#389—591no—
386#387—1181no—
387#387—1181no—
388#380—591no—
389#380—1181no—
390#380—1181no—
391#380C2×D5911yescharacteristic
392C4×C4⋊C4—591no—
393C4×C4⋊C4—591no—
394C4×C22⋊C4—591no—
395C4×C22⋊C4—1181no—
396C4×C22⋊C4—1181no—
397#340—591no—
398#340—591no—
399#340—591no—
400#340—591no—
401#340—591no—
402#340—591no—
403#336—591no—
404#336—591no—
405C59#2257911yescharacteristic, complemented, Sylow
406C2×#41—22no—
407C2×#41—22no—
408C2×#41—22no—
409C2×#41—41no—
410C2×#41—41no—
411C2×#41C4×D5914yes—
412C22×Q8C22×D5911yescharacteristic
413C22×D4—21no—
414C22×D4C22×D5911yescharacteristic
415C23×C4—21no—
416C23×C4—1181no—
417C23×C4C22×D5911yescharacteristic
418C2×M16—594no—
419C2×M16—2361no—
420C2×M16—2361no—
421C22×C8—591no—
422C22×C8—591no—
423C22×C8C22×D5911yescharacteristic
424C22×C8C22×D5911yescharacteristic
425C4×Q8—1182no—
426C4×Q8—1182no—
427C4×Q8—2361no—
428C4×Q8—2361no—
429C2×C4⋊C4—594no—
430C2×C4⋊C4—591no—
431C2×C4⋊C4—591no—
432C2×C4⋊C4—591no—
433C2×C4⋊C4—591no—
434C2×C4⋊C4—1181no—
435C2×C4⋊C4—1182no—
436C2×C4⋊C4—1181no—
437C2×C4⋊C4—1181no—
438C2×C4⋊C4—1181no—
439C2×C4⋊C4—1181no—
440C2×C4⋊C4C22×D5911yescharacteristic
441C2×C4⋊C4C22×D5911yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—591no—
444C2×C22⋊C4—591no—
445C2×C22⋊C4—1181no—
446C2×C22⋊C4—1181no—
447C2×C22⋊C4—1181no—
448C2×C22⋊C4—1181no—
449C2×C22⋊C4—1181no—
450C2×C42—591no—
451C2×C42—591no—
452C2×C42—1181no—
453C2×C42—1181no—
454C2×C42—1181no—
455C2×C42—1181no—
456#107—2361no—
457#106—2361no—
458#103—22no—
459#103—22no—
460#103—594no—
461#103—594no—
462#102—41no—
463#102—41no—
464#102—594no—
465#102—594no—
466C22⋊C8—22no—
467#97—2361no—
468C4×C8—2361no—
469#95—21no—
470#95—591no—
471#95—591no—
472#95—1181no—
473#95—1181no—
474#95—1181no—
475#95—1181no—
476#95—1181no—
477C24—21no—
478C24D4×D5911yescharacteristic
479#41—28no—
480#41—44no—
481#41—44no—
482#41—44no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8C2×C4×D5912yes—
487C2×Q8#1174568712yes—
488C2×Q8#1174568712yes—
489C2×D4—24no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—42no—
496C2×D4—42no—
497C2×D4C2×C4×D5912yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—24no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—42no—
507C22×C4—42no—
508C22×C4—591no—
509C22×C4—591no—
510C22×C4—591no—
511C22×C4—591no—
512C22×C4—1184no—
513C22×C4—1181no—
514C22×C4—1181no—
515C22×C4—1181no—
516C22×C4—1181no—
517C22×C4—4721no—
518C22×C4—4721no—
519C22×C4C2×C4×D5912yes—
520C22×C4D4×D5911yescharacteristic
521C22×C4#1174568712yes—
522C22×C4C23×D5911yescharacteristic
523M16—2364no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—592no—
531C2×C8—592no—
532C2×C8—1181no—
533C2×C8—1181no—
534C2×C8—1181no—
535C2×C8—1181no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—592no—
541C4⋊C4—592no—
542C4⋊C4—1184no—
543C4⋊C4—1181no—
544C4⋊C4—1181no—
545C4⋊C4—1181no—
546C4⋊C4—1181no—
547C4⋊C4—1181no—
548C4⋊C4—1181no—
549C4⋊C4—4721no—
550C4⋊C4—4721no—
551C4⋊C4—2362no—
552C4⋊C4—2362no—
553C22⋊C4—42no—
554C22⋊C4—1182no—
555C22⋊C4—4721no—
556C22⋊C4—4721no—
557C22⋊C4—4721no—
558C22⋊C4—2362no—
559C22⋊C4—2361no—
560C22⋊C4—2362no—
561C42—1181no—
562C42—1181no—
563C42—1181no—
564C42—1181no—
565C42—4721no—
566C42—4721no—
567C42—2362no—
568C42—2362no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23(C22⋊D59)⋊C412yes—
583C23C2×D4×D5911yescharacteristic
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8#1427368314yes—
588D4—22no—
589D4—22no—
590D4—48no—
591D4—41no—
592D4—41no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—48no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—592no—
610C2×C4—1181no—
611C2×C4—1181no—
612C2×C4—1181no—
613C2×C4—1181no—
614C2×C4—1181no—
615C2×C4—1181no—
616C2×C4—1181no—
617C2×C4—1181no—
618C2×C4—1181no—
619C2×C4—1181no—
620C2×C4—1181no—
621C2×C4—1182no—
622C2×C4—1182no—
623C2×C4—4724no—
624C2×C4—2361no—
625C2×C4—2361no—
626C2×C4—2361no—
627C2×C4—2361no—
628C2×C4—2361no—
629C2×C4—2361no—
630C2×C4—2361no—
631C2×C4—2361no—
632C2×C4—2361no—
633C2×C4—2361no—
634C2×C4(C22⋊D59)⋊C412yes—
635C2×C4C22×C4×D5911yescharacteristic
636C2×C4C2×#1174568711yescharacteristic
637C2×C4C2×#1174568711yescharacteristic
638C2×C4C2×#1174568711yescharacteristic
639C2×C4#1427377211yescharacteristic
640C2×C4#1427377211yescharacteristic
641C8—41no—
642C8—41no—
643C8—1182no—
644C8—1182no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—42no—
653C22—42no—
654C22—42no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22C2×(C22⋊D59)⋊C411yescharacteristic
660C22#1683803911yescharacteristic
661C22#1683803911yescharacteristic
662C22#137×D5911yescharacteristic
663C22#1683819811yescharacteristic
664C22#1683820411yescharacteristic
665C22#1683820911yescharacteristic
666C4—22no—
667C4—24no—
668C4—81no—
669C4—81no—
670C4—1182no—
671C4—4721no—
672C4—4721no—
673C4—2361no—
674C4—2362no—
675C4—2362no—
676C4—2361no—
677C4—2361no—
678C4C2×#1427368311yescharacteristic
679C4#1683787511yescharacteristic
680C4#1683808311yescharacteristic
681C4#1683808311yescharacteristic
682C2—42no—
683C2—82no—
684C2#2142694711yescharacteristic
685C2#2143292011yescharacteristic
686C2#2143293111yescharacteristic
687C2#2143609511yescharacteristic
688C2#2143620011yescharacteristic
689C2#2143634211yescharacteristic
690C2#2143642711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1788C2
socleC23×C59C22×D4
prime radicalC22×#41×C59C22
cosocleC22×C4×C59C24
hypercenter#1788D59
hypocenterC59#22579
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1788

Complements

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

NH ≅ G/Ntype
2. #21426975C2semidirect
4. #21424790C2semidirect
5. #21424776C2semidirect
6. #21424726C2semidirect
7. #21424719C2semidirect
8. #21424362C2semidirect
9. #21424348C2semidirect
10. #21424183C2semidirect
11. #21424127C2semidirect
12. #21423928C2semidirect
13. #103×Dic59C2semidirect
14. #21423882C2semidirect
22. #16837403C4semidirect
405. C59#22579semidirect

Maximal subgroups

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

subgroupindex
C2×#168374032
#214269752
#214269372
#214247902
#214247762
#214247262
#214247192
#214243622
#214243482
#214241832
#214241272
#214239282
#103×Dic592
#214238822
C59×#17882
#2257959