Finite Group Databasebeta

← #28594354

Subgroups — #28594354

Order 15104.

1520 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837541C211yesmaximal, characteristic
2C2×#16837500C212yesmaximal, complemented
3C4×#14273701C211yesmaximal, characteristic
4#21424451C212yesmaximal, complemented
5#21424413C211yesmaximal, characteristic
6#21424350C212yesmaximal, complemented
7#21424284C211yesmaximal, characteristic, complemented
8#21424031C211yesmaximal, characteristic, complemented
9#21423926C212yesmaximal, complemented
10#21423858C211yesmaximal, characteristic, complemented
11C22×C4×Q8×C59C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C2×#14273701C2211yescharacteristic
14C2×C4×D236C2211yescharacteristic
15C2×C4×C22⋊D59—21no—
16#16837541C418yescomplemented
17#16837500—28no—
18C2×#14273660C2212yes—
19C2×#14273651C2212yes—
20C2×#14273651C2212yes—
21C2×#14273651C2212yes—
22C2×#14273647C2212yes—
23C2×C22⋊(C59⋊C8)C2211yescharacteristic
24C2×C22⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273638C2212yes—
26#16837304C2211yescharacteristic
27#16837300C2211yescharacteristic
28#16837293—21no—
29#16837281—21no—
30#16837262C2211yescharacteristic
31C4×C4⋊Dic59C2211yescharacteristic
32C4×(C2×D59)⋊C4—21no—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C23×Q8×C59C2211yescharacteristic
37C2×C4×Q8×C59—28no—
38C2×C4×Q8×C59—21no—
39C2×C4×Q8×C59—21no—
40C2×C4×Q8×C59C2212yes—
41C2×C4×Q8×C59C2212yes—
42C22×C4⋊C4×C59—21no—
43C22×C4⋊C4×C59C2211yescharacteristic
44C22×C42×C59—21no—
45C22×C42×C59C2211yescharacteristic
46C22×D236C2311yescharacteristic
47C22×C22⋊D59—21no—
48C22×C4×D59—21no—
49C2×#11745671—24no—
50C2×#11745671—24no—
51C2×#11745671C2×C418yescomplemented
52C22×C59⋊C8C2312yes—
53C22×C59⋊C8C2312yes—
54#14273701—22no—
55#14273701C2×C414yes—
56C4×D236—24no—
57C4×C22⋊D59—44no—
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×C4⋊Dic59C2311yescharacteristic
61C2×Dic59⋊C4—21no—
62C2×C22⋊Dic59—21no—
63C2×(C2×D59)⋊C4—21no—
64C2×(C2×D59)⋊C4—21no—
65C2×(C2×D59)⋊C4—21no—
66C2×(C2×D59)⋊C4—21no—
67C2×C4×Dic59—21no—
68#14273660—24no—
69#14273651—24no—
70#14273651—24no—
71#14273651C2×C418yescomplemented
72#14273647—24no—
73C22⋊(C59⋊C8)—22no—
74C22⋊(C59⋊C8)C2×C414yescomplemented
75#14273638—24no—
76C59⋊#95—21no—
77C59⋊#95—21no—
78C59⋊#95—21no—
79C22×Q8×C59—24no—
80C22×Q8×C59—24no—
81C22×Q8×C59—21no—
82C22×Q8×C59—22no—
83C22×Q8×C59C2×C414yescomplemented
84C22×Q8×C59C2312yes—
85C23×C4×C59—21no—
86C23×C4×C59—21no—
87C23×C4×C59D411yescharacteristic
88C23×C4×C59D411yescharacteristic
89C23×C4×C59C2311yescharacteristic
90C4×Q8×C59—28no—
91C4×Q8×C59—28no—
92C4×Q8×C59—24no—
93C4×Q8×C59—24no—
94C4×Q8×C59—24no—
95C4×Q8×C59D418yescomplemented
96C2×C4⋊C4×C59—28no—
97C2×C4⋊C4×C59—22no—
98C2×C4⋊C4×C59—22no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—24no—
101C2×C4⋊C4×C59C2312yes—
102C2×C42×C59—28no—
103C2×C42×C59—22no—
104C2×C42×C59—22no—
105C2×C42×C59—24no—
106C2×C42×C59D411yescharacteristic
107C2×C42×C59D411yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C24×C59C2×D411yescharacteristic
111C23×D59—21no—
112C2×Q8×C59—24no—
113C2×Q8×C59—216no—
114C2×Q8×C59—24no—
115C2×Q8×C59—216no—
116C2×Q8×C59—24no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C2×Q8×C59—22no—
120C2×Q8×C59C22×C414yes—
121C2×Q8×C59C2×D414yes—
122C2×Q8×C59#4114yes—
123C22×C4×C59—24no—
124C22×C4×C59—24no—
125C22×C4×C59—24no—
126C22×C4×C59—22no—
127C22×C4×C59—22no—
128C22×C4×C59—22no—
129C22×C4×C59—22no—
130C22×C4×C59—22no—
131C22×C4×C59—22no—
132C22×C4×C59—24no—
133C22×C4×C59—22no—
134C22×C4×C59—22no—
135C22×C4×C59—22no—
136C22×C4×C59—41no—
137C22×C4×C59—42no—
138C22×C4×C59—42no—
139C22×C4×C59C22×C412yes—
140C22×C4×C59C2×D411yescharacteristic
141C22×C4×C59C2×D411yescharacteristic
142C22×C4×C59C2×D411yescharacteristic
143C22×C4×C59C2×D411yescharacteristic
144C22×C4×C59C2×D411yescharacteristic
145C22×C4×C59#4112yes—
146C22×C4×C59C2411yescharacteristic
147C2×C22⋊D59—24no—
148C2×C22⋊D59—42no—
149C2×C22⋊D59—42no—
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C2×D236—22no—
154C2×D236—22no—
155C2×D236—21no—
156C2×D236C22×C412yes—
157C2×C4×D59—42no—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C4⋊C4×C59—28no—
161C4⋊C4×C59—24no—
162C4⋊C4×C59—28no—
163C4⋊C4×C59—22no—
164C4⋊C4×C59C2×D414yes—
165C42×C59—28no—
166C42×C59—28no—
167C42×C59—22no—
168C42×C59—22no—
169C42×C59—22no—
170C42×C59C2×D414yes—
171C22⋊Dic59—42no—
172#11745671—216no—
173(C2×D59)⋊C4—24no—
174(C2×D59)⋊C4—42no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177C4⋊Dic59—21no—
178C4⋊Dic59—22no—
179C4⋊Dic59—22no—
180C4⋊Dic59C22×C412yes—
181Dic59⋊C4—42no—
182C4×Dic59—42no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C2×C59⋊C8—22no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8C22×C414yes—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59—21no—
193C23×C59C4×D412yes—
194C23×C59AGL(1,ℤ/8)12yes—
195C23×C59#13712yes—
196C23×C59C22×D411yescharacteristic
197C22×D59—22no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
201C22×D59—41no—
202C22×D59—41no—
203C22×D59—41no—
204Q8×C59—216no—
205Q8×C59—24no—
206Q8×C59—24no—
207Q8×C59—24no—
208Q8×C59C4×D418yescomplemented
209C2×C4×C59—28no—
210C2×C4×C59—216no—
211C2×C4×C59—24no—
212C2×C4×C59—21no—
213C2×C4×C59—24no—
214C2×C4×C59—21no—
215C2×C4×C59—24no—
216C2×C4×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—24no—
223C2×C4×C59—24no—
224C2×C4×C59—21no—
225C2×C4×C59—24no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—44no—
229C2×C4×C59—48no—
230C2×C4×C59—42no—
231C2×C4×C59—42no—
232C2×C4×C59C4×D412yes—
233C2×C4×C59#12012yes—
234C2×C4×C59#12012yes—
235C2×C4×C59#13712yes—
236C2×C4×C59C23×C411yescharacteristic
237C2×C4×C59C22×D412yes—
238C2×C4×C59C2×#4112yes—
239C2×C4×C59C2×#4111yescharacteristic
240C22⋊D59—48no—
241C2×Dic59—22no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250D236—24no—
251D236—42no—
252C4×D59—84no—
253C59⋊C8—24no—
254C59⋊C8—42no—
255#21454—591nomaximal, Sylow
256C22×C59—22no—
257C22×C59—22no—
258C22×C59—22no—
259C22×C59—22no—
260C22×C59—22no—
261C22×C59—22no—
262C22×C59#44014yescomplemented
263C22×C59C2×C4×D411yescharacteristic
264C22×C59C2×#12011yescharacteristic
265C22×C59#52511yescharacteristic
266C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C59C2×#13711yescharacteristic
268C22×C59C2×#13711yescharacteristic
269C22×C59#57511yescharacteristic
270C2×D59—42no—
271C2×D59—42no—
272C2×D59—42no—
273C2×D59—81no—
274C2×D59—81no—
275C2×D59—81no—
276C2×D59—81no—
277C4×C59—22no—
278C4×C59—22no—
279C4×C59—22no—
280C4×C59—28no—
281C4×C59—22no—
282C4×C59—41no—
283C4×C59—44no—
284C4×C59C2×C4×D412yes—
285C4×C59#52512yes—
286Dic59—42no—
287Dic59—81no—
288Dic59—81no—
289C22×C4×Q8D5911yescharacteristic, complemented
290C2×#448—591no—
291C2×#440—592no—
292C4×#121—591no—
293#1880—592no—
294#1874—591no—
295#1863—592no—
296#1852—591no—
297#1787—591no—
298#1763—592no—
299#1749—591no—
300C2×C59—24no—
301C2×C59C4×#12011yescharacteristic
302C2×C59C2×#44011yescharacteristic
303C2×C59C4×#13711yescharacteristic
304C2×C59#300411yescharacteristic
305C2×C59#300611yescharacteristic
306C2×C59#311211yescharacteristic
307C2×C59#311711yescharacteristic
308D59—82no—
309C23×Q8C2×D5911yescharacteristic
310C22×SD16—592no—
311C2×#121—591no—
312C2×C4×Q8—28no—
313C2×C4×Q8—21no—
314C2×C4×Q8—21no—
315C2×C4×Q8C2×D5912yes—
316C2×C4×Q8C2×D5912yes—
317C2×C4×D4—591no—
318C2×C4×D4—1181no—
319C22×C4⋊C4—21no—
320C22×C4⋊C4C2×D5911yescharacteristic
321C22×C42—21no—
322C22×C42C2×D5911yescharacteristic
323#448—598no—
324#440—1188no—
325C2×#107—592no—
326C2×#103—592no—
327C2×#103—592no—
328C2×#103—592no—
329C2×#102—592no—
330C2×C22⋊C8—591no—
331C2×C22⋊C8—591no—
332C2×#97—592no—
333#390—591no—
334#389—591no—
335#388—1181no—
336#386—1181no—
337#380—591no—
338C4×C4⋊C4—591no—
339C4×C22⋊C4—1181no—
340#340—592no—
341#340—592no—
342#336—592no—
343C59#2145411yescharacteristic, complemented, Sylow
344C22×Q8—24no—
345C22×Q8—24no—
346C22×Q8—22no—
347C22×Q8—21no—
348C22×Q8C4×D5914yescomplemented
349C22×Q8C22×D5912yes—
350C22×D4—591no—
351C22×D4—1181no—
352C23×C4—21no—
353C23×C4—21no—
354C23×C4—1181no—
355C23×C4C22⋊D5911yescharacteristic
356C23×C4C22⋊D5911yescharacteristic
357C23×C4C22×D5911yescharacteristic
358C2×SD16—598no—
359C2×SD16—1184no—
360C2×SD16—1184no—
361C22×C8—592no—
362C22×C8—592no—
363#121—594no—
364#121—1182no—
365C4×Q8—28no—
366C4×Q8—28no—
367C4×Q8—24no—
368C4×Q8—24no—
369C4×Q8—24no—
370C4×Q8C22⋊D5918yescomplemented
371C4×D4—1184no—
372C4×D4—2364no—
373C2×C4⋊C4—28no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—24no—
378C2×C4⋊C4—591no—
379C2×C4⋊C4—591no—
380C2×C4⋊C4—591no—
381C2×C4⋊C4—1181no—
382C2×C4⋊C4C22×D5912yes—
383C2×C22⋊C4—1181no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C42—28no—
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—24no—
392C2×C42—1181no—
393C2×C42C22⋊D5911yescharacteristic
394C2×C42C22⋊D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396C2×C42C22×D5911yescharacteristic
397#107—1184no—
398#103—598no—
399#103—1184no—
400#103—1184no—
401#102—1184no—
402C22⋊C8—594no—
403C22⋊C8—1182no—
404#97—1184no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408C24—1181no—
409C24C2×C22⋊D5911yescharacteristic
410C2×Q8—24no—
411C2×Q8—216no—
412C2×Q8—24no—
413C2×Q8—216no—
414C2×Q8—24no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8C2×C4×D5914yes—
419C2×Q8#1174568514yes—
420C2×Q8C2×C22⋊D5914yes—
421C2×D4—592no—
422C2×D4—1184no—
423C2×D4—1182no—
424C2×D4—1182no—
425C2×D4—1181no—
426C2×D4—2362no—
427C2×D4—2362no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—41no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—1181no—
445C22×C4—1181no—
446C22×C4—1181no—
447C22×C4—2362no—
448C22×C4—2362no—
449C22×C4—2362no—
450C22×C4C2×C4×D5912yes—
451C22×C4#1174568512yes—
452C22×C4C2×C22⋊D5911yescharacteristic
453C22×C4C2×C22⋊D5911yescharacteristic
454C22×C4C2×C22⋊D5911yescharacteristic
455C22×C4C2×C22⋊D5911yescharacteristic
456C22×C4C2×C22⋊D5911yescharacteristic
457C22×C4C23×D5911yescharacteristic
458SD16—11816no—
459C2×C8—594no—
460C2×C8—1182no—
461C2×C8—1182no—
462C2×C8—1182no—
463C2×C8—1182no—
464C2×C8—1182no—
465C4⋊C4—28no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—22no—
469C4⋊C4—592no—
470C4⋊C4—1181no—
471C4⋊C4—1182no—
472C4⋊C4—1182no—
473C4⋊C4—2362no—
474C4⋊C4C2×C22⋊D5914yes—
475C22⋊C4—1184no—
476C22⋊C4—2362no—
477C22⋊C4—2362no—
478C22⋊C4—2362no—
479C22⋊C4—2362no—
480C42—28no—
481C42—28no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—2362no—
486C42C2×C22⋊D5914yes—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—1182no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23—2361no—
496C23—2361no—
497C23—2361no—
498C23C4×C22⋊D5912yes—
499C23#1427375212yes—
500C23#1427375812yes—
501C23C22×C22⋊D5911yescharacteristic
502Q8—216no—
503Q8—24no—
504Q8—24no—
505Q8—24no—
506Q8C4×C22⋊D5918yescomplemented
507D4—1184no—
508D4—2368no—
509D4—2362no—
510C2×C4—28no—
511C2×C4—216no—
512C2×C4—21no—
513C2×C4—24no—
514C2×C4—21no—
515C2×C4—24no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—21no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—44no—
530C2×C4—48no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—1182no—
534C2×C4—4724no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4—2361no—
538C2×C4—2361no—
539C2×C4—2361no—
540C2×C4—2361no—
541C2×C4—2361no—
542C2×C4—2361no—
543C2×C4C4×C22⋊D5912yes—
544C2×C4#1427369612yes—
545C2×C4#1427369612yes—
546C2×C4#1427375812yes—
547C2×C4C22×C4×D5911yescharacteristic
548C2×C4C22×C22⋊D5912yes—
549C2×C4C2×#1174568511yescharacteristic
550C2×C4C2×#1174568512yes—
551C8—1184no—
552C8—2362no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—4721no—
560C22—4721no—
561C22—4721no—
562C22—4721no—
563C22—2362no—
564C22—2362no—
565C22—2362no—
566C22#1683750014yescomplemented
567C22C2×C4×C22⋊D5911yescharacteristic
568C22C2×#1427369611yescharacteristic
569C22#1683791311yescharacteristic
570C22C2×#1427375211yescharacteristic
571C22C2×#1427375811yescharacteristic
572C22C2×#1427375811yescharacteristic
573C22#1683820011yescharacteristic
574C4—22no—
575C4—28no—
576C4—22no—
577C4—22no—
578C4—22no—
579C4—41no—
580C4—44no—
581C4—4721no—
582C4—4721no—
583C4—2362no—
584C4C2×C4×C22⋊D5912yes—
585C4#1683791312yes—
586C2—24no—
587C2—4722no—
588C2C4×#1427369611yescharacteristic
589C2C2×#1683750011yescharacteristic
590C2C4×#1427375811yescharacteristic
591C2#2143333011yescharacteristic
592C2#2143334811yescharacteristic
593C2#2143421811yescharacteristic
594C2#2143426711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC22×C4×Q8×C59C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenterC22×C4×Q8D59
hypocenterC59#21454
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#16837500C2semidirect
4. #21424451C2semidirect
6. #21424350C2semidirect
7. #21424284C2semidirect
8. #21424031C2semidirect
9. #21423926C2semidirect
10. #21423858C2semidirect
11. C22×C4×Q8×C59C2semidirect
16. #16837541C4semidirect
51. C2×#11745671C2×C4semidirect
71. #14273651C2×C4semidirect
74. C22⋊(C59⋊C8)C2×C4semidirect
83. C22×Q8×C59C2×C4semidirect
95. C4×Q8×C59D4semidirect
208. Q8×C59C4×D4semidirect
262. C22×C59#440semidirect
289. C22×C4×Q8D59semidirect
343. C59#21454semidirect
348. C22×Q8C4×D59semidirect
370. C4×Q8C22⋊D59semidirect
506. Q8C4×C22⋊D59semidirect
566. C22#16837500semidirect

Maximal subgroups

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

subgroupindex
C2×#168375412
C2×#168375002
C4×#142737012
#214244512
#214244132
#214243502
#214242842
#214240312
#214239262
#214238582
C22×C4×Q8×C592
#2145459