Finite Group Databasebeta

← #28629356

Subgroups — #28629356

Order 15104.

960 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837604C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic, complemented
3C4×#14273701C211yesmaximal, characteristic
4#21425296C212yesmaximal, complemented
5#21425097C212yesmaximal
6#21425020C211yesmaximal, characteristic, complemented
7#21424980C212yesmaximal, complemented
8#21424852C211yesmaximal, characteristic, complemented
9#21424789C212yesmaximal, complemented
10#21424093C211yesmaximal, characteristic, complemented
11C59×#2857C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C2×#14273701C2211yescharacteristic
14C2×C4×D236C2211yescharacteristic
15C2×C4×C22⋊D59—21no—
16#16837604—24no—
17#16837459—24no—
18C2×#14273660C2212yes—
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2212yes—
22C2×#14273651C2212yes—
23C2×#14273651C2212yes—
24C2×#14273647C2212yes—
25C2×C22⋊(C59⋊C8)C2212yes—
26C2×#14273638C2212yes—
27#16837304C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837293—21no—
30#16837281—21no—
31#16837262C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×(C2×D59)⋊C4—21no—
34#16837164C2212yes—
35#16837164C2212yes—
36C2×C4⋊Q8×C59C2211yescharacteristic
37C2×C22⋊Q8×C59C2212yes—
38C22×C42×C59C2211yescharacteristic
39C59×#391C2212yes—
40C59×#389—21no—
41C59×#389—21no—
42C59×#384C2211yescharacteristic
43C59×#380—21no—
44C59×#380—21no—
45C22×D236C2311yescharacteristic
46C22×C22⋊D59—21no—
47C22×C4×D59—21no—
48C2×#11745671—24no—
49C2×#11745671—24no—
50C2×#11745671—42no—
51C22×C59⋊C8C2312yes—
52C22×C59⋊C8C2312yes—
53#14273701—22no—
54#14273701—22no—
55C4×D236—22no—
56C4×D236—22no—
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—42no—
69C4⋊(C59⋊C8)—22no—
70C4⋊(C59⋊C8)D414yescomplemented
71#14273651—24no—
72#14273651—24no—
73#14273651—42no—
74#14273647—42no—
75C22⋊(C59⋊C8)—24no—
76#14273638—24no—
77C59⋊#95—21no—
78C59⋊#95—21no—
79C59⋊#95—21no—
80C22×Q8×C59C2312yes—
81C23×C4×C59D411yescharacteristic
82C23×C4×C59D411yescharacteristic
83C23×C4×C59C2311yescharacteristic
84C4⋊Q8×C59—22no—
85C4⋊Q8×C59—22no—
86C22⋊Q8×C59—42no—
87C22⋊Q8×C59—42no—
88C2×C4⋊C4×C59—21no—
89C2×C4⋊C4×C59—21no—
90C2×C4⋊C4×C59—21no—
91C2×C4⋊C4×C59—21no—
92C2×C4⋊C4×C59—22no—
93C2×C4⋊C4×C59C2312yes—
94C2×C22⋊C4×C59—22no—
95C2×C42×C59—22no—
96C2×C42×C59—22no—
97C2×C42×C59D411yescharacteristic
98C2×C42×C59D411yescharacteristic
99C2×C42×C59C2311yescharacteristic
100C2×C42×C59C2311yescharacteristic
101#95×C59—22no—
102#95×C59—22no—
103C24×C59C2×D411yescharacteristic
104C23×D59—21no—
105C2×Q8×C59—22no—
106C2×Q8×C59—22no—
107C2×Q8×C59—22no—
108C2×Q8×C59—22no—
109C2×Q8×C59—22no—
110C2×Q8×C59—22no—
111C22×C4×C59—22no—
112C22×C4×C59—22no—
113C22×C4×C59—22no—
114C22×C4×C59—22no—
115C22×C4×C59—22no—
116C22×C4×C59—21no—
117C22×C4×C59—21no—
118C22×C4×C59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—41no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C2×C22⋊D59—24no—
132C2×C22⋊D59—42no—
133C2×C22⋊D59—42no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236#4112yes—
143C2×C4×D59—42no—
144C2×C4×D59—42no—
145C2×C4×D59—42no—
146C4⋊C4×C59—22no—
147C4⋊C4×C59—22no—
148C4⋊C4×C59—42no—
149C4⋊C4×C59—44no—
150C4⋊C4×C59—42no—
151C4⋊C4×C59—42no—
152C4⋊C4×C59—42no—
153C22⋊C4×C59—44no—
154C42×C59—22no—
155C42×C59—22no—
156C42×C59—21no—
157C42×C59—21no—
158C42×C59C2×D412yes—
159C42×C59#4112yes—
160C22⋊Dic59—42no—
161#11745671—48no—
162(C2×D59)⋊C4—24no—
163(C2×D59)⋊C4—42no—
164(C2×D59)⋊C4—42no—
165(C2×D59)⋊C4—42no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59#4112yes—
172Dic59⋊C4—42no—
173C4×Dic59—42no—
174C2×C59⋊C8—22no—
175C2×C59⋊C8—22no—
176C2×C59⋊C8—22no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8C2×D414yes—
180C23×C59—21no—
181C23×C59—21no—
182C23×C59—21no—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59#13712yes—
187C23×C59C22×D411yescharacteristic
188C22×D59—22no—
189C22×D59—41no—
190C22×D59—41no—
191C22×D59—41no—
192C22×D59—41no—
193C22×D59—41no—
194C22×D59—41no—
195Q8×C59—42no—
196Q8×C59—42no—
197Q8×C59—42no—
198Q8×C59—42no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—44no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59—41no—
215C2×C4×C59—44no—
216C2×C4×C59—42no—
217C2×C4×C59—42no—
218C2×C4×C59—42no—
219C2×C4×C59—42no—
220C2×C4×C59—42no—
221C2×C4×C59—42no—
222C2×C4×C59#12112yes—
223C2×C4×C59#12112yes—
224C2×C4×C59AGL(1,ℤ/8)12yes—
225C2×C4×C59#13712yes—
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C2×#4111yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59H(5,𝔽2)11yescharacteristic
231C2×C4×C59H(5,𝔽2)11yescharacteristic
232C22⋊D59—48no—
233C2×Dic59—22no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242D236—22no—
243D236—22no—
244D236—41no—
245D236—41no—
246C4×D59—84no—
247C59⋊C8—24no—
248C59⋊C8—42no—
249#24659—591nomaximal, Sylow
250C22×C59—22no—
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—41no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59C2×#12111yescharacteristic
258C22×C59#52511yescharacteristic
259C22×C59#53511yescharacteristic
260C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
261C22×C59C2×#13711yescharacteristic
262C22×C59C2×#13711yescharacteristic
263C22×C59#57511yescharacteristic
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—42no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C2×D59—81no—
271C4×C59—22no—
272C4×C59—41no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—82no—
276C4×C59—82no—
277C4×C59#46814yes—
278C4×C59#54611yescharacteristic
279C4×C59#55011yescharacteristic
280C4×C59#55311yescharacteristic
281C4×C59#55511yescharacteristic
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285C2×#461—591no—
286C2×#432—591no—
287#2857D5911yescharacteristic, complemented
288C4×#121—591no—
289#2036—592no—
290#2000—592no—
291#1981—591no—
292#1973—592no—
293#1947—591no—
294#1935—592no—
295#1802—591no—
296C2×C59—42no—
297C2×C59#241411yescharacteristic
298C2×C59C2×#46811yescharacteristic
299C2×C59#305611yescharacteristic
300C2×C59#311011yescharacteristic
301C2×C59#311611yescharacteristic
302C2×C59#321411yescharacteristic
303C2×C59#321811yescharacteristic
304D59—82no—
305C22×SD16—592no—
306C2×C4⋊Q8C2×D5911yescharacteristic
307C2×C22⋊Q8C2×D5912yes—
308C2×#121—591no—
309C2×C4×D4—591no—
310C2×C4×D4—1181no—
311C22×C42C2×D5911yescharacteristic
312#461—1184no—
313#432—1184no—
314C2×#107—592no—
315C2×C4⋊C8—591no—
316C2×C4⋊C8—591no—
317C2×#103—592no—
318C2×#103—592no—
319C2×#103—592no—
320C2×#102—592no—
321C2×C22⋊C8—592no—
322C2×#97—592no—
323#391C2×D5912yes—
324#390—591no—
325#389—21no—
326#389—21no—
327#389—591no—
328#388—1181no—
329#386—1181no—
330#384C2×D5911yescharacteristic
331#380—21no—
332#380—21no—
333#380—591no—
334C4×C4⋊C4—591no—
335C4×C22⋊C4—1181no—
336#340—592no—
337#340—592no—
338C59#2465911yescharacteristic, complemented, Sylow
339C22×Q8C22×D5912yes—
340C22×D4—591no—
341C22×D4—1181no—
342C23×C4—1181no—
343C23×C4C22⋊D5911yescharacteristic
344C23×C4C22⋊D5911yescharacteristic
345C23×C4C22×D5911yescharacteristic
346C2×SD16—1184no—
347C2×SD16—1184no—
348C2×SD16—2362no—
349C22×C8—592no—
350C22×C8—592no—
351C4⋊Q8—22no—
352C4⋊Q8—22no—
353C22⋊Q8—42no—
354C22⋊Q8—42no—
355#121—1182no—
356#121—1182no—
357C4×D4—1182no—
358C4×D4—1182no—
359C4×D4—2364no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—21no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4C22×D5912yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—1181no—
372C2×C22⋊C4—1181no—
373C2×C22⋊C4—1181no—
374C2×C22⋊C4—1181no—
375C2×C22⋊C4—1181no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—1181no—
379C2×C42C22⋊D5911yescharacteristic
380C2×C42C22⋊D5911yescharacteristic
381C2×C42C22×D5911yescharacteristic
382C2×C42C22×D5911yescharacteristic
383#107—2362no—
384C4⋊C8—594no—
385C4⋊C8—1182no—
386#103—1184no—
387#103—1184no—
388#103—2362no—
389#102—2362no—
390C22⋊C8—1184no—
391#97—1184no—
392#95—22no—
393#95—22no—
394#95—1181no—
395#95—1181no—
396#95—1181no—
397C24—1181no—
398C24C2×C22⋊D5911yescharacteristic
399C2×Q8—22no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×D4—592no—
406C2×D4—1184no—
407C2×D4—1181no—
408C2×D4—1181no—
409C2×D4—1181no—
410C2×D4—1181no—
411C2×D4—1181no—
412C2×D4—2362no—
413C2×D4—2362no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—41no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—2362no—
432C22×C4—2362no—
433C22×C4—2362no—
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C2×C22⋊D5911yescharacteristic
439C22×C4C23×D5911yescharacteristic
440SD16—2368no—
441C2×C8—594no—
442C2×C8—1182no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C2×C8—1182no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—592no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—2362no—
461C22⋊C4—44no—
462C22⋊C4—1184no—
463C22⋊C4—2362no—
464C22⋊C4—2362no—
465C22⋊C4—2362no—
466C22⋊C4—2362no—
467C42—22no—
468C42—22no—
469C42—21no—
470C42—21no—
471C42—2362no—
472C42D4×D5912yes—
473C42#1174568712yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23—1182no—
481C23—2361no—
482C23—2361no—
483C23—2361no—
484C23—2361no—
485C23—2361no—
486C23—2361no—
487C23#1427375812yes—
488C23C22×C22⋊D5911yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—1182no—
494D4—1182no—
495D4—2368no—
496D4—2361no—
497D4—2361no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—1182no—
522C2×C4—4724no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4—2361no—
526C2×C4—2361no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4#1427369912yes—
532C2×C4#1427369912yes—
533C2×C4#1427375212yes—
534C2×C4#1427375812yes—
535C2×C4C2×D4×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4C2×#1174568711yescharacteristic
539C2×C4#1427377211yescharacteristic
540C2×C4#1427377211yescharacteristic
541C8—1184no—
542C8—2362no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22C2×#1427369911yescharacteristic
558C22#1683790911yescharacteristic
559C22#1683796911yescharacteristic
560C22C2×#1427375211yescharacteristic
561C22C2×#1427375811yescharacteristic
562C22C2×#1427375811yescharacteristic
563C22#1683820011yescharacteristic
564C4—22no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—82no—
569C4—82no—
570C4—4721no—
571C4—4721no—
572C4—2362no—
573C4#1683765014yes—
574C4#1683804911yescharacteristic
575C4#1683807011yescharacteristic
576C4#1683809911yescharacteristic
577C4#1683811611yescharacteristic
578C2—42no—
579C2—4722no—
580C2#2142751511yescharacteristic
581C2C2×#1683765011yescharacteristic
582C2#2143372211yescharacteristic
583C2#2143419111yescharacteristic
584C2#2143425211yescharacteristic
585C2#2143509111yescharacteristic
586C2#2143512311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2857C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
hypercenter#2857D59
hypocenterC59#24659
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
1. C2×#16837604C2semidirect
2. C2×#16837459C2semidirect
4. #21425296C2semidirect
6. #21425020C2semidirect
7. #21424980C2semidirect
8. #21424852C2semidirect
9. #21424789C2semidirect
10. #21424093C2semidirect
11. C59×#2857C2semidirect
70. C4⋊(C59⋊C8)D4semidirect
287. #2857D59semidirect
338. C59#24659semidirect

Maximal subgroups

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

subgroupindex
C2×#168376042
C2×#168374592
C4×#142737012
#214252962
#214250972
#214250202
#214249802
#214248522
#214247892
#214240932
C59×#28572
#2465959