Finite Group Databasebeta

← #28619120

Subgroups — #28619120

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426840C211yesmaximal, characteristic, complemented
2#21426734C211yesmaximal, characteristic
3#21425666C211yesmaximal, characteristic, complemented
4#21425614C211yesmaximal, characteristic
5#21425599C211yesmaximal, characteristic, complemented
6#21425590C211yesmaximal, characteristic, complemented
7#21425485C211yesmaximal, characteristic
8#21425479C211yesmaximal, characteristic
9#21425422C211yesmaximal, characteristic, complemented
10#21425231C211yesmaximal, characteristic
11#21423994C211yesmaximal, characteristic, complemented
12#21423954C211yesmaximal, characteristic, complemented
13#21423924C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C4×#107×C59C211yesmaximal, characteristic, complemented
16C2×#14273726C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273660C2211yescharacteristic
19C2×#14273658C2211yescharacteristic
20C2×Dic59⋊C8C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273649C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×#14273638C2211yescharacteristic
27#16837308—21no—
28#16837306—21no—
29#16837303C2211yescharacteristic
30#16837303C2211yescharacteristic
31#16837301C2211yescharacteristic
32#16837298C2211yescharacteristic
33#16837298C2211yescharacteristic
34#16837298C2211yescharacteristic
35#16837293—21no—
36#16837293—21no—
37#16837289—21no—
38#16837286—21no—
39#16837267C2211yescharacteristic
40#16837264—21no—
41#16837261—21no—
42C4×C4⋊Dic59C2211yescharacteristic
43#16837248C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#107×C59C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×C4×C8×C59C2211yescharacteristic
55C4×C4⋊C4×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C59×#340C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C22×Dic118—21no—
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273726—22no—
65#14273726—22no—
66C4×D236—22no—
67C4×D236Q814yescomplemented
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×Dic59⋊C4—21no—
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×(C2×D59)⋊C4—21no—
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×C4×Dic59—21no—
82C2×C4×Dic59C2311yescharacteristic
83C2×C4×Dic59C2311yescharacteristic
84#14273660—22no—
85#14273658—22no—
86Dic59⋊C8—22no—
87Dic59⋊C8—22no—
88#14273649—22no—
89#14273649—22no—
90#14273647—22no—
91#14273647—22no—
92#14273638—22no—
93C59⋊#95—21no—
94C59⋊#95—21no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95D411yescharacteristic
101C59⋊#95D411yescharacteristic
102C59⋊#95D411yescharacteristic
103C59⋊#95D411yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C2×C4⋊C4×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C42×C59—21no—
111C2×C42×C59—21no—
112C2×C42×C59C2311yescharacteristic
113#107×C59—22no—
114#107×C59—22no—
115C4×C8×C59—22no—
116#95×C59—21no—
117#95×C59—21no—
118C23×D59—21no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2411yescharacteristic
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59C2×D411yescharacteristic
130C22×Dic59C2×D411yescharacteristic
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C2×D236—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236C2×Q812yes—
139C2×C4×D59—24no—
140C2×C4×D59—81no—
141C2×C4×D59—81no—
142C2×Dic118—24no—
143C2×Dic118—81no—
144C2×Dic118—81no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59#4112yes—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—21no—
157C4⋊C4×C59—21no—
158C4⋊C4×C59#4112yes—
159C42×C59—81no—
160C42×C59—81no—
161C42×C59C2×Q812yes—
162C42×C59#4112yes—
163(C2×D59)⋊C4—24no—
164(C2×D59)⋊C4—81no—
165(C2×D59)⋊C4—81no—
166(C2×D59)⋊C4—81no—
167(C2×D59)⋊C4—81no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—81no—
173C4⋊Dic59C2×Q812yes—
174C4⋊Dic59#4112yes—
175Dic59⋊C4—24no—
176Dic59⋊C4—81no—
177Dic59⋊C4—81no—
178Dic59⋊C4—81no—
179C4×Dic59—21no—
180C4×Dic59—21no—
181C4×Dic59—21no—
182C4×Dic59—21no—
183C4×Dic59—81no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8#4112yes—
190C23×C59C22×D411yescharacteristic
191C22×D59—22no—
192C22×D59—41no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C2×C4×C59—21no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—22no—
203C2×C4×C59—41no—
204C2×C4×C59—41no—
205C2×C4×C59—41no—
206C2×C4×C59—41no—
207C2×C4×C59—41no—
208C2×C4×C59—41no—
209C2×C4×C59—41no—
210C2×C4×C59—41no—
211C2×C4×C59—41no—
212C2×C4×C59—41no—
213C2×C4×C59—41no—
214C2×C4×C59C22⋊Q812yes—
215C2×C4×C59C22⋊Q812yes—
216C2×C4×C59C22×Q811yescharacteristic
217C2×C4×C59C2×#4111yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×Dic59—22no—
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—22no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243D236—22no—
244D236—22no—
245D236—41no—
246D236—41no—
247C4×D59—84no—
248Dic118—84no—
249C8×C59—22no—
250C8×C59—41no—
251C59⋊C8—22no—
252C59⋊C8—41no—
253#23787—591nomaximal, Sylow
254C22×C59C2×C22⋊Q811yescharacteristic
255C22×C59#52511yescharacteristic
256C22×C59#53211yescharacteristic
257C22×C59#57611yescharacteristic
258C22×C59#57611yescharacteristic
259C22×C59#57711yescharacteristic
260C22×C59#57811yescharacteristic
261C2×D59—42no—
262C2×D59—42no—
263C2×D59—42no—
264C2×D59—81no—
265C2×D59—81no—
266C2×D59—81no—
267C2×D59—81no—
268C4×C59—22no—
269C4×C59—22no—
270C4×C59—42no—
271C4×C59—81no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59#52911yescharacteristic
275C4×C59#52911yescharacteristic
276C4×C59#53311yescharacteristic
277C4×C59#53311yescharacteristic
278Dic59—42no—
279Dic59—41no—
280Dic59—41no—
281Dic59—42no—
282Dic59—41no—
283Dic59—41no—
284Dic59—81no—
285#2331—591no—
286#2319—591no—
287#2101—591no—
288#2092—591no—
289#2088—591no—
290#2087—591no—
291#2067—591no—
292#2067—591no—
293#2057—591no—
294#2023—591no—
295#1776—591no—
296C4×#107D5911yescharacteristic, complemented
297#1768—591no—
298#1762—591no—
299C4×#102—591no—
300C2×C59#239611yescharacteristic
301C2×C59#308111yescharacteristic
302C2×C59#308311yescharacteristic
303C2×C59#312111yescharacteristic
304C2×C59#312211yescharacteristic
305C2×C59#317511yescharacteristic
306C2×C59#317911yescharacteristic
307D59—82no—
308C2×C4⋊Q8—591no—
309C2×C4×D4—591no—
310C2×#107—591no—
311C2×#107C2×D5911yescharacteristic
312C2×#107C2×D5911yescharacteristic
313C2×#106—591no—
314C2×C4⋊C8—591no—
315C2×C4⋊C8—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#97—591no—
321C2×C4×C8C2×D5911yescharacteristic
322#391—1181no—
323#391—1181no—
324#390—591no—
325#390—591no—
326#389—591no—
327#389—591no—
328#389—591no—
329#389—591no—
330#388—1181no—
331#388—1181no—
332#387—1181no—
333#387—1181no—
334#382—591no—
335#381—1181no—
336#380—1181no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4C2×D5911yescharacteristic
339C4×C4⋊C4C2×D5911yescharacteristic
340#376—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340C2×D5911yescharacteristic
348#340C2×D5911yescharacteristic
349#336—591no—
350#336—591no—
351C59#2378711yescharacteristic, complemented, Sylow
352C22×Q8—1181no—
353C22×D4—591no—
354C23×C4—1181no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359C4⋊Q8—1182no—
360C4⋊Q8—1182no—
361C4×D4—594no—
362C4×D4—1182no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—1181no—
367C2×C4⋊C4—1181no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4C22×D5911yescharacteristic
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C42—21no—
381C2×C42—21no—
382C2×C42—591no—
383C2×C42—591no—
384C2×C42—1181no—
385C2×C42C22×D5911yescharacteristic
386#107—22no—
387#107—22no—
388#107—1182no—
389#106—1182no—
390C4⋊C8—1182no—
391C4⋊C8—1182no—
392#102—1182no—
393#102—1182no—
394#102—1182no—
395#102—1182no—
396#97—1182no—
397C4×C8—22no—
398#95—21no—
399#95—21no—
400#95—591no—
401#95—591no—
402#95—591no—
403#95—591no—
404#95—1181no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411C24—1181no—
412C2×Q8—1184no—
413C2×Q8—4721no—
414C2×Q8—4721no—
415C2×D4—592no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—591no—
428C22×C4—591no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1184no—
432C22×C4—1181no—
433C22×C4—4721no—
434C22×C4—4721no—
435C22×C4D4×D5911yescharacteristic
436C22×C4D4×D5911yescharacteristic
437C22×C4C23×D5911yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—592no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8#1174568512yes—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—592no—
458C4⋊C4—592no—
459C4⋊C4—1184no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—4721no—
465C4⋊C4—4721no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C4⋊C4#1174568512yes—
469C22⋊C4—1184no—
470C22⋊C4—4721no—
471C22⋊C4—4721no—
472C22⋊C4—4721no—
473C22⋊C4—4721no—
474C42—81no—
475C42—81no—
476C42—1181no—
477C42—1181no—
478C42—1181no—
479C42—1181no—
480C42—4721no—
481C42#1174568512yes—
482C42Q8×D5912yes—
483C23—1182no—
484C23—2361no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23C2×D4×D5911yescharacteristic
491Q8—4724no—
492D4—1182no—
493D4—1182no—
494D4—2361no—
495D4—2361no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—1182no—
513C2×C4—1181no—
514C2×C4—1181no—
515C2×C4—1182no—
516C2×C4—1181no—
517C2×C4—1181no—
518C2×C4—1181no—
519C2×C4—1181no—
520C2×C4—1181no—
521C2×C4—1181no—
522C2×C4—1181no—
523C2×C4—1181no—
524C2×C4—1181no—
525C2×C4—1181no—
526C2×C4—4724no—
527C2×C4—2361no—
528C2×C4—2361no—
529C2×C4—2361no—
530C2×C4—2361no—
531C2×C4—2361no—
532C2×C4—2361no—
533C2×C4—2361no—
534C2×C4#1427370512yes—
535C2×C4#1427370512yes—
536C2×C4C2×Q8×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4#41×D5911yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C2×C4C2×#1174568511yescharacteristic
541C2×C4C2×#1174568511yescharacteristic
542C8—22no—
543C8—41no—
544C8—1182no—
545C8—2361no—
546C22—4721no—
547C22—4721no—
548C22—4721no—
549C22—4721no—
550C22—2362no—
551C22—2362no—
552C22—2362no—
553C22C2×#1427370511yescharacteristic
554C22#1683791111yescharacteristic
555C22#1683794911yescharacteristic
556C22#1683820311yescharacteristic
557C22#1683820411yescharacteristic
558C22#1683820911yescharacteristic
559C22#1683821511yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—4721no—
567C4—2362no—
568C4—2361no—
569C4—2361no—
570C4—2361no—
571C4—2361no—
572C4—2362no—
573C4#1683793011yescharacteristic
574C4#1683793611yescharacteristic
575C4#1683795611yescharacteristic
576C4#1683795611yescharacteristic
577C2—4722no—
578C2#2142737211yescharacteristic
579C2#2143393011yescharacteristic
580C2#2143395211yescharacteristic
581C2#2143433111yescharacteristic
582C2#2143433611yescharacteristic
583C2#2143473711yescharacteristic
584C2#2143478311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×#107×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenterC4×#107D59
hypocenterC59#23787
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#107

Complements

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

NH ≅ G/Ntype
1. #21426840C2semidirect
3. #21425666C2semidirect
5. #21425599C2semidirect
6. #21425590C2semidirect
9. #21425422C2semidirect
11. #21423994C2semidirect
12. #21423954C2semidirect
15. C4×#107×C59C2semidirect
67. C4×D236Q8semidirect
296. C4×#107D59semidirect
351. C59#23787semidirect

Maximal subgroups

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

subgroupindex
#214268402
#214267342
#214256662
#214256142
#214255992
#214255902
#214254852
#214254792
#214254222
#214252312
#214239942
#214239542
#214239242
C4×#142736492
C4×#107×C592
#2378759