Finite Group Databasebeta

← #28619172

Subgroups — #28619172

Order 15104.

704 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21426837C211yesmaximal, characteristic, complemented
2#21426740C211yesmaximal, characteristic
3#21425660C211yesmaximal, characteristic, complemented
4#21425623C211yesmaximal, characteristic
5#21425589C211yesmaximal, characteristic, complemented
6#21425584C211yesmaximal, characteristic, complemented
7#21425483C211yesmaximal, characteristic
8#21425469C211yesmaximal, characteristic
9#21425435C211yesmaximal, characteristic, complemented
10#21425243C211yesmaximal, characteristic
11#21423993C211yesmaximal, characteristic, complemented
12C4×#14273661C211yesmaximal, characteristic, complemented
13#21423922C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C59×#1768C211yesmaximal, characteristic, complemented
16C2×#14273719C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273661C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273649C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26#16837304C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837300C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837299—21no—
33#16837299—21no—
34#16837291—21no—
35#16837291—21no—
36#16837289—21no—
37#16837289—21no—
38#16837272C2211yescharacteristic
39#16837264—21no—
40#16837261—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837165C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52C2×C4⋊C8×C59C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C59×#376C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61#14273719—22no—
62#14273719—22no—
63C4×D236—22no—
64C4×D236Q814yes—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88#14273661—22no—
89#14273661—22no—
90#14273661—22no—
91#14273659—22no—
92#14273649—22no—
93#14273649—22no—
94#14273649—22no—
95#14273649—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C22×C8×C59C2311yescharacteristic
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C2×C42×C59C2311yescharacteristic
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C4⋊C8×C59—22no—
111C4⋊C8×C59—22no—
112#97×C59—22no—
113C4×C8×C59—22no—
114#95×C59D411yescharacteristic
115#95×C59D411yescharacteristic
116#95×C59D411yescharacteristic
117#95×C59D411yescharacteristic
118C23×D59—21no—
119C22×C4×C59C2×D411yescharacteristic
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2411yescharacteristic
126C22×Dic59—21no—
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
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×C8×C59—21no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59—21no—
149C2×C8×C59—21no—
150C2×C8×C59—21no—
151C2×C8×C59—21no—
152C2×C8×C59#4112yes—
153C2×C8×C59#4112yes—
154C42×C59—21no—
155C42×C59—21no—
156C42×C59—21no—
157C42×C59—21no—
158C42×C59C2×Q812yes—
159C42×C59#4112yes—
160(C2×D59)⋊C4—24no—
161(C2×D59)⋊C4—81no—
162(C2×D59)⋊C4—81no—
163(C2×D59)⋊C4—81no—
164(C2×D59)⋊C4—81no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—81no—
177C4⋊Dic59—81no—
178C4⋊Dic59C2×Q812yes—
179C4⋊Dic59#4112yes—
180C4⋊Dic59#4112yes—
181Dic59⋊C4—24no—
182Dic59⋊C4—24no—
183Dic59⋊C4—81no—
184Dic59⋊C4—81no—
185Dic59⋊C4—81no—
186Dic59⋊C4—81no—
187C4×Dic59—81no—
188C4×Dic59—81no—
189C4×Dic59—81no—
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—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—21no—
211C2×C4×C59—21no—
212C2×C4×C59—21no—
213C2×C4×C59—21no—
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—22no—
224C2×Dic59—22no—
225C2×Dic59—41no—
226C2×Dic59—41no—
227C2×Dic59—41no—
228C2×Dic59—41no—
229C2×Dic59—41no—
230C2×Dic59—41no—
231C2×Dic59—41no—
232C2×Dic59—41no—
233C2×Dic59—41no—
234C2×Dic59—41no—
235C2×Dic59—41no—
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—
248C8×C59—22no—
249C8×C59—22no—
250C8×C59—41no—
251C8×C59—41no—
252#23791—591nomaximal, Sylow
253C22×C59C2×C22⋊Q811yescharacteristic
254C22×C59#52511yescharacteristic
255C22×C59#53211yescharacteristic
256C22×C59#57611yescharacteristic
257C22×C59#57611yescharacteristic
258C22×C59#57711yescharacteristic
259C22×C59#57811yescharacteristic
260C2×D59—42no—
261C2×D59—42no—
262C2×D59—42no—
263C2×D59—81no—
264C2×D59—81no—
265C2×D59—81no—
266C2×D59—81no—
267C4×C59—22no—
268C4×C59—22no—
269C4×C59—41no—
270C4×C59—41no—
271C4×C59—41no—
272C4×C59—41no—
273C4×C59#52911yescharacteristic
274C4×C59#52911yescharacteristic
275C4×C59#53311yescharacteristic
276C4×C59#53311yescharacteristic
277Dic59—42no—
278Dic59—42no—
279Dic59—42no—
280Dic59—81no—
281Dic59—81no—
282Dic59—81no—
283Dic59—81no—
284#2330—591no—
285#2319—591no—
286#2100—591no—
287#2093—591no—
288#2087—591no—
289#2086—591no—
290#2067—591no—
291#2065—591no—
292#2059—591no—
293#2025—591no—
294#1776—591no—
295C4×#107—591no—
296#1768D5911yescharacteristic, complemented
297#1762—591no—
298C4×#102—591no—
299C2×C59#239611yescharacteristic
300C2×C59#308111yescharacteristic
301C2×C59#308311yescharacteristic
302C2×C59#311911yescharacteristic
303C2×C59#312411yescharacteristic
304C2×C59#317611yescharacteristic
305C2×C59#317811yescharacteristic
306D59—82no—
307C2×#125—591no—
308C2×C4×D4—591no—
309C2×#107—591no—
310C2×#107—591no—
311C2×#107—591no—
312C2×#106—591no—
313C2×C4⋊C8C2×D5911yescharacteristic
314C2×C4⋊C8C2×D5911yescharacteristic
315C2×#102—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#97C2×D5911yescharacteristic
320C2×C4×C8C2×D5911yescharacteristic
321#390—591no—
322#390—591no—
323#389—591no—
324#389—591no—
325#389—591no—
326#389—591no—
327#389—1181no—
328#389—1181no—
329#388—1181no—
330#388—1181no—
331#387—1181no—
332#387—1181no—
333#384—591no—
334#381—1181no—
335#380—1181no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339#376C2×D5911yescharacteristic
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#336C2×D5911yescharacteristic
349#336C2×D5911yescharacteristic
350C59#2379111yescharacteristic, complemented, Sylow
351C22×D4—591no—
352C23×C4—1181no—
353C22×C8C22×D5911yescharacteristic
354C22×C8C22×D5911yescharacteristic
355C22×C8C22×D5911yescharacteristic
356C22×C8C22×D5911yescharacteristic
357#125—1182no—
358#125—1182no—
359C4×D4—594no—
360C4×D4—1182no—
361C2×C4⋊C4—591no—
362C2×C4⋊C4—591no—
363C2×C4⋊C4—591no—
364C2×C4⋊C4—591no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—1181no—
369C2×C4⋊C4—1181no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C42—1181no—
382C2×C42—1181no—
383C2×C42—1181no—
384C2×C42C22×D5911yescharacteristic
385C2×C42C22×D5911yescharacteristic
386C2×C42C22×D5911yescharacteristic
387#107—1182no—
388#107—1182no—
389#107—1182no—
390#106—1182no—
391C4⋊C8—22no—
392C4⋊C8—22no—
393#102—1182no—
394#102—1182no—
395#102—1182no—
396#102—1182no—
397#97—22no—
398C4×C8—22no—
399#95—1181no—
400#95—1181no—
401#95—1181no—
402#95—1181no—
403#95—1181no—
404#95—1181no—
405#95—1181no—
406#95D23611yescharacteristic
407#95D23611yescharacteristic
408#95D23611yescharacteristic
409#95D23611yescharacteristic
410C24—1181no—
411C2×D4—592no—
412C2×D4—1181no—
413C2×D4—1181no—
414C2×D4—1181no—
415C2×D4—1181no—
416C2×D4—1181no—
417C22×C4—1181no—
418C22×C4—1181no—
419C22×C4—1181no—
420C22×C4—1184no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1181no—
425C22×C4—4721no—
426C22×C4—4721no—
427C22×C4C2×D23611yescharacteristic
428C22×C4C2×D23611yescharacteristic
429C22×C4C2×D23611yescharacteristic
430C22×C4C2×D23611yescharacteristic
431C22×C4C2×D23611yescharacteristic
432C22×C4C2×D23611yescharacteristic
433C22×C4C23×D5911yescharacteristic
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8#1174568512yes—
445C2×C8#1174568512yes—
446C4⋊C4—592no—
447C4⋊C4—592no—
448C4⋊C4—592no—
449C4⋊C4—1184no—
450C4⋊C4—1184no—
451C4⋊C4—1181no—
452C4⋊C4—1181no—
453C4⋊C4—1181no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—4721no—
463C4⋊C4—4721no—
464C4⋊C4—4721no—
465C4⋊C4—4721no—
466C4⋊C4—4721no—
467C4⋊C4—4721no—
468C22⋊C4—1184no—
469C22⋊C4—4721no—
470C22⋊C4—4721no—
471C22⋊C4—4721no—
472C22⋊C4—4721no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—4721no—
478C42—4721no—
479C42—4721no—
480C42#1174568712yes—
481C42Q8×D5912yes—
482C23—1182no—
483C23—2361no—
484C23—2361no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23C22×D23611yescharacteristic
490D4—1182no—
491D4—1182no—
492D4—2361no—
493D4—2361no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
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—21no—
510C2×C4—1182no—
511C2×C4—1182no—
512C2×C4—1182no—
513C2×C4—4724no—
514C2×C4—2361no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
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—2361no—
532C2×C4#1427370712yes—
533C2×C4#1427370712yes—
534C2×C4C2×Q8×D5911yescharacteristic
535C2×C4#41×D5911yescharacteristic
536C2×C4#41×D5911yescharacteristic
537C2×C4C2×#1174568711yescharacteristic
538C2×C4C2×#1174568511yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C8—22no—
541C8—22no—
542C8—41no—
543C8—41no—
544C22—4721no—
545C22—4721no—
546C22—4721no—
547C22—4721no—
548C22—2362no—
549C22—2362no—
550C22—2362no—
551C22C2×#1427370711yescharacteristic
552C22#1683791411yescharacteristic
553C22#1683795211yescharacteristic
554C22#1683820711yescharacteristic
555C22#1683820711yescharacteristic
556C22#1683821411yescharacteristic
557C22#1683821911yescharacteristic
558C4—22no—
559C4—22no—
560C4—41no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—4721no—
565C4—4721no—
566C4—4721no—
567C4—4721no—
568C4—2362no—
569C4—2362no—
570C4—2362no—
571C4#1683793211yescharacteristic
572C4#1683793211yescharacteristic
573C4#1683795611yescharacteristic
574C4#1683795611yescharacteristic
575C2—4722no—
576C2#2142736811yescharacteristic
577C2#2143392811yescharacteristic
578C2#2143394911yescharacteristic
579C2#2143429711yescharacteristic
580C2#2143436011yescharacteristic
581C2#2143474611yescharacteristic
582C2#2143476411yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1768C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#1768D59
hypocenterC59#23791
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ #376 ◁ #1768

Complements

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

NH ≅ G/Ntype
1. #21426837C2semidirect
3. #21425660C2semidirect
5. #21425589C2semidirect
6. #21425584C2semidirect
9. #21425435C2semidirect
11. #21423993C2semidirect
12. C4×#14273661C2semidirect
15. C59×#1768C2semidirect
296. #1768D59semidirect
350. C59#23791semidirect

Maximal subgroups

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

subgroupindex
#214268372
#214267402
#214256602
#214256232
#214255892
#214255842
#214254832
#214254692
#214254352
#214252432
#214239932
C4×#142736612
#214239222
C4×#142736492
C59×#17682
#2379159