Finite Group Databasebeta

← #28619067

Subgroups — #28619067

Order 15104.

756 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273719C211yesmaximal, characteristic, complemented
2#21426560C211yesmaximal, characteristic
3#21425665C211yesmaximal, characteristic, complemented
4#21425616C211yesmaximal, characteristic
5#21425594C211yesmaximal, characteristic, complemented
6#21425584C211yesmaximal, characteristic, complemented
7#21425483C211yesmaximal, characteristic
8#21425469C211yesmaximal, characteristic
9#21425435C211yesmaximal, characteristic, complemented
10#21425243C211yesmaximal, characteristic
11#21423993C211yesmaximal, characteristic, complemented
12#21423993C211yesmaximal, characteristic, complemented
13#21423922C211yesmaximal, characteristic
14#21423922C211yesmaximal, characteristic
15C59×#1767C211yesmaximal, characteristic, complemented
16C2×#14273719C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273684—21no—
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273659C2211yescharacteristic
22C2×#14273659C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273649C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27#16837304C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837300C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837293—21no—
34#16837293—21no—
35#16837286—21no—
36#16837286—21no—
37#16837268C2211yescharacteristic
38C4×Dic59⋊C4—21no—
39C4×Dic59⋊C4—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×(C2×D59)⋊C4—21no—
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×#97×C59C2211yescharacteristic
56C43×C59C2211yescharacteristic
57C59×#336C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C22×D236C2311yescharacteristic
60C22×C4×D59—21no—
61#14273719—22no—
62#14273719—22no—
63C4×D236—22no—
64C4×D236Q814yes—
65#14273684—44no—
66C2×C4⋊Dic59C2311yescharacteristic
67C2×C4⋊Dic59C2311yescharacteristic
68C2×C4⋊Dic59C2311yescharacteristic
69C2×C4⋊Dic59C2311yescharacteristic
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4—21no—
76C2×Dic59⋊C4—21no—
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87#14273661—22no—
88#14273661—22no—
89#14273659—22no—
90#14273659—22no—
91#14273649—22no—
92#14273649—22no—
93#14273649—22no—
94#14273649—22no—
95C59⋊#95—21no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C22×C8×C59C2311yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C22×C8×C59C2311yescharacteristic
108C2×C42×C59D411yescharacteristic
109C2×C42×C59D411yescharacteristic
110C2×C42×C59D411yescharacteristic
111C2×C42×C59D411yescharacteristic
112C2×C42×C59C2311yescharacteristic
113C2×C42×C59C2311yescharacteristic
114C2×C42×C59C2311yescharacteristic
115C4⋊C8×C59—22no—
116C4⋊C8×C59—22no—
117#97×C59—22no—
118#97×C59—22no—
119C23×D59—21no—
120C22×C4×C59C2×D411yescharacteristic
121C22×C4×C59C2×D411yescharacteristic
122C22×C4×C59C2×D411yescharacteristic
123C22×C4×C59C2×D411yescharacteristic
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59—21no—
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236C2×Q812yes—
140C2×C4×D59—24no—
141C2×C4×D59—42no—
142C2×C4×D59—42no—
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—21no—
153C2×C8×C59#4112yes—
154C2×C8×C59#4112yes—
155C42×C59—22no—
156C42×C59—21no—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—41no—
161C42×C59—41no—
162C42×C59—41no—
163C42×C59C2×Q812yes—
164C42×C59#4112yes—
165(C2×D59)⋊C4—24no—
166(C2×D59)⋊C4—42no—
167(C2×D59)⋊C4—42no—
168(C2×D59)⋊C4—42no—
169(C2×D59)⋊C4—42no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59—21no—
175C4⋊Dic59—21no—
176C4⋊Dic59—21no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59C2×Q812yes—
182C4⋊Dic59#4112yes—
183C4⋊Dic59#4112yes—
184Dic59⋊C4—24no—
185Dic59⋊C4—24no—
186Dic59⋊C4—42no—
187Dic59⋊C4—42no—
188Dic59⋊C4—42no—
189Dic59⋊C4—42no—
190C4×Dic59—42no—
191C4×Dic59—42no—
192C4×Dic59—42no—
193C23×C59C22×D411yescharacteristic
194C22×D59—22no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C22×D59—41no—
200C22×D59—41no—
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×C59—21no—
215C2×C4×C59—21no—
216C2×C4×C59C22⋊Q812yes—
217C2×C4×C59C22⋊Q812yes—
218C2×C4×C59AGL(1,ℤ/8)12yes—
219C2×C4×C59C22×Q811yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C59C2×#4111yescharacteristic
222C2×C4×C59C2×#4111yescharacteristic
223C2×C4×C59C2×#4111yescharacteristic
224C2×C4×C59C2×#4111yescharacteristic
225C2×Dic59—22no—
226C2×Dic59—22no—
227C2×Dic59—22no—
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—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
246D236—22no—
247D236—22no—
248D236—41no—
249D236—41no—
250C4×D59—48no—
251C8×C59—22no—
252C8×C59—22no—
253C8×C59—41no—
254C8×C59—41no—
255#23784—591nomaximal, Sylow
256C22×C59C2×C22⋊Q811yescharacteristic
257C22×C59#52511yescharacteristic
258C22×C59#53211yescharacteristic
259C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
260C22×C59#57511yescharacteristic
261C22×C59#57511yescharacteristic
262C22×C59#57511yescharacteristic
263C2×D59—42no—
264C2×D59—42no—
265C2×D59—42no—
266C2×D59—81no—
267C2×D59—81no—
268C2×D59—81no—
269C2×D59—81no—
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—41no—
274C4×C59—41no—
275C4×C59—41no—
276C4×C59#52911yescharacteristic
277C4×C59#52911yescharacteristic
278C4×C59#53311yescharacteristic
279C4×C59#53311yescharacteristic
280Dic59—42no—
281Dic59—42no—
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287C4×#125—591no—
288#2293—591no—
289#2101—591no—
290#2092—591no—
291#2088—591no—
292#2086—591no—
293#2067—591no—
294#2065—591no—
295#2059—591no—
296#2025—591no—
297#1776—591no—
298#1776—591no—
299#1767D5911yescharacteristic, complemented
300#1762—591no—
301#1762—591no—
302C2×C59#239611yescharacteristic
303C2×C59#307711yescharacteristic
304C2×C59#308011yescharacteristic
305C2×C59#311111yescharacteristic
306C2×C59#311511yescharacteristic
307C2×C59#318011yescharacteristic
308C2×C59#318311yescharacteristic
309D59—82no—
310C2×#125—591no—
311C2×C4×D4—591no—
312C2×#117—1181no—
313C2×#107—591no—
314C2×#107—591no—
315C2×#106—591no—
316C2×#106—591no—
317C2×C4⋊C8C2×D5911yescharacteristic
318C2×C4⋊C8C2×D5911yescharacteristic
319C2×#102—591no—
320C2×#102—591no—
321C2×#102—591no—
322C2×#102—591no—
323C2×#97C2×D5911yescharacteristic
324C2×#97C2×D5911yescharacteristic
325#390—591no—
326#390—591no—
327#389—591no—
328#389—591no—
329#389—591no—
330#389—591no—
331#388—1181no—
332#388—1181no—
333#387—1181no—
334#387—1181no—
335#382—591no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—1181no—
340C4×C4⋊C4—1181no—
341C4×C22⋊C4—1181no—
342C43C2×D5911yescharacteristic
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#336C2×D5911yescharacteristic
352#336C2×D5911yescharacteristic
353C59#2378411yescharacteristic, complemented, Sylow
354C22×D4—591no—
355C23×C4—1181no—
356C22×C8C22×D5911yescharacteristic
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359C22×C8C22×D5911yescharacteristic
360#125—1182no—
361#125—1182no—
362C4×D4—594no—
363C4×D4—1182no—
364#117—2364no—
365C2×C4⋊C4—591no—
366C2×C4⋊C4—591no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C22⋊C4—1181no—
382C2×C22⋊C4—1181no—
383C2×C42—1181no—
384C2×C42—1181no—
385C2×C42—1181no—
386C2×C42D23611yescharacteristic
387C2×C42D23611yescharacteristic
388C2×C42D23611yescharacteristic
389C2×C42D23611yescharacteristic
390C2×C42C22×D5911yescharacteristic
391C2×C42C22×D5911yescharacteristic
392C2×C42C22×D5911yescharacteristic
393#107—1182no—
394#107—1182no—
395#106—1182no—
396#106—1182no—
397C4⋊C8—22no—
398C4⋊C8—22no—
399#102—1182no—
400#102—1182no—
401#102—1182no—
402#102—1182no—
403#97—22no—
404#97—22no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414C24—1181no—
415C2×D4—592no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C2×D4—1181no—
420C2×D4—1181no—
421C22×C4—1181no—
422C22×C4—1181no—
423C22×C4—1181no—
424C22×C4—1184no—
425C22×C4—1181no—
426C22×C4—1181no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—2362no—
430C22×C4—2362no—
431C22×C4C2×D23611yescharacteristic
432C22×C4C2×D23611yescharacteristic
433C22×C4C2×D23611yescharacteristic
434C22×C4C2×D23611yescharacteristic
435C22×C4C2×D23611yescharacteristic
436C22×C4C2×D23611yescharacteristic
437C22×C4C23×D5911yescharacteristic
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8#1174568512yes—
449C2×C8#1174568512yes—
450C4⋊C4—592no—
451C4⋊C4—592no—
452C4⋊C4—592no—
453C4⋊C4—1184no—
454C4⋊C4—1184no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—2362no—
467C4⋊C4—2362no—
468C4⋊C4—2362no—
469C4⋊C4—2362no—
470C22⋊C4—1184no—
471C22⋊C4—2362no—
472C22⋊C4—2362no—
473C22⋊C4—2362no—
474C22⋊C4—2362no—
475C42—22no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—2362no—
484C42—2362no—
485C42—2362no—
486C42#1174568712yes—
487C42Q8×D5912yes—
488C23—1182no—
489C23—2361no—
490C23—2361no—
491C23—2361no—
492C23—2361no—
493C23—2361no—
494C23—2361no—
495C23C22×D23611yescharacteristic
496D4—1182no—
497D4—1182no—
498D4—2361no—
499D4—2361no—
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—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—1182no—
516C2×C4—1182no—
517C2×C4—1182no—
518C2×C4—2368no—
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—2361no—
533C2×C4—2361no—
534C2×C4—2361no—
535C2×C4—2361no—
536C2×C4—2361no—
537C2×C4#1427370712yes—
538C2×C4#1427370712yes—
539C2×C4#1427375512yes—
540C2×C4C2×Q8×D5911yescharacteristic
541C2×C4#41×D5911yescharacteristic
542C2×C4#41×D5911yescharacteristic
543C2×C4C2×#1174568711yescharacteristic
544C2×C4C2×#1174568511yescharacteristic
545C2×C4C2×#1174568511yescharacteristic
546C8—22no—
547C8—22no—
548C8—41no—
549C8—41no—
550C22—4721no—
551C22—4721no—
552C22—4721no—
553C22—4721no—
554C22—2362no—
555C22—2362no—
556C22—2362no—
557C22C2×#1427370711yescharacteristic
558C22#1683791411yescharacteristic
559C22#1683795211yescharacteristic
560C22C2×#1427375511yescharacteristic
561C22#1683820211yescharacteristic
562C22#1683820211yescharacteristic
563C22#1683820211yescharacteristic
564C4—22no—
565C4—22no—
566C4—22no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—4721no—
571C4—4721no—
572C4—4721no—
573C4—4721no—
574C4—2362no—
575C4—2362no—
576C4—2362no—
577C4#1683793211yescharacteristic
578C4#1683793211yescharacteristic
579C4#1683795611yescharacteristic
580C4#1683795611yescharacteristic
581C2—4722no—
582C2#2142736811yescharacteristic
583C2#2143389511yescharacteristic
584C2#2143391911yescharacteristic
585C2#2143420711yescharacteristic
586C2#2143424811yescharacteristic
587C2#2143479411yescharacteristic
588C2#2143482511yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1767C2
socleC23×C59C22×D4
prime radicalC22×D236C23
hypercenter#1767D59
hypocenterC59#23784
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C43 ◁ #1767

Complements

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

NH ≅ G/Ntype
1. C4×#14273719C2semidirect
3. #21425665C2semidirect
5. #21425594C2semidirect
6. #21425584C2semidirect
9. #21425435C2semidirect
11. #21423993C2semidirect
12. #21423993C2semidirect
15. C59×#1767C2semidirect
299. #1767D59semidirect
353. C59#23784semidirect

Maximal subgroups

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

subgroupindex
C4×#142737192
#214265602
#214256652
#214256162
#214255942
#214255842
#214254832
#214254692
#214254352
#214252432
#214239932
#214239222
C59×#17672
#2378459