Finite Group Databasebeta

← #28611840

Subgroups — #28611840

Order 15104.

774 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
1C2×#16837701C211yesmaximal, characteristic
2#21427194C211yesmaximal, characteristic, complemented
3#21427065C211yesmaximal, characteristic
4#21425059C211yesmaximal, characteristic, complemented
5#21424865C211yesmaximal, characteristic, complemented
6#21424780C211yesmaximal, characteristic
7#21424779C211yesmaximal, characteristic
8#21424722C211yesmaximal, characteristic
9#21424716C211yesmaximal, characteristic
10#21424688C211yesmaximal, characteristic, complemented
11#21424676C211yesmaximal, characteristic, complemented
12#21423988C211yesmaximal, characteristic, complemented
13C8×Dic59⋊C4C211yesmaximal, characteristic, complemented
14#21423910C211yesmaximal, characteristic
15C4×#106×C59C211yesmaximal, characteristic, complemented
16C2×#14273718C2211yescharacteristic
17C2×#14273689C2211yescharacteristic
18#16837701C418yescomplemented
19C2×#14273660C2211yescharacteristic
20C2×#14273660C2211yescharacteristic
21C2×Dic59⋊C8C2211yescharacteristic
22C2×#14273649C2211yescharacteristic
23C2×#14273649C2211yescharacteristic
24C2×#14273647C2211yescharacteristic
25C2×#14273647C2211yescharacteristic
26C2×C8×Dic59C2211yescharacteristic
27C2×C8×Dic59C2211yescharacteristic
28#16837304C2211yescharacteristic
29#16837303C2211yescharacteristic
30#16837301C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837299—21no—
33#16837293—21no—
34#16837291—21no—
35#16837286—21no—
36#16837285—21no—
37#16837267C2211yescharacteristic
38#16837264—21no—
39C4⋊C4×Dic59C2211yescharacteristic
40C4⋊C4×Dic59C2211yescharacteristic
41C4⋊C4×Dic59C2211yescharacteristic
42C4×Dic59⋊C4C2211yescharacteristic
43C4×C4⋊Dic59—21no—
44C4×(C2×D59)⋊C4—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837157C2211yescharacteristic
52#16837156C2211yescharacteristic
53C2×#106×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4×C8×C59C2211yescharacteristic
56C4×C4⋊C4×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C59⋊C8C2311yescharacteristic
63C22×C59⋊C8C2311yescharacteristic
64#14273718—22no—
65#14273718C2×C414yes—
66#14273689—22no—
67#14273689C2×C414yes—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×Dic59⋊C4—21no—
74C2×Dic59⋊C4—21no—
75C2×Dic59⋊C4D411yescharacteristic
76C2×Dic59⋊C4D411yescharacteristic
77C2×(C2×D59)⋊C4—21no—
78C2×(C2×D59)⋊C4—21no—
79C2×(C2×D59)⋊C4—21no—
80C2×(C2×D59)⋊C4—21no—
81C2×(C2×D59)⋊C4—21no—
82C2×C4×Dic59—21no—
83C2×C4×Dic59—21no—
84C2×C4×Dic59—21no—
85C2×C4×Dic59C2311yescharacteristic
86C2×C4×Dic59C2311yescharacteristic
87#14273660—22no—
88#14273660C2×C414yes—
89Dic59⋊C8C2×C414yes—
90#14273649—22no—
91#14273649C2×C414yes—
92#14273647—22no—
93#14273647C2×C414yes—
94C8×Dic59—22no—
95C8×Dic59—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×C8×C59C2311yescharacteristic
106C22×C8×C59C2311yescharacteristic
107C2×C4⋊C4×C59C2311yescharacteristic
108C2×C4⋊C4×C59C2311yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C4⋊C4×C59C2311yescharacteristic
111C2×C42×C59—21no—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114#106×C59—22no—
115#106×C59C2×C414yes—
116C4×C8×C59—22no—
117#95×C59—21no—
118#95×C59—21no—
119C23×D59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59C2×D411yescharacteristic
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2411yescharacteristic
127C22×Dic59—21no—
128C22×Dic59—21no—
129C22×Dic59—21no—
130C22×Dic59C2×D411yescharacteristic
131C22×Dic59C2×D411yescharacteristic
132C22×Dic59C2×D411yescharacteristic
133C22×Dic59C2×D411yescharacteristic
134C2×D236—21no—
135C2×D236—21no—
136C2×D236—21no—
137C2×D236—21no—
138C2×D236—21no—
139C2×D236C22×C412yes—
140C2×C4×D59—24no—
141C2×C4×D59—81no—
142C2×C4×D59—81no—
143C2×C8×C59—21no—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59C22×C412yes—
148C2×C8×C59#4112yes—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59C22×C412yes—
155C4⋊C4×C59C22×C412yes—
156C4⋊C4×C59#4112yes—
157C42×C59—21no—
158C42×C59—21no—
159C42×C59—42no—
160C42×C59—81no—
161(C2×D59)⋊C4—24no—
162(C2×D59)⋊C4—42no—
163(C2×D59)⋊C4—42no—
164(C2×D59)⋊C4—81no—
165(C2×D59)⋊C4—81no—
166C4⋊Dic59—21no—
167C4⋊Dic59—24no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—42no—
173C4⋊Dic59C22×C412yes—
174Dic59⋊C4—22no—
175Dic59⋊C4—24no—
176Dic59⋊C4—22no—
177Dic59⋊C4—81no—
178C4×Dic59—21no—
179C4×Dic59—21no—
180C4×Dic59—81no—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59C22×C412yes—
184C4×Dic59#4112yes—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8C22×C412yes—
191C23×C59C22×D411yescharacteristic
192C22×D59—22no—
193C22×D59—41no—
194C22×D59—41no—
195C22×D59—41no—
196C22×D59—41no—
197C22×D59—41no—
198C22×D59—41no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—22no—
202C2×C4×C59—22no—
203C2×C4×C59—21no—
204C2×C4×C59—22no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
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×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59#13512yes—
217C2×C4×C59C23×C411yescharacteristic
218C2×C4×C59C2×#4111yescharacteristic
219C2×C4×C59C2×#4111yescharacteristic
220C2×C4×C59C2×#4111yescharacteristic
221C2×C4×C592−1+411yescharacteristic
222C2×C4×C592−1+411yescharacteristic
223C2×Dic59—21no—
224C2×Dic59—21no—
225C2×Dic59—21no—
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—22no—
230C2×Dic59—21no—
231C2×Dic59—21no—
232C2×Dic59—21no—
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×Dic59C4×D412yes—
242C2×Dic59C4×D412yes—
243C2×Dic59#13512yes—
244D236—22no—
245D236—22no—
246D236—41no—
247D236—41no—
248C4×D59—84no—
249C8×C59—22no—
250C8×C59—22no—
251C59⋊C8—22no—
252C59⋊C8—41no—
253#23065—591nomaximal, Sylow
254C22×C59C2×C4×D411yescharacteristic
255C22×C59#54711yescharacteristic
256C22×C59#54811yescharacteristic
257C22×C59C2×#13511yescharacteristic
258C22×C59C2×#13511yescharacteristic
259C22×C59#57611yescharacteristic
260C22×C59#57711yescharacteristic
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—41no—
270C4×C59—42no—
271C4×C59—42no—
272C4×C59—42no—
273C4×C59—81no—
274C4×C59C4×#4111yescharacteristic
275C4×C59#51911yescharacteristic
276C4×C59#55211yescharacteristic
277C4×C59#55611yescharacteristic
278Dic59—22no—
279Dic59—22no—
280Dic59—22no—
281Dic59—41no—
282Dic59—42no—
283Dic59—81no—
284Dic59—81no—
285C2×#478—591no—
286#2370—591no—
287#2357—591no—
288#1991—591no—
289#1949—591no—
290#1934—591no—
291#1934—591no—
292#1926—591no—
293#1925—591no—
294#1921—591no—
295#1919—591no—
296C4×#107—591no—
297C4×#106D5911yescharacteristic, complemented
298C8×C4⋊C4—591no—
299C4×#102—591no—
300C2×C59#236211yescharacteristic
301C2×C59C4×#13511yescharacteristic
302C2×C59#294611yescharacteristic
303C2×C59#333711yescharacteristic
304C2×C59#333911yescharacteristic
305C2×C59#336111yescharacteristic
306C2×C59#336511yescharacteristic
307D59—82no—
308C2×#125—591no—
309C2×C4×D4—591no—
310#478—598no—
311C2×#107—591no—
312C2×#107—591no—
313C2×#106C2×D5911yescharacteristic
314C2×#106C2×D5911yescharacteristic
315C2×C4⋊C8—591no—
316C2×#102—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×C4×C8—591no—
321C2×C4×C8—591no—
322C2×C4×C8C2×D5911yescharacteristic
323#390—591no—
324#390—591no—
325#389—591no—
326#389—591no—
327#389—1181no—
328#388—1181no—
329#388—1181no—
330#387—1181no—
331#387—1181no—
332#382—591no—
333#381—1181no—
334C4×C4⋊C4—591no—
335C4×C4⋊C4—591no—
336C4×C4⋊C4—591no—
337C4×C4⋊C4—591no—
338C4×C4⋊C4—1181no—
339C4×C4⋊C4C2×D5911yescharacteristic
340C4×C4⋊C4C2×D5911yescharacteristic
341C4×C22⋊C4—1181no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340C2×D5911yescharacteristic
349#340C2×D5911yescharacteristic
350#336—591no—
351#336—591no—
352C59#2306511yescharacteristic, complemented, Sylow
353C22×D4—591no—
354C23×C4—1181no—
355C22×C8—591no—
356C22×C8—591no—
357C22×C8C22×D5911yescharacteristic
358C22×C8C22×D5911yescharacteristic
359#125—594no—
360#125—1182no—
361C4×D4—594no—
362C4×D4—1182no—
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⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C4⋊C4C22×D5911yescharacteristic
376C2×C22⋊C4—1181no—
377C2×C22⋊C4—1181no—
378C2×C22⋊C4—1181no—
379C2×C22⋊C4—1181no—
380C2×C22⋊C4—1181no—
381C2×C42—21no—
382C2×C42—21no—
383C2×C42—591no—
384C2×C42—591no—
385C2×C42—1181no—
386C2×C42—1181no—
387C2×C42—1181no—
388C2×C42C22×D5911yescharacteristic
389#107—594no—
390#107—1182no—
391#106—22no—
392#106C4×D5914yes—
393C4⋊C8—594no—
394#102—594no—
395#102—594no—
396#102—1182no—
397#102—1182no—
398C4×C8—22no—
399C4×C8—1182no—
400C4×C8—1182no—
401#95—21no—
402#95—21no—
403#95—591no—
404#95—591no—
405#95—1181no—
406#95—1181no—
407#95—1181no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412C24—1181no—
413C2×D4—592no—
414C2×D4—1181no—
415C2×D4—1181no—
416C2×D4—1181no—
417C2×D4—1181no—
418C2×D4—1181no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—591no—
427C22×C4—1184no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—4721no—
432C22×C4—4721no—
433C22×C4D4×D5911yescharacteristic
434C22×C4D4×D5911yescharacteristic
435C22×C4C23×D5911yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—592no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8C2×C4×D5912yes—
447C2×C8#1174568512yes—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—592no—
454C4⋊C4—1182no—
455C4⋊C4—1184no—
456C4⋊C4—1184no—
457C4⋊C4—1182no—
458C4⋊C4—1181no—
459C4⋊C4—1181no—
460C4⋊C4—1181no—
461C4⋊C4—1181no—
462C4⋊C4—1181no—
463C4⋊C4—4721no—
464C4⋊C4—2362no—
465C4⋊C4C2×C4×D5912yes—
466C4⋊C4C2×C4×D5912yes—
467C4⋊C4#1174568512yes—
468C22⋊C4—1184no—
469C22⋊C4—4721no—
470C22⋊C4—4721no—
471C22⋊C4—2362no—
472C22⋊C4—2362no—
473C42—21no—
474C42—21no—
475C42—42no—
476C42—81no—
477C42—592no—
478C42—592no—
479C42—1181no—
480C42—1181no—
481C42—4721no—
482C42—4721no—
483C42—4721no—
484C23—1182no—
485C23—2361no—
486C23—2361no—
487C23—2361no—
488C23—2361no—
489C23—2361no—
490C23—2361no—
491C23C2×D4×D5911yescharacteristic
492D4—1182no—
493D4—1182no—
494D4—2361no—
495D4—2361no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
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—41no—
513C2×C4—592no—
514C2×C4—592no—
515C2×C4—592no—
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—1182no—
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—2361no—
535C2×C4#1427374812yes—
536C2×C4C22×C4×D5911yescharacteristic
537C2×C4#41×D5911yescharacteristic
538C2×C4C2×#1174568511yescharacteristic
539C2×C4C2×#1174568511yescharacteristic
540C2×C4#1427377411yescharacteristic
541C2×C4#1427377511yescharacteristic
542C8—22no—
543C8—22no—
544C8—1182no—
545C8—2361no—
546C22—4721no—
547C22—4721no—
548C22—4721no—
549C22—4721no—
550C22—2362no—
551C22—2362no—
552C22—2362no—
553C22C4×D4×D5911yescharacteristic
554C22#1683805311yescharacteristic
555C22#1683806111yescharacteristic
556C22#135×D5911yescharacteristic
557C22C2×#1427374811yescharacteristic
558C22#1683820411yescharacteristic
559C22#1683820811yescharacteristic
560C4—22no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—1182no—
567C4—1182no—
568C4—1182no—
569C4—4721no—
570C4—4721no—
571C4—2361no—
572C4—2362no—
573C4C4×#1174568511yescharacteristic
574C4#1683787811yescharacteristic
575C4#1683808511yescharacteristic
576C4#1683812411yescharacteristic
577C2—4722no—
578C2#2142712411yescharacteristic
579C2C4×#1427374811yescharacteristic
580C2#2143292311yescharacteristic
581C2#2143647111yescharacteristic
582C2#2143649811yescharacteristic
583C2#2143675411yescharacteristic
584C2#2143679311yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC4×#106×C59C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenterC4×#106D59
hypocenterC59#23065
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
2. #21427194C2semidirect
4. #21425059C2semidirect
5. #21424865C2semidirect
10. #21424688C2semidirect
11. #21424676C2semidirect
12. #21423988C2semidirect
13. C8×Dic59⋊C4C2semidirect
15. C4×#106×C59C2semidirect
18. #16837701C4semidirect
297. C4×#106D59semidirect
352. C59#23065semidirect

Maximal subgroups

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

subgroupindex
C2×#168377012
#214271942
#214270652
#214250592
#214248652
#214247802
#214247792
#214247222
#214247162
#214246882
#214246762
#214239882
C8×Dic59⋊C42
#214239102
C4×#106×C592
#2306559