Finite Group Databasebeta

← #28599102

Subgroups — #28599102

Order 15104.

754 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
1C2×#16837427C211yesmaximal, characteristic
2#21426706C211yesmaximal, characteristic
3#21426701C211yesmaximal, characteristic
4#21424867C211yesmaximal, characteristic, complemented
5#21424867C211yesmaximal, characteristic, complemented
6#21424862C211yesmaximal, characteristic, complemented
7#21424859C211yesmaximal, characteristic, complemented
8#21424856C211yesmaximal, characteristic, complemented
9#21424855C211yesmaximal, characteristic, complemented
10#21424851C211yesmaximal, characteristic, complemented
11#21424128C211yesmaximal, characteristic
12#21424126C211yesmaximal, characteristic
13#21424122C211yesmaximal, characteristic
14#21424120C211yesmaximal, characteristic
15C59×#1949C211yesmaximal, characteristic, complemented
16C22×#11745659C2211yescharacteristic
17C22×C8×D59C2211yescharacteristic
18C2×#14273684C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273661C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25#16837427C418yescomplemented
26C2×C4⋊(C59⋊C8)C2211yescharacteristic
27C2×Dic59⋊C8C2211yescharacteristic
28C2×Dic59⋊C8C2211yescharacteristic
29#16837294—21no—
30#16837294—21no—
31#16837293—21no—
32#16837291—21no—
33#16837289—21no—
34#16837289—21no—
35#16837286—21no—
36#16837284—21no—
37#16837272C2211yescharacteristic
38#16837271C2211yescharacteristic
39#16837267C2211yescharacteristic
40#16837259C2211yescharacteristic
41#16837259C2211yescharacteristic
42#16837259C2211yescharacteristic
43#16837259C2211yescharacteristic
44C4⋊C4×Dic59C2211yescharacteristic
45C4⋊C4×Dic59C2211yescharacteristic
46C4×C4⋊Dic59C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837164C2211yescharacteristic
52#16837164C2211yescharacteristic
53C2×#107×C59C2211yescharacteristic
54C2×#106×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C59×#382C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#340C2211yescharacteristic
59C59×#340C2211yescharacteristic
60C22×C4×D59C2311yescharacteristic
61C2×#11745659—41no—
62C2×#11745659—41no—
63C2×#11745659C2×C414yes—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C2×C8×D59—41no—
67C2×C8×D59—41no—
68C2×C8×D59C2×C414yes—
69#14273684C2×C414yes—
70#14273684Q814yes—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×Dic59⋊C4—21no—
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4D411yescharacteristic
80C2×Dic59⋊C4D411yescharacteristic
81C2×(C2×D59)⋊C4—21no—
82C2×(C2×D59)⋊C4—21no—
83C2×(C2×D59)⋊C4—21no—
84C2×(C2×D59)⋊C4—21no—
85C2×(C2×D59)⋊C4D411yescharacteristic
86C2×(C2×D59)⋊C4D411yescharacteristic
87C2×C4×Dic59—21no—
88C2×C4×Dic59—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59C2311yescharacteristic
91#14273661—41no—
92#14273661—41no—
93#14273660—41no—
94#14273658—41no—
95#14273658—41no—
96#14273658—41no—
97C4⋊(C59⋊C8)C2×C414yes—
98Dic59⋊C8C2×C414yes—
99Dic59⋊C8C2×C414yes—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111C2×C4⋊C4×C59C2311yescharacteristic
112C2×C4⋊C4×C59C2311yescharacteristic
113C2×C4⋊C4×C59C2311yescharacteristic
114C2×C4⋊C4×C59C2311yescharacteristic
115C2×C42×C59—21no—
116C2×C42×C59C2311yescharacteristic
117#107×C59—41no—
118#106×C59—41no—
119C4⋊C8×C59C2×C414yes—
120#95×C59—21no—
121#95×C59—21no—
122#95×C59—21no—
123C23×D59C2×D411yescharacteristic
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2411yescharacteristic
131C22×Dic59—21no—
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59C2×D411yescharacteristic
136C22×Dic59C2×D411yescharacteristic
137C22×Dic59C2×D411yescharacteristic
138C2×C4×D59—21no—
139C2×C4×D59—21no—
140C2×C4×D59—21no—
141C2×C4×D59—21no—
142C2×C4×D59C22×C412yes—
143C2×C4×D59C2×Q812yes—
144C2×C8×C59—21no—
145C2×C8×C59—21no—
146C2×C8×C59—21no—
147C2×C8×C59—21no—
148C2×C8×C59C22×C412yes—
149C2×C8×C59C22×C412yes—
150C4⋊C4×C59—21no—
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—
158C42×C59—81no—
159C42×C59C22×C412yes—
160C42×C59C2×Q812yes—
161(C2×D59)⋊C4—81no—
162(C2×D59)⋊C4—81no—
163(C2×D59)⋊C4—81no—
164(C2×D59)⋊C4—81no—
165(C2×D59)⋊C4C4⋊C414yes—
166(C2×D59)⋊C4C4⋊C414yes—
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—81no—
176C4⋊Dic59—81no—
177Dic59⋊C4—81no—
178Dic59⋊C4—81no—
179Dic59⋊C4C4⋊C414yes—
180Dic59⋊C4C4⋊C414yes—
181C4×Dic59—81no—
182C4×Dic59—81no—
183C4×Dic59—81no—
184C4×Dic59C22×C412yes—
185C4×Dic59C2×Q812yes—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8C22×C412yes—
191C2×C59⋊C8C22×C412yes—
192#11745659—44no—
193C8×D59—44no—
194C23×C59C22×D411yescharacteristic
195C22×D59—21no—
196C22×D59—21no—
197C22×D59—21no—
198C22×D59—21no—
199C22×D59—21no—
200C22×D59—21no—
201C22×D59C2×C4⋊C412yes—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
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×C59—41no—
215C2×C4×C59—41no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59C2×C4⋊C412yes—
219C2×C4×C59C2×C4⋊C412yes—
220C2×C4×C59C23×C411yescharacteristic
221C2×C4×C59C22×Q811yescharacteristic
222C2×C4×C59H(5,𝔽2)11yescharacteristic
223C2×C4×C59H(5,𝔽2)11yescharacteristic
224C2×C4×C592−1+411yescharacteristic
225C2×C4×C592−1+411yescharacteristic
226C2×Dic59—21no—
227C2×Dic59—21no—
228C2×Dic59—21no—
229C2×Dic59—21no—
230C2×Dic59—21no—
231C2×Dic59—21no—
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×Dic59C2×C4⋊C412yes—
245C2×Dic59C2×C4⋊C412yes—
246C2×Dic59C2×C4⋊C412yes—
247C4×D59—22no—
248C4×D59—22no—
249C4×D59—22no—
250C4×D59—22no—
251C8×C59—22no—
252C8×C59—22no—
253C59⋊C8—22no—
254C59⋊C8—22no—
255#21966—591nomaximal, Sylow
256C22×C59C22×C4⋊C411yescharacteristic
257C22×C59#53711yescharacteristic
258C22×C59#53711yescharacteristic
259C22×C59#57611yescharacteristic
260C22×C59#57711yescharacteristic
261C22×C59#57711yescharacteristic
262C22×C59#57811yescharacteristic
263C2×D59—22no—
264C2×D59—22no—
265C2×D59—22no—
266C2×D59—41no—
267C2×D59—41no—
268C2×D59—41no—
269C2×D59—41no—
270C4×C59—22no—
271C4×C59—22no—
272C4×C59—81no—
273C4×C59—81no—
274C4×C59—81no—
275C4×C59—81no—
276C4×C59#51811yescharacteristic
277C4×C59#51911yescharacteristic
278C4×C59#54411yescharacteristic
279C4×C59#54411yescharacteristic
280Dic59—22no—
281Dic59—22no—
282Dic59—22no—
283Dic59—81no—
284Dic59—81no—
285Dic59—81no—
286Dic59—81no—
287C2×#424—591no—
288#2314—591no—
289#2314—591no—
290#1949—591no—
291#1949—591no—
292#1949—591no—
293#1949D5911yescharacteristic, complemented
294#1948—591no—
295#1948—591no—
296#1947—591no—
297#1947—591no—
298#1809—591no—
299#1809—591no—
300#1808—591no—
301#1807—591no—
302C2×C59#231611yescharacteristic
303C2×C59#291211yescharacteristic
304C2×C59#291311yescharacteristic
305C2×C59#325711yescharacteristic
306C2×C59#325711yescharacteristic
307C2×C59#325811yescharacteristic
308C2×C59#325811yescharacteristic
309D59—42no—
310C22×M16—591no—
311C23×C8—591no—
312C2×#117—591no—
313C2×#107—591no—
314C2×#107—591no—
315C2×#107—591no—
316C2×#107C2×D5911yescharacteristic
317C2×#106—591no—
318C2×#106—591no—
319C2×#106—591no—
320C2×#106C2×D5911yescharacteristic
321#424—598no—
322C2×C4⋊C8—591no—
323C2×C4⋊C8—591no—
324C2×C4⋊C8—591no—
325C2×C4⋊C8C2×D5911yescharacteristic
326#388—1181no—
327#388—1181no—
328#388—1181no—
329#388—1181no—
330#387—1181no—
331#387—1181no—
332#387—1181no—
333#387—1181no—
334#384—591no—
335#384—591no—
336#382—591no—
337#382C2×D5911yescharacteristic
338#380—591no—
339#380—591no—
340#380—591no—
341#380—591no—
342C4×C4⋊C4—591no—
343C4×C4⋊C4—591no—
344C4×C4⋊C4—591no—
345C4×C4⋊C4C2×D5911yescharacteristic
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#340—591no—
351#340—591no—
352#340C2×D5911yescharacteristic
353#340C2×D5911yescharacteristic
354C59#2196611yescharacteristic, complemented, Sylow
355C23×C4—591no—
356C2×M16—594no—
357C2×M16—2361no—
358C2×M16—2361no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—594no—
362C22×C8—2361no—
363C22×C8—2361no—
364C22×C8C22×D5911yescharacteristic
365C22×C8C22×D5911yescharacteristic
366#117—594no—
367#117—594no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—591no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4C22×D5911yescharacteristic
379C2×C4⋊C4C22×D5911yescharacteristic
380C2×C4⋊C4C22×D5911yescharacteristic
381C2×C4⋊C4C22×D5911yescharacteristic
382C2×C22⋊C4—591no—
383C2×C22⋊C4—591no—
384C2×C22⋊C4—1181no—
385C2×C22⋊C4—1181no—
386C2×C22⋊C4—1181no—
387C2×C22⋊C4—1181no—
388C2×C42—21no—
389C2×C42—591no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42C22×D5911yescharacteristic
394#107—41no—
395#107—2361no—
396#107—2361no—
397#107—2361no—
398#106—41no—
399#106—2361no—
400#106—2361no—
401#106—2361no—
402C4⋊C8—594no—
403C4⋊C8—594no—
404C4⋊C8—594no—
405C4⋊C8C4×D5914yes—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—1181no—
410#95—1181no—
411#95—1181no—
412#95—1181no—
413#95—1181no—
414#95—1181no—
415#95—1181no—
416#95—1181no—
417#95—1181no—
418C24—591no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—591no—
424C22×C4—591no—
425C22×C4—591no—
426C22×C4—592no—
427C22×C4—592no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4—1181no—
433C22×C4—1181no—
434C22×C4—1181no—
435C22×C4—1181no—
436C22×C4D4×D5911yescharacteristic
437C22×C4D4×D5911yescharacteristic
438C22×C4C23×D5911yescharacteristic
439M16—2364no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—592no—
445C2×C8—592no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—2364no—
451C2×C8C2×C4×D5912yes—
452C2×C8C2×C4×D5912yes—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—594no—
462C4⋊C4—594no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—1181no—
470C4⋊C4—1181no—
471C4⋊C4—4721no—
472C4⋊C4—4721no—
473C4⋊C4—4721no—
474C4⋊C4—4721no—
475C22⋊C4—594no—
476C22⋊C4—594no—
477C22⋊C4—4721no—
478C22⋊C4—4721no—
479C22⋊C4—4721no—
480C22⋊C4—4721no—
481C42—81no—
482C42—592no—
483C42—592no—
484C42—4721no—
485C42—4721no—
486C42—4721no—
487C42C2×C4×D5912yes—
488C42Q8×D5912yes—
489C23—592no—
490C23—1181no—
491C23—1181no—
492C23—1181no—
493C23—1181no—
494C23—1181no—
495C23—1181no—
496C23C2×D4×D5911yescharacteristic
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
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—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—1182no—
523C2×C4—1182no—
524C2×C4—1182no—
525C2×C4—1182no—
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—2361no—
538C2×C4C4⋊C4×D5912yes—
539C2×C4C4⋊C4×D5912yes—
540C2×C4C22×C4×D5911yescharacteristic
541C2×C4C2×Q8×D5911yescharacteristic
542C2×C4#1427377211yescharacteristic
543C2×C4#1427377311yescharacteristic
544C2×C4#1427377411yescharacteristic
545C2×C4#1427377511yescharacteristic
546C8—22no—
547C8—22no—
548C8—1182no—
549C8—1182no—
550C22—1182no—
551C22—1182no—
552C22—1182no—
553C22—2361no—
554C22—2361no—
555C22—2361no—
556C22—2361no—
557C22C2×C4⋊C4×D5911yescharacteristic
558C22#1683798111yescharacteristic
559C22#1683798211yescharacteristic
560C22#1683820411yescharacteristic
561C22#1683821111yescharacteristic
562C22#1683821211yescharacteristic
563C22#1683821611yescharacteristic
564C4—22no—
565C4—22no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—81no—
570C4—1182no—
571C4—1182no—
572C4—1182no—
573C4—4721no—
574C4—4721no—
575C4—4721no—
576C4—4721no—
577C4#1683787411yescharacteristic
578C4#1683787811yescharacteristic
579C4#1683802911yescharacteristic
580C4#1683803211yescharacteristic
581C2—2362no—
582C2#2142671111yescharacteristic
583C2#2143269011yescharacteristic
584C2#2143269311yescharacteristic
585C2#2143550411yescharacteristic
586C2#2143551311yescharacteristic
587C2#2143552611yescharacteristic
588C2#2143552711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1949C2
socleC23×C59C22×D4
prime radicalC23×D59C2×D4
cosocleC22×C4×C59C24
hypercenter#1949D59
hypocenterC59#21966
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #21424867C2semidirect
5. #21424867C2semidirect
6. #21424862C2semidirect
7. #21424859C2semidirect
8. #21424856C2semidirect
9. #21424855C2semidirect
10. #21424851C2semidirect
15. C59×#1949C2semidirect
25. #16837427C4semidirect
293. #1949D59semidirect
354. C59#21966semidirect

Maximal subgroups

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

subgroupindex
C2×#168374272
#214267062
#214267012
#214248672
#214248622
#214248592
#214248562
#214248552
#214248512
#214241282
#214241262
#214241222
#214241202
C59×#19492
#2196659