Finite Group Databasebeta

← #28651548

Subgroups — #28651548

Order 15104.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837710C211yesmaximal, characteristic, complemented
2C2×#16837444C211yesmaximal, characteristic
3#21427809C211yesmaximal, characteristic
4#21427099C211yesmaximal, characteristic
5#21425633C211yesmaximal, characteristic, complemented
6#21425584C211yesmaximal, characteristic, complemented
7#21425476C211yesmaximal, characteristic, complemented
8#21425435C211yesmaximal, characteristic, complemented
9#21424855C211yesmaximal, characteristic, complemented
10#21424675C211yesmaximal, characteristic, complemented
11#21424529C211yesmaximal, characteristic
12#21424406C211yesmaximal, characteristic
13#21424353C211yesmaximal, characteristic, complemented
14#21424286C211yesmaximal, characteristic
15C59×#1851C211yesmaximal, characteristic
16C2×#14273719C2211yescharacteristic
17C2×#14273706C2211yescharacteristic
18C2×C4×Dic118C2211yescharacteristic
19C2×#14273686C2211yescharacteristic
20C22×C4⋊Dic59C2211yescharacteristic
21#16837710—24no—
22#16837444—24no—
23C2×#14273661C2211yescharacteristic
24C2×#14273661C2211yescharacteristic
25C2×#14273661C2211yescharacteristic
26C2×#14273661C2211yescharacteristic
27C2×#14273659C2211yescharacteristic
28C2×#14273659C2211yescharacteristic
29C2×#14273652C2211yescharacteristic
30C2×#14273652C2211yescharacteristic
31C2×#14273652C2211yescharacteristic
32C2×#14273652C2211yescharacteristic
33#16837338—21no—
34#16837325—21no—
35#16837308C2211yescharacteristic
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837300C2211yescharacteristic
39#16837299—21no—
40#16837299—21no—
41#16837289—21no—
42#16837285—21no—
43#16837274—21no—
44#16837272C2211yescharacteristic
45#16837262C2211yescharacteristic
46C4×C4⋊Dic59C2211yescharacteristic
47C22⋊C4×Dic59—21no—
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53#16837165C2211yescharacteristic
54C22×M16×C59C2211yescharacteristic
55C2×C4⋊C8×C59C2211yescharacteristic
56C2×C4⋊C8×C59C2211yescharacteristic
57C2×C22⋊C8×C59C2211yescharacteristic
58C4×C22⋊C4×C59C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C59×#336C2211yescharacteristic
61C22×Dic118C2311yescharacteristic
62C23×Dic59—21no—
63#14273719—22no—
64#14273719—41no—
65#14273706—22no—
66#14273706Q814yes—
67C4×Dic118—22no—
68C4×Dic118—41no—
69#14273686—22no—
70#14273686—41no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—21no—
75C2×C4⋊Dic59—41no—
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×C4⋊Dic59C2311yescharacteristic
82C2×C4⋊Dic59C2311yescharacteristic
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C2×Dic59⋊C4—21no—
88C2×Dic59⋊C4—21no—
89C2×Dic59⋊C4—21no—
90C2×C22⋊Dic59—21no—
91C2×C22⋊Dic59—21no—
92C2×C22⋊Dic59—21no—
93C2×C4×Dic59—21no—
94C2×C4×Dic59—21no—
95C2×C4×Dic59—21no—
96#14273661—22no—
97#14273661—22no—
98#14273661—22no—
99#14273661—41no—
100#14273659—22no—
101#14273659D414yes—
102#14273652—22no—
103#14273652—22no—
104#14273652—22no—
105#14273652—41no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C59⋊#95—21no—
109C59⋊#95—21no—
110C59⋊#95—21no—
111C59⋊#95—21no—
112C59⋊#95—21no—
113C23×C4×C59C2311yescharacteristic
114C2×M16×C59—22no—
115C2×M16×C59—22no—
116C2×M16×C59—22no—
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22×C8×C59C2311yescharacteristic
120C22×C8×C59C2311yescharacteristic
121C2×C22⋊C4×C59D411yescharacteristic
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C42×C59C2311yescharacteristic
124C2×C42×C59C2311yescharacteristic
125C4⋊C8×C59—22no—
126C4⋊C8×C59—22no—
127C22⋊C8×C59—22no—
128#95×C59D411yescharacteristic
129#95×C59D411yescharacteristic
130C24×C59C2×D411yescharacteristic
131C22×C4×C59—21no—
132C22×C4×C59—21no—
133C22×C4×C59—21no—
134C22×C4×C59—21no—
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2×Q812yes—
141C22×C4×C59#4112yes—
142C22×C4×C59C2411yescharacteristic
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—21no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—21no—
151C22×Dic59—42no—
152C22×Dic59—42no—
153C22×Dic59—81no—
154C2×Dic118—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118C2×Q812yes—
160M16×C59—28no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59C2×D412yes—
172C2×C8×C59#4112yes—
173C22⋊C4×C59—22no—
174C22⋊C4×C59—41no—
175C42×C59—21no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C22⋊Dic59—24no—
180C22⋊Dic59—42no—
181C22⋊Dic59—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—42no—
194C4⋊Dic59—42no—
195C4⋊Dic59—81no—
196C4⋊Dic59C2×D412yes—
197C4⋊Dic59C2×D412yes—
198C4⋊Dic59C2×Q812yes—
199Dic59⋊C4—24no—
200Dic59⋊C4—42no—
201Dic59⋊C4—42no—
202Dic59⋊C4—42no—
203Dic59⋊C4—42no—
204Dic59⋊C4—81no—
205Dic59⋊C4—81no—
206C4×Dic59—42no—
207C4×Dic59—42no—
208C4×Dic59—81no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—21no—
215C23×C59C22⋊Q812yes—
216C23×C59C22×D411yescharacteristic
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—22no—
222C2×C4×C59—22no—
223C2×C4×C59—22no—
224C2×C4×C59—22no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—21no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—21no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59C22⋊Q812yes—
235C2×C4×C59#13712yes—
236C2×C4×C59C22×D411yescharacteristic
237C2×C4×C59C22×Q811yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59C2×#4111yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C2×C4×C592−1+411yescharacteristic
242C2×Dic59—22no—
243C2×Dic59—22no—
244C2×Dic59—22no—
245C2×Dic59—22no—
246C2×Dic59—41no—
247C2×Dic59—41no—
248C2×Dic59—41no—
249C2×Dic59—41no—
250C2×Dic59—41no—
251C2×Dic59—41no—
252C2×Dic59—41no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—84no—
267Dic118—22no—
268Dic118—22no—
269Dic118—41no—
270Dic118—41no—
271C8×C59—22no—
272C8×C59—22no—
273C8×C59—41no—
274C8×C59—41no—
275#26362—591nomaximal, Sylow
276C22×C59—22no—
277C22×C59—22no—
278C22×C59—22no—
279C22×C59—41no—
280C22×C59—41no—
281C22×C59—41no—
282C22×C59—41no—
283C22×C59C2×C22⋊Q811yescharacteristic
284C22×C59D4211yescharacteristic
285C22×C59#54711yescharacteristic
286C22×C59C2×#13711yescharacteristic
287C22×C59#57511yescharacteristic
288C22×C59#57611yescharacteristic
289C22×C59#57811yescharacteristic
290C4×C59—22no—
291C4×C59—22no—
292C4×C59—41no—
293C4×C59—41no—
294C4×C59—41no—
295C4×C59#52511yescharacteristic
296C4×C59#53911yescharacteristic
297C4×C59D4×Q811yescharacteristic
298C4×C59#55411yescharacteristic
299Dic59—42no—
300Dic59—42no—
301Dic59—42no—
302Dic59—42no—
303Dic59—81no—
304Dic59—81no—
305Dic59—81no—
306Dic59—81no—
307C2×#479—591no—
308C2×#428—591no—
309#2441—591no—
310#2360—591no—
311#2095—591no—
312#2086—591no—
313#2066—591no—
314#2059—591no—
315#1947—591no—
316#1919—591no—
317#1894—591no—
318#1873—591no—
319#1863—591no—
320#1852—591no—
321#1851D5911yescharacteristic
322C2×C59—42no—
323C2×C59#244911yescharacteristic
324C2×C59#307811yescharacteristic
325C2×C59#308311yescharacteristic
326C2×C59#326811yescharacteristic
327C2×C59#327611yescharacteristic
328C2×C59#334611yescharacteristic
329C2×C59#335211yescharacteristic
330C22×M16C2×D5911yescharacteristic
331C2×#125—591no—
332C2×C22⋊Q8—591no—
333C2×C4×Q8—591no—
334C2×#117—591no—
335C22×C4⋊C4—591no—
336#479—1184no—
337#428—1184no—
338C2×#107—591no—
339C2×#107—591no—
340C2×#107—591no—
341C2×#107—591no—
342C2×#106—591no—
343C2×#106—591no—
344C2×C4⋊C8C2×D5911yescharacteristic
345C2×C4⋊C8C2×D5911yescharacteristic
346C2×#103—591no—
347C2×#103—591no—
348C2×#103—591no—
349C2×#103—591no—
350C2×C22⋊C8C2×D5911yescharacteristic
351#399—1181no—
352#396—1181no—
353#391—591no—
354#389—591no—
355#389—591no—
356#389—591no—
357#389—1181no—
358#389—1181no—
359#387—1181no—
360#387—1181no—
361#385—1181no—
362#384—591no—
363#380—591no—
364C4×C4⋊C4—591no—
365C4×C22⋊C4—1181no—
366C4×C22⋊C4C2×D5911yescharacteristic
367#340—591no—
368#340—591no—
369#340—591no—
370#340—591no—
371#340—591no—
372#340—591no—
373#336C2×D5911yescharacteristic
374#336C2×D5911yescharacteristic
375C59#2636211yescharacteristic, complemented, Sylow
376C22×Q8—591no—
377C23×C4—1181no—
378C23×C4C22×D5911yescharacteristic
379C2×M16—22no—
380C2×M16—22no—
381C2×M16—22no—
382C22×C8C22×D5911yescharacteristic
383C22×C8C22×D5911yescharacteristic
384C22×C8C22×D5911yescharacteristic
385C22×C8C22×D5911yescharacteristic
386#125—1182no—
387#125—2361no—
388C22⋊Q8—594no—
389C22⋊Q8—1182no—
390C4×Q8—1182no—
391C4×Q8—2361no—
392#117—1182no—
393#117—2361no—
394C2×C4⋊C4—591no—
395C2×C4⋊C4—591no—
396C2×C4⋊C4—591no—
397C2×C4⋊C4—591no—
398C2×C4⋊C4—591no—
399C2×C4⋊C4—591no—
400C2×C4⋊C4—591no—
401C2×C4⋊C4—1182no—
402C2×C4⋊C4—1181no—
403C2×C4⋊C4—1181no—
404C2×C4⋊C4—1181no—
405C2×C4⋊C4—1181no—
406C2×C4⋊C4—1181no—
407C2×C4⋊C4—1181no—
408C2×C4⋊C4—1181no—
409C2×C4⋊C4—1181no—
410C2×C4⋊C4—1181no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—2361no—
413C2×C22⋊C4—1181no—
414C2×C22⋊C4—1181no—
415C2×C22⋊C4—1181no—
416C2×C22⋊C4D23611yescharacteristic
417C2×C22⋊C4D23611yescharacteristic
418C2×C42—1181no—
419C2×C42—1181no—
420C2×C42—1181no—
421C2×C42C22×D5911yescharacteristic
422C2×C42C22×D5911yescharacteristic
423#107—1182no—
424#107—1182no—
425#107—1182no—
426#107—2361no—
427#106—594no—
428#106—1182no—
429C4⋊C8—22no—
430C4⋊C8—22no—
431#103—1182no—
432#103—1182no—
433#103—1182no—
434#103—2361no—
435C22⋊C8—22no—
436#95—1181no—
437#95—1181no—
438#95—1181no—
439#95—1181no—
440#95—1181no—
441#95—1181no—
442#95—1181no—
443#95D23611yescharacteristic
444#95D23611yescharacteristic
445C24C2×D23611yescharacteristic
446C2×Q8—592no—
447C2×Q8—1181no—
448C2×Q8—1181no—
449C2×Q8—1181no—
450C2×Q8—1181no—
451C2×Q8—1181no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—4721no—
465C22×C4—2362no—
466C22×C4—2362no—
467C22×C4C2×D23611yescharacteristic
468C22×C4C2×D23611yescharacteristic
469C22×C4C2×D23611yescharacteristic
470C22×C4C2×D23611yescharacteristic
471C22×C4C2×D23611yescharacteristic
472C22×C4#1174568712yes—
473C22×C4Q8×D5912yes—
474C22×C4C23×D5911yescharacteristic
475M16—28no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8D4×D5912yes—
487C2×C8#1174568712yes—
488C4⋊C4—592no—
489C4⋊C4—592no—
490C4⋊C4—592no—
491C4⋊C4—1184no—
492C4⋊C4—1181no—
493C4⋊C4—1181no—
494C4⋊C4—1181no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C4⋊C4—1181no—
502C4⋊C4—1181no—
503C4⋊C4—4721no—
504C4⋊C4—4721no—
505C4⋊C4—4721no—
506C4⋊C4—2362no—
507C4⋊C4—2362no—
508C4⋊C4—2362no—
509C4⋊C4—2362no—
510C4⋊C4—2362no—
511C4⋊C4—2362no—
512C22⋊C4—22no—
513C22⋊C4—41no—
514C22⋊C4—1184no—
515C22⋊C4—4721no—
516C22⋊C4—2362no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—4721no—
522C42—2362no—
523C42—2362no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1427370712yes—
531C23C22×D23611yescharacteristic
532Q8—1182no—
533Q8—1182no—
534Q8—2361no—
535Q8—2361no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—1182no—
554C2×C4—1182no—
555C2×C4—1182no—
556C2×C4—1182no—
557C2×C4—4724no—
558C2×C4—2361no—
559C2×C4—2361no—
560C2×C4—2361no—
561C2×C4—2361no—
562C2×C4—2361no—
563C2×C4—2361no—
564C2×C4—2361no—
565C2×C4—2361no—
566C2×C4—2361no—
567C2×C4—2361no—
568C2×C4—2361no—
569C2×C4—2361no—
570C2×C4—2361no—
571C2×C4—2361no—
572C2×C4—2361no—
573C2×C4—2361no—
574C2×C4—2361no—
575C2×C4—2361no—
576C2×C4—2361no—
577C2×C4—2361no—
578C2×C4#1427370712yes—
579C2×C4#1427376012yes—
580C2×C4C2×D4×D5911yescharacteristic
581C2×C4C2×Q8×D5911yescharacteristic
582C2×C4C2×#1174568711yescharacteristic
583C2×C4C2×#1174568711yescharacteristic
584C2×C4#1427377311yescharacteristic
585C2×C4#1427377411yescharacteristic
586C8—22no—
587C8—22no—
588C8—41no—
589C8—41no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1427370711yescharacteristic
598C22D4×D23611yescharacteristic
599C22#1683805611yescharacteristic
600C22C2×#1427376011yescharacteristic
601C22#1683820211yescharacteristic
602C22#1683820711yescharacteristic
603C22#1683821911yescharacteristic
604C4—22no—
605C4—22no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—4721no—
610C4—4721no—
611C4—4721no—
612C4—4721no—
613C4—2362no—
614C4—2362no—
615C4—2362no—
616C4—2362no—
617C4#1683790811yescharacteristic
618C4#1683799611yescharacteristic
619C4#1683806511yescharacteristic
620C4#1683811011yescharacteristic
621C2—42no—
622C2#2142790311yescharacteristic
623C2#2143390311yescharacteristic
624C2#2143394911yescharacteristic
625C2#2143564111yescharacteristic
626C2#2143574411yescharacteristic
627C2#2143658311yescharacteristic
628C2#2143667011yescharacteristic
subgroupG/H
centerC23C22×D236
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#1851C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenter#1851D59
hypocenterC59#26362
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#16837710C2semidirect
5. #21425633C2semidirect
6. #21425584C2semidirect
7. #21425476C2semidirect
8. #21425435C2semidirect
9. #21424855C2semidirect
10. #21424675C2semidirect
13. #21424353C2semidirect
375. C59#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168377102
C2×#168374442
#214278092
#214270992
#214256332
#214255842
#214254762
#214254352
#214248552
#214246752
#214245292
#214244062
#214243532
#214242862
C59×#18512
#2636259