Finite Group Databasebeta

← #28610700

Subgroups — #28610700

Order 15104.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837703C211yesmaximal, characteristic
2#21427190C211yesmaximal, characteristic, complemented
3C4⋊C4×D236C211yesmaximal, characteristic
4#21425065C211yesmaximal, characteristic, complemented
5#21424860C211yesmaximal, characteristic, complemented
6#21424834C211yesmaximal, characteristic, complemented
7#21424824C211yesmaximal, characteristic, complemented
8#21424682C211yesmaximal, characteristic, complemented
9#21424668C211yesmaximal, characteristic, complemented
10#21424360C211yesmaximal, characteristic
11#21424360C211yesmaximal, characteristic
12#21423922C211yesmaximal, characteristic
13#21423922C211yesmaximal, characteristic
14C4×#14273649C211yesmaximal, characteristic
15C59×#1771C211yesmaximal, characteristic, complemented
16C2×#14273719C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×C4×D236C2211yescharacteristic
19C2×C4×D236C2211yescharacteristic
20C2×C4⋊C4×D59—21no—
21#16837703C418yescomplemented
22C2×#14273661C2211yescharacteristic
23C2×#14273659C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273659C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29C2×#14273649C2211yescharacteristic
30#16837300C2211yescharacteristic
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837297—21no—
35#16837284—21no—
36#16837283—21no—
37#16837276—21no—
38#16837276—21no—
39#16837272C2211yescharacteristic
40#16837259—21no—
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C4×C4⋊Dic59C2211yescharacteristic
44#16837253—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C2×C4⋊C8×C59C2211yescharacteristic
54C2×#97×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C4⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C4×D59—21no—
62C22×C4×D59—21no—
63C22×C4×D59—21no—
64#14273719—22no—
65#14273719C2×C414yescomplemented
66C4×D236—22no—
67C4×D236—22no—
68C4×D236—22no—
69C4×D236—41no—
70C4×D236—41no—
71C4×D236C2×C414yes—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×Dic59⋊C4—21no—
85C2×Dic59⋊C4—21no—
86C2×Dic59⋊C4—21no—
87C4⋊C4×D59—44no—
88C2×(C2×D59)⋊C4—21no—
89C2×(C2×D59)⋊C4—21no—
90C2×(C2×D59)⋊C4—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93C2×C4×Dic59—21no—
94#14273661C2×C414yescomplemented
95#14273659—41no—
96#14273659—41no—
97#14273659C2×C414yescomplemented
98#14273649—22no—
99#14273649C2×C414yes—
100#14273649C2×C414yes—
101#14273649Q814yes—
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
111C22×C8×C59C2311yescharacteristic
112C22×C8×C59C2311yescharacteristic
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C42×C59C2311yescharacteristic
116C2×C42×C59C2311yescharacteristic
117C2×C42×C59C2311yescharacteristic
118C4⋊C8×C59C2×C414yescomplemented
119#97×C59—22no—
120#97×C59—41no—
121C4×C8×C59—41no—
122#95×C59D411yescharacteristic
123#95×C59D411yescharacteristic
124C23×D59—21no—
125C22×C4×C59C2×D411yescharacteristic
126C22×C4×C59C2×D411yescharacteristic
127C22×C4×C59C2×D411yescharacteristic
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2411yescharacteristic
132C22×Dic59—21no—
133C22×Dic59—21no—
134C22×Dic59—21no—
135C22×Dic59—21no—
136C22×Dic59—21no—
137C22×Dic59—21no—
138C22×Dic59—21no—
139C2×D236—21no—
140C2×D236—21no—
141C2×D236—21no—
142C2×D236C22×C412yes—
143C2×D236C2×Q812yes—
144C2×D236#4112yes—
145C2×C4×D59—24no—
146C2×C4×D59—42no—
147C2×C4×D59—42no—
148C2×C4×D59—42no—
149C2×C4×D59—42no—
150C2×C4×D59—42no—
151C2×C4×D59—42no—
152C2×C4×D59—81no—
153C2×C4×D59—81no—
154C2×C8×C59—21no—
155C2×C8×C59—21no—
156C2×C8×C59—21no—
157C2×C8×C59—21no—
158C2×C8×C59—21no—
159C2×C8×C59—21no—
160C2×C8×C59—21no—
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59C22×C412yes—
164C2×C8×C59C22×C412yes—
165C2×C8×C59C2×Q812yes—
166C4⋊C4×C59—22no—
167C4⋊C4×C59—41no—
168C42×C59—21no—
169C42×C59—21no—
170C42×C59—21no—
171C42×C59—21no—
172C42×C59C22×C412yes—
173C42×C59#4112yes—
174(C2×D59)⋊C4—24no—
175(C2×D59)⋊C4—42no—
176(C2×D59)⋊C4—42no—
177C4⋊Dic59—21no—
178C4⋊Dic59—21no—
179C4⋊Dic59—21no—
180C4⋊Dic59—21no—
181C4⋊Dic59—21no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—42no—
188C4⋊Dic59—81no—
189C4⋊Dic59C22×C412yes—
190C4⋊Dic59C22×C412yes—
191C4⋊Dic59C22×C412yes—
192C4⋊Dic59C2×Q812yes—
193Dic59⋊C4—24no—
194Dic59⋊C4—24no—
195Dic59⋊C4—42no—
196Dic59⋊C4—42no—
197Dic59⋊C4—42no—
198Dic59⋊C4—81no—
199C4×Dic59—42no—
200C4×Dic59—81no—
201C4×Dic59—81no—
202C23×C59C22×D411yescharacteristic
203C22×D59—22no—
204C22×D59—22no—
205C22×D59—22no—
206C22×D59—41no—
207C22×D59—41no—
208C22×D59—41no—
209C22×D59—41no—
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×C59—21no—
217C2×C4×C59—21no—
218C2×C4×C59—21no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59C4×D412yes—
226C2×C4×C59C4×D412yes—
227C2×C4×C59AGL(1,ℤ/8)12yes—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×Q811yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C2×C4×C59C2×#4111yescharacteristic
232C2×C4×C59H(5,𝔽2)11yescharacteristic
233C2×C4×C592−1+411yescharacteristic
234C2×Dic59—22no—
235C2×Dic59—22no—
236C2×Dic59—22no—
237C2×Dic59—22no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244C2×Dic59—41no—
245C2×Dic59—41no—
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—
255D236—22no—
256D236—22no—
257D236—22no—
258D236C4×Q814yes—
259C4×D59—48no—
260C4×D59—84no—
261C4×D59—84no—
262C8×C59—22no—
263C8×C59—22no—
264C8×C59—22no—
265C8×C59—41no—
266#22978—591nomaximal, Sylow
267C22×C59C2×C4×D411yescharacteristic
268C22×C59#54611yescharacteristic
269C22×C59D4×Q811yescharacteristic
270C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C59#57511yescharacteristic
272C22×C59#57711yescharacteristic
273C22×C59#57711yescharacteristic
274C2×D59—24no—
275C2×D59—42no—
276C2×D59—42no—
277C2×D59—42no—
278C2×D59—42no—
279C2×D59—42no—
280C2×D59—42no—
281C4×C59—22no—
282C4×C59—22no—
283C4×C59—22no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59C2×C4×Q811yescharacteristic
288C4×C59#52011yescharacteristic
289C4×C59#55211yescharacteristic
290C4×C59#55411yescharacteristic
291Dic59—42no—
292Dic59—42no—
293Dic59—42no—
294Dic59—42no—
295Dic59—81no—
296Dic59—81no—
297Dic59—81no—
298C2×#478—591no—
299#2370—591no—
300D4×C4⋊C4—591no—
301#1992—591no—
302#1948—591no—
303#1943—591no—
304#1941—591no—
305#1920—591no—
306#1918—591no—
307#1864—591no—
308#1864—591no—
309#1771D5911yescharacteristic, complemented
310#1762—591no—
311#1762—591no—
312C4×#102—591no—
313C2×C59#236011yescharacteristic
314C2×C59C4×AGL(1,ℤ/8)11yescharacteristic
315C2×C59#295211yescharacteristic
316C2×C59#330711yescharacteristic
317C2×C59#331011yescharacteristic
318C2×C59#338411yescharacteristic
319C2×C59#338811yescharacteristic
320D59—44no—
321C2×#125—591no—
322C2×C4×D4—591no—
323C2×C4×D4—591no—
324C2×C4×D4—591no—
325C22×C4⋊C4—1181no—
326#478—598no—
327C2×#107—591no—
328C2×#106—591no—
329C2×#106—591no—
330C2×#106—591no—
331C2×C4⋊C8C2×D5911yescharacteristic
332C2×#102—591no—
333C2×#102—591no—
334C2×#102—591no—
335C2×#102—591no—
336C2×#97C2×D5911yescharacteristic
337C2×#97C2×D5911yescharacteristic
338C2×C4×C8C2×D5911yescharacteristic
339#389—591no—
340#389—591no—
341#389—591no—
342#389—591no—
343#389—1181no—
344#387—1181no—
345#387—1181no—
346#385—1181no—
347#385—1181no—
348#384—591no—
349#380—1181no—
350C4×C4⋊C4—591no—
351C4×C4⋊C4—591no—
352C4×C4⋊C4—591no—
353C4×C4⋊C4—1181no—
354C4×C4⋊C4C2×D5911yescharacteristic
355#340—591no—
356#340—591no—
357#340—591no—
358#340—591no—
359#340—591no—
360#340—591no—
361#340—591no—
362#340—591no—
363#336C2×D5911yescharacteristic
364#336C2×D5911yescharacteristic
365C59#2297811yescharacteristic, complemented, Sylow
366C22×D4—591no—
367C23×C4—1181no—
368C23×C4—1181no—
369C23×C4—1181no—
370C22×C8C22×D5911yescharacteristic
371C22×C8C22×D5911yescharacteristic
372C22×C8C22×D5911yescharacteristic
373C22×C8C22×D5911yescharacteristic
374#125—594no—
375#125—1182no—
376C4×D4—594no—
377C4×D4—1182no—
378C4×D4—1182no—
379C4×D4—1182no—
380C4×D4—2361no—
381C4×D4—2361no—
382C2×C4⋊C4—591no—
383C2×C4⋊C4—591no—
384C2×C4⋊C4—591no—
385C2×C4⋊C4—591no—
386C2×C4⋊C4—591no—
387C2×C4⋊C4—591no—
388C2×C4⋊C4—591no—
389C2×C4⋊C4—1181no—
390C2×C4⋊C4—1181no—
391C2×C4⋊C4—1181no—
392C2×C4⋊C4—1181no—
393C2×C4⋊C4—1181no—
394C2×C4⋊C4—1181no—
395C2×C4⋊C4—1181no—
396C2×C4⋊C4—1181no—
397C2×C4⋊C4—2364no—
398C2×C4⋊C4D23611yescharacteristic
399C2×C4⋊C4D23611yescharacteristic
400C2×C22⋊C4—1181no—
401C2×C22⋊C4—1181no—
402C2×C22⋊C4—1181no—
403C2×C42—1181no—
404C2×C42—1181no—
405C2×C42—1181no—
406C2×C42C22×D5911yescharacteristic
407C2×C42C22×D5911yescharacteristic
408C2×C42C22×D5911yescharacteristic
409#107—594no—
410#106—594no—
411#106—2361no—
412#106—2361no—
413C4⋊C8C4×D5914yescomplemented
414#102—594no—
415#102—594no—
416#102—594no—
417#102—1182no—
418#97—22no—
419#97—41no—
420C4×C8—41no—
421#95—1181no—
422#95—1181no—
423#95—1181no—
424#95—1181no—
425#95—1181no—
426#95—1181no—
427#95—1181no—
428#95D23611yescharacteristic
429#95D23611yescharacteristic
430C24—1181no—
431C2×D4—592no—
432C2×D4—592no—
433C2×D4—592no—
434C2×D4—1181no—
435C2×D4—1181no—
436C2×D4—1181no—
437C22×C4—1181no—
438C22×C4—1181no—
439C22×C4—1181no—
440C22×C4—1181no—
441C22×C4—1184no—
442C22×C4—1181no—
443C22×C4—1181no—
444C22×C4—1181no—
445C22×C4—4721no—
446C22×C4—4721no—
447C22×C4—2362no—
448C22×C4—2362no—
449C22×C4—2362no—
450C22×C4—2362no—
451C22×C4—2362no—
452C22×C4—2362no—
453C22×C4C2×D23611yescharacteristic
454C22×C4C2×D23611yescharacteristic
455C22×C4C2×D23611yescharacteristic
456C22×C4C2×D23611yescharacteristic
457C22×C4C2×D23611yescharacteristic
458C22×C4C2×D23611yescharacteristic
459C22×C4C23×D5911yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8C2×C4×D5912yes—
470C2×C8C2×C4×D5912yes—
471C2×C8Q8×D5912yes—
472C4⋊C4—22no—
473C4⋊C4—41no—
474C4⋊C4—592no—
475C4⋊C4—592no—
476C4⋊C4—592no—
477C4⋊C4—592no—
478C4⋊C4—1184no—
479C4⋊C4—1184no—
480C4⋊C4—1181no—
481C4⋊C4—1181no—
482C4⋊C4—1181no—
483C4⋊C4—1181no—
484C4⋊C4—1181no—
485C4⋊C4—1181no—
486C4⋊C4—1181no—
487C4⋊C4—1181no—
488C4⋊C4—1181no—
489C4⋊C4—1181no—
490C4⋊C4—4721no—
491C4⋊C4—4721no—
492C4⋊C4—2362no—
493C4⋊C4—2362no—
494C4⋊C4—2362no—
495C4⋊C4—2362no—
496C22⋊C4—1184no—
497C22⋊C4—2362no—
498C22⋊C4—2362no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—4721no—
504C42—4721no—
505C42—2362no—
506C42C2×C4×D5912yes—
507C42#1174568512yes—
508C23—1182no—
509C23—1182no—
510C23—1182no—
511C23—2361no—
512C23—2361no—
513C23—2361no—
514C23—2361no—
515C23C22×D23611yescharacteristic
516D4—594no—
517D4—1182no—
518D4—1182no—
519D4—1182no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—1182no—
536C2×C4—1182no—
537C2×C4—1182no—
538C2×C4—1182no—
539C2×C4—4724no—
540C2×C4—4724no—
541C2×C4—2368no—
542C2×C4—2361no—
543C2×C4—2361no—
544C2×C4—2361no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4—2361no—
555C2×C4—2361no—
556C2×C4—2361no—
557C2×C4—2361no—
558C2×C4—2361no—
559C2×C4C4×D23612yes—
560C2×C4C4×D23612yes—
561C2×C4#1427375512yes—
562C2×C4C22×C4×D5911yescharacteristic
563C2×C4C2×Q8×D5911yescharacteristic
564C2×C4#41×D5911yescharacteristic
565C2×C4C2×#1174568511yescharacteristic
566C2×C4#1427377211yescharacteristic
567C2×C4#1427377411yescharacteristic
568C8—22no—
569C8—22no—
570C8—22no—
571C8—41no—
572C22—1184no—
573C22—2362no—
574C22—2362no—
575C22—2362no—
576C22—2362no—
577C22—2362no—
578C22—2362no—
579C22C2×C4×D23611yescharacteristic
580C22#1683804711yescharacteristic
581C22Q8×D23611yescharacteristic
582C22C2×#1427375511yescharacteristic
583C22#1683820211yescharacteristic
584C22#1683821411yescharacteristic
585C22#1683821411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—4721no—
593C4—4721no—
594C4—4721no—
595C4—2362no—
596C4—2362no—
597C4—2362no—
598C4—2362no—
599C4C4×Q8×D5911yescharacteristic
600C4#1683788111yescharacteristic
601C4#1683809111yescharacteristic
602C4#1683810711yescharacteristic
603C2—2364no—
604C2#2142710311yescharacteristic
605C2C4×#1427375511yescharacteristic
606C2#2143298111yescharacteristic
607C2#2143607411yescharacteristic
608C2#2143611911yescharacteristic
609C2#2143699711yescharacteristic
610C2#2143704311yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1771C2
socleC23×C59C22×D4
prime radicalC22×D236C23
cosocleC22×C4×C59C24
hypercenter#1771D59
hypocenterC59#22978
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. #21427190C2semidirect
4. #21425065C2semidirect
5. #21424860C2semidirect
6. #21424834C2semidirect
7. #21424824C2semidirect
8. #21424682C2semidirect
9. #21424668C2semidirect
15. C59×#1771C2semidirect
21. #16837703C4semidirect
65. #14273719C2×C4semidirect
94. #14273661C2×C4semidirect
97. #14273659C2×C4semidirect
118. C4⋊C8×C59C2×C4semidirect
309. #1771D59semidirect
365. C59#22978semidirect
413. C4⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377032
#214271902
C4⋊C4×D2362
#214250652
#214248602
#214248342
#214248242
#214246822
#214246682
#214243602
#214239222
C4×#142736492
C59×#17712
#2297859