Finite Group Databasebeta

← #28651604

Subgroups — #28651604

Order 15104.

896 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837702C211yesmaximal, characteristic, complemented
2C2×#16837442C211yesmaximal, characteristic, complemented
3#21427809C211yesmaximal, characteristic, complemented
4#21425629C211yesmaximal, characteristic, complemented
5#21425598C211yesmaximal, characteristic, complemented
6#21425484C211yesmaximal, characteristic, complemented
7#21425434C211yesmaximal, characteristic, complemented
8#21424859C211yesmaximal, characteristic, complemented
9#21424667C211yesmaximal, characteristic, complemented
10#21424535C211yesmaximal, characteristic
11#21424419C211yesmaximal, characteristic
12#21424357C211yesmaximal, characteristic, complemented
13#21424289C211yesmaximal, characteristic, complemented
14#21424281C211yesmaximal, characteristic, complemented
15C59×#2361C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×#14273719C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19#16837702—24no—
20#16837442—24no—
21C2×#14273660C2211yescharacteristic
22C2×#14273660C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×#14273658C2211yescharacteristic
25C2×#14273658C2211yescharacteristic
26C2×#14273658C2211yescharacteristic
27C2×C4⋊(C59⋊C8)C2211yescharacteristic
28C2×C4⋊(C59⋊C8)C2211yescharacteristic
29C2×#14273648C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×C22⋊(C59⋊C8)C2211yescharacteristic
34#16837338—21no—
35#16837325—21no—
36#16837300C2211yescharacteristic
37#16837300C2211yescharacteristic
38#16837299—21no—
39#16837285—21no—
40#16837272C2211yescharacteristic
41#16837262C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#121×C59C2211yescharacteristic
51C2×C4×D4×C59—21no—
52C2×C4×D4×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#390C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#388—21no—
57C59×#386—21no—
58C59×#385—21no—
59C4×C4⋊C4×C59C2211yescharacteristic
60C4×C22⋊C4×C59C2211yescharacteristic
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664—22no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66C22×C59⋊C8C2311yescharacteristic
67C22×C59⋊C8C2311yescharacteristic
68#14273719—22no—
69#14273719—41no—
70#14273686—22no—
71#14273686—41no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×Dic59⋊C4—21no—
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×C22⋊Dic59—21no—
83C2×C22⋊Dic59—21no—
84C2×C4×Dic59—21no—
85#14273660—22no—
86#14273660D414yescomplemented
87#14273658—22no—
88#14273658—22no—
89#14273658—22no—
90#14273658—41no—
91C4⋊(C59⋊C8)—22no—
92C4⋊(C59⋊C8)—22no—
93#14273648—22no—
94#14273648—22no—
95#14273648—22no—
96#14273648—41no—
97C22⋊(C59⋊C8)—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C22×D4×C59—21no—
102C22×D4×C59C2311yescharacteristic
103C23×C4×C59—21no—
104C23×C4×C59—21no—
105C23×C4×C59C2311yescharacteristic
106#121×C59—22no—
107#121×C59Q814yes—
108C4×D4×C59—22no—
109C4×D4×C59—41no—
110C4×D4×C59—44no—
111C2×C4⋊C4×C59—22no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—41no—
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59D411yescharacteristic
123C2×C22⋊C4×C59D411yescharacteristic
124C2×C42×C59—21no—
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59D411yescharacteristic
130#95×C59D411yescharacteristic
131C24×C59—21no—
132C24×C59C2×D411yescharacteristic
133C2×D4×C59—24no—
134C2×D4×C59—21no—
135C2×D4×C59—21no—
136C2×D4×C59—21no—
137C2×D4×C59—21no—
138C2×D4×C59—21no—
139C2×D4×C59—42no—
140C2×D4×C59—42no—
141C2×D4×C59C2×Q812yes—
142C22×C4×C59—21no—
143C22×C4×C59—21no—
144C22×C4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—42no—
150C22×C4×C59—42no—
151C22×C4×C59—42no—
152C22×C4×C59—42no—
153C22×C4×C59—81no—
154C22×C4×C59—81no—
155C22×C4×C59C2×D411yescharacteristic
156C22×C4×C59C2×D411yescharacteristic
157C22×C4×C59C2×D411yescharacteristic
158C22×C4×C59C2×D411yescharacteristic
159C22×C4×C59C2×D411yescharacteristic
160C22×C4×C59C2×Q812yes—
161C22×C4×C59#4112yes—
162C22×C4×C59C2411yescharacteristic
163C22×Dic59—21no—
164C22×Dic59—21no—
165C22×Dic59—21no—
166C22×Dic59—21no—
167C4⋊C4×C59—21no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—42no—
172C4⋊C4×C59—42no—
173C4⋊C4×C59C2×D412yes—
174C4⋊C4×C59C2×Q812yes—
175C22⋊C4×C59—24no—
176C22⋊C4×C59—22no—
177C22⋊C4×C59—41no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—81no—
181C42×C59—21no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—42no—
186C22⋊Dic59—42no—
187C22⋊Dic59—81no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—42no—
196C4⋊Dic59—81no—
197C4⋊Dic59C2×D412yes—
198Dic59⋊C4—42no—
199Dic59⋊C4—42no—
200Dic59⋊C4—42no—
201Dic59⋊C4—81no—
202C4×Dic59—42no—
203#11745664—28no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8C2×D412yes—
215C2×C59⋊C8#4112yes—
216C23×C59—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59—21no—
222C23×C59—22no—
223C23×C59—41no—
224C23×C59—41no—
225C23×C59—41no—
226C23×C59—41no—
227C23×C59—41no—
228C23×C59—41no—
229C23×C59C22⋊Q812yes—
230C23×C59C22×D411yescharacteristic
231D4×C59—22no—
232D4×C59—22no—
233D4×C59—48no—
234D4×C59—41no—
235D4×C59—41no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—22no—
241C2×C4×C59—22no—
242C2×C4×C59—22no—
243C2×C4×C59—22no—
244C2×C4×C59—22no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—22no—
248C2×C4×C59—21no—
249C2×C4×C59—21no—
250C2×C4×C59—21no—
251C2×C4×C59—21no—
252C2×C4×C59—21no—
253C2×C4×C59—21no—
254C2×C4×C59—21no—
255C2×C4×C59—41no—
256C2×C4×C59—41no—
257C2×C4×C59—41no—
258C2×C4×C59—41no—
259C2×C4×C59—41no—
260C2×C4×C59—41no—
261C2×C4×C59—41no—
262C2×C4×C59—84no—
263C2×C4×C59—84no—
264C2×C4×C59C22⋊Q812yes—
265C2×C4×C59AGL(1,ℤ/8)12yes—
266C2×C4×C59C22×D411yescharacteristic
267C2×C4×C59C22×Q811yescharacteristic
268C2×C4×C59C2×#4111yescharacteristic
269C2×C4×C59C2×#4111yescharacteristic
270C2×C4×C59H(5,𝔽2)11yescharacteristic
271C2×C4×C592−1+411yescharacteristic
272C2×Dic59—22no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—41no—
281C2×Dic59—41no—
282C2×Dic59—41no—
283C2×Dic59—41no—
284C59⋊C8—22no—
285C59⋊C8—22no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#26366—591nomaximal, Sylow
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—41no—
293C22×C59—41no—
294C22×C59—41no—
295C22×C59—41no—
296C22×C59—42no—
297C22×C59—42no—
298C22×C59—42no—
299C22×C59—81no—
300C22×C59—81no—
301C22×C59—81no—
302C22×C59—81no—
303C22×C59C2×C22⋊Q811yescharacteristic
304C22×C59D4211yescharacteristic
305C22×C59#54711yescharacteristic
306C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
307C22×C59#57511yescharacteristic
308C22×C59#57711yescharacteristic
309C22×C59#57711yescharacteristic
310C4×C59—22no—
311C4×C59—22no—
312C4×C59—42no—
313C4×C59—41no—
314C4×C59—41no—
315C4×C59—41no—
316C4×C59—42no—
317C4×C59—81no—
318C4×C59#52511yescharacteristic
319C4×C59#53911yescharacteristic
320C4×C59D4×Q811yescharacteristic
321C4×C59#55411yescharacteristic
322Dic59—42no—
323Dic59—81no—
324Dic59—81no—
325Dic59—81no—
326C2×#478—591no—
327C2×#428—591no—
328#2441—591no—
329#2361D5911yescharacteristic
330#2094—591no—
331#2088—591no—
332#2067—591no—
333#2059—591no—
334#1948—591no—
335#1918—591no—
336#1895—591no—
337#1875—591no—
338#1864—591no—
339#1853—591no—
340#1851—591no—
341C2×C59—42no—
342C2×C59—82no—
343C2×C59#244911yescharacteristic
344C2×C59#307711yescharacteristic
345C2×C59#308411yescharacteristic
346C2×C59#327111yescharacteristic
347C2×C59#327211yescharacteristic
348C2×C59#334511yescharacteristic
349C2×C59#335111yescharacteristic
350C22×M16—591no—
351C2×#125—591no—
352C2×#121C2×D5911yescharacteristic
353C2×C4×D4—21no—
354C2×C4×D4C2×D5911yescharacteristic
355C2×#117—591no—
356C22×C4⋊C4C2×D5911yescharacteristic
357#478—1184no—
358#428—1184no—
359C2×#107—591no—
360C2×#107—591no—
361C2×#106—591no—
362C2×#106—591no—
363C2×#106—591no—
364C2×#106—591no—
365C2×C4⋊C8—591no—
366C2×C4⋊C8—591no—
367C2×#102—591no—
368C2×#102—591no—
369C2×#102—591no—
370C2×#102—591no—
371C2×C22⋊C8—591no—
372#399—1181no—
373#396—1181no—
374#390C2×D5911yescharacteristic
375#389—591no—
376#389—591no—
377#389—1181no—
378#389C2×D5911yescharacteristic
379#388—21no—
380#387—1181no—
381#386—21no—
382#385—21no—
383#384—591no—
384#380—591no—
385C4×C4⋊C4C2×D5911yescharacteristic
386C4×C22⋊C4C2×D5911yescharacteristic
387#340—591no—
388#340—591no—
389#340—591no—
390#340—591no—
391#340—591no—
392#340—591no—
393#336—591no—
394#336—591no—
395C59#2636611yescharacteristic, complemented, Sylow
396C22×D4—21no—
397C22×D4C22×D5911yescharacteristic
398C23×C4—21no—
399C23×C4—21no—
400C23×C4C22×D5911yescharacteristic
401C2×M16—1182no—
402C2×M16—1182no—
403C2×M16—1182no—
404C22×C8—591no—
405C22×C8—591no—
406C22×C8—591no—
407C22×C8—591no—
408#125—1182no—
409#125—2361no—
410#121—22no—
411#121Dic11814yes—
412C4×D4—22no—
413C4×D4—41no—
414C4×D4—44no—
415#117—1182no—
416#117—2361no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—41no—
421C2×C4⋊C4—591no—
422C2×C4⋊C4—591no—
423C2×C4⋊C4—591no—
424C2×C4⋊C4—591no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—1181no—
428C2×C4⋊C4—1181no—
429C2×C4⋊C4—1181no—
430C2×C4⋊C4—1181no—
431C2×C4⋊C4C22×D5911yescharacteristic
432C2×C4⋊C4C22×D5911yescharacteristic
433C2×C4⋊C4C22×D5911yescharacteristic
434C2×C22⋊C4—21no—
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—21no—
438C2×C22⋊C4—1181no—
439C2×C22⋊C4—1181no—
440C2×C22⋊C4C22⋊D5911yescharacteristic
441C2×C22⋊C4C22⋊D5911yescharacteristic
442C2×C42—21no—
443C2×C42—1181no—
444C2×C42C22×D5911yescharacteristic
445C2×C42C22×D5911yescharacteristic
446#107—594no—
447#107—1182no—
448#106—1182no—
449#106—1182no—
450#106—1182no—
451#106—2361no—
452C4⋊C8—1182no—
453C4⋊C8—1182no—
454#102—1182no—
455#102—1182no—
456#102—1182no—
457#102—2361no—
458C22⋊C8—1182no—
459#95—21no—
460#95—21no—
461#95—1181no—
462#95—1181no—
463#95—1181no—
464#95C22⋊D5911yescharacteristic
465#95C22⋊D5911yescharacteristic
466C24—21no—
467C24C2×C22⋊D5911yescharacteristic
468C2×D4—24no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4C2×Dic11812yes—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—1181no—
491C22×C4—1181no—
492C22×C4—1181no—
493C22×C4—1181no—
494C22×C4C2×Dic11812yes—
495C22×C4#1174568512yes—
496C22×C4C2×C22⋊D5911yescharacteristic
497C22×C4C2×C22⋊D5911yescharacteristic
498C22×C4C2×C22⋊D5911yescharacteristic
499C22×C4C2×C22⋊D5911yescharacteristic
500C22×C4C2×C22⋊D5911yescharacteristic
501C22×C4C23×D5911yescharacteristic
502M16—1188no—
503C2×C8—592no—
504C2×C8—592no—
505C2×C8—1181no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8—1181no—
513C2×C8—1181no—
514C2×C8—1181no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—592no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—1181no—
527C4⋊C4—1181no—
528C4⋊C4—1181no—
529C4⋊C4—4721no—
530C4⋊C4—4721no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4—2362no—
535C4⋊C4C2×Dic11812yes—
536C4⋊C4D4×D5912yes—
537C22⋊C4—24no—
538C22⋊C4—22no—
539C22⋊C4—41no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—81no—
543C22⋊C4—4721no—
544C22⋊C4—2362no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—42no—
550C42—2362no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—22no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23#1427370612yes—
565C23C22×C22⋊D5911yescharacteristic
566D4—22no—
567D4—22no—
568D4—48no—
569D4—41no—
570D4—41no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—84no—
598C2×C4—84no—
599C2×C4—1182no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4—2361no—
606C2×C4—2361no—
607C2×C4—2361no—
608C2×C4—2361no—
609C2×C4—2361no—
610C2×C4—2361no—
611C2×C4#1427370612yes—
612C2×C4#1427375312yes—
613C2×C4C2×D4×D5911yescharacteristic
614C2×C4C22×Dic11811yescharacteristic
615C2×C4#41×D5911yescharacteristic
616C2×C4C2×#1174568511yescharacteristic
617C2×C4#1427377211yescharacteristic
618C2×C4#1427377411yescharacteristic
619C8—1182no—
620C8—1182no—
621C8—2361no—
622C8—2361no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—42no—
631C22—42no—
632C22—42no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22C2×#1427370611yescharacteristic
638C22D4×C22⋊D5911yescharacteristic
639C22#1683805111yescharacteristic
640C22C2×#1427375311yescharacteristic
641C22#1683820011yescharacteristic
642C22#1683821011yescharacteristic
643C22#1683821311yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—42no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—81no—
652C4—4721no—
653C4—4721no—
654C4—4721no—
655C4—2362no—
656C4#1683791111yescharacteristic
657C4#1683799411yescharacteristic
658C4D4×Dic11811yescharacteristic
659C4#1683811111yescharacteristic
660C2—42no—
661C2—82no—
662C2#2142789911yescharacteristic
663C2#2143389311yescharacteristic
664C2#2143395811yescharacteristic
665C2#2143568211yescharacteristic
666C2#2143569211yescharacteristic
667C2#2143656611yescharacteristic
668C2#2143665011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2361C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
hypercenter#2361D59
hypocenterC59#26366
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2361

Complements

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

NH ≅ G/Ntype
1. C2×#16837702C2semidirect
2. C2×#16837442C2semidirect
3. #21427809C2semidirect
4. #21425629C2semidirect
5. #21425598C2semidirect
6. #21425484C2semidirect
7. #21425434C2semidirect
8. #21424859C2semidirect
9. #21424667C2semidirect
12. #21424357C2semidirect
13. #21424289C2semidirect
14. #21424281C2semidirect
86. #14273660D4semidirect
395. C59#26366semidirect

Maximal subgroups

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

subgroupindex
C2×#168377022
C2×#168374422
#214278092
#214256292
#214255982
#214254842
#214254342
#214248592
#214246672
#214245352
#214244192
#214243572
#214242892
#214242812
C59×#23612
#2636659