Finite Group Databasebeta

← #28633980

Subgroups — #28633980

Order 15104.

716 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊Dic59C211yesmaximal, characteristic, complemented
2#21425659C211yesmaximal, characteristic, complemented
3#21425654C211yesmaximal, characteristic, complemented
4#21425499C211yesmaximal, characteristic
5#21425499C211yesmaximal, characteristic
6#21425037C211yesmaximal, characteristic, complemented
7#21424862C211yesmaximal, characteristic, complemented
8#21424720C211yesmaximal, characteristic
9#21424713C211yesmaximal, characteristic
10#21424685C211yesmaximal, characteristic, complemented
11#21424678C211yesmaximal, characteristic, complemented
12#21424229C211yesmaximal, characteristic, complemented
13#21424096C211yesmaximal, characteristic
14#21423921C211yesmaximal, characteristic
15C59×#2580C211yesmaximal, characteristic
16C2×Q8×Dic59—21no—
17C2×#14273660C2211yescharacteristic
18C2×#14273658C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273648C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273648C2211yescharacteristic
24C2×#14273638C2211yescharacteristic
25#16837301—21no—
26#16837301—21no—
27#16837301—21no—
28#16837272C2211yescharacteristic
29#16837272C2211yescharacteristic
30#16837272C2211yescharacteristic
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C4⋊Dic59C2211yescharacteristic
33C4×C4⋊Dic59C2211yescharacteristic
34#16837164C2211yescharacteristic
35#16837164C2211yescharacteristic
36#16837164C2211yescharacteristic
37#16837164C2211yescharacteristic
38#16837164C2211yescharacteristic
39#16837164C2211yescharacteristic
40#16837164C2211yescharacteristic
41#16837164C2211yescharacteristic
42#16837164C2211yescharacteristic
43#16837164C2211yescharacteristic
44#16837157C2211yescharacteristic
45#16837157C2211yescharacteristic
46#16837157C2211yescharacteristic
47#16837157C2211yescharacteristic
48C2×C4×Q8×C59C2211yescharacteristic
49C2×C4×D4×C59C2211yescharacteristic
50C59×#399—21no—
51C59×#393—21no—
52C59×#390C2211yescharacteristic
53C59×#390C2211yescharacteristic
54C59×#389C2211yescharacteristic
55C59×#389C2211yescharacteristic
56C59×#385—21no—
57C59×#385—21no—
58C59×#382C2211yescharacteristic
59C22×C59⋊C8C2311yescharacteristic
60C22×C59⋊C8C2311yescharacteristic
61C22×C59⋊C8C2311yescharacteristic
62C22×C59⋊C8C2311yescharacteristic
63Q8×Dic59—44no—
64C2×C4⋊Dic59—21no—
65C2×C4⋊Dic59—21no—
66C2×C4⋊Dic59—21no—
67C2×C4⋊Dic59—21no—
68C2×C4⋊Dic59—21no—
69C2×C4⋊Dic59—21no—
70C2×C4⋊Dic59C2311yescharacteristic
71C2×C4⋊Dic59C2311yescharacteristic
72C2×C4⋊Dic59C2311yescharacteristic
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4×Dic59—21no—
75C2×C4×Dic59—21no—
76C2×C4×Dic59—21no—
77#14273660—22no—
78#14273658—22no—
79C4⋊(C59⋊C8)—22no—
80#14273648—22no—
81#14273648—22no—
82#14273648—22no—
83#14273648—22no—
84#14273638—22no—
85C59⋊#95—21no—
86C59⋊#95—21no—
87C59⋊#95—21no—
88C22×Q8×C59D411yescharacteristic
89C22×D4×C59C2311yescharacteristic
90C23×C4×C59—21no—
91C4×Q8×C59—22no—
92C4×Q8×C59—22no—
93C4×D4×C59—22no—
94C4×D4×C59—22no—
95C2×C4⋊C4×C59—21no—
96C2×C4⋊C4×C59—21no—
97C2×C4⋊C4×C59D411yescharacteristic
98C2×C4⋊C4×C59D411yescharacteristic
99C2×C4⋊C4×C59D411yescharacteristic
100C2×C4⋊C4×C59C2311yescharacteristic
101C2×C4⋊C4×C59C2311yescharacteristic
102C2×C4⋊C4×C59C2311yescharacteristic
103C2×C22⋊C4×C59—21no—
104C2×C22⋊C4×C59—21no—
105C2×C22⋊C4×C59—21no—
106C2×C22⋊C4×C59—21no—
107C2×C22⋊C4×C59—21no—
108C2×C42×C59C2311yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111#95×C59—21no—
112#95×C59—21no—
113#95×C59—21no—
114#95×C59—21no—
115C24×C59—21no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—22no—
119C2×D4×C59—21no—
120C2×D4×C59—21no—
121C2×D4×C59—21no—
122C2×D4×C59—21no—
123C2×D4×C59—21no—
124C2×D4×C59#4112yes—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—42no—
129C22×C4×C59—81no—
130C22×C4×C59—81no—
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2411yescharacteristic
138C22×Dic59—21no—
139C22×Dic59—21no—
140C22×Dic59—21no—
141C22×Dic59—21no—
142C4⋊C4×C59—22no—
143C4⋊C4×C59—22no—
144C4⋊C4×C59—22no—
145C4⋊C4×C59—21no—
146C4⋊C4×C59—21no—
147C4⋊C4×C59—21no—
148C4⋊C4×C59—21no—
149C4⋊C4×C59—21no—
150C4⋊C4×C59—21no—
151C4⋊C4×C59—81no—
152C4⋊C4×C59—81no—
153C22⋊C4×C59—42no—
154C22⋊C4×C59—42no—
155C22⋊C4×C59—42no—
156C22⋊C4×C59—81no—
157C22⋊C4×C59—81no—
158C42×C59—21no—
159C42×C59—21no—
160C42×C59—21no—
161C42×C59—21no—
162C42×C59#4112yes—
163C42×C59#4112yes—
164C4⋊Dic59—21no—
165C4⋊Dic59—21no—
166C4⋊Dic59—21no—
167C4⋊Dic59—21no—
168C4⋊Dic59—21no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—42no—
172C4⋊Dic59—42no—
173C4⋊Dic59—42no—
174C4⋊Dic59—81no—
175C4⋊Dic59—81no—
176C4⋊Dic59—81no—
177C4⋊Dic59#4112yes—
178C4×Dic59—42no—
179C4×Dic59—42no—
180C4×Dic59—42no—
181C2×C59⋊C8—21no—
182C2×C59⋊C8—21no—
183C2×C59⋊C8—21no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—21no—
188C2×C59⋊C8—21no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—21no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8—21no—
193C23×C59—22no—
194C23×C59—41no—
195C23×C59—41no—
196C23×C59—41no—
197C23×C59—41no—
198C23×C59—41no—
199C23×C59—41no—
200C23×C59C22×D411yescharacteristic
201Q8×C59—28no—
202D4×C59—22no—
203D4×C59—22no—
204D4×C59—41no—
205D4×C59—41no—
206C2×C4×C59—21no—
207C2×C4×C59—21no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
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—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—41no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—84no—
230C2×C4×C59#12312yes—
231C2×C4×C59#12312yes—
232C2×C4×C59AGL(1,ℤ/8)12yes—
233C2×C4×C59#13712yes—
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59C2×#4111yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C592−1+411yescharacteristic
239C2×C4×C592−1+411yescharacteristic
240C2×Dic59—22no—
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—
252C59⋊C8—41no—
253C59⋊C8—41no—
254C59⋊C8—41no—
255C59⋊C8—41no—
256#25002—591nomaximal, Sylow
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59—81no—
261C22×C59—81no—
262C22×C59—81no—
263C22×C59—81no—
264C22×C59C2×#12311yescharacteristic
265C22×C59#54811yescharacteristic
266C22×C59#54811yescharacteristic
267C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C59C2×#13711yescharacteristic
269C22×C59#57611yescharacteristic
270C22×C59#57711yescharacteristic
271C4×C59—22no—
272C4×C59—22no—
273C4×C59—22no—
274C4×C59—22no—
275C4×C59—41no—
276C4×C59—41no—
277C4×C59—81no—
278C4×C59—81no—
279C4×C59—81no—
280C4×C59#52911yescharacteristic
281C4×C59#54111yescharacteristic
282C4×C59#55211yescharacteristic
283C4×C59#55611yescharacteristic
284Dic59—42no—
285Dic59—81no—
286Dic59—81no—
287Dic59—81no—
288#2580D5911yescharacteristic
289Q8×C4⋊C4—591no—
290#2100—591no—
291#2099—591no—
292#2071—591no—
293#2071—591no—
294#1985—591no—
295#1949—591no—
296#1926—591no—
297#1925—591no—
298#1921—591no—
299#1920—591no—
300#1836—591no—
301#1802—591no—
302#1762—591no—
303C2×C59—82no—
304C2×C59#242111yescharacteristic
305C2×C59#309211yescharacteristic
306C2×C59#309611yescharacteristic
307C2×C59#336611yescharacteristic
308C2×C59#337211yescharacteristic
309C2×C59#337311yescharacteristic
310C2×C59#337511yescharacteristic
311C2×C4×Q8—1181no—
312C2×C4×Q8C2×D5911yescharacteristic
313C2×C4×D4C2×D5911yescharacteristic
314C2×#107—591no—
315C2×#106—591no—
316C2×C4⋊C8—591no—
317C2×#102—591no—
318C2×#102—591no—
319C2×#102—591no—
320C2×#102—591no—
321C2×#97—591no—
322#399—21no—
323#393—21no—
324#390C2×D5911yescharacteristic
325#390C2×D5911yescharacteristic
326#389—1181no—
327#389—1181no—
328#389—1181no—
329#389C2×D5911yescharacteristic
330#389C2×D5911yescharacteristic
331#385—21no—
332#385—21no—
333#384—591no—
334#384—591no—
335#384—591no—
336#382C2×D5911yescharacteristic
337C4×C4⋊C4—591no—
338C4×C4⋊C4—591no—
339C4×C4⋊C4—591no—
340#340—591no—
341#340—591no—
342#340—591no—
343#340—591no—
344#340—591no—
345#340—591no—
346#340—591no—
347#340—591no—
348#340—591no—
349#340—591no—
350#336—591no—
351#336—591no—
352#336—591no—
353#336—591no—
354C59#2500211yescharacteristic, complemented, Sylow
355C22×Q8C22⋊D5911yescharacteristic
356C22×D4C22×D5911yescharacteristic
357C23×C4—21no—
358C22×C8—591no—
359C22×C8—591no—
360C22×C8—591no—
361C22×C8—591no—
362C4×Q8—22no—
363C4×Q8—22no—
364C4×Q8—2364no—
365C4×D4—22no—
366C4×D4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—591no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—591no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1181no—
375C2×C4⋊C4—1181no—
376C2×C4⋊C4—1181no—
377C2×C4⋊C4—1181no—
378C2×C4⋊C4—1181no—
379C2×C4⋊C4C22⋊D5911yescharacteristic
380C2×C4⋊C4C22⋊D5911yescharacteristic
381C2×C4⋊C4C22⋊D5911yescharacteristic
382C2×C4⋊C4C22×D5911yescharacteristic
383C2×C4⋊C4C22×D5911yescharacteristic
384C2×C4⋊C4C22×D5911yescharacteristic
385C2×C22⋊C4—21no—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—21no—
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C42—1181no—
391C2×C42—1181no—
392C2×C42—1181no—
393C2×C42C22×D5911yescharacteristic
394C2×C42C22×D5911yescharacteristic
395C2×C42C22×D5911yescharacteristic
396#107—1182no—
397#106—1182no—
398C4⋊C8—1182no—
399#102—1182no—
400#102—1182no—
401#102—1182no—
402#102—1182no—
403#97—1182no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—1181no—
409#95—1181no—
410#95—1181no—
411C24—21no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4#1174568912yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—42no—
425C22×C4—81no—
426C22×C4—81no—
427C22×C4—1181no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4C2×C22⋊D5911yescharacteristic
432C22×C4C2×C22⋊D5911yescharacteristic
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C23×D5911yescharacteristic
438C2×C8—1181no—
439C2×C8—1181no—
440C2×C8—1181no—
441C2×C8—1181no—
442C2×C8—1181no—
443C2×C8—1181no—
444C2×C8—1181no—
445C2×C8—1181no—
446C2×C8—1181no—
447C2×C8—1181no—
448C2×C8—1181no—
449C2×C8—1181no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—592no—
462C4⋊C4—1181no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1181no—
466C4⋊C4—1181no—
467C4⋊C4—1181no—
468C4⋊C4—1181no—
469C4⋊C4—4721no—
470C4⋊C4—4721no—
471C4⋊C4—4721no—
472C4⋊C4—2362no—
473C4⋊C4—2362no—
474C4⋊C4—2362no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—81no—
479C22⋊C4—81no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—2362no—
485C42—2362no—
486C42—2362no—
487C42#1174568712yes—
488C42#1174568712yes—
489C23—22no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23C22×C22⋊D5911yescharacteristic
497Q8—28no—
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—84no—
526C2×C4—1182no—
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×C4#1427370812yes—
539C2×C4#1427370812yes—
540C2×C4#1427375212yes—
541C2×C4#1427375712yes—
542C2×C4#41×D5911yescharacteristic
543C2×C4C2×#1174568711yescharacteristic
544C2×C4C2×#1174568711yescharacteristic
545C2×C4C2×#1174568911yescharacteristic
546C2×C4#1427377411yescharacteristic
547C2×C4#1427377511yescharacteristic
548C8—2361no—
549C8—2361no—
550C8—2361no—
551C8—2361no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×#1427370811yescharacteristic
560C22#1683805911yescharacteristic
561C22#1683806311yescharacteristic
562C22C2×#1427375211yescharacteristic
563C22C2×#1427375711yescharacteristic
564C22#1683820511yescharacteristic
565C22#1683821311yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—41no—
571C4—41no—
572C4—81no—
573C4—81no—
574C4—81no—
575C4—4721no—
576C4—4721no—
577C4—4721no—
578C4—2362no—
579C4#1683793811yescharacteristic
580C4#1683801211yescharacteristic
581C4#1683809411yescharacteristic
582C4#1683812311yescharacteristic
583C2—82no—
584C2#2142759511yescharacteristic
585C2#2143401611yescharacteristic
586C2#2143404811yescharacteristic
587C2#2143680311yescharacteristic
588C2#2143687711yescharacteristic
589C2#2143688411yescharacteristic
590C2#2143690611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2580C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2580D59
hypocenterC59#25002
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊Dic59C2semidirect
2. #21425659C2semidirect
3. #21425654C2semidirect
6. #21425037C2semidirect
7. #21424862C2semidirect
10. #21424685C2semidirect
11. #21424678C2semidirect
12. #21424229C2semidirect
354. C59#25002semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Dic592
#214256592
#214256542
#214254992
#214250372
#214248622
#214247202
#214247132
#214246852
#214246782
#214242292
#214240962
#214239212
C59×#25802
#2500259