Finite Group Databasebeta

← #28629377

Subgroups — #28629377

Order 15104.

1016 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837602C211yesmaximal, characteristic, complemented
2C2×#16837459C211yesmaximal, characteristic, complemented
3#21432441C211yesmaximal, characteristic
4#21425278C212yesmaximal, complemented
5#21425097C212yesmaximal, complemented
6#21425023C211yesmaximal, characteristic, complemented
7#21424980C212yesmaximal, complemented
8#21424854C211yesmaximal, characteristic, complemented
9#21424792C212yesmaximal, complemented
10#21424101C211yesmaximal, characteristic, complemented
11C4×C22⋊Q8×C59C211yesmaximal, characteristic, complemented
12C22×#11745671C2212yes—
13C2×#14273724C2211yescharacteristic
14C2×#14273701—21no—
15C2×#14273701—21no—
16C2×#14273701C2212yes—
17#16837602—24no—
18#16837459—24no—
19C2×#14273660C2212yes—
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×C4⋊(C59⋊C8)C2211yescharacteristic
22C2×#14273651C2212yes—
23C2×#14273647C2212yes—
24C2×#14273647C2212yes—
25C2×#14273647C2212yes—
26C2×C22⋊(C59⋊C8)C2212yes—
27C2×#14273638C2212yes—
28#16837304—21no—
29#16837304—21no—
30#16837304C2212yes—
31#16837272C2211yescharacteristic
32#16837164C2212yes—
33#16837164C2212yes—
34C2×C22⋊Q8×C59C2211yescharacteristic
35C2×C4×Q8×C59C2211yescharacteristic
36C22×C42×C59C2211yescharacteristic
37C59×#391C2211yescharacteristic
38C59×#389C2211yescharacteristic
39C59×#387—21no—
40C59×#385—21no—
41C59×#380C2211yescharacteristic
42C4×C4⋊C4×C59—21no—
43C4×C4⋊C4×C59C2211yescharacteristic
44C4×C22⋊C4×C59—21no—
45C22×D236—21no—
46C22×D236—21no—
47C22×D236C2312yes—
48C22×C22⋊D59—22no—
49C2×#11745671—24no—
50C2×#11745671—24no—
51C2×#11745671—42no—
52C22×C59⋊C8C2312yes—
53C22×C59⋊C8C2312yes—
54#14273724—22no—
55#14273724—22no—
56#14273701—42no—
57#14273701—42no—
58#14273701—44no—
59#14273701—44no—
60C2×C4⋊Dic59—21no—
61C2×C4⋊Dic59—21no—
62C2×C4⋊Dic59C2312yes—
63C2×(C2×D59)⋊C4—22no—
64C2×(C2×D59)⋊C4—22no—
65C2×(C2×D59)⋊C4—22no—
66#14273660—42no—
67C4⋊(C59⋊C8)—22no—
68C4⋊(C59⋊C8)D414yescomplemented
69#14273651—42no—
70#14273647—24no—
71#14273647—24no—
72#14273647—42no—
73C22⋊(C59⋊C8)—24no—
74#14273638—24no—
75C22×Q8×C59C2311yescharacteristic
76C23×C4×C59D411yescharacteristic
77C23×C4×C59D411yescharacteristic
78C23×C4×C59C2311yescharacteristic
79C22⋊Q8×C59—22no—
80C22⋊Q8×C59—22no—
81C4×Q8×C59—22no—
82C4×Q8×C59—22no—
83C2×C4⋊C4×C59—21no—
84C2×C4⋊C4×C59—21no—
85C2×C4⋊C4×C59—21no—
86C2×C4⋊C4×C59C2311yescharacteristic
87C2×C4⋊C4×C59C2311yescharacteristic
88C2×C4⋊C4×C59C2311yescharacteristic
89C2×C22⋊C4×C59—21no—
90C2×C22⋊C4×C59—21no—
91C2×C42×C59—22no—
92C2×C42×C59—22no—
93C2×C42×C59—21no—
94C2×C42×C59—21no—
95C2×C42×C59D411yescharacteristic
96C2×C42×C59D411yescharacteristic
97C2×C42×C59C2311yescharacteristic
98C2×C42×C59C2311yescharacteristic
99#95×C59—21no—
100#95×C59—21no—
101#95×C59—21no—
102#95×C59—21no—
103#95×C59—21no—
104C24×C59C2×D411yescharacteristic
105C23×D59—22no—
106C2×Q8×C59—21no—
107C2×Q8×C59—21no—
108C2×Q8×C59—21no—
109C2×Q8×C59—21no—
110C2×Q8×C59—21no—
111C2×Q8×C59#4112yes—
112C22×C4×C59—22no—
113C22×C4×C59—21no—
114C22×C4×C59—21no—
115C22×C4×C59—22no—
116C22×C4×C59—22no—
117C22×C4×C59—22no—
118C22×C4×C59—22no—
119C22×C4×C59—21no—
120C22×C4×C59—21no—
121C22×C4×C59—21no—
122C22×C4×C59—21no—
123C22×C4×C59—21no—
124C22×C4×C59—21no—
125C22×C4×C59—21no—
126C22×C4×C59—21no—
127C22×C4×C59—41no—
128C22×C4×C59C2×D411yescharacteristic
129C22×C4×C59C2×D411yescharacteristic
130C22×C4×C59C2×D411yescharacteristic
131C22×C4×C59C2×D411yescharacteristic
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2411yescharacteristic
134C2×C22⋊D59—44no—
135C2×C22⋊D59—44no—
136C2×C22⋊D59—44no—
137C22×Dic59—22no—
138C2×D236—22no—
139C2×D236—22no—
140C2×D236—22no—
141C2×D236—22no—
142C2×D236—22no—
143C2×D236—22no—
144C2×D236—42no—
145C2×D236—42no—
146C2×D236—42no—
147C2×D236—42no—
148C2×D236—42no—
149C2×D236—42no—
150C4⋊C4×C59—24no—
151C4⋊C4×C59—21no—
152C4⋊C4×C59—21no—
153C4⋊C4×C59—21no—
154C4⋊C4×C59—21no—
155C4⋊C4×C59—21no—
156C4⋊C4×C59—42no—
157C4⋊C4×C59—42no—
158C4⋊C4×C59#4112yes—
159C22⋊C4×C59—24no—
160C22⋊C4×C59—42no—
161C42×C59—22no—
162C42×C59—22no—
163C42×C59—21no—
164C42×C59—21no—
165C42×C59—42no—
166C42×C59—42no—
167C42×C59C2×D412yes—
168C42×C59#4112yes—
169#11745671—48no—
170(C2×D59)⋊C4—44no—
171(C2×D59)⋊C4—44no—
172(C2×D59)⋊C4—44no—
173C4⋊Dic59—22no—
174C4⋊Dic59—22no—
175C4⋊Dic59—42no—
176C4⋊Dic59—42no—
177C2×C59⋊C8—22no—
178C2×C59⋊C8—22no—
179C2×C59⋊C8—22no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8C2×D414yes—
183C23×C59—21no—
184C23×C59—21no—
185C23×C59—21no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59AGL(1,ℤ/8)12yes—
190C23×C59C22×D411yescharacteristic
191C22×D59—42no—
192C22×D59—42no—
193C22×D59—42no—
194C22×D59—42no—
195C22×D59—42no—
196C22×D59—42no—
197C22×D59—42no—
198Q8×C59—22no—
199Q8×C59—22no—
200Q8×C59—41no—
201Q8×C59—41no—
202C2×C4×C59—21no—
203C2×C4×C59—21no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—21no—
207C2×C4×C59—22no—
208C2×C4×C59—22no—
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—44no—
216C2×C4×C59—41no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59—41no—
222C2×C4×C59—41no—
223C2×C4×C59—41no—
224C2×C4×C59—44no—
225C2×C4×C59—41no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59—41no—
229C2×C4×C59—41no—
230C2×C4×C59—41no—
231C2×C4×C59#12112yes—
232C2×C4×C59#12112yes—
233C2×C4×C59AGL(1,ℤ/8)12yes—
234C2×C4×C59#13712yes—
235C2×C4×C59C22×D411yescharacteristic
236C2×C4×C59C2×#4111yescharacteristic
237C2×C4×C59C2×#4111yescharacteristic
238C2×C4×C59C2×#4111yescharacteristic
239C2×C4×C59H(5,𝔽2)11yescharacteristic
240C2×C4×C59H(5,𝔽2)11yescharacteristic
241C22⋊D59—88no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245D236—42no—
246D236—42no—
247D236—42no—
248D236—42no—
249D236—84no—
250D236—84no—
251C59⋊C8—24no—
252C59⋊C8—42no—
253#24661—591nomaximal, Sylow
254C22×C59—22no—
255C22×C59—22no—
256C22×C59—22no—
257C22×C59—41no—
258C22×C59—41no—
259C22×C59—41no—
260C22×C59—41no—
261C22×C59C2×#12111yescharacteristic
262C22×C59#52511yescharacteristic
263C22×C59#53511yescharacteristic
264C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
265C22×C59C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C59C2×#13711yescharacteristic
267C22×C59#57511yescharacteristic
268C2×D59—82no—
269C2×D59—82no—
270C2×D59—82no—
271C2×D59—82no—
272C2×D59—82no—
273C2×D59—82no—
274C2×D59—82no—
275C4×C59—22no—
276C4×C59—42no—
277C4×C59—41no—
278C4×C59—42no—
279C4×C59—41no—
280C4×C59—41no—
281C4×C59—81no—
282C4×C59—81no—
283C4×C59#46814yes—
284C4×C59#54611yescharacteristic
285C4×C59#55011yescharacteristic
286C4×C59#55311yescharacteristic
287C4×C59#55511yescharacteristic
288Dic59—82no—
289C2×#460—591no—
290C2×#432—591no—
291#2856—591no—
292C4×C22⋊Q8D5911yescharacteristic, complemented
293#2033—592no—
294#2000—592no—
295#1982—591no—
296#1973—592no—
297#1947—591no—
298#1936—592no—
299#1803—591no—
300C2×C59—42no—
301C2×C59#241411yescharacteristic
302C2×C59C2×#46811yescharacteristic
303C2×C59#305511yescharacteristic
304C2×C59#311211yescharacteristic
305C2×C59#311511yescharacteristic
306C2×C59#321611yescharacteristic
307C2×C59#321711yescharacteristic
308D59—162no—
309C22×SD16—592no—
310C2×#127—591no—
311C2×C22⋊Q8C2×D5911yescharacteristic
312C2×#121—592no—
313C2×#121—1181no—
314C2×#121—1181no—
315C2×C4×Q8C2×D5911yescharacteristic
316C22×C42C2×D5911yescharacteristic
317#460—1184no—
318#432—1184no—
319C2×#107—592no—
320C2×C4⋊C8—591no—
321C2×C4⋊C8—591no—
322C2×#103—592no—
323C2×#102—592no—
324C2×#102—592no—
325C2×#102—592no—
326C2×C22⋊C8—592no—
327C2×#97—592no—
328#391C2×D5911yescharacteristic
329#390—592no—
330#390—1181no—
331#390—1181no—
332#389C2×D5911yescharacteristic
333#387—21no—
334#385—21no—
335#384—591no—
336#380C2×D5911yescharacteristic
337C4×C4⋊C4—21no—
338C4×C4⋊C4C2×D5911yescharacteristic
339C4×C22⋊C4—21no—
340#340—592no—
341#340—592no—
342C59#2466111yescharacteristic, complemented, Sylow
343C22×Q8C22×D5911yescharacteristic
344C22×D4—592no—
345C22×D4—1182no—
346C22×D4—1181no—
347C22×D4—1181no—
348C23×C4C22⋊D5911yescharacteristic
349C23×C4C22⋊D5911yescharacteristic
350C23×C4C22×D5911yescharacteristic
351C2×SD16—1184no—
352C2×SD16—1184no—
353C2×SD16—2362no—
354C22×C8—592no—
355C22×C8—592no—
356#127—1182no—
357#127—1182no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360#121—2362no—
361#121—2362no—
362#121—2364no—
363#121—2364no—
364C4×Q8—22no—
365C4×Q8—22no—
366C2×C4⋊C4—21no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—592no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4C22×D5911yescharacteristic
373C2×C4⋊C4C22×D5911yescharacteristic
374C2×C4⋊C4C22×D5911yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—21no—
377C2×C22⋊C4—1182no—
378C2×C22⋊C4—1182no—
379C2×C22⋊C4—1182no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—21no—
383C2×C42—21no—
384C2×C42C22⋊D5911yescharacteristic
385C2×C42C22⋊D5911yescharacteristic
386C2×C42C22×D5911yescharacteristic
387C2×C42C22×D5911yescharacteristic
388#107—2362no—
389C4⋊C8—594no—
390C4⋊C8—1182no—
391#103—2362no—
392#102—1184no—
393#102—1184no—
394#102—2362no—
395C22⋊C8—1184no—
396#97—1184no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401#95—21no—
402C24—1182no—
403C24C2×C22⋊D5911yescharacteristic
404C2×Q8—21no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8#1174568512yes—
410C2×D4—1182no—
411C2×D4—1182no—
412C2×D4—1182no—
413C2×D4—1182no—
414C2×D4—1182no—
415C2×D4—1182no—
416C2×D4—2362no—
417C2×D4—2362no—
418C2×D4—2364no—
419C2×D4—2364no—
420C2×D4—2364no—
421C2×D4—2362no—
422C2×D4—2362no—
423C2×D4—2362no—
424C2×D4—2362no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—41no—
441C22×C4—1182no—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C2×C22⋊D5911yescharacteristic
445C22×C4C2×C22⋊D5911yescharacteristic
446C22×C4C2×C22⋊D5911yescharacteristic
447C22×C4C23×D5911yescharacteristic
448SD16—2368no—
449C2×C8—594no—
450C2×C8—1182no—
451C2×C8—1182no—
452C2×C8—1182no—
453C2×C8—1182no—
454C2×C8—1182no—
455C4⋊C4—24no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—1182no—
464C4⋊C4—1182no—
465C4⋊C4—2362no—
466C4⋊C4—2362no—
467C4⋊C4#1174568512yes—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—2364no—
471C22⋊C4—2364no—
472C22⋊C4—2364no—
473C42—22no—
474C42—22no—
475C42—21no—
476C42—21no—
477C42—42no—
478C42—42no—
479C42D4×D5912yes—
480C42#1174568712yes—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—2362no—
488C23—2362no—
489C23—2362no—
490C23—2362no—
491C23—2362no—
492C23—2362no—
493C23—2362no—
494C23#1427375212yes—
495C23C22×C22⋊D5911yescharacteristic
496Q8—22no—
497Q8—22no—
498Q8—41no—
499Q8—41no—
500D4—4724no—
501D4—4728no—
502D4—4724no—
503D4—2362no—
504D4—2362no—
505D4—2362no—
506D4—2362no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—2362no—
537C2×C4—2362no—
538C2×C4—2362no—
539C2×C4#1427369912yes—
540C2×C4#1427369912yes—
541C2×C4#1427375212yes—
542C2×C4#1427375812yes—
543C2×C4C2×D4×D5911yescharacteristic
544C2×C4C2×#1174568711yescharacteristic
545C2×C4C2×#1174568511yescharacteristic
546C2×C4C2×#1174568511yescharacteristic
547C2×C4#1427377311yescharacteristic
548C2×C4#1427377311yescharacteristic
549C8—1184no—
550C8—2362no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—4722no—
559C22—4722no—
560C22—4722no—
561C22—4722no—
562C22—4722no—
563C22—4722no—
564C22—4722no—
565C22C2×#1427369911yescharacteristic
566C22#1683791311yescharacteristic
567C22#1683796611yescharacteristic
568C22C2×#1427375211yescharacteristic
569C22C2×#1427375211yescharacteristic
570C22C2×#1427375811yescharacteristic
571C22#1683820011yescharacteristic
572C4—22no—
573C4—41no—
574C4—42no—
575C4—41no—
576C4—41no—
577C4—42no—
578C4—81no—
579C4—81no—
580C4—4722no—
581C4#1683765014yes—
582C4#1683804811yescharacteristic
583C4#1683807311yescharacteristic
584C4#1683809811yescharacteristic
585C4#1683811711yescharacteristic
586C2—42no—
587C2—9442no—
588C2#2142751411yescharacteristic
589C2C2×#1683765011yescharacteristic
590C2#2143371711yescharacteristic
591C2#2143421811yescharacteristic
592C2#2143424711yescharacteristic
593C2#2143510311yescharacteristic
594C2#2143511111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC4×C22⋊Q8×C59C2
socleC23×C59C22×D4
prime radical#21432441C2
hypercenterC4×C22⋊Q8D59
hypocenterC59#24661
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837602C2semidirect
2. C2×#16837459C2semidirect
4. #21425278C2semidirect
5. #21425097C2semidirect
6. #21425023C2semidirect
7. #21424980C2semidirect
8. #21424854C2semidirect
9. #21424792C2semidirect
10. #21424101C2semidirect
11. C4×C22⋊Q8×C59C2semidirect
68. C4⋊(C59⋊C8)D4semidirect
292. C4×C22⋊Q8D59semidirect
342. C59#24661semidirect

Maximal subgroups

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

subgroupindex
C2×#168376022
C2×#168374592
#214324412
#214252782
#214250972
#214250232
#214249802
#214248542
#214247922
#214241012
C4×C22⋊Q8×C592
#2466159