Finite Group Databasebeta

← #28624249

Subgroups — #28624249

Order 15104.

1392 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837609C211yesmaximal, characteristic, complemented
2C2×#16837558C212yesmaximal, complemented
3C2×#16837507C211yesmaximal, characteristic, complemented
4#21429005C211yesmaximal, characteristic
5#21425289C212yesmaximal, complemented
6#21425104C212yesmaximal, complemented
7#21424917C211yesmaximal, characteristic, complemented
8#21424895C211yesmaximal, characteristic
9#21424111C211yesmaximal, characteristic, complemented
10#21424031C212yesmaximal, complemented
11C2×D4×Q8×C59C211yesmaximal, characteristic
12C22×#11745672C2212yes—
13C22×#11745672C2211yescharacteristic
14C2×#14273706C2212yes—
15C2×C4×Dic118C2211yescharacteristic
16#16837609—24no—
17#16837558—28no—
18#16837507—24no—
19C2×#14273658C2211yescharacteristic
20C2×C4⋊(C59⋊C8)C2211yescharacteristic
21C2×#14273651C2212yes—
22C2×#14273651C2211yescharacteristic
23C2×#14273651C2212yes—
24C2×#14273651C2212yes—
25C2×#14273651C2212yes—
26C2×#14273650C2211yescharacteristic
27C2×C22⋊(C59⋊C8)C2212yes—
28C2×C22⋊(C59⋊C8)C2212yes—
29C2×#14273638C2211yescharacteristic
30#16837333—21no—
31#16837316—21no—
32#16837290—22no—
33#16837268C2211yescharacteristic
34#16837262C2212yes—
35#16837164C2212yes—
36C23×Q8×C59C2212yes—
37D4×Q8×C59—24no—
38D4×Q8×C59—24no—
39C2×C4⋊Q8×C59—21no—
40C2×C4⋊Q8×C59C2211yescharacteristic
41C2×C22⋊Q8×C59—22no—
42C2×C22⋊Q8×C59C2212yes—
43C2×C4×Q8×C59C2211yescharacteristic
44C2×C4×D4×C59—21no—
45C2×C4×D4×C59C2211yescharacteristic
46C22×Dic118C2311yescharacteristic
47C2×#11745672—22no—
48C2×#11745672—22no—
49C2×#11745672—24no—
50C2×#11745672—24no—
51C2×#11745672—42no—
52C2×#11745672D414yescomplemented
53C22×C59⋊C8C2311yescharacteristic
54C22×C59⋊C8C2312yes—
55C22×C59⋊C8C2311yescharacteristic
56#14273706—24no—
57#14273706—42no—
58C4×Dic118—22no—
59C4×Dic118—22no—
60C2×C4⋊Dic59C2312yes—
61C2×C4⋊Dic59C2311yescharacteristic
62C2×Dic59⋊C4—22no—
63C2×Dic59⋊C4—21no—
64C2×C22⋊Dic59—22no—
65C2×C22⋊Dic59—22no—
66C2×C4×Dic59—21no—
67#14273658—22no—
68C4⋊(C59⋊C8)—22no—
69#14273651—22no—
70#14273651—24no—
71#14273651—24no—
72#14273651—24no—
73#14273651—42no—
74#14273650—22no—
75C22⋊(C59⋊C8)—24no—
76C22⋊(C59⋊C8)—42no—
77#14273638—22no—
78C59⋊#95—22no—
79C59⋊#95—21no—
80C59⋊#95—21no—
81C22×Q8×C59—24no—
82C22×Q8×C59—24no—
83C22×Q8×C59—22no—
84C22×Q8×C59—24no—
85C22×Q8×C59—24no—
86C22×Q8×C59—42no—
87C22×Q8×C59—42no—
88C22×Q8×C59C2312yes—
89C22×Q8×C59C2311yescharacteristic
90C22×D4×C59D411yescharacteristic
91C23×C4×C59—22no—
92C23×C4×C59C2312yes—
93C4⋊Q8×C59—28no—
94C4⋊Q8×C59—22no—
95C4⋊Q8×C59D414yes—
96C22⋊Q8×C59—216no—
97C22⋊Q8×C59—24no—
98C22⋊Q8×C59—24no—
99C4×Q8×C59—22no—
100C4×Q8×C59D414yes—
101C4×D4×C59—28no—
102C4×D4×C59—22no—
103C4×D4×C59—22no—
104C2×C4⋊C4×C59—21no—
105C2×C4⋊C4×C59—21no—
106C2×C4⋊C4×C59—21no—
107C2×C4⋊C4×C59—22no—
108C2×C4⋊C4×C59D411yescharacteristic
109C2×C4⋊C4×C59C2311yescharacteristic
110C2×C22⋊C4×C59—22no—
111C2×C22⋊C4×C59D412yes—
112C2×C42×C59—21no—
113C2×C42×C59C2311yescharacteristic
114C24×C59C2×D412yes—
115C2×Q8×C59—28no—
116C2×Q8×C59—21no—
117C2×Q8×C59—22no—
118C2×Q8×C59—21no—
119C2×Q8×C59—21no—
120C2×Q8×C59—22no—
121C2×Q8×C59—22no—
122C2×Q8×C59—22no—
123C2×Q8×C59—22no—
124C2×Q8×C59—44no—
125C2×Q8×C59—44no—
126C2×Q8×C59—48no—
127C2×Q8×C59—48no—
128C2×Q8×C59—48no—
129C2×Q8×C59—48no—
130C2×Q8×C59C2×D414yes—
131C2×Q8×C59C2×D412yes—
132C2×Q8×C59C2×D412yes—
133C2×Q8×C59#4112yes—
134C2×D4×C59—24no—
135C2×D4×C59—22no—
136C22×C4×C59—21no—
137C22×C4×C59—28no—
138C22×C4×C59—21no—
139C22×C4×C59—22no—
140C22×C4×C59—22no—
141C22×C4×C59—22no—
142C22×C4×C59—22no—
143C22×C4×C59—22no—
144C22×C4×C59—22no—
145C22×C4×C59—22no—
146C22×C4×C59—44no—
147C22×C4×C59—44no—
148C22×C4×C59C2×D412yes—
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2411yescharacteristic
152C22×Dic59—21no—
153C22×Dic59—22no—
154C22×Dic59—21no—
155C2×Dic118—21no—
156C2×Dic118—21no—
157C2×Dic118—21no—
158C2×Dic118—21no—
159C2×Dic118—21no—
160C2×Dic118C2×D412yes—
161C4⋊C4×C59—28no—
162C4⋊C4×C59—24no—
163C4⋊C4×C59—24no—
164C4⋊C4×C59—24no—
165C4⋊C4×C59—22no—
166C4⋊C4×C59—21no—
167C4⋊C4×C59C2×D412yes—
168C22⋊C4×C59—28no—
169C22⋊C4×C59—24no—
170C42×C59—24no—
171C42×C59C2×D412yes—
172C42×C59C2×D412yes—
173C22⋊Dic59—44no—
174C22⋊Dic59—44no—
175#11745672—28no—
176#11745672—48no—
177C4⋊Dic59—21no—
178C4⋊Dic59—22no—
179C4⋊Dic59—21no—
180C4⋊Dic59—22no—
181Dic59⋊C4—44no—
182Dic59⋊C4—42no—
183C4×Dic59—42no—
184C2×C59⋊C8—21no—
185C2×C59⋊C8—21no—
186C2×C59⋊C8—21no—
187C2×C59⋊C8—22no—
188C2×C59⋊C8—22no—
189C2×C59⋊C8—21no—
190C2×C59⋊C8—22no—
191C2×C59⋊C8—21no—
192C2×C59⋊C8C2×D412yes—
193C23×C59—22no—
194C23×C59—22no—
195C23×C59—22no—
196C23×C59—22no—
197C23×C59—22no—
198C23×C59—22no—
199C23×C59#13714yes—
200C23×C59C22×D411yescharacteristic
201Q8×C59—22no—
202Q8×C59—22no—
203Q8×C59—22no—
204Q8×C59—24no—
205Q8×C59—24no—
206Q8×C59—416no—
207Q8×C59—42no—
208Q8×C59—42no—
209Q8×C59#12114yes—
210D4×C59—28no—
211C2×C4×C59—22no—
212C2×C4×C59—22no—
213C2×C4×C59—22no—
214C2×C4×C59—22no—
215C2×C4×C59—22no—
216C2×C4×C59—22no—
217C2×C4×C59—24no—
218C2×C4×C59—22no—
219C2×C4×C59—22no—
220C2×C4×C59—22no—
221C2×C4×C59—416no—
222C2×C4×C59—42no—
223C2×C4×C59—42no—
224C2×C4×C59—42no—
225C2×C4×C59—42no—
226C2×C4×C59—42no—
227C2×C4×C59—42no—
228C2×C4×C59—42no—
229C2×C4×C59#12012yes—
230C2×C4×C59#12012yes—
231C2×C4×C59C22×D411yescharacteristic
232C2×C4×C59C22×D411yescharacteristic
233C2×C4×C59C22×D411yescharacteristic
234C2×C4×C59C2×#4111yescharacteristic
235C2×C4×C59H(5,𝔽2)11yescharacteristic
236C2×C4×C59H(5,𝔽2)11yescharacteristic
237C2×Dic59—22no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—42no—
242C2×Dic59—42no—
243C2×Dic59—42no—
244C2×Dic59—42no—
245Dic118—22no—
246Dic118—22no—
247Dic118—41no—
248Dic118—41no—
249C59⋊C8—22no—
250C59⋊C8—41no—
251C59⋊C8—42no—
252#24255—591nomaximal, Sylow
253C22×C59—24no—
254C22×C59—24no—
255C22×C59—24no—
256C22×C59—42no—
257C22×C59—42no—
258C22×C59—42no—
259C22×C59—42no—
260C22×C59C2×#12011yescharacteristic
261C22×C59D4211yescharacteristic
262C22×C59#54611yescharacteristic
263C22×C59C2×#13712yes—
264C22×C59#57812yes—
265C4×C59—24no—
266C4×C59—22no—
267C4×C59—22no—
268C4×C59—42no—
269C4×C59—44no—
270C4×C59—42no—
271C4×C59C2×#12111yescharacteristic
272C4×C59#53511yescharacteristic
273C4×C59D4211yescharacteristic
274C4×C59#54611yescharacteristic
275Dic59—42no—
276Dic59—81no—
277Dic59—82no—
278C2×D4×Q8D5911yescharacteristic
279C2×#462—591no—
280C2×#451—592no—
281C2×#441—591no—
282#2536—591no—
283#2035—592no—
284#2001—592no—
285#1960—591no—
286#1955—591no—
287#1805—591no—
288#1787—592no—
289C2×C59—44no—
290C2×C59#240811yescharacteristic
291C2×C59#300411yescharacteristic
292C2×C59#301111yescharacteristic
293C2×C59#327012yes—
294C2×C59#331212yes—
295C23×Q8C2×D5912yes—
296C22×Q16—591no—
297C22×Q16—592no—
298D4×Q8—24no—
299D4×Q8—24no—
300C2×C4⋊Q8—21no—
301C2×C4⋊Q8C2×D5911yescharacteristic
302C2×C22⋊Q8—22no—
303C2×C22⋊Q8—592no—
304C2×C22⋊Q8C2×D5912yes—
305C2×C4×Q8—591no—
306C2×C4×Q8C2×D5911yescharacteristic
307C2×C4×D4—21no—
308C2×C4×D4C2×D5911yescharacteristic
309#462—1184no—
310#451—1188no—
311#441—1184no—
312C2×#106—591no—
313C2×C4⋊C8—591no—
314C2×#103—592no—
315C2×#103—591no—
316C2×#103—592no—
317C2×#103—592no—
318C2×#103—591no—
319C2×#103—592no—
320C2×C22⋊C8—592no—
321C2×C22⋊C8—592no—
322C2×#97—591no—
323#398—1181no—
324#394—1181no—
325#388—1182no—
326#382—591no—
327#380—592no—
328#340—592no—
329C59#2425511yescharacteristic, complemented, Sylow
330C22×Q8—24no—
331C22×Q8—24no—
332C22×Q8—22no—
333C22×Q8—24no—
334C22×Q8—24no—
335C22×Q8—42no—
336C22×Q8—42no—
337C22×Q8—591no—
338C22×Q8C22×D5912yes—
339C22×Q8C22×D5911yescharacteristic
340C22×D4C22⋊D5911yescharacteristic
341C23×C4—22no—
342C23×C4C22×D5912yes—
343C2×Q16—594no—
344C2×Q16—1182no—
345C2×Q16—1182no—
346C2×Q16—1184no—
347C2×Q16—1184no—
348C2×Q16—2362no—
349C22×C8—591no—
350C22×C8—592no—
351C22×C8—591no—
352C4⋊Q8—28no—
353C4⋊Q8—22no—
354C4⋊Q8C22⋊D5914yes—
355C22⋊Q8—216no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—1184no—
359C22⋊Q8—2362no—
360C4×Q8—22no—
361C4×Q8—1182no—
362C4×Q8—1182no—
363C4×Q8C22⋊D5914yes—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—591no—
372C2×C4⋊C4—592no—
373C2×C4⋊C4—1181no—
374C2×C4⋊C4—1182no—
375C2×C4⋊C4C22⋊D5911yescharacteristic
376C2×C4⋊C4C22×D5911yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—1182no—
379C2×C22⋊C4—1182no—
380C2×C22⋊C4C22⋊D5912yes—
381C2×C42—21no—
382C2×C42—1181no—
383C2×C42C22×D5911yescharacteristic
384#106—1182no—
385C4⋊C8—1182no—
386#103—1184no—
387#103—1184no—
388#103—1182no—
389#103—1184no—
390#103—1182no—
391#103—2362no—
392C22⋊C8—1184no—
393C22⋊C8—2362no—
394#97—1182no—
395#95—1182no—
396#95—1181no—
397#95—1181no—
398C24C2×C22⋊D5912yes—
399C2×Q8—28no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—592no—
415C2×Q8—1181no—
416C2×Q8—1181no—
417C2×Q8—1181no—
418C2×Q8—1181no—
419C2×Q8—1181no—
420C2×Q8D4×D5912yes—
421C2×Q8#1174568712yes—
422C2×Q8C2×C22⋊D5914yes—
423C2×Q8C2×C22⋊D5912yes—
424C2×D4—24no—
425C2×D4—22no—
426C22×C4—21no—
427C22×C4—28no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—1181no—
439C22×C4—1182no—
440C22×C4—1181no—
441C22×C4C2×C22⋊D5912yes—
442C22×C4C2×C22⋊D5911yescharacteristic
443C22×C4C2×C22⋊D5911yescharacteristic
444C22×C4C23×D5911yescharacteristic
445Q16—1188no—
446Q16—2368no—
447C2×C8—592no—
448C2×C8—1181no—
449C2×C8—1181no—
450C2×C8—1182no—
451C2×C8—1181no—
452C2×C8—1182no—
453C2×C8—1181no—
454C2×C8—1182no—
455C2×C8—1181no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—1182no—
463C4⋊C4—1181no—
464C4⋊C4—1181no—
465C4⋊C4—1182no—
466C4⋊C4—2364no—
467C4⋊C4—2362no—
468C4⋊C4C2×C22⋊D5912yes—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—2364no—
472C22⋊C4—2364no—
473C42—24no—
474C42—2362no—
475C42C2×C22⋊D5912yes—
476C42C2×C22⋊D5912yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23#1427375814yes—
484C23C22×C22⋊D5911yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—24no—
489Q8—24no—
490Q8—416no—
491Q8—42no—
492Q8—42no—
493Q8—1182no—
494Q8—1182no—
495Q8—2361no—
496Q8—2361no—
497Q8#1427369814yes—
498D4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—1182no—
518C2×C4—2361no—
519C2×C4—2362no—
520C2×C4—2362no—
521C2×C4—2362no—
522C2×C4—2362no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4#1427369612yes—
526C2×C4#1427369612yes—
527C2×C4C2×D4×D5911yescharacteristic
528C2×C4C22×C22⋊D5911yescharacteristic
529C2×C4C22×C22⋊D5911yescharacteristic
530C2×C4C2×#1174568711yescharacteristic
531C2×C4#1427377211yescharacteristic
532C2×C4#1427377211yescharacteristic
533C8—1182no—
534C8—2361no—
535C8—2362no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22C2×#1427369611yescharacteristic
544C22D4×C22⋊D5911yescharacteristic
545C22#1683804611yescharacteristic
546C22C2×#1427375812yes—
547C22#1683821712yes—
548C4—22no—
549C4—22no—
550C4—24no—
551C4—42no—
552C4—42no—
553C4—44no—
554C4—4722no—
555C4—4721no—
556C4—2362no—
557C4C2×#1427369811yescharacteristic
558C4#1683796911yescharacteristic
559C4D4×C22⋊D5911yescharacteristic
560C4#1683804611yescharacteristic
561C2—44no—
562C2#2142744611yescharacteristic
563C2#2143333011yescharacteristic
564C2#2143339111yescharacteristic
565C2#2143566712yes—
566C2#2143614712yes—
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×D4×Q8×C59C2
socleC23×C59C22×D4
prime radicalC22×D4×C59D4
hypercenterC2×D4×Q8D59
hypocenterC59#24255
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16837609C2semidirect
2. C2×#16837558C2semidirect
3. C2×#16837507C2semidirect
5. #21425289C2semidirect
6. #21425104C2semidirect
7. #21424917C2semidirect
9. #21424111C2semidirect
10. #21424031C2semidirect
52. C2×#11745672D4semidirect
329. C59#24255semidirect

Maximal subgroups

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

subgroupindex
C2×#168376092
C2×#168375582
C2×#168375072
#214290052
#214252892
#214251042
#214249172
#214248952
#214241112
#214240312
C2×D4×Q8×C592
#2425559