Finite Group Databasebeta

← #28623045

Subgroups — #28623045

Order 15104.

1114 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837694C211yesmaximal, characteristic, complemented
2C2×#16837609C211yesmaximal, characteristic, complemented
3C4×#14273706C211yesmaximal, characteristic
4#21425296C212yesmaximal
5#21425104C212yesmaximal
6#21425045C211yesmaximal, characteristic, complemented
7#21425020C211yesmaximal, characteristic, complemented
8#21424437C212yesmaximal, complemented
9#21424350C212yesmaximal, complemented
10#21424076C211yesmaximal, characteristic
11C2×C59×#533C211yesmaximal, characteristic
12C22×#11745672C2212yes—
13C2×#14273706C2211yescharacteristic
14C2×C4×Dic118C2211yescharacteristic
15#16837694—24no—
16#16837609—24no—
17C2×#14273660C2212yes—
18C2×C4⋊(C59⋊C8)C2211yescharacteristic
19C2×C4⋊(C59⋊C8)C2211yescharacteristic
20C2×#14273651C2212yes—
21C2×#14273651C2212yes—
22C2×#14273651C2212yes—
23C2×#14273650C2212yes—
24C2×C22⋊(C59⋊C8)C2212yes—
25#16837308C2211yescharacteristic
26#16837300C2211yescharacteristic
27#16837286—21no—
28#16837275—21no—
29#16837262C2211yescharacteristic
30C4×Dic59⋊C4—21no—
31C4×C4⋊Dic59C2211yescharacteristic
32C4×C22⋊Dic59—21no—
33#16837164C2212yes—
34#16837164C2212yes—
35#16837157C2212yes—
36C59×#533—24no—
37C59×#533—24no—
38C2×C4⋊Q8×C59C2211yescharacteristic
39C2×C4⋊Q8×C59C2211yescharacteristic
40C2×#125×C59—21no—
41C2×C22⋊Q8×C59—21no—
42C2×C22⋊Q8×C59C2212yes—
43C2×C4×Q8×C59—21no—
44C2×C4×Q8×C59C2212yes—
45C2×#117×C59—21no—
46C22×C42×C59C2211yescharacteristic
47C22×Dic118C2311yescharacteristic
48C2×#11745672—24no—
49C2×#11745672—24no—
50C2×#11745672—42no—
51C22×C59⋊C8C2312yes—
52C22×C59⋊C8C2312yes—
53#14273706—22no—
54#14273706—22no—
55C4×Dic118—22no—
56C4×Dic118—22no—
57C2×C4⋊Dic59C2311yescharacteristic
58C2×C4⋊Dic59C2311yescharacteristic
59C2×C4⋊Dic59C2311yescharacteristic
60C2×Dic59⋊C4—21no—
61C2×Dic59⋊C4—21no—
62C2×Dic59⋊C4—21no—
63C2×C22⋊Dic59—21no—
64C2×C22⋊Dic59—21no—
65C2×C4×Dic59—21no—
66C2×C4×Dic59—21no—
67#14273660—24no—
68C4⋊(C59⋊C8)—22no—
69C4⋊(C59⋊C8)—22no—
70#14273651—24no—
71#14273651—24no—
72#14273651—42no—
73#14273650—24no—
74C22⋊(C59⋊C8)—42no—
75C59⋊#95—21no—
76C59⋊#95—21no—
77C59⋊#95—21no—
78C59⋊#95—21no—
79C59⋊#95—21no—
80C22×Q8×C59—21no—
81C22×Q8×C59C2312yes—
82C23×C4×C59D411yescharacteristic
83C23×C4×C59D411yescharacteristic
84C23×C4×C59C2311yescharacteristic
85C4⋊Q8×C59—22no—
86C4⋊Q8×C59—22no—
87C4⋊Q8×C59D414yes—
88C4⋊Q8×C59D414yes—
89#125×C59—28no—
90C22⋊Q8×C59—28no—
91C22⋊Q8×C59—24no—
92C22⋊Q8×C59—24no—
93C4×Q8×C59—28no—
94C4×Q8×C59—24no—
95C4×Q8×C59—24no—
96#117×C59—28no—
97C2×C4⋊C4×C59—21no—
98C2×C4⋊C4×C59—21no—
99C2×C4⋊C4×C59—21no—
100C2×C4⋊C4×C59—22no—
101C2×C4⋊C4×C59—22no—
102C2×C4⋊C4×C59C2312yes—
103C2×C22⋊C4×C59—22no—
104C2×C42×C59—22no—
105C2×C42×C59—22no—
106C2×C42×C59—22no—
107C2×C42×C59D411yescharacteristic
108C2×C42×C59D411yescharacteristic
109C2×C42×C59C2311yescharacteristic
110C2×C42×C59C2311yescharacteristic
111C24×C59C2×D411yescharacteristic
112C2×Q8×C59—24no—
113C2×Q8×C59—22no—
114C2×Q8×C59—22no—
115C2×Q8×C59—22no—
116C2×Q8×C59—22no—
117C2×Q8×C59—22no—
118C2×Q8×C59—42no—
119C2×Q8×C59—42no—
120C2×Q8×C59C2×D414yes—
121C22×C4×C59—22no—
122C22×C4×C59—22no—
123C22×C4×C59—22no—
124C22×C4×C59—22no—
125C22×C4×C59—22no—
126C22×C4×C59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59—21no—
132C22×C4×C59—41no—
133C22×C4×C59—41no—
134C22×C4×C59Q1614yescomplemented
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59C2×D411yescharacteristic
138C22×C4×C59C2×D411yescharacteristic
139C22×C4×C59C2×D411yescharacteristic
140C22×C4×C59C2411yescharacteristic
141C22×Dic59—21no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C2×Dic118—21no—
146C2×Dic118—21no—
147C2×Dic118—21no—
148C2×Dic118—21no—
149C2×Dic118—21no—
150C2×Dic118#4112yes—
151C4⋊C4×C59—28no—
152C4⋊C4×C59—28no—
153C4⋊C4×C59—24no—
154C4⋊C4×C59—24no—
155C4⋊C4×C59—24no—
156C4⋊C4×C59—22no—
157C4⋊C4×C59C2×D414yes—
158C22⋊C4×C59—28no—
159C42×C59—28no—
160C42×C59—22no—
161C42×C59—22no—
162C42×C59—21no—
163C42×C59—21no—
164C42×C59C2×D412yes—
165C42×C59C2×D412yes—
166C22⋊Dic59—24no—
167C22⋊Dic59—42no—
168#11745672—48no—
169C4⋊Dic59—21no—
170C4⋊Dic59—21no—
171C4⋊Dic59—21no—
172C4⋊Dic59—21no—
173C4⋊Dic59—21no—
174C4⋊Dic59#4112yes—
175Dic59⋊C4—24no—
176Dic59⋊C4—42no—
177Dic59⋊C4—42no—
178C4×Dic59—42no—
179C4×Dic59—42no—
180C2×C59⋊C8—22no—
181C2×C59⋊C8—22no—
182C2×C59⋊C8—22no—
183C2×C59⋊C8—22no—
184C2×C59⋊C8—22no—
185C2×C59⋊C8—22no—
186C23×C59—21no—
187C23×C59—21no—
188C23×C59—21no—
189C23×C59—21no—
190C23×C59—21no—
191C23×C59—21no—
192C23×C59C2×Q1612yes—
193C23×C59C22×D411yescharacteristic
194Q8×C59—24no—
195Q8×C59—24no—
196Q8×C59—48no—
197Q8×C59—42no—
198Q8×C59—42no—
199C2×C4×C59—21no—
200C2×C4×C59—21no—
201C2×C4×C59—21no—
202C2×C4×C59—21no—
203C2×C4×C59—28no—
204C2×C4×C59—21no—
205C2×C4×C59—21no—
206C2×C4×C59—24no—
207C2×C4×C59—24no—
208C2×C4×C59—21no—
209C2×C4×C59—21no—
210C2×C4×C59—21no—
211C2×C4×C59—21no—
212C2×C4×C59—44no—
213C2×C4×C59—42no—
214C2×C4×C59—42no—
215C2×C4×C59—42no—
216C2×C4×C59—42no—
217C2×C4×C59—41no—
218C2×C4×C59—41no—
219C2×C4×C59—41no—
220C2×C4×C59—41no—
221C2×C4×C59#12012yes—
222C2×C4×C59#12012yes—
223C2×C4×C59C2×Q1612yes—
224C2×C4×C59C2×Q1612yes—
225C2×C4×C59#13712yes—
226C2×C4×C59C22×D411yescharacteristic
227C2×C4×C59C22×D411yescharacteristic
228C2×C4×C59C2×#4111yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C2×C4×C59H(5,𝔽2)11yescharacteristic
231C2×C4×C592−1+411yescharacteristic
232C2×Dic59—22no—
233C2×Dic59—22no—
234C2×Dic59—41no—
235C2×Dic59—41no—
236C2×Dic59—41no—
237C2×Dic59—41no—
238C2×Dic59—41no—
239C2×Dic59—41no—
240C2×Dic59—41no—
241C2×Dic59—41no—
242C2×Dic59—41no—
243C2×Dic59—41no—
244Dic118—22no—
245Dic118—22no—
246Dic118—41no—
247Dic118—41no—
248C59⋊C8—42no—
249C59⋊C8—42no—
250#24155—591nomaximal, Sylow
251C22×C59—22no—
252C22×C59—22no—
253C22×C59—22no—
254C22×C59—41no—
255C22×C59—41no—
256C22×C59—41no—
257C22×C59—41no—
258C22×C59C2×#12011yescharacteristic
259C22×C59#52511yescharacteristic
260C22×C59#53711yescharacteristic
261C22×C59C22×Q1611yescharacteristic
262C22×C59C2×#13711yescharacteristic
263C22×C59#57511yescharacteristic
264C22×C59#57511yescharacteristic
265C4×C59—22no—
266C4×C59—22no—
267C4×C59—41no—
268C4×C59—44no—
269C4×C59—41no—
270C4×C59—44no—
271C4×C59#45114yes—
272C4×C59#54611yescharacteristic
273C4×C59#54711yescharacteristic
274C4×C59#54811yescharacteristic
275C4×C59#55011yescharacteristic
276Dic59—42no—
277Dic59—42no—
278Dic59—81no—
279Dic59—81no—
280C2×#533D5911yescharacteristic
281C2×#477—591no—
282C2×#462—591no—
283C4×C22⋊Q8—591no—
284#2036—592no—
285#2001—592no—
286#1987—591no—
287#1981—591no—
288#1878—592no—
289#1863—592no—
290#1797—591no—
291C2×C59—42no—
292C2×C59#240711yescharacteristic
293C2×C59C2×#45111yescharacteristic
294C2×C59#300511yescharacteristic
295C2×C59#310411yescharacteristic
296C2×C59#311611yescharacteristic
297C2×C59#324911yescharacteristic
298C2×C59#325211yescharacteristic
299C22×Q16—592no—
300#533—24no—
301#533—24no—
302C2×C4⋊Q8C2×D5911yescharacteristic
303C2×C4⋊Q8C2×D5911yescharacteristic
304C2×#125—21no—
305C2×C22⋊Q8—21no—
306C2×C22⋊Q8—591no—
307C2×C22⋊Q8C2×D5912yes—
308C2×C4×Q8—21no—
309C2×C4×Q8—591no—
310C2×C4×Q8C2×D5912yes—
311C2×#117—21no—
312C22×C42C2×D5911yescharacteristic
313#477—1184no—
314#462—1184no—
315C2×#107—592no—
316C2×C4⋊C8—591no—
317C2×C4⋊C8—591no—
318C2×#103—592no—
319C2×#103—592no—
320C2×#103—592no—
321C2×#103—592no—
322C2×C22⋊C8—592no—
323#391—591no—
324#389—591no—
325#387—1181no—
326#385—1181no—
327#380—591no—
328C4×C4⋊C4—591no—
329C4×C4⋊C4—1181no—
330C4×C22⋊C4—1181no—
331#340—592no—
332#340—592no—
333#336—592no—
334C59#2415511yescharacteristic, complemented, Sylow
335C22×Q8—21no—
336C22×Q8—591no—
337C22×Q8C22×D5912yes—
338C23×C4C22⋊D5911yescharacteristic
339C23×C4C22⋊D5911yescharacteristic
340C23×C4C22×D5911yescharacteristic
341C2×Q16—1184no—
342C2×Q16—1184no—
343C2×Q16—2362no—
344C22×C8—592no—
345C22×C8—592no—
346C4⋊Q8—22no—
347C4⋊Q8—22no—
348C4⋊Q8C22⋊D5914yes—
349C4⋊Q8C22⋊D5914yes—
350#125—28no—
351C22⋊Q8—28no—
352C22⋊Q8—24no—
353C22⋊Q8—24no—
354C22⋊Q8—1182no—
355C22⋊Q8—1182no—
356C4×Q8—28no—
357C4×Q8—24no—
358C4×Q8—24no—
359C4×Q8—1182no—
360C4×Q8—1182no—
361#117—28no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—591no—
368C2×C4⋊C4—591no—
369C2×C4⋊C4—591no—
370C2×C4⋊C4—1181no—
371C2×C4⋊C4—1181no—
372C2×C4⋊C4—1181no—
373C2×C4⋊C4C22×D5912yes—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—1181no—
376C2×C22⋊C4—1181no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—1181no—
381C2×C42—1181no—
382C2×C42C22⋊D5911yescharacteristic
383C2×C42C22⋊D5911yescharacteristic
384C2×C42C22×D5911yescharacteristic
385C2×C42C22×D5911yescharacteristic
386#107—1184no—
387C4⋊C8—1182no—
388C4⋊C8—1182no—
389#103—1184no—
390#103—1184no—
391#103—1184no—
392#103—2362no—
393C22⋊C8—2362no—
394#95—1181no—
395#95—1181no—
396#95—1181no—
397#95—1181no—
398#95—1181no—
399C24C2×C22⋊D5911yescharacteristic
400C2×Q8—24no—
401C2×Q8—22no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—42no—
407C2×Q8—42no—
408C2×Q8—592no—
409C2×Q8—1181no—
410C2×Q8—1181no—
411C2×Q8—1181no—
412C2×Q8—1181no—
413C2×Q8—1181no—
414C2×Q8C2×C22⋊D5914yes—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—1181no—
429C22×C4—1181no—
430C22×C4—1181no—
431C22×C4—1181no—
432C22×C4#1174567214yescomplemented
433C22×C4C2×C22⋊D5911yescharacteristic
434C22×C4C2×C22⋊D5911yescharacteristic
435C22×C4C2×C22⋊D5911yescharacteristic
436C22×C4C2×C22⋊D5911yescharacteristic
437C22×C4C2×C22⋊D5911yescharacteristic
438C22×C4C23×D5911yescharacteristic
439Q16—2368no—
440C2×C8—1182no—
441C2×C8—1182no—
442C2×C8—1182no—
443C2×C8—1182no—
444C2×C8—1182no—
445C2×C8—1182no—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—24no—
451C4⋊C4—22no—
452C4⋊C4—592no—
453C4⋊C4—1184no—
454C4⋊C4—1181no—
455C4⋊C4—1181no—
456C4⋊C4—1181no—
457C4⋊C4—1181no—
458C4⋊C4—1181no—
459C4⋊C4—2362no—
460C4⋊C4—2362no—
461C4⋊C4C2×C22⋊D5914yes—
462C22⋊C4—28no—
463C22⋊C4—1184no—
464C22⋊C4—2362no—
465C42—28no—
466C42—22no—
467C42—22no—
468C42—21no—
469C42—21no—
470C42—2362no—
471C42—2362no—
472C42C2×C22⋊D5912yes—
473C42C2×C22⋊D5912yes—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—21no—
480C23C2×#1174567212yes—
481C23C22×C22⋊D5911yescharacteristic
482Q8—24no—
483Q8—24no—
484Q8—48no—
485Q8—42no—
486Q8—42no—
487Q8—1182no—
488Q8—1182no—
489Q8—2361no—
490Q8—2361no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—28no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—1182no—
514C2×C4—1182no—
515C2×C4—2361no—
516C2×C4—2361no—
517C2×C4—2361no—
518C2×C4—2361no—
519C2×C4—2361no—
520C2×C4—2361no—
521C2×C4—2361no—
522C2×C4—2361no—
523C2×C4—2361no—
524C2×C4—2361no—
525C2×C4#1427369612yes—
526C2×C4#1427369612yes—
527C2×C4C2×#1174567212yes—
528C2×C4C2×#1174567212yes—
529C2×C4#1427375812yes—
530C2×C4C22×C22⋊D5911yescharacteristic
531C2×C4C22×C22⋊D5911yescharacteristic
532C2×C4#41×D5911yescharacteristic
533C2×C4#41×D5911yescharacteristic
534C2×C4#1427377311yescharacteristic
535C2×C4#1427377411yescharacteristic
536C8—2362no—
537C8—2362no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22C2×#1427369611yescharacteristic
546C22#1683790911yescharacteristic
547C22#1683798011yescharacteristic
548C22C22×#1174567211yescharacteristic
549C22C2×#1427375811yescharacteristic
550C22#1683820011yescharacteristic
551C22#1683820011yescharacteristic
552C4—22no—
553C4—22no—
554C4—41no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—4721no—
559C4—4721no—
560C4—2362no—
561C4—2362no—
562C4#1683755814yes—
563C4#1683804011yescharacteristic
564C4#1683805111yescharacteristic
565C4#1683805911yescharacteristic
566C4#1683807111yescharacteristic
567C2—42no—
568C2#2142743511yescharacteristic
569C2C2×#1683755811yescharacteristic
570C2#2143333811yescharacteristic
571C2#2143413111yescharacteristic
572C2#2143425211yescharacteristic
573C2#2143543811yescharacteristic
574C2#2143546311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#533C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#533D59
hypocenterC59#24155
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. C2×#16837694C2semidirect
2. C2×#16837609C2semidirect
6. #21425045C2semidirect
7. #21425020C2semidirect
8. #21424437C2semidirect
9. #21424350C2semidirect
134. C22×C4×C59Q16semidirect
334. C59#24155semidirect
432. C22×C4#11745672semidirect

Maximal subgroups

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

subgroupindex
C2×#168376942
C2×#168376092
C4×#142737062
#214252962
#214251042
#214250452
#214250202
#214244372
#214243502
#214240762
C2×C59×#5332
#2415559