Finite Group Databasebeta

← #28481430

Subgroups — #28481430

Order 15104.

698 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
1#121×Dic59C211yesmaximal, characteristic
2#21424074C211yesmaximal, characteristic, complemented
3C2×#16837136C211yesmaximal, characteristic
4#21422322C211yesmaximal, characteristic, complemented
5#21421976C211yesmaximal, characteristic, complemented
6#21421975C211yesmaximal, characteristic, complemented
7C2×C59×#327C211yesmaximal, characteristic
8C2×D4×Dic59—21no—
9C2×D4×Dic59C411yescharacteristic, complemented
10C22×C4×Dic59C2211yescharacteristic
11C2×Dic59⋊C8—21no—
12C2×C22⋊(C59⋊C8)C2211yescharacteristic
13#16837305C411yescharacteristic, complemented
14#16837301C411yescharacteristic, complemented
15#16837295—21no—
16#16837282—21no—
17#16837262C2211yescharacteristic
18C4⋊C4×Dic59C411yescharacteristic, complemented
19C22⋊C4×Dic59—21no—
20C2×C59⋊#95—21no—
21C2×C59⋊#95—21no—
22C2×C59⋊#95C2211yescharacteristic
23#16837156—21no—
24#16837136—22no—
25C2×#121×C59C2211yescharacteristic
26C2×C22⋊C8×C59C2211yescharacteristic
27C2×#95×C59C2211yescharacteristic
28C59×#327—22no—
29C23×Dic59—21no—
30C23×Dic59—21no—
31C23×Dic59D411yescharacteristic
32C23×Dic59D411yescharacteristic
33C22×C59⋊C8—21no—
34D4×Dic59—42no—
35D4×Dic59—42no—
36D4×Dic59—42no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59C2×C411yescharacteristic
39C2×C4⋊Dic59C2×C411yescharacteristic
40C2×C22⋊Dic59—21no—
41C2×C22⋊Dic59—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44C2×C4×Dic59—21no—
45C2×C4×Dic59—42no—
46C2×C4×Dic59C2×C411yescharacteristic
47C2×C4×Dic59C2×C411yescharacteristic
48C2×C4×Dic59D411yescharacteristic
49C2×C4×Dic59D411yescharacteristic
50Dic59⋊C8—42no—
51C22⋊(C59⋊C8)—21no—
52C22⋊(C59⋊C8)—21no—
53C59⋊#95—24no—
54C59⋊#95—21no—
55C59⋊#95—21no—
56C59⋊#95—21no—
57C59⋊#95—21no—
58C59⋊#95—21no—
59C59⋊#95—42no—
60C22×D4×C59—21no—
61C22×D4×C59C2×C411yescharacteristic
62C23×C4×C59—21no—
63C23×C4×C59C2311yescharacteristic
64C22×C8×C59—21no—
65#121×C59—21no—
66#121×C59—21no—
67#121×C59D412yes—
68#121×C59Q812yes—
69C2×C4⋊C4×C59C2×C411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C22⋊C8×C59—21no—
72C22⋊C8×C59—21no—
73#95×C59—21no—
74#95×C59—21no—
75C24×C59—21no—
76C24×C59C2×D411yescharacteristic
77C2×D4×C59—22no—
78C2×D4×C59—22no—
79C2×D4×C59—21no—
80C2×D4×C59—41no—
81C2×D4×C59—41no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—41no—
85C2×D4×C59—41no—
86C2×D4×C59C4⋊C412yes—
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—41no—
94C22×C4×C59—41no—
95C22×C4×C59—41no—
96C22×C4×C59—41no—
97C22×C4×C59C22×C411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59C2×Q811yescharacteristic
101C22×C4×C59#4111yescharacteristic
102C22×C4×C59#4111yescharacteristic
103C22×Dic59—21no—
104C22×Dic59—21no—
105C22×Dic59—21no—
106C22×Dic59—21no—
107C22×Dic59—22no—
108C22×Dic59—22no—
109C22×Dic59—22no—
110C22×Dic59—21no—
111C22×Dic59—21no—
112C22×Dic59—41no—
113C22×Dic59—41no—
114C22×Dic59—42no—
115C22×Dic59—42no—
116C22×Dic59—41no—
117C22×Dic59—41no—
118C22×Dic59—41no—
119C22×Dic59—41no—
120C22×Dic59—41no—
121C22×Dic59—41no—
122C22×Dic59—42no—
123C22×Dic59C22⋊C411yescharacteristic
124C22×Dic59C22⋊C411yescharacteristic
125C22×Dic59C22⋊C411yescharacteristic
126C22×Dic59C22⋊C411yescharacteristic
127C2×C8×C59—41no—
128C2×C8×C59—41no—
129C2×C8×C59—41no—
130C4⋊C4×C59—21no—
131C4⋊C4×C59C4⋊C412yes—
132C22⋊C4×C59—22no—
133C22⋊C4×C59—22no—
134C22⋊Dic59—42no—
135C22⋊Dic59—42no—
136C22⋊Dic59—41no—
137C22⋊Dic59—41no—
138C22⋊Dic59—42no—
139C4⋊Dic59—42no—
140C4⋊Dic59—41no—
141C4⋊Dic59—41no—
142C4×Dic59—22no—
143C4×Dic59—22no—
144C4×Dic59—41no—
145C4×Dic59—41no—
146C4×Dic59—42no—
147C2×C59⋊C8—41no—
148C2×C59⋊C8—41no—
149C2×C59⋊C8—41no—
150C23×C59—21no—
151C23×C59—21no—
152C23×C59—21no—
153C23×C59—21no—
154C23×C59—22no—
155C23×C59—41no—
156C23×C59—41no—
157C23×C59—41no—
158C23×C59—41no—
159C23×C59—41no—
160C23×C59—41no—
161C23×C59C2×C22⋊C411yescharacteristic
162C23×C59#12111yescharacteristic
163C23×C59C22⋊Q811yescharacteristic
164C23×C59#13511yescharacteristic
165C23×C59#13511yescharacteristic
166C23×C59AGL(1,ℤ/8)11yescharacteristic
167C23×C59AGL(1,ℤ/8)11yescharacteristic
168D4×C59—42no—
169D4×C59—42no—
170D4×C59—42no—
171D4×C59—42no—
172D4×C59—42no—
173D4×C59—42no—
174C2×C4×C59—21no—
175C2×C4×C59—21no—
176C2×C4×C59—21no—
177C2×C4×C59—21no—
178C2×C4×C59—22no—
179C2×C4×C59—41no—
180C2×C4×C59—41no—
181C2×C4×C59—41no—
182C2×C4×C59—41no—
183C2×C4×C59—41no—
184C2×C4×C59—41no—
185C2×C4×C59—42no—
186C2×C4×C59—42no—
187C2×C4×C59—42no—
188C2×C4×C59—81no—
189C2×C4×C59—81no—
190C2×C4×C59—81no—
191C2×C4×C59—81no—
192C2×C4×C59C2×C4⋊C411yescharacteristic
193C2×C4×C59#11711yescharacteristic
194C2×C4×C59#12111yescharacteristic
195C2×C4×C59C22⋊Q811yescharacteristic
196C2×Dic59—21no—
197C2×Dic59—21no—
198C2×Dic59—21no—
199C2×Dic59—21no—
200C2×Dic59—21no—
201C2×Dic59—21no—
202C2×Dic59—21no—
203C2×Dic59—21no—
204C2×Dic59—21no—
205C2×Dic59—21no—
206C2×Dic59—42no—
207C2×Dic59—42no—
208C2×Dic59—42no—
209C2×Dic59—42no—
210C2×Dic59—42no—
211C2×Dic59—42no—
212C2×Dic59—42no—
213C2×Dic59—41no—
214C2×Dic59—41no—
215C2×Dic59—41no—
216C2×Dic59—41no—
217C2×Dic59—41no—
218C2×Dic59—41no—
219C2×Dic59—84no—
220C2×Dic59—81no—
221C2×Dic59—81no—
222C2×Dic59—81no—
223C2×Dic59—81no—
224C2×Dic59#9912yes—
225C2×Dic59#10412yes—
226C8×C59—81no—
227C59⋊C8—81no—
228#8575—591nomaximal, Sylow
229C22×C59—21no—
230C22×C59—21no—
231C22×C59—21no—
232C22×C59—21no—
233C22×C59—21no—
234C22×C59—21no—
235C22×C59—21no—
236C22×C59—21no—
237C22×C59—21no—
238C22×C59—21no—
239C22×C59—21no—
240C22×C59—21no—
241C22×C59—42no—
242C22×C59—42no—
243C22×C59—42no—
244C22×C59—81no—
245C22×C59—81no—
246C22×C59—81no—
247C22×C59—81no—
248C22×C59#38011yescharacteristic
249C22×C59C2×#9911yescharacteristic
250C22×C59#41011yescharacteristic
251C22×C59#41611yescharacteristic
252C22×C59#41811yescharacteristic
253C22×C59C2×#10411yescharacteristic
254C22×C59#42111yescharacteristic
255C22×C59#46311yescharacteristic
256C22×C59#46311yescharacteristic
257C22×C59#47811yescharacteristic
258C22×C59#47811yescharacteristic
259C4×C59—21no—
260C4×C59—21no—
261C4×C59—21no—
262C4×C59—21no—
263C4×C59—42no—
264C4×C59—82no—
265Dic59—22no—
266Dic59—22no—
267Dic59—41no—
268Dic59—41no—
269Dic59—82no—
270Dic59—81no—
271Dic59—81no—
272C4×#121—591no—
273#1797—591no—
274C2×#327—591no—
275C2×#327D5911yescharacteristic
276#1434—591no—
277#1294—591no—
278#1294—591no—
279C2×C59—21no—
280C2×C59—21no—
281C2×C59—21no—
282C2×C59—21no—
283C2×C59—82no—
284C2×C59#150111yescharacteristic
285C2×C59#150511yescharacteristic
286C2×C59#151011yescharacteristic
287C2×C59#152111yescharacteristic
288C2×C59#179911yescharacteristic
289C2×C59#180411yescharacteristic
290C2×C59#180611yescharacteristic
291C2×#121C2×D5911yescharacteristic
292C2×C4×D4—591no—
293C2×C4×D4—1181no—
294C22×C42—591no—
295C2×C4⋊C8—1181no—
296C2×C22⋊C8—591no—
297C2×C22⋊C8C2×D5911yescharacteristic
298#390—591no—
299#389—591no—
300#388—1181no—
301#386—1181no—
302#380—591no—
303C4×C4⋊C4—591no—
304C4×C22⋊C4—1181no—
305C2×#95—591no—
306C2×#95—1181no—
307C2×#95—1181no—
308C2×#95C2×D5911yescharacteristic
309#336—1181no—
310#327—22no—
311#327—1182no—
312C59#857511yescharacteristic, complemented, Sylow
313C22×D4—21no—
314C22×D4C4×D5911yescharacteristic
315C23×C4—21no—
316C23×C4—591no—
317C23×C4—591no—
318C23×C4—1181no—
319C23×C4—1181no—
320C23×C4C22×D5911yescharacteristic
321C22×C8—21no—
322C22×C8—1181no—
323#121—21no—
324#121—21no—
325#121Dic11812yes—
326#121C22⋊D5912yes—
327C4×D4—2362no—
328C4×D4—2362no—
329C4×D4—2362no—
330C2×C4⋊C4—591no—
331C2×C4⋊C4—591no—
332C2×C4⋊C4—1181no—
333C2×C4⋊C4C4×D5911yescharacteristic
334C2×C22⋊C4—21no—
335C2×C22⋊C4—1181no—
336C2×C22⋊C4—1181no—
337C2×C22⋊C4—1181no—
338C2×C22⋊C4—1181no—
339C2×C42—591no—
340C2×C42—591no—
341C2×C42—591no—
342C2×C42—591no—
343C2×C42—1181no—
344C2×C42—2362no—
345C4⋊C8—2362no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—1181no—
349C22⋊C8—1181no—
350#95—21no—
351#95—21no—
352#95—1181no—
353#95—1181no—
354#95—1184no—
355#95—1181no—
356#95—1181no—
357#95—1181no—
358#95—2362no—
359C24—21no—
360C24D4×D5911yescharacteristic
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4Dic59⋊C412yes—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—591no—
382C22×C4—591no—
383C22×C4—591no—
384C22×C4—591no—
385C22×C4—1182no—
386C22×C4—1182no—
387C22×C4—1181no—
388C22×C4—1181no—
389C22×C4—1181no—
390C22×C4—1181no—
391C22×C4—1182no—
392C22×C4—1181no—
393C22×C4—1181no—
394C22×C4—2362no—
395C22×C4—2361no—
396C22×C4—2361no—
397C22×C4—2362no—
398C22×C4—2362no—
399C22×C4—2361no—
400C22×C4—2361no—
401C22×C4—2361no—
402C22×C4—2361no—
403C22×C4—2361no—
404C22×C4—2361no—
405C22×C4C2×Dic11811yescharacteristic
406C22×C4C2×C4×D5911yescharacteristic
407C22×C4D4×D5911yescharacteristic
408C22×C4#1174568711yescharacteristic
409C22×C4#1174568711yescharacteristic
410C22×C4C2×C22⋊D5911yescharacteristic
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—2361no—
415C2×C8—2361no—
416C2×C8—2361no—
417C4⋊C4—21no—
418C4⋊C4—2361no—
419C4⋊C4—2361no—
420C4⋊C4—2362no—
421C4⋊C4Dic59⋊C412yes—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—2362no—
425C22⋊C4—2362no—
426C22⋊C4—2362no—
427C22⋊C4—2361no—
428C22⋊C4—2361no—
429C42—1182no—
430C42—1182no—
431C42—2361no—
432C42—2361no—
433C42—2362no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—22no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23C22⋊C4×D5911yescharacteristic
446C23#1427369911yescharacteristic
447C23#1427370411yescharacteristic
448C23#1427374611yescharacteristic
449C23#1427374811yescharacteristic
450C23#1427375111yescharacteristic
451C23#1427375411yescharacteristic
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—42no—
457D4—42no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—592no—
477C2×C4—592no—
478C2×C4—1181no—
479C2×C4—1181no—
480C2×C4—1181no—
481C2×C4—1181no—
482C2×C4—1181no—
483C2×C4—1181no—
484C2×C4—1181no—
485C2×C4—1181no—
486C2×C4—1181no—
487C2×C4—1181no—
488C2×C4—4724no—
489C2×C4—4721no—
490C2×C4—4721no—
491C2×C4—4721no—
492C2×C4—4721no—
493C2×C4—2362no—
494C2×C4—2362no—
495C2×C4—2362no—
496C2×C4—2362no—
497C2×C4—2362no—
498C2×C4—2362no—
499C2×C4—2362no—
500C2×C4—2361no—
501C2×C4—2361no—
502C2×C4—2361no—
503C2×C4—2361no—
504C2×C4—2361no—
505C2×C4—2361no—
506C2×C4C2×Dic59⋊C411yescharacteristic
507C2×C4#1427368311yescharacteristic
508C2×C4#1427369911yescharacteristic
509C2×C4#1427370411yescharacteristic
510C8—81no—
511C8—4721no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22#1683726011yescharacteristic
532C22#99×D5911yescharacteristic
533C22#1683737411yescharacteristic
534C22#1683739511yescharacteristic
535C22#1683740511yescharacteristic
536C22#104×D5911yescharacteristic
537C22#1683741711yescharacteristic
538C22#1683761511yescharacteristic
539C22#1683761911yescharacteristic
540C22#1683770011yescharacteristic
541C22#1683770411yescharacteristic
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—42no—
547C4—82no—
548C4—1182no—
549C4—1182no—
550C4—4722no—
551C4—4721no—
552C4—4721no—
553C4—2361no—
554C4—2361no—
555C2—21no—
556C2—21no—
557C2—21no—
558C2—21no—
559C2—82no—
560C2#2142258411yescharacteristic
561C2#2142260211yescharacteristic
562C2#2142262311yescharacteristic
563C2#2142267011yescharacteristic
564C2#2142408211yescharacteristic
565C2#2142410411yescharacteristic
566C2#2142411411yescharacteristic
subgroupG/H
centerC23C22⋊C4×D59
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC2×C59×#327C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#121×C59C22
cosocleC23×C4×C59C23
hypercenterC2×#327D59
hypocenterC59#8575
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#327

Complements

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

NH ≅ G/Ntype
2. #21424074C2semidirect
4. #21422322C2semidirect
5. #21421976C2semidirect
6. #21421975C2semidirect
9. C2×D4×Dic59C4semidirect
13. #16837305C4semidirect
14. #16837301C4semidirect
18. C4⋊C4×Dic59C4semidirect
312. C59#8575semidirect

Maximal subgroups

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

subgroupindex
#121×Dic592
#214240742
C2×#168371362
#214223222
#214219762
#214219752
C2×C59×#3272
#857559