Finite Group Databasebeta

← #28636938

Subgroups — #28636938

Order 15104.

858 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837735C211yesmaximal, characteristic
2#21428575C211yesmaximal, characteristic
3#21427105C211yesmaximal, characteristic
4#21425570C211yesmaximal, characteristic, complemented
5#21425507C211yesmaximal, characteristic, complemented
6#21425487C211yesmaximal, characteristic, complemented
7#21425416C211yesmaximal, characteristic, complemented
8#21424851C211yesmaximal, characteristic, complemented
9#21424740C211yesmaximal, characteristic, complemented
10#21424439C211yesmaximal, characteristic, complemented
11#21424404C211yesmaximal, characteristic
12#21424274C211yesmaximal, characteristic
13#21424099C211yesmaximal, characteristic, complemented
14C2×#16837165C211yesmaximal, characteristic
15C2×C59×#467C211yesmaximal, characteristic
16C2×Q8×Dic59C2211yescharacteristic
17C22×C4⋊Dic59C2211yescharacteristic
18C22×C4⋊Dic59C2211yescharacteristic
19#16837735—24no—
20C2×#14273660C2211yescharacteristic
21C2×#14273660C2211yescharacteristic
22C2×Dic59⋊C8C2211yescharacteristic
23C2×Dic59⋊C8C2211yescharacteristic
24C2×#14273651C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×C22⋊(C59⋊C8)C2211yescharacteristic
29#16837332—21no—
30#16837332—21no—
31#16837309C2211yescharacteristic
32#16837309C2211yescharacteristic
33#16837309C2211yescharacteristic
34#16837301—21no—
35#16837301C2211yescharacteristic
36#16837288—21no—
37#16837277—21no—
38#16837274—21no—
39#16837274—21no—
40#16837271C2211yescharacteristic
41#16837271C2211yescharacteristic
42C4⋊C4×Dic59C2211yescharacteristic
43C22⋊C4×Dic59—21no—
44C4×C22⋊Dic59C2211yescharacteristic
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837164C2211yescharacteristic
51#16837156C2211yescharacteristic
52#16837156C2211yescharacteristic
53C22×Q16×C59C2211yescharacteristic
54C23×C8×C59C2211yescharacteristic
55C2×C22⋊Q8×C59C2211yescharacteristic
56C2×C22⋊Q8×C59C2211yescharacteristic
57C59×#467—24no—
58C2×#107×C59C2211yescharacteristic
59C2×#103×C59C2211yescharacteristic
60C2×#103×C59C2211yescharacteristic
61C23×Dic59—21no—
62C23×Dic59—21no—
63C22×C59⋊C8C2311yescharacteristic
64C22×C59⋊C8C2311yescharacteristic
65Q8×Dic59—22no—
66Q8×Dic59—41no—
67C2×C4⋊Dic59—22no—
68C2×C4⋊Dic59—22no—
69C2×C4⋊Dic59—22no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59—21no—
74C2×C4⋊Dic59—41no—
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×Dic59⋊C4—21no—
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C22⋊Dic59D411yescharacteristic
84C2×C22⋊Dic59D411yescharacteristic
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59C2311yescharacteristic
88C2×C4×Dic59C2311yescharacteristic
89#14273660—22no—
90#14273660—22no—
91Dic59⋊C8—22no—
92Dic59⋊C8—41no—
93#14273651—22no—
94#14273651—22no—
95#14273651—22no—
96#14273651—41no—
97C22⋊(C59⋊C8)—22no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95D411yescharacteristic
107C59⋊#95D411yescharacteristic
108C22×Q8×C59C2311yescharacteristic
109C22×Q8×C59C2311yescharacteristic
110C23×C4×C59C2311yescharacteristic
111C2×Q16×C59—22no—
112C2×Q16×C59—22no—
113C2×Q16×C59—41no—
114C22×C8×C59—22no—
115C22×C8×C59—22no—
116C22×C8×C59—41no—
117C22×C8×C59C2311yescharacteristic
118C22×C8×C59C2311yescharacteristic
119C22⋊Q8×C59—22no—
120C22⋊Q8×C59—22no—
121C22⋊Q8×C59—22no—
122C22⋊Q8×C59D414yes—
123C2×C4⋊C4×C59—21no—
124C2×C4⋊C4×C59—21no—
125C2×C4⋊C4×C59C2311yescharacteristic
126C2×C4⋊C4×C59C2311yescharacteristic
127C2×C22⋊C4×C59—21no—
128C2×C22⋊C4×C59—21no—
129#107×C59—22no—
130#103×C59—22no—
131#103×C59—22no—
132C24×C59C2×D411yescharacteristic
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59—21no—
139C2×Q8×C59—21no—
140C2×Q8×C59—21no—
141C2×Q8×C59—21no—
142C2×Q8×C59—21no—
143C2×Q8×C59—21no—
144C2×Q8×C59C2×D412yes—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59—21no—
149C22×C4×C59—21no—
150C22×C4×C59—21no—
151C22×C4×C59—21no—
152C22×C4×C59—21no—
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2×D412yes—
155C22×C4×C59#4112yes—
156C22×C4×C59C2411yescharacteristic
157C22×Dic59—21no—
158C22×Dic59—21no—
159C22×Dic59—21no—
160C22×Dic59—21no—
161C22×Dic59—42no—
162C22×Dic59—42no—
163C22×Dic59—42no—
164C22×Dic59—81no—
165C22×Dic59—81no—
166C22×Dic59—81no—
167C22×Dic59C2×D411yescharacteristic
168C22×Dic59C2×D411yescharacteristic
169C22×Dic59C2×D411yescharacteristic
170C22×Dic59C2×D411yescharacteristic
171Q16×C59—44no—
172C2×C8×C59—21no—
173C2×C8×C59—21no—
174C2×C8×C59—21no—
175C2×C8×C59—21no—
176C2×C8×C59—21no—
177C2×C8×C59—44no—
178C2×C8×C59#4112yes—
179C4⋊C4×C59—24no—
180C4⋊C4×C59—21no—
181C4⋊C4×C59—21no—
182C4⋊C4×C59—21no—
183C4⋊C4×C59—42no—
184C4⋊C4×C59C2×D412yes—
185C22⋊C4×C59—24no—
186C22⋊C4×C59—42no—
187C22⋊Dic59—22no—
188C22⋊Dic59Q1614yescomplemented
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—42no—
197C4⋊Dic59—42no—
198C4⋊Dic59—42no—
199C4⋊Dic59—81no—
200C4⋊Dic59#4112yes—
201Dic59⋊C4—81no—
202Dic59⋊C4—81no—
203Dic59⋊C4—81no—
204Dic59⋊C4—81no—
205C4×Dic59—21no—
206C4×Dic59—21no—
207C4×Dic59—21no—
208C4×Dic59—21no—
209C4×Dic59—42no—
210C4×Dic59—81no—
211C2×C59⋊C8—21no—
212C2×C59⋊C8—21no—
213C2×C59⋊C8—21no—
214C2×C59⋊C8—21no—
215C2×C59⋊C8—21no—
216C2×C59⋊C8—21no—
217C23×C59—21no—
218C23×C59—21no—
219C23×C59—21no—
220C23×C59—21no—
221C23×C59#12112yes—
222C23×C59C2×Q1612yes—
223C23×C59#13712yes—
224C23×C59C22×D411yescharacteristic
225Q8×C59—22no—
226Q8×C59—22no—
227Q8×C59—41no—
228Q8×C59—41no—
229Q8×C59—41no—
230Q8×C59—41no—
231Q8×C59—41no—
232Q8×C59—41no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—22no—
238C2×C4×C59—22no—
239C2×C4×C59—22no—
240C2×C4×C59—22no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—41no—
244C2×C4×C59—41no—
245C2×C4×C59—41no—
246C2×C4×C59—41no—
247C2×C4×C59—41no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59#12112yes—
252C2×C4×C59C22×D411yescharacteristic
253C2×C4×C59C2×#4111yescharacteristic
254C2×C4×C59C2×#4111yescharacteristic
255C2×C4×C59C2×#4111yescharacteristic
256C2×C4×C59H(5,𝔽2)11yescharacteristic
257C2×C4×C592−1+411yescharacteristic
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—22no—
264C2×Dic59—21no—
265C2×Dic59—21no—
266C2×Dic59—21no—
267C2×Dic59—21no—
268C2×Dic59—21no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—41no—
275C2×Dic59—41no—
276C2×Dic59—41no—
277C2×Dic59—41no—
278C2×Dic59—41no—
279C2×Dic59—41no—
280C2×Dic59—84no—
281C2×Dic59—84no—
282C2×Dic59C2×Q1612yes—
283C2×Dic59C2×Q1612yes—
284C8×C59—22no—
285C8×C59—41no—
286C59⋊C8—41no—
287C59⋊C8—41no—
288#25206—591nomaximal, Sylow
289C22×C59—22no—
290C22×C59—22no—
291C22×C59—22no—
292C22×C59—22no—
293C22×C59—22no—
294C22×C59—22no—
295C22×C59#46214yes—
296C22×C59C2×#12111yescharacteristic
297C22×C59#54611yescharacteristic
298C22×C59#54711yescharacteristic
299C22×C59C22×Q1611yescharacteristic
300C22×C59C2×#13711yescharacteristic
301C22×C59#57611yescharacteristic
302C22×C59#57711yescharacteristic
303C4×C59—22no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59—81no—
307C4×C59—81no—
308C4×C59#52911yescharacteristic
309C4×C59#53911yescharacteristic
310C4×C59#54611yescharacteristic
311C4×C59#54711yescharacteristic
312Dic59—22no—
313Dic59—22no—
314Dic59—41no—
315Dic59—41no—
316Dic59—42no—
317Dic59—81no—
318Dic59—81no—
319Dic59—81no—
320C2×#484—591no—
321C2×#467D5911yescharacteristic
322#2502—591no—
323#2360—591no—
324#2083—591no—
325#2072—591no—
326#2068—591no—
327#2056—591no—
328#1947—591no—
329#1928—591no—
330#1878—591no—
331#1872—591no—
332#1850—591no—
333#1803—591no—
334C2×#340—591no—
335C2×C59—24no—
336C2×C59#242511yescharacteristic
337C2×C59C2×#46211yescharacteristic
338C2×C59#304511yescharacteristic
339C2×C59#329911yescharacteristic
340C2×C59#331411yescharacteristic
341C2×C59#333811yescharacteristic
342C2×C59#334411yescharacteristic
343C22×Q16C2×D5911yescharacteristic
344C23×C8C2×D5911yescharacteristic
345C2×C22⋊Q8C2×D5911yescharacteristic
346C2×C22⋊Q8C2×D5911yescharacteristic
347C2×C4×Q8—591no—
348C22×C4⋊C4—591no—
349C22×C4⋊C4—591no—
350#484—1184no—
351#467—24no—
352C2×#107—591no—
353C2×#107—591no—
354C2×#107C2×D5911yescharacteristic
355C2×C4⋊C8—591no—
356C2×C4⋊C8—591no—
357C2×#103—591no—
358C2×#103—591no—
359C2×#103—591no—
360C2×#103—591no—
361C2×#103C2×D5911yescharacteristic
362C2×#103C2×D5911yescharacteristic
363C2×C22⋊C8—591no—
364#398—1181no—
365#398—1181no—
366#391—591no—
367#391—591no—
368#391—591no—
369#389—591no—
370#389—1181no—
371#387—1181no—
372#385—1181no—
373#385—1181no—
374#385—1181no—
375#384—591no—
376#384—591no—
377C4×C4⋊C4—591no—
378C4×C22⋊C4—591no—
379C4×C22⋊C4—1181no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#340—591no—
385#340—591no—
386#336—591no—
387#336—591no—
388C59#2520611yescharacteristic, complemented, Sylow
389C22×Q8C22×D5911yescharacteristic
390C22×Q8C22×D5911yescharacteristic
391C23×C4—1181no—
392C23×C4—1181no—
393C23×C4C22×D5911yescharacteristic
394C2×Q16—22no—
395C2×Q16—22no—
396C2×Q16—41no—
397C22×C8—22no—
398C22×C8—22no—
399C22×C8—41no—
400C22×C8—591no—
401C22×C8—591no—
402C22×C8C22×D5911yescharacteristic
403C22×C8C22×D5911yescharacteristic
404C22⋊Q8—22no—
405C22⋊Q8—22no—
406C22⋊Q8—22no—
407C22⋊Q8C22⋊D5914yes—
408C4×Q8—1182no—
409C4×Q8—2361no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—21no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—591no—
414C2×C4⋊C4—591no—
415C2×C4⋊C4—591no—
416C2×C4⋊C4—1182no—
417C2×C4⋊C4—1182no—
418C2×C4⋊C4—1182no—
419C2×C4⋊C4—1181no—
420C2×C4⋊C4—1181no—
421C2×C4⋊C4—1181no—
422C2×C4⋊C4—1181no—
423C2×C4⋊C4—1181no—
424C2×C4⋊C4—1181no—
425C2×C4⋊C4—1181no—
426C2×C4⋊C4—1181no—
427C2×C4⋊C4—2361no—
428C2×C4⋊C4C22×D5911yescharacteristic
429C2×C4⋊C4C22×D5911yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—591no—
433C2×C22⋊C4—591no—
434C2×C42—591no—
435C2×C42—591no—
436C2×C42—1181no—
437C2×C42—1181no—
438#107—22no—
439#107—1182no—
440#107—1182no—
441C4⋊C8—1182no—
442C4⋊C8—2361no—
443#103—22no—
444#103—22no—
445#103—1182no—
446#103—1182no—
447#103—1182no—
448#103—2361no—
449C22⋊C8—1182no—
450#95—591no—
451#95—591no—
452#95—1181no—
453#95—1181no—
454#95—1181no—
455#95—1181no—
456#95—1181no—
457#95—1181no—
458#95—1181no—
459#95—1181no—
460C24D4×D5911yescharacteristic
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8C2×C22⋊D5912yes—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—591no—
482C22×C4—591no—
483C22×C4—591no—
484C22×C4—591no—
485C22×C4—1181no—
486C22×C4—1181no—
487C22×C4—1181no—
488C22×C4—1181no—
489C22×C4—4721no—
490C22×C4—4721no—
491C22×C4—4721no—
492C22×C4—2362no—
493C22×C4—2362no—
494C22×C4—2362no—
495C22×C4D4×D5911yescharacteristic
496C22×C4#1174568712yes—
497C22×C4C2×C22⋊D5912yes—
498C22×C4C23×D5911yescharacteristic
499Q16—44no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—44no—
506C2×C8—1181no—
507C2×C8—1181no—
508C2×C8—1181no—
509C2×C8—1181no—
510C2×C8—1181no—
511C2×C8—1181no—
512C2×C8#1174568712yes—
513C4⋊C4—24no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—42no—
518C4⋊C4—592no—
519C4⋊C4—1181no—
520C4⋊C4—1181no—
521C4⋊C4—1181no—
522C4⋊C4—1181no—
523C4⋊C4—1181no—
524C4⋊C4—1181no—
525C4⋊C4—1181no—
526C4⋊C4—4721no—
527C4⋊C4—4721no—
528C4⋊C4—4721no—
529C4⋊C4—4721no—
530C4⋊C4—4721no—
531C4⋊C4—2362no—
532C4⋊C4—2362no—
533C4⋊C4—2362no—
534C4⋊C4C2×C22⋊D5912yes—
535C22⋊C4—24no—
536C22⋊C4—42no—
537C22⋊C4—594no—
538C22⋊C4—1182no—
539C42—1181no—
540C42—1181no—
541C42—1181no—
542C42—1181no—
543C42—4721no—
544C42—2362no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23#1427369912yes—
550C23Q16×D5912yes—
551C23#1427375912yes—
552C23C2×D4×D5911yescharacteristic
553Q8—22no—
554Q8—22no—
555Q8—41no—
556Q8—41no—
557Q8—41no—
558Q8—41no—
559Q8—41no—
560Q8—41no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—592no—
580C2×C4—592no—
581C2×C4—1181no—
582C2×C4—1181no—
583C2×C4—1181no—
584C2×C4—1181no—
585C2×C4—1181no—
586C2×C4—1181no—
587C2×C4—1181no—
588C2×C4—1181no—
589C2×C4—1181no—
590C2×C4—1182no—
591C2×C4—1181no—
592C2×C4—4724no—
593C2×C4—4724no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4—2361no—
597C2×C4—2361no—
598C2×C4—2361no—
599C2×C4—2361no—
600C2×C4—2361no—
601C2×C4—2361no—
602C2×C4—2361no—
603C2×C4—2361no—
604C2×C4—2361no—
605C2×C4#1427369912yes—
606C2×C4C22×C22⋊D5911yescharacteristic
607C2×C4#41×D5911yescharacteristic
608C2×C4C2×#1174568711yescharacteristic
609C2×C4C2×#1174568711yescharacteristic
610C2×C4#1427377211yescharacteristic
611C2×C4#1427377411yescharacteristic
612C8—22no—
613C8—41no—
614C8—2361no—
615C8—2361no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—22no—
621C22—22no—
622C22#1683761014yes—
623C22C2×#1427369911yescharacteristic
624C22#1683804111yescharacteristic
625C22#1683805211yescharacteristic
626C22C2×Q16×D5911yescharacteristic
627C22C2×#1427375911yescharacteristic
628C22#1683820411yescharacteristic
629C22#1683820911yescharacteristic
630C4—22no—
631C4—42no—
632C4—42no—
633C4—81no—
634C4—81no—
635C4—1182no—
636C4—1182no—
637C4—4721no—
638C4—4721no—
639C4—4721no—
640C4—2361no—
641C4—2362no—
642C4—2361no—
643C4#1683793311yescharacteristic
644C4#1683799711yescharacteristic
645C4#1683804611yescharacteristic
646C4#1683805111yescharacteristic
647C2—24no—
648C2#2142765311yescharacteristic
649C2C2×#1683761011yescharacteristic
650C2#2143363311yescharacteristic
651C2#2143596711yescharacteristic
652C2#2143618311yescharacteristic
653C2#2143648511yescharacteristic
654C2#2143655711yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC2×C59×#467C2
socleC23×C59C22×D4
prime radicalC24×C59C2×D4
hypercenterC2×#467D59
hypocenterC59#25206
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
4. #21425570C2semidirect
5. #21425507C2semidirect
6. #21425487C2semidirect
7. #21425416C2semidirect
8. #21424851C2semidirect
9. #21424740C2semidirect
10. #21424439C2semidirect
13. #21424099C2semidirect
188. C22⋊Dic59Q16semidirect
388. C59#25206semidirect

Maximal subgroups

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

subgroupindex
C2×#168377352
#214285752
#214271052
#214255702
#214255072
#214254872
#214254162
#214248512
#214247402
#214244392
#214244042
#214242742
#214240992
C2×#168371652
C2×C59×#4672
#2520659