Finite Group Databasebeta

← #28610195

Subgroups — #28610195

Order 15104.

874 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21427138C211yesmaximal, characteristic
2#21426973C211yesmaximal, characteristic, complemented
3#21424835C211yesmaximal, characteristic, complemented
4#21424781C211yesmaximal, characteristic, complemented
5#21424725C211yesmaximal, characteristic, complemented
6#21424690C211yesmaximal, characteristic, complemented
7#21424669C211yesmaximal, characteristic, complemented
8#21424535C211yesmaximal, characteristic
9#21424345C211yesmaximal, characteristic, complemented
10#21424297C211yesmaximal, characteristic, complemented
11#107×Dic59C211yesmaximal, characteristic, complemented
12#21423923C211yesmaximal, characteristic, complemented
13#21423882C211yesmaximal, characteristic, complemented
14#21423815C211yesmaximal, characteristic, complemented
15C2×C59×#480C211yesmaximal, characteristic
16C22×#11745664C2211yescharacteristic
17C2×D4×Dic59C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C2×#14273661C2211yescharacteristic
20C2×#14273659C2211yescharacteristic
21C2×#14273648C2211yescharacteristic
22C2×#14273648C2211yescharacteristic
23C2×#14273637C2211yescharacteristic
24C2×C8×Dic59C2211yescharacteristic
25#16837305—21no—
26#16837305C2211yescharacteristic
27#16837301C2211yescharacteristic
28#16837301C2211yescharacteristic
29#16837301C2211yescharacteristic
30#16837295—21no—
31#16837288—21no—
32#16837288—21no—
33#16837282—21no—
34#16837262—21no—
35#16837262C2211yescharacteristic
36C4⋊C4×Dic59C2211yescharacteristic
37C4⋊C4×Dic59C2211yescharacteristic
38C4⋊C4×Dic59C2211yescharacteristic
39C22⋊C4×Dic59—21no—
40C4×C22⋊Dic59—21no—
41C4×C22⋊Dic59C2211yescharacteristic
42#16837165C2211yescharacteristic
43#16837165C2211yescharacteristic
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837164C2211yescharacteristic
49#16837164C2211yescharacteristic
50#16837156C2211yescharacteristic
51#16837156C2211yescharacteristic
52C2×#121×C59C2211yescharacteristic
53C22×C4⋊C4×C59C2211yescharacteristic
54C59×#480C418yescomplemented
55C2×#107×C59C2211yescharacteristic
56C2×#107×C59C2211yescharacteristic
57C2×#102×C59C2211yescharacteristic
58C2×#102×C59C2211yescharacteristic
59C2×C22⋊C8×C59C2211yescharacteristic
60C23×Dic59—21no—
61C2×#11745664—22no—
62C2×#11745664—22no—
63C2×#11745664—41no—
64C22×C59⋊C8C2311yescharacteristic
65C22×C59⋊C8C2311yescharacteristic
66D4×Dic59—22no—
67D4×Dic59—41no—
68#14273686—22no—
69#14273686—41no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—21no—
72C2×C4⋊Dic59—21no—
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C22⋊Dic59—21no—
78C2×C22⋊Dic59—21no—
79C2×C22⋊Dic59—21no—
80C2×C22⋊Dic59—21no—
81C2×C22⋊Dic59—21no—
82C2×C22⋊Dic59D411yescharacteristic
83C2×C22⋊Dic59D411yescharacteristic
84C2×C4×Dic59—21no—
85C2×C4×Dic59—21no—
86C2×C4×Dic59—21no—
87C2×C4×Dic59—21no—
88C2×C4×Dic59C2311yescharacteristic
89C2×C4×Dic59C2311yescharacteristic
90#14273661—22no—
91#14273659—41no—
92#14273648—22no—
93#14273648—41no—
94#14273637—41no—
95C8×Dic59—22no—
96C59⋊#95—21no—
97C59⋊#95—21no—
98C59⋊#95—21no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95D411yescharacteristic
104C59⋊#95D411yescharacteristic
105C22×D4×C59—21no—
106C22×D4×C59C2311yescharacteristic
107C23×C4×C59—21no—
108C23×C4×C59C2311yescharacteristic
109C22×C8×C59C2311yescharacteristic
110C22×C8×C59C2311yescharacteristic
111#121×C59—22no—
112#121×C59C2×C414yes—
113C2×C4⋊C4×C59—21no—
114C2×C4⋊C4×C59—22no—
115C2×C4⋊C4×C59C2×C414yescomplemented
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C4⋊C4×C59C2311yescharacteristic
118C2×C4⋊C4×C59C2311yescharacteristic
119C2×C22⋊C4×C59—21no—
120#107×C59C2×C414yescomplemented
121#107×C59C2×C414yescomplemented
122#102×C59C2×C414yes—
123#102×C59C2×C414yes—
124C22⋊C8×C59C2×C414yescomplemented
125C24×C59—21no—
126C24×C59C2×D411yescharacteristic
127C2×D4×C59—24no—
128C2×D4×C59—21no—
129C2×D4×C59—21no—
130C2×D4×C59—21no—
131C2×D4×C59—21no—
132C2×D4×C59—21no—
133C2×D4×C59—81no—
134C2×D4×C59—81no—
135C2×D4×C59C22×C412yes—
136C22×C4×C59—21no—
137C22×C4×C59—21no—
138C22×C4×C59—21no—
139C22×C4×C59—21no—
140C22×C4×C59—21no—
141C22×C4×C59—21no—
142C22×C4×C59—21no—
143C22×C4×C59—24no—
144C22×C4×C59—81no—
145C22×C4×C59—81no—
146C22×C4×C59C22×C412yes—
147C22×C4×C59C2×D411yescharacteristic
148C22×C4×C59#4112yes—
149C22×C4×C59C2411yescharacteristic
150C22×Dic59—21no—
151C22×Dic59—21no—
152C22×Dic59—21no—
153C22×Dic59—21no—
154C22×Dic59—42no—
155C22×Dic59—42no—
156C22×Dic59—81no—
157C22×Dic59C2×D411yescharacteristic
158C22×Dic59C2×D411yescharacteristic
159C22×Dic59C2×D411yescharacteristic
160C22×Dic59C2×D411yescharacteristic
161C2×C8×C59—21no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59C22×C412yes—
166C2×C8×C59C22×C412yes—
167C4⋊C4×C59—24no—
168C4⋊C4×C59—21no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59C22×C412yes—
172C4⋊C4×C59C22×C412yes—
173C4⋊C4×C59C22×C412yes—
174C22⋊C4×C59—24no—
175C22⋊Dic59—22no—
176C22⋊Dic59—41no—
177C22⋊Dic59—42no—
178C22⋊Dic59—42no—
179C22⋊Dic59—42no—
180C22⋊Dic59—42no—
181C22⋊Dic59—81no—
182C4⋊Dic59—21no—
183C4⋊Dic59—21no—
184C4⋊Dic59—21no—
185C4⋊Dic59—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—81no—
190C4⋊Dic59—81no—
191C4⋊Dic59—81no—
192C4⋊Dic59#4112yes—
193C4×Dic59—21no—
194C4×Dic59—21no—
195C4×Dic59—21no—
196C4×Dic59—21no—
197C4×Dic59—42no—
198C4×Dic59—81no—
199C4×Dic59—81no—
200C4×Dic59—81no—
201#11745664—44no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8#4112yes—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59—22no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59C4×D412yes—
222C23×C59C22×D411yescharacteristic
223D4×C59—22no—
224D4×C59—22no—
225D4×C59—41no—
226D4×C59—41no—
227D4×C59—84no—
228C2×C4×C59—21no—
229C2×C4×C59—21no—
230C2×C4×C59—22no—
231C2×C4×C59—22no—
232C2×C4×C59—22no—
233C2×C4×C59—22no—
234C2×C4×C59—22no—
235C2×C4×C59—22no—
236C2×C4×C59—22no—
237C2×C4×C59—41no—
238C2×C4×C59—41no—
239C2×C4×C59—41no—
240C2×C4×C59—41no—
241C2×C4×C59—41no—
242C2×C4×C59—41no—
243C2×C4×C59—84no—
244C2×C4×C59C4×D412yes—
245C2×C4×C59C23×C411yescharacteristic
246C2×C4×C59C2×#4111yescharacteristic
247C2×C4×C59C2×#4111yescharacteristic
248C2×C4×C59C2×#4111yescharacteristic
249C2×C4×C59H(5,𝔽2)11yescharacteristic
250C2×C4×C592−1+411yescharacteristic
251C2×Dic59—21no—
252C2×Dic59—21no—
253C2×Dic59—21no—
254C2×Dic59—21no—
255C2×Dic59—22no—
256C2×Dic59—21no—
257C2×Dic59—21no—
258C2×Dic59—21no—
259C2×Dic59—21no—
260C2×Dic59—21no—
261C2×Dic59—21no—
262C2×Dic59—21no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—41no—
271C2×Dic59—41no—
272C2×Dic59—41no—
273C2×Dic59—41no—
274C2×Dic59—84no—
275C2×Dic59#13712yes—
276C8×C59—22no—
277C8×C59—22no—
278C59⋊C8—22no—
279C59⋊C8—41no—
280#22941—591nomaximal, Sylow
281C22×C59—22no—
282C22×C59—22no—
283C22×C59—22no—
284C22×C59—41no—
285C22×C59—41no—
286C22×C59—41no—
287C22×C59—41no—
288C22×C59—42no—
289C22×C59—42no—
290C22×C59—42no—
291C22×C59—81no—
292C22×C59—81no—
293C22×C59—81no—
294C22×C59—81no—
295C22×C59C2×C4×D411yescharacteristic
296C22×C59#54611yescharacteristic
297C22×C59#54711yescharacteristic
298C22×C59C2×#13711yescharacteristic
299C22×C59#57511yescharacteristic
300C22×C59#57611yescharacteristic
301C22×C59#57611yescharacteristic
302C4×C59—22no—
303C4×C59—42no—
304C4×C59—42no—
305C4×C59—42no—
306C4×C59C4×#4111yescharacteristic
307C4×C59#52011yescharacteristic
308C4×C59#55111yescharacteristic
309C4×C59#55211yescharacteristic
310Dic59—22no—
311Dic59—41no—
312Dic59—41no—
313Dic59—41no—
314Dic59—42no—
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318C2×#480D5911yescharacteristic
319#2363—591no—
320#2345—591no—
321#1943—591no—
322#1934—591no—
323#1926—591no—
324#1921—591no—
325#1918—591no—
326#1895—591no—
327#1862—591no—
328#1854—591no—
329C4×#107—591no—
330#1762—591no—
331#1753—591no—
332#1736—591no—
333C2×C59—42no—
334C2×C59—82no—
335C2×C59#235911yescharacteristic
336C2×C59C4×#13711yescharacteristic
337C2×C59#294811yescharacteristic
338C2×C59#332211yescharacteristic
339C2×C59#333311yescharacteristic
340C2×C59#334611yescharacteristic
341C2×C59#334711yescharacteristic
342C22×M16—591no—
343C2×#121C2×D5911yescharacteristic
344C2×C4×D4—591no—
345C2×#117—591no—
346C22×C4⋊C4C2×D5911yescharacteristic
347#480Dic5918yescomplemented
348C2×#107—591no—
349C2×#107C2×D5911yescharacteristic
350C2×#107C2×D5911yescharacteristic
351C2×#106—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102C2×D5911yescharacteristic
355C2×#102C2×D5911yescharacteristic
356C2×C22⋊C8C2×D5911yescharacteristic
357C2×#97—591no—
358C2×C4×C8—591no—
359#390—591no—
360#390—1181no—
361#389—591no—
362#389—591no—
363#389—591no—
364#388—1181no—
365#387—1181no—
366#387—1181no—
367#386—1181no—
368#380—591no—
369#380—1181no—
370C4×C4⋊C4—591no—
371C4×C4⋊C4—591no—
372C4×C4⋊C4—591no—
373C4×C22⋊C4—591no—
374C4×C22⋊C4—1181no—
375C4×C22⋊C4—1181no—
376#340—591no—
377#340—591no—
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#340—591no—
383#340—591no—
384#336—591no—
385#336—591no—
386C59#2294111yescharacteristic, complemented, Sylow
387C22×D4—21no—
388C22×D4C22×D5911yescharacteristic
389C23×C4—21no—
390C23×C4—1181no—
391C23×C4C22×D5911yescharacteristic
392C2×M16—1182no—
393C2×M16—1182no—
394C2×M16—2361no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8C22×D5911yescharacteristic
398C22×C8C22×D5911yescharacteristic
399#121—22no—
400#121C2×Dic5914yes—
401C4×D4—1182no—
402C4×D4—2361no—
403#117—1182no—
404#117—2361no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—22no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—1181no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4C2×Dic5914yescomplemented
415C2×C4⋊C4C22×D5911yescharacteristic
416C2×C4⋊C4C22×D5911yescharacteristic
417C2×C4⋊C4C22×D5911yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—591no—
420C2×C22⋊C4—591no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4—1181no—
425C2×C22⋊C4—1181no—
426C2×C42—591no—
427C2×C42—591no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42—1181no—
431C2×C42—1181no—
432#107—1182no—
433#107C2×Dic5914yescomplemented
434#107C2×Dic5914yescomplemented
435#106—2361no—
436#102—1182no—
437#102—2361no—
438#102C2×Dic5914yes—
439#102C2×Dic5914yes—
440C22⋊C8C2×Dic5914yescomplemented
441#97—2361no—
442C4×C8—1182no—
443#95—591no—
444#95—591no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452C24—21no—
453C24D4×D5911yescharacteristic
454C2×D4—24no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—81no—
461C2×D4—81no—
462C2×D4C22×Dic5912yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—591no—
474C22×C4—591no—
475C22×C4—591no—
476C22×C4—591no—
477C22×C4—1181no—
478C22×C4—1181no—
479C22×C4—1181no—
480C22×C4—1181no—
481C22×C4—4721no—
482C22×C4—2362no—
483C22×C4—2362no—
484C22×C4D4×D5911yescharacteristic
485C22×C4#1174568712yes—
486C22×C4C22×Dic5912yes—
487C22×C4C23×D5911yescharacteristic
488M16—2364no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—592no—
494C2×C8—1181no—
495C2×C8—1181no—
496C2×C8—1181no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8C22×Dic5912yes—
500C2×C8C22×Dic5912yes—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—592no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—4721no—
514C4⋊C4—4721no—
515C4⋊C4—4721no—
516C4⋊C4C22×Dic5912yes—
517C4⋊C4C22×Dic5912yes—
518C4⋊C4C22×Dic5912yes—
519C22⋊C4—24no—
520C22⋊C4—1182no—
521C22⋊C4—4721no—
522C22⋊C4—2362no—
523C22⋊C4—2361no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C42—1181no—
528C42—1181no—
529C42—1181no—
530C42—1181no—
531C42—4721no—
532C42—4721no—
533C42—4721no—
534C42—2362no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23D4×Dic5912yes—
549C23C2×D4×D5911yescharacteristic
550D4—22no—
551D4—22no—
552D4—41no—
553D4—41no—
554D4—84no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—84no—
571C2×C4—592no—
572C2×C4—1181no—
573C2×C4—1181no—
574C2×C4—1181no—
575C2×C4—1181no—
576C2×C4—1181no—
577C2×C4—1181no—
578C2×C4—1181no—
579C2×C4—1181no—
580C2×C4—1181no—
581C2×C4—1181no—
582C2×C4—1182no—
583C2×C4—1181no—
584C2×C4—4724no—
585C2×C4—2361no—
586C2×C4—2361no—
587C2×C4—2361no—
588C2×C4—2361no—
589C2×C4—2361no—
590C2×C4—2361no—
591C2×C4—2361no—
592C2×C4—2361no—
593C2×C4—2361no—
594C2×C4—2361no—
595C2×C4—2361no—
596C2×C4D4×Dic5912yes—
597C2×C4C23×Dic5911yescharacteristic
598C2×C4#41×D5911yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4C2×#1174568711yescharacteristic
601C2×C4#1427377311yescharacteristic
602C2×C4#1427377411yescharacteristic
603C8—22no—
604C8—22no—
605C8—1182no—
606C8—2361no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—42no—
615C22—42no—
616C22—42no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22C2×D4×Dic5911yescharacteristic
622C22#1683804411yescharacteristic
623C22#1683805011yescharacteristic
624C22#137×D5911yescharacteristic
625C22#1683820111yescharacteristic
626C22#1683820311yescharacteristic
627C22#1683820311yescharacteristic
628C4—22no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—1182no—
633C4—4721no—
634C4—4721no—
635C4—4721no—
636C4—2361no—
637C4—2361no—
638C4—2362no—
639C4—2361no—
640C4#41×Dic5911yescharacteristic
641C4#1683788511yescharacteristic
642C4#1683807511yescharacteristic
643C4#1683808911yescharacteristic
644C2—42no—
645C2—82no—
646C2#2142709211yescharacteristic
647C2#137×Dic5911yescharacteristic
648C2#2143294811yescharacteristic
649C2#2143630011yescharacteristic
650C2#2143643211yescharacteristic
651C2#2143658411yescharacteristic
652C2#2143659911yescharacteristic
subgroupG/H
centerC23C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#480C2
socleC23×C59C22×D4
prime radicalC2×#121×C59C22
cosocleC22×C4×C59C24
hypercenterC2×#480D59
hypocenterC59#22941
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#480

Complements

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

NH ≅ G/Ntype
2. #21426973C2semidirect
3. #21424835C2semidirect
4. #21424781C2semidirect
5. #21424725C2semidirect
6. #21424690C2semidirect
7. #21424669C2semidirect
9. #21424345C2semidirect
10. #21424297C2semidirect
11. #107×Dic59C2semidirect
12. #21423923C2semidirect
13. #21423882C2semidirect
14. #21423815C2semidirect
54. C59×#480C4semidirect
115. C2×C4⋊C4×C59C2×C4semidirect
120. #107×C59C2×C4semidirect
121. #107×C59C2×C4semidirect
124. C22⋊C8×C59C2×C4semidirect
347. #480Dic59semidirect
386. C59#22941semidirect
414. C2×C4⋊C4C2×Dic59semidirect
433. #107C2×Dic59semidirect
434. #107C2×Dic59semidirect
440. C22⋊C8C2×Dic59semidirect

Maximal subgroups

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

subgroupindex
#214271382
#214269732
#214248352
#214247812
#214247252
#214246902
#214246692
#214245352
#214243452
#214242972
#107×Dic592
#214239232
#214238822
#214238152
C2×C59×#4802
#2294159