Finite Group Databasebeta

← #28610193

Subgroups — #28610193

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
1C2×#16837716C211yesmaximal, characteristic
2#21427132C211yesmaximal, characteristic
3#21426971C211yesmaximal, characteristic, complemented
4#21424834C211yesmaximal, characteristic, complemented
5#21424779C211yesmaximal, characteristic, complemented
6#21424724C211yesmaximal, characteristic, complemented
7#21424689C211yesmaximal, characteristic, complemented
8#21424668C211yesmaximal, characteristic, complemented
9#21424534C211yesmaximal, characteristic
10#21424346C211yesmaximal, characteristic, complemented
11#21424297C211yesmaximal, characteristic, complemented
12C4×#14273661C211yesmaximal, characteristic, complemented
13#21423922C211yesmaximal, characteristic, complemented
14#21423816C211yesmaximal, characteristic, complemented
15C59×#1753C211yesmaximal, characteristic, complemented
16C2×#14273701C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18C2×#14273686C2211yescharacteristic
19C22×C4⋊Dic59C2211yescharacteristic
20#16837716C418yescomplemented
21C2×#14273661C2211yescharacteristic
22C2×#14273661C2211yescharacteristic
23C2×#14273661C2211yescharacteristic
24C2×#14273659C2211yescharacteristic
25C2×#14273649C2211yescharacteristic
26C2×#14273649C2211yescharacteristic
27C2×#14273649C2211yescharacteristic
28C2×#14273649C2211yescharacteristic
29#16837304C2211yescharacteristic
30#16837302—21no—
31#16837300C2211yescharacteristic
32#16837300C2211yescharacteristic
33#16837300C2211yescharacteristic
34#16837291—21no—
35#16837285—21no—
36#16837285—21no—
37#16837278—21no—
38#16837262C2211yescharacteristic
39#16837260—21no—
40C4×C4⋊Dic59C2211yescharacteristic
41C4×C4⋊Dic59C2211yescharacteristic
42C4×C4⋊Dic59C2211yescharacteristic
43C22⋊C4×Dic59—21no—
44#16837249—21no—
45#16837165C2211yescharacteristic
46#16837165C2211yescharacteristic
47#16837165C2211yescharacteristic
48#16837165C2211yescharacteristic
49#16837165C2211yescharacteristic
50#16837165C2211yescharacteristic
51#16837165C2211yescharacteristic
52#16837165C2211yescharacteristic
53C22×M16×C59C2211yescharacteristic
54C2×C22⋊C8×C59C2211yescharacteristic
55C2×#97×C59C2211yescharacteristic
56C2×C4×C8×C59C2211yescharacteristic
57C4×C22⋊C4×C59C2211yescharacteristic
58C59×#336C2211yescharacteristic
59C59×#336C2211yescharacteristic
60C22×D236C2311yescharacteristic
61C22×C22⋊D59—21no—
62C23×Dic59—21no—
63C22×C4×D59—21no—
64#14273701—22no—
65#14273701C2×C414yes—
66C4×D236—22no—
67C4×D236—41no—
68#14273686—22no—
69#14273686—41no—
70C2×C4⋊Dic59—21no—
71C2×C4⋊Dic59—22no—
72C2×C4⋊Dic59C2×C414yescomplemented
73C2×C4⋊Dic59C2311yescharacteristic
74C2×C4⋊Dic59C2311yescharacteristic
75C2×C4⋊Dic59C2311yescharacteristic
76C2×C4⋊Dic59C2311yescharacteristic
77C2×C4⋊Dic59C2311yescharacteristic
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×Dic59⋊C4—21no—
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×C22⋊Dic59—21no—
84C2×C22⋊Dic59—21no—
85C2×(C2×D59)⋊C4—21no—
86C2×(C2×D59)⋊C4—21no—
87C2×(C2×D59)⋊C4—21no—
88C2×(C2×D59)⋊C4—21no—
89C2×C4×Dic59—21no—
90C2×C4×Dic59—21no—
91C2×C4×Dic59—21no—
92C2×C4×Dic59—21no—
93#14273661—22no—
94#14273661C2×C414yescomplemented
95#14273661C2×C414yescomplemented
96#14273659—41no—
97#14273649—22no—
98#14273649—41no—
99#14273649C2×C414yes—
100#14273649C2×C414yes—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C59⋊#95—21no—
106C59⋊#95—21no—
107C59⋊#95—21no—
108C23×C4×C59C2311yescharacteristic
109C2×M16×C59—22no—
110C2×M16×C59—22no—
111C2×M16×C59—41no—
112C22×C8×C59C2311yescharacteristic
113C22×C8×C59C2311yescharacteristic
114C22×C8×C59C2311yescharacteristic
115C22×C8×C59C2311yescharacteristic
116C2×C22⋊C4×C59D411yescharacteristic
117C2×C22⋊C4×C59D411yescharacteristic
118C2×C42×C59C2311yescharacteristic
119C2×C42×C59C2311yescharacteristic
120C22⋊C8×C59C2×C414yescomplemented
121#97×C59—41no—
122C4×C8×C59—22no—
123#95×C59D411yescharacteristic
124#95×C59D411yescharacteristic
125C24×C59C2×D411yescharacteristic
126C23×D59—21no—
127C22×C4×C59—21no—
128C22×C4×C59—21no—
129C22×C4×C59—21no—
130C22×C4×C59—21no—
131C22×C4×C59C22×C412yes—
132C22×C4×C59C2×D411yescharacteristic
133C22×C4×C59C2×D411yescharacteristic
134C22×C4×C59C2×D411yescharacteristic
135C22×C4×C59C2×D411yescharacteristic
136C22×C4×C59C2×D411yescharacteristic
137C22×C4×C59#4112yes—
138C22×C4×C59C2411yescharacteristic
139C2×C22⋊D59—24no—
140C2×C22⋊D59—81no—
141C2×C22⋊D59—81no—
142C22×Dic59—21no—
143C22×Dic59—21no—
144C22×Dic59—21no—
145C22×Dic59—21no—
146C22×Dic59—24no—
147C22×Dic59—21no—
148C22×Dic59—21no—
149C22×Dic59—21no—
150C22×Dic59—81no—
151C22×Dic59—81no—
152C2×D236—21no—
153C2×D236—21no—
154C2×D236—21no—
155C2×D236—21no—
156C2×D236—21no—
157C2×D236C22×C412yes—
158C2×C4×D59—42no—
159C2×C4×D59—42no—
160C2×C4×D59—81no—
161M16×C59—44no—
162C2×C8×C59—21no—
163C2×C8×C59—21no—
164C2×C8×C59—21no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59—21no—
170C2×C8×C59—21no—
171C2×C8×C59C22×C412yes—
172C2×C8×C59C22×C412yes—
173C2×C8×C59#4112yes—
174C22⋊C4×C59—22no—
175C22⋊C4×C59—41no—
176C42×C59—21no—
177C42×C59—21no—
178C42×C59—21no—
179C42×C59—21no—
180C22⋊Dic59—42no—
181C22⋊Dic59—42no—
182(C2×D59)⋊C4—24no—
183(C2×D59)⋊C4—42no—
184(C2×D59)⋊C4—42no—
185(C2×D59)⋊C4—81no—
186C4⋊Dic59—24no—
187C4⋊Dic59—21no—
188C4⋊Dic59—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59—21no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—21no—
197C4⋊Dic59C22×C412yes—
198C4⋊Dic59C22×C412yes—
199C4⋊Dic59C22×C412yes—
200C4⋊Dic59#4112yes—
201Dic59⋊C4—81no—
202Dic59⋊C4—81no—
203Dic59⋊C4—81no—
204C4×Dic59—42no—
205C4×Dic59—81no—
206C4×Dic59—81no—
207C4×Dic59—81no—
208C23×C59—21no—
209C23×C59—21no—
210C23×C59—21no—
211C23×C59—21no—
212C23×C59—21no—
213C23×C59—21no—
214C23×C59C4×D412yes—
215C23×C59C22×D411yescharacteristic
216C22×D59—22no—
217C22×D59—41no—
218C22×D59—41no—
219C22×D59—41no—
220C22×D59—41no—
221C22×D59—41no—
222C22×D59—41no—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—21no—
227C2×C4×C59—22no—
228C2×C4×C59—22no—
229C2×C4×C59—22no—
230C2×C4×C59—22no—
231C2×C4×C59—21no—
232C2×C4×C59—21no—
233C2×C4×C59—21no—
234C2×C4×C59—21no—
235C2×C4×C59—21no—
236C2×C4×C59—21no—
237C2×C4×C59—21no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59C4×D412yes—
241C2×C4×C59#13712yes—
242C2×C4×C59C23×C411yescharacteristic
243C2×C4×C59C2×#4111yescharacteristic
244C2×C4×C59C2×#4111yescharacteristic
245C2×C4×C59C2×#4111yescharacteristic
246C2×C4×C59H(5,𝔽2)11yescharacteristic
247C2×C4×C592−1+411yescharacteristic
248C22⋊D59—84no—
249C2×Dic59—22no—
250C2×Dic59—22no—
251C2×Dic59—22no—
252C2×Dic59—22no—
253C2×Dic59—41no—
254C2×Dic59—41no—
255C2×Dic59—41no—
256C2×Dic59—41no—
257C2×Dic59—41no—
258C2×Dic59—41no—
259C2×Dic59—41no—
260C2×Dic59—41no—
261C2×Dic59—41no—
262C2×Dic59—41no—
263C2×Dic59—41no—
264C2×Dic59—41no—
265C2×Dic59—41no—
266C2×Dic59—41no—
267C2×Dic59—41no—
268C2×Dic59—41no—
269C2×Dic59—41no—
270C2×Dic59—84no—
271D236—22no—
272D236—22no—
273D236—41no—
274D236—41no—
275C4×D59—84no—
276C8×C59—22no—
277C8×C59—22no—
278C8×C59—22no—
279C8×C59—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×C59C2×C4×D411yescharacteristic
289C22×C59#54611yescharacteristic
290C22×C59#54711yescharacteristic
291C22×C59C2×#13711yescharacteristic
292C22×C59#57511yescharacteristic
293C22×C59#57611yescharacteristic
294C22×C59#57611yescharacteristic
295C2×D59—42no—
296C2×D59—42no—
297C2×D59—42no—
298C2×D59—81no—
299C2×D59—81no—
300C2×D59—81no—
301C2×D59—81no—
302C4×C59—22no—
303C4×C59—22no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—41no—
307C4×C59C4×#4111yescharacteristic
308C4×C59#52011yescharacteristic
309C4×C59#55111yescharacteristic
310C4×C59#55211yescharacteristic
311Dic59—42no—
312Dic59—42no—
313Dic59—42no—
314Dic59—42no—
315Dic59—81no—
316Dic59—81no—
317Dic59—81no—
318C2×#480—591no—
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#1753D5911yescharacteristic, complemented
332#1736—591no—
333C2×C59—42no—
334C2×C59#235911yescharacteristic
335C2×C59C4×#13711yescharacteristic
336C2×C59#294811yescharacteristic
337C2×C59#332211yescharacteristic
338C2×C59#333311yescharacteristic
339C2×C59#334611yescharacteristic
340C2×C59#334711yescharacteristic
341D59—82no—
342C22×M16C2×D5911yescharacteristic
343C2×#121—591no—
344C2×C4×D4—591no—
345C2×#117—591no—
346C22×C4⋊C4—591no—
347#480—598no—
348C2×#107—591no—
349C2×#107—591no—
350C2×#107—591no—
351C2×#106—591no—
352C2×#102—591no—
353C2×#102—591no—
354C2×#102—591no—
355C2×#102—591no—
356C2×C22⋊C8C2×D5911yescharacteristic
357C2×#97C2×D5911yescharacteristic
358C2×C4×C8C2×D5911yescharacteristic
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—1181no—
374C4×C22⋊C4—1181no—
375C4×C22⋊C4C2×D5911yescharacteristic
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#336C2×D5911yescharacteristic
385#336C2×D5911yescharacteristic
386C59#2294111yescharacteristic, complemented, Sylow
387C22×D4—591no—
388C22×D4—1181no—
389C23×C4—1181no—
390C23×C4—1181no—
391C23×C4C22×D5911yescharacteristic
392C2×M16—22no—
393C2×M16—22no—
394C2×M16—41no—
395C22×C8C22×D5911yescharacteristic
396C22×C8C22×D5911yescharacteristic
397C22×C8C22×D5911yescharacteristic
398C22×C8C22×D5911yescharacteristic
399#121—594no—
400#121—1182no—
401C4×D4—1182no—
402C4×D4—2361no—
403#117—1182no—
404#117—2361no—
405C2×C4⋊C4—594no—
406C2×C4⋊C4—591no—
407C2×C4⋊C4—591no—
408C2×C4⋊C4—591no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—591no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1182no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4—1181no—
417C2×C4⋊C4—1181no—
418C2×C22⋊C4—1181no—
419C2×C22⋊C4—1181no—
420C2×C22⋊C4—1181no—
421C2×C22⋊C4—1181no—
422C2×C22⋊C4—1181no—
423C2×C22⋊C4—1181no—
424C2×C22⋊C4D23611yescharacteristic
425C2×C22⋊C4D23611yescharacteristic
426C2×C42—1181no—
427C2×C42—1181no—
428C2×C42—1181no—
429C2×C42—1181no—
430C2×C42C22×D5911yescharacteristic
431C2×C42C22×D5911yescharacteristic
432#107—594no—
433#107—594no—
434#107—1182no—
435#106—2361no—
436#102—594no—
437#102—594no—
438#102—1182no—
439#102—2361no—
440C22⋊C8C4×D5914yescomplemented
441#97—41no—
442C4×C8—22no—
443#95—1181no—
444#95—1181no—
445#95—1181no—
446#95—1181no—
447#95—1181no—
448#95—1181no—
449#95—1181no—
450#95D23611yescharacteristic
451#95D23611yescharacteristic
452C24—1181no—
453C24C2×D23611yescharacteristic
454C2×D4—592no—
455C2×D4—1184no—
456C2×D4—1181no—
457C2×D4—1181no—
458C2×D4—1181no—
459C2×D4—1181no—
460C2×D4—1181no—
461C2×D4—4721no—
462C2×D4—4721no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1184no—
468C22×C4—1181no—
469C22×C4—1181no—
470C22×C4—1181no—
471C22×C4—1181no—
472C22×C4—1181no—
473C22×C4—1181no—
474C22×C4—1181no—
475C22×C4—4721no—
476C22×C4—4721no—
477C22×C4—4721no—
478C22×C4—2362no—
479C22×C4—2362no—
480C22×C4C2×C4×D5912yes—
481C22×C4C2×D23611yescharacteristic
482C22×C4C2×D23611yescharacteristic
483C22×C4C2×D23611yescharacteristic
484C22×C4C2×D23611yescharacteristic
485C22×C4C2×D23611yescharacteristic
486C22×C4#1174568512yes—
487C22×C4C23×D5911yescharacteristic
488M16—44no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8C2×C4×D5912yes—
499C2×C8C2×C4×D5912yes—
500C2×C8#1174568712yes—
501C4⋊C4—592no—
502C4⋊C4—592no—
503C4⋊C4—592no—
504C4⋊C4—592no—
505C4⋊C4—1184no—
506C4⋊C4—1181no—
507C4⋊C4—1181no—
508C4⋊C4—1181no—
509C4⋊C4—1181no—
510C4⋊C4—1181no—
511C4⋊C4—1181no—
512C4⋊C4—1181no—
513C4⋊C4—1181no—
514C4⋊C4—1181no—
515C4⋊C4—1181no—
516C4⋊C4—4721no—
517C4⋊C4—4721no—
518C4⋊C4—4721no—
519C22⋊C4—22no—
520C22⋊C4—41no—
521C22⋊C4—1184no—
522C22⋊C4—4721no—
523C22⋊C4—2362no—
524C22⋊C4—2362no—
525C22⋊C4—2362no—
526C22⋊C4—2362no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—4721no—
532C42—4721no—
533C42—4721no—
534C42—2362no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—1182no—
542C23—2361no—
543C23—2361no—
544C23—2361no—
545C23—2361no—
546C23—2361no—
547C23—2361no—
548C23C4×D23612yes—
549C23C22×D23611yescharacteristic
550D4—1182no—
551D4—1182no—
552D4—4724no—
553D4—2361no—
554D4—2361no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—1182no—
573C2×C4—1182no—
574C2×C4—1182no—
575C2×C4—1182no—
576C2×C4—4724no—
577C2×C4—4724no—
578C2×C4—2361no—
579C2×C4—2361no—
580C2×C4—2361no—
581C2×C4—2361no—
582C2×C4—2361no—
583C2×C4—2361no—
584C2×C4—2361no—
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×C4C4×D23612yes—
596C2×C4#1427376012yes—
597C2×C4C22×C4×D5911yescharacteristic
598C2×C4#41×D5911yescharacteristic
599C2×C4C2×#1174568711yescharacteristic
600C2×C4C2×#1174568511yescharacteristic
601C2×C4#1427377211yescharacteristic
602C2×C4#1427377411yescharacteristic
603C8—22no—
604C8—22no—
605C8—22no—
606C8—41no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—4721no—
615C22—4721no—
616C22—4721no—
617C22—4721no—
618C22—2362no—
619C22—2362no—
620C22—2362no—
621C22C2×C4×D23611yescharacteristic
622C22#1683804711yescharacteristic
623C22#1683805611yescharacteristic
624C22C2×#1427376011yescharacteristic
625C22#1683820211yescharacteristic
626C22#1683820711yescharacteristic
627C22#1683820711yescharacteristic
628C4—22no—
629C4—22no—
630C4—41no—
631C4—41no—
632C4—41no—
633C4—4721no—
634C4—4721no—
635C4—4721no—
636C4—2362no—
637C4—2362no—
638C4—2362no—
639C4—2362no—
640C4C4×#1174568711yescharacteristic
641C4#1683788111yescharacteristic
642C4#1683807811yescharacteristic
643C4#1683808511yescharacteristic
644C2—42no—
645C2—4722no—
646C2#2142709011yescharacteristic
647C2C4×#1427376011yescharacteristic
648C2#2143294911yescharacteristic
649C2#2143629911yescharacteristic
650C2#2143643511yescharacteristic
651C2#2143658311yescharacteristic
652C2#2143659811yescharacteristic
subgroupG/H
centerC23C22×D236
commutatorC2×C4×C59C23×C4
FrattiniC22×C4C23×D59
FittingC59×#1753C2
socleC23×C59C22×D4
prime radicalC2×#14273701C22
cosocleC22×C4×C59C24
hypercenter#1753D59
hypocenterC59#22941
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
3. #21426971C2semidirect
4. #21424834C2semidirect
5. #21424779C2semidirect
6. #21424724C2semidirect
7. #21424689C2semidirect
8. #21424668C2semidirect
10. #21424346C2semidirect
11. #21424297C2semidirect
12. C4×#14273661C2semidirect
13. #21423922C2semidirect
14. #21423816C2semidirect
15. C59×#1753C2semidirect
20. #16837716C4semidirect
72. C2×C4⋊Dic59C2×C4semidirect
94. #14273661C2×C4semidirect
95. #14273661C2×C4semidirect
120. C22⋊C8×C59C2×C4semidirect
331. #1753D59semidirect
386. C59#22941semidirect
440. C22⋊C8C4×D59semidirect

Maximal subgroups

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

subgroupindex
C2×#168377162
#214271322
#214269712
#214248342
#214247792
#214247242
#214246892
#214246682
#214245342
#214243462
#214242972
C4×#142736612
#214239222
#214238162
C59×#17532
#2294159