Finite Group Databasebeta

← #28558747

Subgroups — #28558747

Order 15104.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434145C211yesmaximal, characteristic
2#21433241C211yesmaximal, characteristic, complemented
3#21433119C211yesmaximal, characteristic, complemented
4#21422959C211yesmaximal, characteristic, complemented
5#21422952C211yesmaximal, characteristic, complemented
6#21422931C211yesmaximal, characteristic, complemented
7#21422924C211yesmaximal, characteristic, complemented
8#21422919C211yesmaximal, characteristic, complemented
9#21422907C211yesmaximal, characteristic, complemented
10#21422895C211yesmaximal, characteristic, complemented
11#21422890C211yesmaximal, characteristic, complemented
12#21422520C211yesmaximal, characteristic, complemented
13#21422502C211yesmaximal, characteristic, complemented
14#21422396C211yesmaximal, characteristic, complemented
15C59×#1574C211yesmaximal, characteristic, complemented
16#16837936C2211yescharacteristic
17C2×C4×D236C2211yescharacteristic
18#16837738C411yescharacteristic
19#16837721C411yescharacteristic
20#16837703C411yescharacteristic
21#16837703C411yescharacteristic
22#16837618C411yescharacteristic
23#16837618C411yescharacteristic
24#16837555C411yescharacteristic
25#16837538C411yescharacteristic
26#16837529C2211yescharacteristic, complemented
27#16837526C2211yescharacteristic, complemented
28C4×#11745660C2211yescharacteristic
29C4×#11745660C2211yescharacteristic
30C4×D472C2211yescharacteristic
31C4×D472C2211yescharacteristic
32#16837479C2211yescharacteristic, complemented
33#16837478—21no—
34#16837471—21no—
35C8×D236—21no—
36C8×D236—21no—
37#16837467C2211yescharacteristic
38#16837467C2211yescharacteristic
39#16837467C2211yescharacteristic
40#16837467C2211yescharacteristic
41#16837467C2211yescharacteristic, complemented
42#16837462—21no—
43#16837455C2211yescharacteristic
44#16837455C2211yescharacteristic
45#16837427—21no—
46C22⋊C8×D59—21no—
47C22⋊C8×D59—21no—
48#16837132C2211yescharacteristic, complemented
49#16837132C2211yescharacteristic, complemented
50#16837129C2211yescharacteristic
51#16837129C2211yescharacteristic
52#16837129C2211yescharacteristic
53#16837129C2211yescharacteristic
54#16837129C2211yescharacteristic, complemented
55#16837129C2211yescharacteristic, complemented
56#16837124C2211yescharacteristic, complemented
57#16837124C2211yescharacteristic, complemented
58#16837123C2211yescharacteristic, complemented
59C8×C59⋊C8C2211yescharacteristic, complemented
60C59×#431C2211yescharacteristic
61C59×#431C2211yescharacteristic
62C59×#423C2211yescharacteristic
63C59×#335C2211yescharacteristic, complemented
64C59×#335C2211yescharacteristic, complemented
65C59×#334C2211yescharacteristic, complemented
66C59×#334C2211yescharacteristic, complemented
67C22×D236C2×C411yescharacteristic
68C22×C4×D59—21no—
69C2×#11745660C2×C411yescharacteristic
70C2×#11745660C2×C411yescharacteristic
71C2×D472C2×C411yescharacteristic
72C2×D472C2×C411yescharacteristic
73C2×#11745659—21no—
74C2×C8×D59—21no—
75C2×C8×D59—21no—
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78C2×C8×D59—21no—
79#14273721—21no—
80#14273719C2×C411yescharacteristic
81#14273715C2×C411yescharacteristic
82#14273710—21no—
83#14273706C2×C411yescharacteristic
84#14273701C2×C411yescharacteristic
85C4×Dic118C2311yescharacteristic
86C4×D236—21no—
87C4×D236—21no—
88C4×D236C2×C411yescharacteristic
89C4×D236C2×C411yescharacteristic
90C4×D236C2311yescharacteristic
91C4×D236C2311yescharacteristic
92C4×D236C2311yescharacteristic
93C4×C22⋊D59—21no—
94#14273684—21no—
95C2×C4⋊Dic59C2×C411yescharacteristic
96C2×(C2×D59)⋊C4—21no—
97#14273661C2×C411yescharacteristic
98#14273661C2×C411yescharacteristic
99#14273659C2×C411yescharacteristic
100#14273659C2×C411yescharacteristic
101C4⋊(C59⋊C8)—21no—
102C4⋊(C59⋊C8)D411yescharacteristic
103C4⋊(C59⋊C8)D411yescharacteristic
104Dic59⋊C8—21no—
105Dic59⋊C8—21no—
106#14273652C2×C411yescharacteristic
107#14273652C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273649C2×C411yescharacteristic
110#14273649C2×C411yescharacteristic
111#14273649C2×C411yescharacteristic
112#14273649C2×C411yescharacteristic
113#14273649C2×C411yescharacteristic
114C22⋊(C59⋊C8)—21no—
115C22⋊(C59⋊C8)D411yescharacteristic
116C22⋊(C59⋊C8)D411yescharacteristic
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8—21no—
121(C2×D59)⋊C8—21no—
122(C2×D59)⋊C8—21no—
123#14273637—21no—
124C4×C59⋊C8C2311yescharacteristic
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×M16×C59—21no—
130C2×C42×C59C2311yescharacteristic
131C4⋊C8×C59—21no—
132C4⋊C8×C59C2×C411yescharacteristic
133C4⋊C8×C59C2×C411yescharacteristic
134C4⋊C8×C59C2311yescharacteristic
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137C4⋊C8×C59C2311yescharacteristic
138C22⋊C8×C59—21no—
139C22⋊C8×C59C2×C411yescharacteristic
140C22⋊C8×C59C2×C411yescharacteristic
141C4×C8×C59—21no—
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C23×D59—21no—
145C22×C4×C59C22×C411yescharacteristic
146C22×C4×C59C2×D411yescharacteristic
147C22×C4×C59#4111yescharacteristic
148C2×C22⋊D59—21no—
149C22×Dic59—21no—
150C2×D236—21no—
151C2×D236—22no—
152C2×D236—22no—
153C2×D236C22×C411yescharacteristic
154C2×D236C22×C411yescharacteristic
155C2×D236C22×C411yescharacteristic
156C2×C4×D59—21no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—21no—
160C2×C4×D59—21no—
161C2×C4×D59—81no—
162C2×C4×D59—81no—
163C2×Dic118C22×C411yescharacteristic
164M16×C59—81no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—21no—
169C2×C8×C59C22×C411yescharacteristic
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C22×C411yescharacteristic
172C2×C8×C59C22×C411yescharacteristic
173C42×C59C22×C411yescharacteristic
174C42×C59C2×D411yescharacteristic
175C42×C59#4111yescharacteristic
176C42×C59C2411yescharacteristic
177C22⋊Dic59—21no—
178(C2×D59)⋊C4—21no—
179(C2×D59)⋊C4—21no—
180(C2×D59)⋊C4—21no—
181(C2×D59)⋊C4—21no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186C4⋊Dic59—21no—
187C4⋊Dic59C22×C411yescharacteristic
188C4⋊Dic59C22×C411yescharacteristic
189C4⋊Dic59C22×C411yescharacteristic
190C4⋊Dic59C22×C411yescharacteristic
191Dic59⋊C4—21no—
192Dic59⋊C4—21no—
193Dic59⋊C4—21no—
194Dic59⋊C4—21no—
195C4×Dic59—21no—
196C2×C59⋊C8—21no—
197C2×C59⋊C8—21no—
198C2×C59⋊C8C2×D411yescharacteristic
199C2×C59⋊C8C2×D411yescharacteristic
200C2×C59⋊C8C2×D411yescharacteristic
201C2×C59⋊C8C2×D411yescharacteristic
202D472—22no—
203D472—22no—
204#11745660—22no—
205#11745660—22no—
206#11745659—81no—
207C8×D59—81no—
208C8×D59—81no—
209C8×D59—81no—
210C8×D59—81no—
211C8×D59—81no—
212C23×C59C4×D411yescharacteristic
213C22×D59—21no—
214C22×D59—21no—
215C22×D59—24no—
216C22×D59—21no—
217C22×D59—81no—
218C22×D59—81no—
219C2×C4×C59—21no—
220C2×C4×C59—21no—
221C2×C4×C59—21no—
222C2×C4×C59—21no—
223C2×C4×C59—41no—
224C2×C4×C59C4×D411yescharacteristic
225C2×C4×C59C4×D411yescharacteristic
226C2×C4×C59C4×D411yescharacteristic
227C2×C4×C59C23×C411yescharacteristic
228C2×C4×C59C22×D411yescharacteristic
229C2×C4×C59C2×#4111yescharacteristic
230C22⋊D59—81no—
231C2×Dic59—21no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—21no—
235C2×Dic59—21no—
236C2×Dic59—81no—
237D236—24no—
238D236—21no—
239D236—21no—
240D236—21no—
241D236—21no—
242D236#13112yes—
243D236#13112yes—
244C4×D59—41no—
245C4×D59—41no—
246C4×D59—41no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—81no—
251C4×D59—81no—
252Dic118—21no—
253Dic118—21no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59—21no—
257C8×C59—21no—
258C8×C59—41no—
259C8×C59—41no—
260C8×C59—41no—
261C8×C59—41no—
262C59⋊C8—21no—
263C59⋊C8—21no—
264C59⋊C8—21no—
265C59⋊C8—21no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#17918—591nomaximal, Sylow
269C22×C59—21no—
270C22×C59—21no—
271C22×C59—21no—
272C22×C59C2×C4×D411yescharacteristic
273C2×D59—22no—
274C2×D59—22no—
275C2×D59—41no—
276C2×D59—41no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—84no—
283C4×C59—21no—
284C4×C59—21no—
285C4×C59—21no—
286C4×C59—21no—
287C4×C59C2×#13111yescharacteristic
288C4×C59C2×#13111yescharacteristic
289C4×C59#57611yescharacteristic
290C4×C59#57711yescharacteristic
291Dic59—41no—
292Dic59—41no—
293Dic59—41no—
294Dic59—41no—
295Dic59—41no—
296#3105—591no—
297#2993—591no—
298#2972—591no—
299#1586—591no—
300#1585—591no—
301#1582—591no—
302#1581—591no—
303#1580—591no—
304SD16⋊C8—591no—
305C8×SD16—591no—
306C8×D8—591no—
307#1574D5911yescharacteristic, complemented
308#1486—591no—
309#1482—591no—
310#1454—591no—
311C2×C59—41no—
312C2×C59#292911yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#295711yescharacteristic
315D59—42no—
316D59—42no—
317D59—81no—
318#529—591no—
319C2×C4×D4—591no—
320#484—591no—
321#481—591no—
322#478—591no—
323#478—591no—
324#463—591no—
325#463—591no—
326#450—591no—
327#447—591no—
328#446—591no—
329C8×Q8—591no—
330C4×SD16—591no—
331C4×SD16—591no—
332C4×D8—591no—
333C4×D8—591no—
334#436—591no—
335#436—1181no—
336#435—1181no—
337C8×D4—591no—
338C8×D4—591no—
339C8×D4—591no—
340C8×D4—591no—
341C8×D4—591no—
342C8×D4—1181no—
343C8×D4—1181no—
344#433—1181no—
345#431—591no—
346#431—591no—
347#431C2×D5911yescharacteristic
348#431C2×D5911yescharacteristic
349#424—1181no—
350#423C2×D5911yescharacteristic
351C2×C22⋊C8—1181no—
352C2×C22⋊C8—1181no—
353#335C2×D5911yescharacteristic, complemented
354#335C2×D5911yescharacteristic, complemented
355#334C2×D5911yescharacteristic, complemented
356#334C2×D5911yescharacteristic, complemented
357Q8⋊C8—591no—
358Q8⋊C8—591no—
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363D4⋊C8—591no—
364D4⋊C8—591no—
365#322—591no—
366#322—591no—
367#322—591no—
368C82—591no—
369C59#1791811yescharacteristic, complemented, Sylow
370C22×D4—591no—
371C23×C4—1181no—
372C2×SD16—591no—
373C2×SD16—591no—
374C2×D8—591no—
375C2×D8—591no—
376C2×M16—21no—
377C2×M16—1181no—
378C22×C8—1181no—
379C22×C8—1181no—
380C22×C8—1181no—
381C22×C8—1181no—
382C22×C8—1181no—
383#126—1181no—
384#125—591no—
385#124—591no—
386#123—1181no—
387C22⋊Q8—591no—
388#121—591no—
389C4×Q8—591no—
390C4×D4—591no—
391C4×D4—591no—
392C4×D4—591no—
393C4×D4—591no—
394C4×D4—591no—
395C4×D4—1181no—
396C4×D4—1181no—
397C4×D4—1181no—
398#117—1181no—
399C2×C4⋊C4—591no—
400C2×C22⋊C4—1181no—
401C2×C42C22×D5911yescharacteristic
402#107—591no—
403#107—591no—
404#106—591no—
405#106—591no—
406C4⋊C8—21no—
407C4⋊C8—591no—
408C4⋊C8—591no—
409C4⋊C8—1181no—
410C4⋊C8—1181no—
411C4⋊C8—1181no—
412C4⋊C8C4×D5911yescharacteristic
413C4⋊C8C4×D5911yescharacteristic
414C4⋊C8C22×D5911yescharacteristic
415C4⋊C8C22×D5911yescharacteristic
416C4⋊C8C22×D5911yescharacteristic
417C4⋊C8C22×D5911yescharacteristic
418#103—591no—
419#103—591no—
420#102—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424#102—591no—
425#102—591no—
426C22⋊C8—21no—
427C22⋊C8—591no—
428C22⋊C8—591no—
429C22⋊C8—1181no—
430C22⋊C8—1181no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433C22⋊C8—1181no—
434C22⋊C8—1181no—
435C22⋊C8—1181no—
436C22⋊C8C4×D5911yescharacteristic
437C22⋊C8C4×D5911yescharacteristic
438#97—1181no—
439C4×C8—21no—
440C4×C8—591no—
441C4×C8—591no—
442C4×C8—591no—
443C4×C8—591no—
444C4×C8—1181no—
445C4×C8C22×D5911yescharacteristic
446C4×C8C22×D5911yescharacteristic
447C24—1181no—
448C2×Q8—591no—
449C2×D4—591no—
450C2×D4—591no—
451C2×D4—591no—
452C2×D4—1182no—
453C2×D4—1182no—
454C2×D4—1181no—
455C2×D4—1181no—
456C22×C4—1181no—
457C22×C4—1181no—
458C22×C4—1181no—
459C22×C4—1181no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—4721no—
463C22×C4—4721no—
464C22×C4C2×C4×D5911yescharacteristic
465C22×C4D4×D5911yescharacteristic
466C22×C4#1174568911yescharacteristic
467SD16—1182no—
468SD16—1182no—
469D8—1182no—
470D8—1182no—
471M16—81no—
472M16—4721no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—591no—
478C2×C8—591no—
479C2×C8—591no—
480C2×C8—591no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—4721no—
484C2×C8—4721no—
485C2×C8—4721no—
486C2×C8—4721no—
487C2×C8—4721no—
488C2×C8C2×C4×D5911yescharacteristic
489C2×C8C2×C4×D5911yescharacteristic
490C2×C8C2×C4×D5911yescharacteristic
491C2×C8C2×C4×D5911yescharacteristic
492C4⋊C4—591no—
493C4⋊C4—591no—
494C4⋊C4—591no—
495C4⋊C4—591no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4—1181no—
500C4⋊C4—1181no—
501C22⋊C4—1181no—
502C22⋊C4—1181no—
503C22⋊C4—1181no—
504C22⋊C4—1181no—
505C22⋊C4—1181no—
506C22⋊C4—1181no—
507C22⋊C4—1181no—
508C22⋊C4—1181no—
509C22⋊C4—1181no—
510C42—1181no—
511C42C2×C4×D5911yescharacteristic
512C42D4×D5911yescharacteristic
513C42#1174568911yescharacteristic
514C42C23×D5911yescharacteristic
515C23—1184no—
516C23—1181no—
517C23—1181no—
518C23—1181no—
519C23—4721no—
520C23—4721no—
521C23#1427368911yescharacteristic
522Q8—1181no—
523Q8—1181no—
524D4—592no—
525D4—592no—
526D4—1184no—
527D4—1181no—
528D4—1181no—
529D4—1181no—
530D4—1181no—
531D4—4721no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—1181no—
538C2×C4—1181no—
539C2×C4—1181no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—4721no—
543C2×C4—4721no—
544C2×C4—4721no—
545C2×C4—2361no—
546C2×C4—2361no—
547C2×C4—2361no—
548C2×C4—2361no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4#1427368911yescharacteristic
552C2×C4#1427368911yescharacteristic
553C2×C4#1427368911yescharacteristic
554C2×C4C22×C4×D5911yescharacteristic
555C2×C4C2×D4×D5911yescharacteristic
556C2×C4C2×#1174568911yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—1181no—
566C8—1181no—
567C8—1181no—
568C8—1181no—
569C8—2361no—
570C8—2361no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—1182no—
575C22—1182no—
576C22—4724no—
577C22—2361no—
578C22—2361no—
579C22—2361no—
580C22—2361no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22C2×#1427368911yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—2361no—
590C4—2361no—
591C4—2361no—
592C4—2361no—
593C4—2361no—
594C4#131×D5911yescharacteristic
595C4#131×D5911yescharacteristic
596C4#1683820611yescharacteristic
597C4#1683821111yescharacteristic
598C2—41no—
599C2—4721no—
600C2—2362no—
601C2—2362no—
602C2#2143278911yescharacteristic
603C2#2143292611yescharacteristic
604C2#2143301211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC59×#1574C2
socleC22×C59C2×C4×D4
prime radical#16837538C4
cosocleC42×C59C24
hypercenter#1574D59
hypocenterC59#17918
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1574

Complements

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

NH ≅ G/Ntype
2. #21433241C2semidirect
3. #21433119C2semidirect
4. #21422959C2semidirect
5. #21422952C2semidirect
6. #21422931C2semidirect
7. #21422924C2semidirect
8. #21422919C2semidirect
9. #21422907C2semidirect
10. #21422895C2semidirect
11. #21422890C2semidirect
12. #21422520C2semidirect
13. #21422502C2semidirect
14. #21422396C2semidirect
15. C59×#1574C2semidirect
26. #16837529C22semidirect
27. #16837526C22semidirect
32. #16837479C22semidirect
41. #16837467C22semidirect
48. #16837132C22semidirect
49. #16837132C22semidirect
54. #16837129C22semidirect
55. #16837129C22semidirect
56. #16837124C22semidirect
57. #16837124C22semidirect
58. #16837123C22semidirect
59. C8×C59⋊C8C22semidirect
63. C59×#335C22semidirect
64. C59×#335C22semidirect
65. C59×#334C22semidirect
66. C59×#334C22semidirect
307. #1574D59semidirect
353. #335C2×D59semidirect
354. #335C2×D59semidirect
355. #334C2×D59semidirect
356. #334C2×D59semidirect
369. C59#17918semidirect

Maximal subgroups

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

subgroupindex
#214341452
#214332412
#214331192
#214229592
#214229522
#214229312
#214229242
#214229192
#214229072
#214228952
#214228902
#214225202
#214225022
#214223962
C59×#15742
#1791859