Finite Group Databasebeta

← #28558754

Subgroups — #28558754

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#21434147C211yesmaximal, characteristic
2#21433244C211yesmaximal, characteristic, complemented
3#21433117C211yesmaximal, characteristic, complemented
4#21422959C211yesmaximal, characteristic, complemented
5#21422954C211yesmaximal, characteristic, complemented
6#21422935C211yesmaximal, characteristic, complemented
7#21422922C211yesmaximal, characteristic, complemented
8#21422918C211yesmaximal, characteristic, complemented
9#21422906C211yesmaximal, characteristic, complemented
10C8×D472C211yesmaximal, characteristic, complemented
11#21422883C211yesmaximal, characteristic, complemented
12#21422519C211yesmaximal, characteristic, complemented
13#21422503C211yesmaximal, characteristic, complemented
14#21422395C211yesmaximal, characteristic, complemented
15C8×SD16×C59C211yesmaximal, characteristic, complemented
16#16837939C2211yescharacteristic
17C4×D4×D59C2211yescharacteristic
18#16837736C411yescharacteristic
19#16837722C411yescharacteristic
20#16837701C411yescharacteristic
21#16837700C411yescharacteristic
22#16837620C411yescharacteristic
23#16837619C411yescharacteristic
24#16837551C411yescharacteristic
25#16837537C411yescharacteristic
26#16837533C2211yescharacteristic, complemented
27C4×#11745671C2211yescharacteristic
28C4×D472C2211yescharacteristic
29C4×#11745669C2211yescharacteristic
30#16837481—21no—
31#16837479C2211yescharacteristic, complemented
32#16837472—21no—
33C8×D236C2211yescharacteristic
34C8×D236C2211yescharacteristic, complemented
35D4×C59⋊C8C2211yescharacteristic
36#16837467C2211yescharacteristic
37C8×C22⋊D59—21no—
38(C22⋊D59)⋊C8—21no—
39#16837461—21no—
40#16837454C2211yescharacteristic
41#16837453C2211yescharacteristic
42#16837453C2211yescharacteristic
43#16837453C2211yescharacteristic
44#16837427—21no—
45#16837424C2211yescharacteristic
46C2×(C2×D59)⋊C8—21no—
47C22⋊C8×D59—21no—
48#16837155C2211yescharacteristic, complemented
49#16837154C2211yescharacteristic, complemented
50#16837152C2211yescharacteristic, complemented
51Q8⋊(C59⋊C8)C2211yescharacteristic, complemented
52#16837131C2211yescharacteristic
53#16837131C2211yescharacteristic, complemented
54D4⋊(C59⋊C8)C2211yescharacteristic
55#16837129C2211yescharacteristic
56#16837129C2211yescharacteristic, complemented
57#16837124C2211yescharacteristic, complemented
58#16837124C2211yescharacteristic, complemented
59#16837124C2211yescharacteristic, complemented
60C8×Q8×C59C2211yescharacteristic, complemented
61C4×SD16×C59C2211yescharacteristic
62C8×D4×C59C2211yescharacteristic
63C59×#334C2211yescharacteristic, complemented
64C59×Q8⋊C8C2211yescharacteristic, complemented
65C59×D4⋊C8C2211yescharacteristic
66C82×C59C2211yescharacteristic, complemented
67C2×D4×D59C2×C411yescharacteristic
68C22×C4×D59—21no—
69C2×#11745671C2×C411yescharacteristic
70C2×D472C2×C411yescharacteristic
71C2×#11745669C2×C411yescharacteristic
72C2×#11745659—21no—
73C2×#11745659—21no—
74C22×C59⋊C8—21no—
75C2×C8×D59—21no—
76C2×C8×D59—21no—
77C2×C8×D59—21no—
78#14273722—21no—
79#14273718C2×C411yescharacteristic
80#14273716C2×C411yescharacteristic
81#14273712—21no—
82#14273703C2×C411yescharacteristic
83#14273702C2×C411yescharacteristic
84C4×D236—21no—
85C4×D236C2311yescharacteristic
86C4×D236C2311yescharacteristic
87D4×Dic59C2×C411yescharacteristic
88#14273689C2×C411yescharacteristic
89C4×C22⋊D59—21no—
90(C22⋊D59)⋊C4—21no—
91#14273684—21no—
92C4⋊C4×D59C2×C411yescharacteristic
93C22⋊C4×D59—21no—
94C42×D59C2311yescharacteristic
95#14273661C2×C411yescharacteristic
96#14273660C2×C411yescharacteristic
97#14273658C2×C411yescharacteristic
98C4⋊(C59⋊C8)—21no—
99C4⋊(C59⋊C8)C2311yescharacteristic
100C4⋊(C59⋊C8)C2311yescharacteristic
101Dic59⋊C8—21no—
102Dic59⋊C8—21no—
103Dic59⋊C8C2×C411yescharacteristic
104Dic59⋊C8C2×C411yescharacteristic
105Dic59⋊C8D411yescharacteristic
106Dic59⋊C8D411yescharacteristic
107#14273651C2×C411yescharacteristic
108#14273649C2×C411yescharacteristic
109#14273649C2×C411yescharacteristic
110#14273648C2×C411yescharacteristic
111#14273647C2×C411yescharacteristic
112#14273647C2×C411yescharacteristic
113C22⋊(C59⋊C8)—21no—
114(C2×D59)⋊C8—21no—
115(C2×D59)⋊C8—21no—
116(C2×D59)⋊C8—21no—
117(C2×D59)⋊C8—21no—
118(C2×D59)⋊C8—21no—
119(C2×D59)⋊C8—21no—
120(C2×D59)⋊C8C2×C411yescharacteristic
121(C2×D59)⋊C8C2×C411yescharacteristic
122(C2×D59)⋊C8D411yescharacteristic
123(C2×D59)⋊C8D411yescharacteristic
124#14273638—21no—
125C4×C59⋊C8C2311yescharacteristic
126C4×C59⋊C8C2311yescharacteristic
127C4×C59⋊C8C2311yescharacteristic
128C8×Dic59—21no—
129C2×SD16×C59C2×C411yescharacteristic
130C22×C8×C59—21no—
131C4×Q8×C59C2311yescharacteristic
132C4×D4×C59C2311yescharacteristic
133#106×C59C2×C411yescharacteristic
134C4⋊C8×C59—21no—
135C4⋊C8×C59C2311yescharacteristic
136C4⋊C8×C59C2311yescharacteristic
137#103×C59C2×C411yescharacteristic
138#102×C59C2×C411yescharacteristic
139C22⋊C8×C59—21no—
140C4×C8×C59—21no—
141C4×C8×C59C2311yescharacteristic
142C4×C8×C59C2311yescharacteristic
143C4×C8×C59C2311yescharacteristic
144C23×D59—21no—
145C2×Q8×C59C22×C411yescharacteristic
146C2×D4×C59C22×C411yescharacteristic
147C22×C4×C59—21no—
148C2×C22⋊D59—21no—
149C22×Dic59—21no—
150D4×D59—22no—
151D4×D59—22no—
152C2×D236—21no—
153C2×D236C22×C411yescharacteristic
154C2×D236C22×C411yescharacteristic
155C2×C4×D59—21no—
156C2×C4×D59—21no—
157C2×C4×D59—21no—
158C2×C4×D59—21no—
159C2×C4×D59—81no—
160C2×C4×D59—81no—
161C2×C4×D59C22×C411yescharacteristic
162C2×C4×D59C2×D411yescharacteristic
163C2×C4×D59#4111yescharacteristic
164SD16×C59—22no—
165C2×C8×C59—21no—
166C2×C8×C59—21no—
167C2×C8×C59—21no—
168C2×C8×C59—81no—
169C2×C8×C59C22×C411yescharacteristic
170C2×C8×C59C22×C411yescharacteristic
171C2×C8×C59C2×D411yescharacteristic
172C2×C8×C59C2×D411yescharacteristic
173C4⋊C4×C59—21no—
174C4⋊C4×C59C22×C411yescharacteristic
175C4⋊C4×C59C22×C411yescharacteristic
176C22⋊C4×C59—21no—
177C42×C59—21no—
178C42×C59C2411yescharacteristic
179C22⋊Dic59—21no—
180#11745671—22no—
181#11745669—22no—
182(C2×D59)⋊C4—21no—
183(C2×D59)⋊C4—21no—
184(C2×D59)⋊C4—21no—
185(C2×D59)⋊C4—21no—
186(C2×D59)⋊C4—21no—
187(C2×D59)⋊C4—21no—
188(C2×D59)⋊C4—21no—
189C4⋊Dic59—21no—
190C4⋊Dic59C22×C411yescharacteristic
191C4⋊Dic59C22×C411yescharacteristic
192Dic59⋊C4—21no—
193Dic59⋊C4—21no—
194Dic59⋊C4—21no—
195C4×Dic59C22×C411yescharacteristic
196C4×Dic59C2×D411yescharacteristic
197C4×Dic59#4111yescharacteristic
198C2×C59⋊C8—21no—
199C2×C59⋊C8—21no—
200C2×C59⋊C8—21no—
201C2×C59⋊C8—81no—
202C2×C59⋊C8C22×C411yescharacteristic
203C2×C59⋊C8C22×C411yescharacteristic
204C2×C59⋊C8C2×D411yescharacteristic
205C2×C59⋊C8C2×D411yescharacteristic
206D472—22no—
207#11745659—81no—
208#11745659—81no—
209C8×D59—81no—
210C8×D59—81no—
211C8×D59—81no—
212C23×C59—21no—
213C22×D59—21no—
214C22×D59—24no—
215C22×D59—21no—
216C22×D59—81no—
217C22×D59—81no—
218C22×D59C4×D411yescharacteristic
219Q8×C59—21no—
220Q8×C59—21no—
221D4×C59—21no—
222D4×C59#13112yes—
223C2×C4×C59—21no—
224C2×C4×C59—21no—
225C2×C4×C59—21no—
226C2×C4×C59—41no—
227C2×C4×C59—41no—
228C2×C4×C59C23×C411yescharacteristic
229C2×C4×C59C22×D411yescharacteristic
230C2×C4×C59C2×#4111yescharacteristic
231C22⋊D59—24no—
232C2×Dic59—21no—
233C2×Dic59—21no—
234C2×Dic59—81no—
235C2×Dic59C4×D411yescharacteristic
236C2×Dic59C4×D411yescharacteristic
237C2×Dic59C4×D411yescharacteristic
238D236—21no—
239D236—21no—
240D236—21no—
241D236—81no—
242D236#13112yes—
243C4×D59—21no—
244C4×D59—21no—
245C4×D59—21no—
246C4×D59—21no—
247C4×D59—41no—
248C4×D59—41no—
249C4×D59—41no—
250C4×D59—41no—
251C4×D59—41no—
252C4×D59—81no—
253C4×D59—81no—
254C8×C59—21no—
255C8×C59—21no—
256C8×C59—21no—
257C8×C59—21no—
258C8×C59—41no—
259C8×C59—41no—
260C8×C59—41no—
261C59⋊C8—21no—
262C59⋊C8—21no—
263C59⋊C8—21no—
264C59⋊C8—21no—
265C59⋊C8—41no—
266C59⋊C8—41no—
267C59⋊C8—41no—
268#17918—591nomaximal, Sylow
269C22×C59—22no—
270C22×C59—41no—
271C22×C59—41no—
272C22×C59C2×C4×D411yescharacteristic
273C2×D59—21no—
274C2×D59—22no—
275C2×D59—21no—
276C2×D59—21no—
277C2×D59—41no—
278C2×D59—41no—
279C2×D59—41no—
280C2×D59—41no—
281C2×D59—41no—
282C2×D59—84no—
283C4×C59—21no—
284C4×C59—41no—
285C4×C59—41no—
286C4×C59—41no—
287C4×C59C2×#13111yescharacteristic
288C4×C59C2×#13111yescharacteristic
289C4×C59#57611yescharacteristic
290C4×C59#57711yescharacteristic
291Dic59—21no—
292Dic59—21no—
293Dic59—21no—
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×SD16D5911yescharacteristic, complemented
306C8×D8—591no—
307#1574—591no—
308#1486—591no—
309#1482—591no—
310#1454—591no—
311C2×C59—42no—
312C2×C59#292911yescharacteristic
313C2×C59#294611yescharacteristic
314C2×C59#295711yescharacteristic
315D59—41no—
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×Q8C2×D5911yescharacteristic, complemented
330C4×SD16—591no—
331C4×SD16C2×D5911yescharacteristic
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—1181no—
342C8×D4—1181no—
343C8×D4C2×D5911yescharacteristic
344#433—1181no—
345#431—591no—
346#431—591no—
347#431—591no—
348#431—591no—
349#424—1181no—
350#423—591no—
351C2×C22⋊C8—1181no—
352C2×C22⋊C8—1181no—
353#335—591no—
354#335—591no—
355#334—591no—
356#334C2×D5911yescharacteristic, complemented
357Q8⋊C8—591no—
358Q8⋊C8C2×D5911yescharacteristic, complemented
359D4⋊C8—591no—
360D4⋊C8—591no—
361D4⋊C8—591no—
362D4⋊C8—591no—
363D4⋊C8—591no—
364D4⋊C8C2×D5911yescharacteristic
365#322—591no—
366#322—591no—
367#322—591no—
368C82C2×D5911yescharacteristic, complemented
369C59#1791811yescharacteristic, complemented, Sylow
370C22×D4—591no—
371C23×C4—1181no—
372C2×SD16—591no—
373C2×SD16C4×D5911yescharacteristic
374C2×D8—591no—
375C2×D8—591no—
376C2×M16—1181no—
377C2×M16—1181no—
378C22×C8—21no—
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×Q8C22×D5911yescharacteristic
390C4×D4—591no—
391C4×D4—591no—
392C4×D4—591no—
393C4×D4—591no—
394C4×D4—1181no—
395C4×D4—1181no—
396C4×D4—1181no—
397C4×D4C22×D5911yescharacteristic
398#117—1181no—
399C2×C4⋊C4—591no—
400C2×C22⋊C4—1181no—
401C2×C42—591no—
402#107—591no—
403#107—591no—
404#106—591no—
405#106C4×D5911yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—591no—
408C4⋊C8—591no—
409C4⋊C8—591no—
410C4⋊C8—591no—
411C4⋊C8—591no—
412C4⋊C8—591no—
413C4⋊C8—1181no—
414C4⋊C8—1181no—
415C4⋊C8—1181no—
416C4⋊C8C22×D5911yescharacteristic
417C4⋊C8C22×D5911yescharacteristic
418#103—591no—
419#103C4×D5911yescharacteristic
420#102—591no—
421#102—591no—
422#102—591no—
423#102—591no—
424#102—591no—
425#102C4×D5911yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—591no—
428C22⋊C8—591no—
429C22⋊C8—591no—
430C22⋊C8—591no—
431C22⋊C8—1181no—
432C22⋊C8—1181no—
433C22⋊C8—1181no—
434C22⋊C8—1181no—
435C22⋊C8—1181no—
436C22⋊C8—1181no—
437C22⋊C8—1181no—
438#97—1181no—
439C4×C8—21no—
440C4×C8—591no—
441C4×C8—591no—
442C4×C8—591no—
443C4×C8—1181no—
444C4×C8C22×D5911yescharacteristic
445C4×C8C22×D5911yescharacteristic
446C4×C8C22×D5911yescharacteristic
447C24—1181no—
448C2×Q8C2×C4×D5911yescharacteristic
449C2×D4—591no—
450C2×D4—591no—
451C2×D4—1181no—
452C2×D4—1181no—
453C2×D4—1182no—
454C2×D4—1182no—
455C2×D4C2×C4×D5911yescharacteristic
456C22×C4—21no—
457C22×C4—591no—
458C22×C4—591no—
459C22×C4—591no—
460C22×C4—1181no—
461C22×C4—1181no—
462C22×C4—1181no—
463C22×C4—1181no—
464C22×C4—1181no—
465C22×C4—4721no—
466C22×C4—4721no—
467SD16—22no—
468SD16—1182no—
469D8—1182no—
470D8—1182no—
471M16—4721no—
472M16—4721no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—81no—
477C2×C8—591no—
478C2×C8—591no—
479C2×C8—591no—
480C2×C8—591no—
481C2×C8—1181no—
482C2×C8—1181no—
483C2×C8—1181no—
484C2×C8—4721no—
485C2×C8—4721no—
486C2×C8—4721no—
487C2×C8—4721no—
488C2×C8C2×C4×D5911yescharacteristic
489C2×C8C2×C4×D5911yescharacteristic
490C2×C8D4×D5911yescharacteristic
491C2×C8D4×D5911yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—591no—
494C4⋊C4—591no—
495C4⋊C4—1181no—
496C4⋊C4—1181no—
497C4⋊C4—1181no—
498C4⋊C4—1181no—
499C4⋊C4C2×C4×D5911yescharacteristic
500C4⋊C4C2×C4×D5911yescharacteristic
501C22⋊C4—21no—
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—21no—
511C42—591no—
512C42—591no—
513C42—591no—
514C42C23×D5911yescharacteristic
515C23—21no—
516C23—591no—
517C23—1184no—
518C23—1181no—
519C23—1181no—
520C23—4721no—
521C23—4721no—
522Q8—21no—
523Q8—21no—
524D4—21no—
525D4—592no—
526D4—1184no—
527D4—1181no—
528D4—1181no—
529D4—1181no—
530D4—4721no—
531D4#1427373312yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—591no—
538C2×C4—591no—
539C2×C4—591no—
540C2×C4—1181no—
541C2×C4—1181no—
542C2×C4—1181no—
543C2×C4—1181no—
544C2×C4—1181no—
545C2×C4—1181no—
546C2×C4—4721no—
547C2×C4—4721no—
548C2×C4—4721no—
549C2×C4—2361no—
550C2×C4—2361no—
551C2×C4—2361no—
552C2×C4—2361no—
553C2×C4—2361no—
554C2×C4C22×C4×D5911yescharacteristic
555C2×C4C2×D4×D5911yescharacteristic
556C2×C4#41×D5911yescharacteristic
557C8—21no—
558C8—21no—
559C8—21no—
560C8—21no—
561C8—41no—
562C8—41no—
563C8—41no—
564C8—1181no—
565C8—1181no—
566C8—1181no—
567C8—1181no—
568C8—2361no—
569C8—2361no—
570C8—2361no—
571C22—22no—
572C22—41no—
573C22—41no—
574C22—1181no—
575C22—1181no—
576C22—1181no—
577C22—1182no—
578C22—4724no—
579C22—2361no—
580C22—2361no—
581C22—2361no—
582C22—2361no—
583C22—2361no—
584C22C4×D4×D5911yescharacteristic
585C4—21no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—1181no—
590C4—1181no—
591C4—1181no—
592C4—2361no—
593C4—2361no—
594C4#131×D5911yescharacteristic
595C4C2×#1427373311yescharacteristic
596C4#1683820411yescharacteristic
597C4#1683820811yescharacteristic
598C2—42no—
599C2—4721no—
600C2—2361no—
601C2—2362no—
602C2#2143278511yescharacteristic
603C2#2143292311yescharacteristic
604C2#2143300811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D59
commutatorC2×C4×C59C23×C4
FrattiniC42C23×D59
FittingC8×SD16×C59C2
socleC22×C59C2×C4×D4
prime radical#16837537C4
cosocleC42×C59C24
hypercenterC8×SD16D59
hypocenterC59#17918
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ C8×SD16

Complements

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

NH ≅ G/Ntype
2. #21433244C2semidirect
3. #21433117C2semidirect
4. #21422959C2semidirect
5. #21422954C2semidirect
6. #21422935C2semidirect
7. #21422922C2semidirect
8. #21422918C2semidirect
9. #21422906C2semidirect
10. C8×D472C2semidirect
11. #21422883C2semidirect
12. #21422519C2semidirect
13. #21422503C2semidirect
14. #21422395C2semidirect
15. C8×SD16×C59C2semidirect
26. #16837533C22semidirect
31. #16837479C22semidirect
34. C8×D236C22semidirect
48. #16837155C22semidirect
49. #16837154C22semidirect
50. #16837152C22semidirect
51. Q8⋊(C59⋊C8)C22semidirect
53. #16837131C22semidirect
56. #16837129C22semidirect
57. #16837124C22semidirect
58. #16837124C22semidirect
59. #16837124C22semidirect
60. C8×Q8×C59C22semidirect
63. C59×#334C22semidirect
64. C59×Q8⋊C8C22semidirect
66. C82×C59C22semidirect
305. C8×SD16D59semidirect
329. C8×Q8C2×D59semidirect
356. #334C2×D59semidirect
358. Q8⋊C8C2×D59semidirect
368. C82C2×D59semidirect
369. C59#17918semidirect

Maximal subgroups

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

subgroupindex
#214341472
#214332442
#214331172
#214229592
#214229542
#214229352
#214229222
#214229182
#214229062
C8×D4722
#214228832
#214225192
#214225032
#214223952
C8×SD16×C592
#1791859