Finite Group Databasebeta

← #28658609

Subgroups — #28658609

Order 15104.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837761C211yesmaximal, characteristic
2C2×#16837507C211yesmaximal, characteristic, complemented
3#21427935C211yesmaximal, characteristic, complemented
4#21425470C211yesmaximal, characteristic
5#21425429C211yesmaximal, characteristic, complemented
6#21425418C211yesmaximal, characteristic, complemented
7#21425190C211yesmaximal, characteristic
8#21424949C211yesmaximal, characteristic
9#21424796C211yesmaximal, characteristic
10#21424785C211yesmaximal, characteristic, complemented
11#21424775C211yesmaximal, characteristic, complemented
12#21424454C211yesmaximal, characteristic
13#21424364C211yesmaximal, characteristic, complemented
14#21423965C211yesmaximal, characteristic, complemented
15C59×#2448C211yesmaximal, characteristic
16C22×#11745672C2211yescharacteristic
17C22×#11745670C2211yescharacteristic
18C2×#14273727C2211yescharacteristic
19C2×C4×Dic118C2211yescharacteristic
20#16837761—24no—
21#16837507—24no—
22C2×#14273658C2211yescharacteristic
23C2×#14273658C2211yescharacteristic
24C2×C4⋊(C59⋊C8)C2211yescharacteristic
25C2×#14273651C2211yescharacteristic
26C2×#14273651C2211yescharacteristic
27C2×#14273651C2211yescharacteristic
28C2×#14273650C2211yescharacteristic
29C2×#14273650C2211yescharacteristic
30C2×#14273648C2211yescharacteristic
31C2×#14273648C2211yescharacteristic
32C2×#14273648C2211yescharacteristic
33C2×#14273638C2211yescharacteristic
34C2×#14273638C2211yescharacteristic
35C2×C4×C59⋊C8C2211yescharacteristic
36#16837336—21no—
37#16837322—21no—
38#16837308C2211yescharacteristic
39#16837308C2211yescharacteristic
40#16837306—21no—
41#16837300C2211yescharacteristic
42#16837300C2211yescharacteristic
43#16837288—21no—
44#16837164C2211yescharacteristic
45#16837164C2211yescharacteristic
46#16837164C2211yescharacteristic
47#16837164C2211yescharacteristic
48#16837157C2211yescharacteristic
49#16837157C2211yescharacteristic
50C2×#124×C59C2211yescharacteristic
51C2×C4×Q8×C59C2211yescharacteristic
52C59×#399—21no—
53C59×#397—21no—
54C59×#391C2211yescharacteristic
55C59×#390C2211yescharacteristic
56C59×#390C2211yescharacteristic
57C59×#389C2211yescharacteristic
58C59×#388—21no—
59C59×#388—21no—
60C4×C4⋊C4×C59C2211yescharacteristic
61C22×Dic118—21no—
62C22×Dic118C2311yescharacteristic
63C2×#11745672—22no—
64C2×#11745672—22no—
65C2×#11745672—22no—
66C2×#11745670—22no—
67C2×#11745670—22no—
68C2×#11745670—41no—
69C22×C59⋊C8C2311yescharacteristic
70C22×C59⋊C8C2311yescharacteristic
71C22×C59⋊C8C2311yescharacteristic
72C22×C59⋊C8C2311yescharacteristic
73#14273727—22no—
74#14273727—41no—
75C4×Dic118—22no—
76C4×Dic118—41no—
77C2×C4⋊Dic59—21no—
78C2×C4⋊Dic59C2311yescharacteristic
79C2×C4⋊Dic59C2311yescharacteristic
80C2×C4⋊Dic59C2311yescharacteristic
81C2×Dic59⋊C4—21no—
82C2×Dic59⋊C4—21no—
83C2×Dic59⋊C4—21no—
84C2×C4×Dic59—21no—
85#14273658—22no—
86#14273658—41no—
87C4⋊(C59⋊C8)—22no—
88#14273651—22no—
89#14273651—22no—
90#14273651D414yes—
91#14273650—22no—
92#14273650—22no—
93#14273648—22no—
94#14273648—22no—
95#14273648—22no—
96#14273638—22no—
97#14273638—22no—
98C4×C59⋊C8—22no—
99C59⋊#95—21no—
100C59⋊#95—21no—
101C59⋊#95—21no—
102C59⋊#95—21no—
103C59⋊#95—21no—
104C59⋊#95—21no—
105C22×Q8×C59C2311yescharacteristic
106C22×D4×C59C2311yescharacteristic
107#124×C59—22no—
108#124×C59—41no—
109C4×Q8×C59—22no—
110C4×Q8×C59—41no—
111C2×C4⋊C4×C59—21no—
112C2×C4⋊C4×C59—21no—
113C2×C4⋊C4×C59D411yescharacteristic
114C2×C4⋊C4×C59D411yescharacteristic
115C2×C4⋊C4×C59C2311yescharacteristic
116C2×C4⋊C4×C59C2311yescharacteristic
117C2×C22⋊C4×C59—21no—
118C2×C22⋊C4×C59—21no—
119C2×C22⋊C4×C59—21no—
120C2×C22⋊C4×C59—21no—
121C2×C22⋊C4×C59—21no—
122C2×C22⋊C4×C59—21no—
123C2×C42×C59—21no—
124C2×C42×C59C2311yescharacteristic
125C2×C42×C59C2311yescharacteristic
126C2×C42×C59C2311yescharacteristic
127#95×C59—21no—
128#95×C59—21no—
129#95×C59—21no—
130#95×C59D411yescharacteristic
131#95×C59D411yescharacteristic
132C24×C59—21no—
133C2×Q8×C59—21no—
134C2×Q8×C59—21no—
135C2×Q8×C59—21no—
136C2×Q8×C59—21no—
137C2×Q8×C59—21no—
138C2×Q8×C59C2×D412yes—
139C2×D4×C59—21no—
140C2×D4×C59—21no—
141C2×D4×C59—21no—
142C2×D4×C59—21no—
143C2×D4×C59—21no—
144C2×D4×C59—21no—
145C22×C4×C59—21no—
146C22×C4×C59—21no—
147C22×C4×C59—21no—
148C22×C4×C59C2×D411yescharacteristic
149C22×C4×C59C2×D411yescharacteristic
150C22×C4×C59C2×D411yescharacteristic
151C22×C4×C59C2×D411yescharacteristic
152C22×C4×C59C2×D411yescharacteristic
153C22×C4×C59C2×D411yescharacteristic
154C22×C4×C59C2411yescharacteristic
155C22×Dic59—21no—
156C22×Dic59—21no—
157C22×Dic59—21no—
158C22×Dic59—21no—
159C2×Dic118—21no—
160C2×Dic118—21no—
161C2×Dic118—21no—
162C2×Dic118—21no—
163C2×Dic118—21no—
164C2×Dic118—42no—
165C2×Dic118—42no—
166C2×Dic118—81no—
167C2×Dic118#4112yes—
168C4⋊C4×C59—22no—
169C4⋊C4×C59—21no—
170C4⋊C4×C59—21no—
171C4⋊C4×C59—21no—
172C4⋊C4×C59—21no—
173C4⋊C4×C59—41no—
174C4⋊C4×C59—42no—
175C4⋊C4×C59—81no—
176C22⋊C4×C59—42no—
177C22⋊C4×C59—42no—
178C22⋊C4×C59—42no—
179C22⋊C4×C59—42no—
180C22⋊C4×C59—81no—
181C22⋊C4×C59—81no—
182C42×C59—21no—
183C42×C59—21no—
184C42×C59—21no—
185C42×C59—21no—
186C42×C59—21no—
187C42×C59—21no—
188C42×C59—42no—
189#11745672—28no—
190#11745670—44no—
191C4⋊Dic59—21no—
192C4⋊Dic59—21no—
193C4⋊Dic59—21no—
194C4⋊Dic59—21no—
195C4⋊Dic59—21no—
196C4⋊Dic59—42no—
197C4⋊Dic59C2×D412yes—
198Dic59⋊C4—42no—
199Dic59⋊C4—81no—
200Dic59⋊C4—81no—
201C4×Dic59—42no—
202C2×C59⋊C8—21no—
203C2×C59⋊C8—21no—
204C2×C59⋊C8—21no—
205C2×C59⋊C8—21no—
206C2×C59⋊C8—21no—
207C2×C59⋊C8—21no—
208C2×C59⋊C8—21no—
209C2×C59⋊C8—21no—
210C2×C59⋊C8—21no—
211C2×C59⋊C8C2×D412yes—
212C2×C59⋊C8#4112yes—
213C2×C59⋊C8#4112yes—
214C23×C59—41no—
215C23×C59—41no—
216C23×C59—41no—
217C23×C59—41no—
218C23×C59—41no—
219C23×C59—41no—
220C23×C59—41no—
221C23×C59C22×D411yescharacteristic
222Q8×C59—22no—
223Q8×C59—22no—
224Q8×C59—41no—
225Q8×C59—41no—
226D4×C59—41no—
227D4×C59—41no—
228D4×C59—41no—
229D4×C59—41no—
230C2×C4×C59—21no—
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—22no—
238C2×C4×C59—21no—
239C2×C4×C59—21no—
240C2×C4×C59—21no—
241C2×C4×C59—21no—
242C2×C4×C59—21no—
243C2×C4×C59—21no—
244C2×C4×C59—21no—
245C2×C4×C59—21no—
246C2×C4×C59—21no—
247C2×C4×C59—21no—
248C2×C4×C59—41no—
249C2×C4×C59—41no—
250C2×C4×C59—41no—
251C2×C4×C59—41no—
252C2×C4×C59—41no—
253C2×C4×C59—41no—
254C2×C4×C59—41no—
255C2×C4×C59—41no—
256C2×C4×C59#13712yes—
257C2×C4×C59C22×D411yescharacteristic
258C2×C4×C59C2×#4111yescharacteristic
259C2×C4×C59C2×#4111yescharacteristic
260C2×C4×C59C2×#4111yescharacteristic
261C2×C4×C59H(5,𝔽2)11yescharacteristic
262C2×C4×C592−1+411yescharacteristic
263C2×Dic59—22no—
264C2×Dic59—22no—
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—41no—
275Dic118—22no—
276Dic118—22no—
277Dic118—41no—
278Dic118—41no—
279Dic118—84no—
280C59⋊C8—22no—
281C59⋊C8—22no—
282C59⋊C8—22no—
283C59⋊C8—41no—
284#26846—591nomaximal, Sylow
285C22×C59—81no—
286C22×C59—81no—
287C22×C59—81no—
288C22×C59—81no—
289C22×C59—81no—
290C22×C59—81no—
291C22×C59—81no—
292C22×C59#53211yescharacteristic
293C22×C59D4211yescharacteristic
294C22×C59#54811yescharacteristic
295C22×C59C2×#13711yescharacteristic
296C22×C59#57511yescharacteristic
297C22×C59#57711yescharacteristic
298C22×C59#57711yescharacteristic
299C4×C59—22no—
300C4×C59—41no—
301C4×C59—42no—
302C4×C59—41no—
303C4×C59—41no—
304C4×C59—41no—
305C4×C59—41no—
306C4×C59—81no—
307C4×C59—81no—
308C4×C59#52511yescharacteristic
309C4×C59#54211yescharacteristic
310C4×C59#54711yescharacteristic
311C4×C59#55111yescharacteristic
312Dic59—42no—
313Dic59—42no—
314Dic59—81no—
315Dic59—81no—
316C2×#489—591no—
317C2×#441—591no—
318#2451—591no—
319#2448D5911yescharacteristic
320#2065—591no—
321#2058—591no—
322#2056—591no—
323#2015—591no—
324#1968—591no—
325#1936—591no—
326#1935—591no—
327#1933—591no—
328#1880—591no—
329#1865—591no—
330#1770—591no—
331C2×C59—161no—
332C2×C59#246111yescharacteristic
333C2×C59#317311yescharacteristic
334C2×C59#318111yescharacteristic
335C2×C59#327511yescharacteristic
336C2×C59#327811yescharacteristic
337C2×C59#336911yescharacteristic
338C2×C59#337111yescharacteristic
339C22×Q16—591no—
340C22×SD16—591no—
341C2×C4⋊Q8—591no—
342C2×#124C2×D5911yescharacteristic
343C2×C4×Q8—591no—
344C2×C4×Q8C2×D5911yescharacteristic
345#489—1184no—
346#441—1184no—
347C2×#106—591no—
348C2×#106—591no—
349C2×C4⋊C8—591no—
350C2×#103—591no—
351C2×#103—591no—
352C2×#103—591no—
353C2×#103—591no—
354C2×#103—591no—
355C2×#102—591no—
356C2×#102—591no—
357C2×#102—591no—
358C2×#97—591no—
359C2×#97—591no—
360C2×C4×C8—591no—
361#399—21no—
362#398—1181no—
363#397—21no—
364#395—1181no—
365#391—591no—
366#391—591no—
367#391—1181no—
368#391C2×D5911yescharacteristic
369#390C2×D5911yescharacteristic
370#390C2×D5911yescharacteristic
371#389—591no—
372#389—591no—
373#389C2×D5911yescharacteristic
374#388—21no—
375#388—21no—
376#387—1181no—
377C4×C4⋊C4C2×D5911yescharacteristic
378#340—591no—
379#340—591no—
380#340—591no—
381#340—591no—
382#336—591no—
383#336—591no—
384C59#2684611yescharacteristic, complemented, Sylow
385C22×Q8—591no—
386C22×Q8—1181no—
387C22×Q8C22×D5911yescharacteristic
388C22×D4C22×D5911yescharacteristic
389C2×Q16—1182no—
390C2×Q16—1182no—
391C2×Q16—1182no—
392C2×SD16—1182no—
393C2×SD16—1182no—
394C2×SD16—2361no—
395C22×C8—591no—
396C22×C8—591no—
397C22×C8—591no—
398C22×C8—591no—
399C4⋊Q8—1182no—
400C4⋊Q8—2361no—
401#124—22no—
402#124—41no—
403C4×Q8—22no—
404C4×Q8—41no—
405C4×Q8—1182no—
406C4×Q8—2361no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—591no—
410C2×C4⋊C4—591no—
411C2×C4⋊C4—591no—
412C2×C4⋊C4—1181no—
413C2×C4⋊C4—1181no—
414C2×C4⋊C4—1181no—
415C2×C4⋊C4—1181no—
416C2×C4⋊C4C22⋊D5911yescharacteristic
417C2×C4⋊C4C22⋊D5911yescharacteristic
418C2×C4⋊C4C22×D5911yescharacteristic
419C2×C4⋊C4C22×D5911yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—21no—
423C2×C22⋊C4—21no—
424C2×C22⋊C4—21no—
425C2×C22⋊C4—21no—
426C2×C42—21no—
427C2×C42—1181no—
428C2×C42C22×D5911yescharacteristic
429C2×C42C22×D5911yescharacteristic
430C2×C42C22×D5911yescharacteristic
431#106—1182no—
432#106—2361no—
433C4⋊C8—1182no—
434#103—594no—
435#103—1182no—
436#103—1182no—
437#103—1182no—
438#103—1182no—
439#102—1182no—
440#102—1182no—
441#102—1182no—
442#97—1182no—
443#97—1182no—
444C4×C8—1182no—
445#95—21no—
446#95—21no—
447#95—21no—
448#95—1181no—
449#95—1181no—
450#95—1181no—
451#95—1181no—
452#95—1181no—
453#95—1181no—
454#95C22⋊D5911yescharacteristic
455#95C22⋊D5911yescharacteristic
456C24—21no—
457C2×Q8—21no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—592no—
463C2×Q8—1181no—
464C2×Q8—1181no—
465C2×Q8—1181no—
466C2×Q8—1181no—
467C2×Q8—1181no—
468C2×Q8—4721no—
469C2×Q8—2362no—
470C2×Q8—2362no—
471C2×Q8D4×D5912yes—
472C2×D4—21no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—21no—
477C2×D4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—1181no—
482C22×C4—1181no—
483C22×C4—1181no—
484C22×C4—1181no—
485C22×C4C2×C22⋊D5911yescharacteristic
486C22×C4C2×C22⋊D5911yescharacteristic
487C22×C4C2×C22⋊D5911yescharacteristic
488C22×C4C2×C22⋊D5911yescharacteristic
489C22×C4C2×C22⋊D5911yescharacteristic
490C22×C4C2×C22⋊D5911yescharacteristic
491C22×C4C23×D5911yescharacteristic
492Q16—1188no—
493SD16—2364no—
494C2×C8—592no—
495C2×C8—592no—
496C2×C8—592no—
497C2×C8—1181no—
498C2×C8—1181no—
499C2×C8—1181no—
500C2×C8—1181no—
501C2×C8—1181no—
502C2×C8—1181no—
503C2×C8—1181no—
504C2×C8—1181no—
505C2×C8—1181no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—41no—
512C4⋊C4—42no—
513C4⋊C4—81no—
514C4⋊C4—592no—
515C4⋊C4—1181no—
516C4⋊C4—1181no—
517C4⋊C4—1181no—
518C4⋊C4—1181no—
519C4⋊C4—1181no—
520C4⋊C4—4721no—
521C4⋊C4—4721no—
522C4⋊C4—2362no—
523C4⋊C4—2362no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—42no—
528C22⋊C4—81no—
529C22⋊C4—81no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—42no—
537C42—2362no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23C22×C22⋊D5911yescharacteristic
546Q8—22no—
547Q8—22no—
548Q8—41no—
549Q8—41no—
550Q8—1182no—
551Q8—1182no—
552Q8—4724no—
553Q8—2361no—
554Q8—2361no—
555D4—41no—
556D4—41no—
557D4—41no—
558D4—41no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
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—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—22no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—1182no—
586C2×C4—1182no—
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×C4—2361no—
597C2×C4#1427375712yes—
598C2×C4C2×D4×D5911yescharacteristic
599C2×C4#41×D5911yescharacteristic
600C2×C4#41×D5911yescharacteristic
601C2×C4#41×D5911yescharacteristic
602C2×C4#1427377211yescharacteristic
603C2×C4#1427377411yescharacteristic
604C8—1182no—
605C8—1182no—
606C8—1182no—
607C8—2361no—
608C22—81no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22#1683794811yescharacteristic
616C22D4×C22⋊D5911yescharacteristic
617C22#1683805911yescharacteristic
618C22C2×#1427375711yescharacteristic
619C22#1683820011yescharacteristic
620C22#1683821011yescharacteristic
621C22#1683821311yescharacteristic
622C4—22no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—42no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—4721no—
632C4—4721no—
633C4—2362no—
634C4—2362no—
635C4#1683791011yescharacteristic
636C4#1683802011yescharacteristic
637C4#1683805011yescharacteristic
638C4#1683808011yescharacteristic
639C2—161no—
640C2#2142806211yescharacteristic
641C2#2143470711yescharacteristic
642C2#2143480611yescharacteristic
643C2#2143572811yescharacteristic
644C2#2143576211yescharacteristic
645C2#2143683211yescharacteristic
646C2#2143685811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D59
commutator, cosocleC22×C4×C59C24
FrattiniC22×C4C23×D59
FittingC59×#2448C2
socleC23×C59C22×D4
prime radicalC22×D4×C59C23
hypercenter#2448D59
hypocenterC59#26846
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2448

Complements

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

NH ≅ G/Ntype
2. C2×#16837507C2semidirect
3. #21427935C2semidirect
5. #21425429C2semidirect
6. #21425418C2semidirect
10. #21424785C2semidirect
11. #21424775C2semidirect
13. #21424364C2semidirect
14. #21423965C2semidirect
384. C59#26846semidirect

Maximal subgroups

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

subgroupindex
C2×#168377612
C2×#168375072
#214279352
#214254702
#214254292
#214254182
#214251902
#214249492
#214247962
#214247852
#214247752
#214244542
#214243642
#214239652
C59×#24482
#2684659