Finite Group Databasebeta

← #27406559

Subgroups — #27406559

Order 13568.

982 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810724C211yesmaximal, characteristic
2C2×SD16×Dic53C211yesmaximal, characteristic
3#20277775C211yesmaximal, characteristic, complemented
4#20277625C211yesmaximal, characteristic
5#20275536C211yesmaximal, characteristic, complemented
6#20275485C211yesmaximal, characteristic, complemented
7#20275407C211yesmaximal, characteristic, complemented
8#20275393C211yesmaximal, characteristic, complemented
9#20275051C211yesmaximal, characteristic
10#20275051C211yesmaximal, characteristic
11#20274985C211yesmaximal, characteristic, complemented
12#20274971C211yesmaximal, characteristic, complemented
13#20274592C211yesmaximal, characteristic, complemented
14#20274540C211yesmaximal, characteristic
15C2×C53×#461C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×D4×Dic53C2211yescharacteristic
19C2×D4×Dic53C2211yescharacteristic
20#16810724—24no—
21SD16×Dic53—24no—
22C2×#13133151C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133138C2211yescharacteristic
29C2×#13133138C2211yescharacteristic
30C2×#13133128C2211yescharacteristic
31C2×#13133127C2211yescharacteristic
32C2×C8×Dic53C2211yescharacteristic
33#16810530C2211yescharacteristic
34#16810530C2211yescharacteristic
35#16810522—21no—
36#16810522C2211yescharacteristic
37#16810522C2211yescharacteristic
38#16810509—21no—
39#16810509—21no—
40#16810498—21no—
41#16810498—21no—
42#16810489C2211yescharacteristic
43#16810483—21no—
44C4⋊C4×Dic53—21no—
45C4×C4⋊Dic53C2211yescharacteristic
46C4×C4⋊Dic53C2211yescharacteristic
47C2×#13133124—21no—
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C22×SD16×C53C2211yescharacteristic
55C22×SD16×C53C2211yescharacteristic
56C2×C4⋊Q8×C53C2211yescharacteristic
57C2×C4×D4×C53C2211yescharacteristic
58C53×#461C418yescomplemented
59C2×C4⋊C8×C53C2211yescharacteristic
60C2×#103×C53C2211yescharacteristic
61C2×#103×C53C2211yescharacteristic
62C23×Dic53—21no—
63C23×Dic53—21no—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66Q8×Dic53—22no—
67Q8×Dic53—22no—
68Q8×Dic53—41no—
69Q8×Dic53—41no—
70D4×Dic53—22no—
71D4×Dic53—22no—
72D4×Dic53—41no—
73D4×Dic53—41no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53D411yescharacteristic
79C2×C4⋊Dic53D411yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C4×Dic53—21no—
87C2×C4×Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53C2311yescharacteristic
90C2×C4×Dic53C2311yescharacteristic
91#13133151—22no—
92#13133149—22no—
93#13133141—22no—
94#13133141—22no—
95#13133141—41no—
96#13133141—41no—
97#13133138—22no—
98#13133138D414yes—
99#13133128—22no—
100#13133127—22no—
101C8×Dic53—22no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124D411yescharacteristic
112#13133124D411yescharacteristic
113C22×Q8×C53C2311yescharacteristic
114C22×Q8×C53C2311yescharacteristic
115C22×D4×C53C2311yescharacteristic
116C23×C4×C53—21no—
117C2×SD16×C53—22no—
118C2×SD16×C53—22no—
119C2×SD16×C53—22no—
120C2×SD16×C53—22no—
121C2×SD16×C53C2×C414yes—
122C2×SD16×C53C2×C414yes—
123C22×C8×C53C2311yescharacteristic
124C22×C8×C53C2311yescharacteristic
125C4⋊Q8×C53—22no—
126C4⋊Q8×C53C2×C414yescomplemented
127C4×D4×C53—22no—
128C4×D4×C53C2×C414yes—
129C2×C4⋊C4×C53—21no—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53C2311yescharacteristic
132C2×C22⋊C4×C53—21no—
133C2×C42×C53C2311yescharacteristic
134C4⋊C8×C53C2×C414yescomplemented
135#103×C53C2×C414yescomplemented
136#103×C53C2×C414yescomplemented
137C24×C53—21no—
138C2×Q8×C53—21no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53C22×C412yes—
149C2×Q8×C53C22×C412yes—
150C2×D4×C53—21no—
151C2×D4×C53—21no—
152C2×D4×C53—21no—
153C2×D4×C53C22×C412yes—
154C2×D4×C53C2×D412yes—
155C2×D4×C53#4112yes—
156C22×C4×C53—21no—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—24no—
160C22×C4×C53—81no—
161C22×C4×C53—81no—
162C22×C4×C53C2×D411yescharacteristic
163C22×C4×C53C2×D411yescharacteristic
164C22×C4×C53C2411yescharacteristic
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—21no—
169C22×Dic53—42no—
170C22×Dic53—42no—
171C22×Dic53—42no—
172C22×Dic53—42no—
173C22×Dic53—81no—
174C22×Dic53—81no—
175C22×Dic53C2×D411yescharacteristic
176C22×Dic53C2×D411yescharacteristic
177C22×Dic53C2×D411yescharacteristic
178C22×Dic53C2×D411yescharacteristic
179SD16×C53—28no—
180SD16×C53—28no—
181C2×C8×C53—21no—
182C2×C8×C53—21no—
183C2×C8×C53—21no—
184C2×C8×C53—21no—
185C2×C8×C53C22×C412yes—
186C2×C8×C53C22×C412yes—
187C4⋊C4×C53—24no—
188C4⋊C4×C53—24no—
189C4⋊C4×C53—21no—
190C4⋊C4×C53C22×C412yes—
191C22⋊C4×C53—24no—
192C42×C53C22×C412yes—
193C42×C53#4112yes—
194C22⋊Dic53—42no—
195C22⋊Dic53—42no—
196C4⋊Dic53—21no—
197C4⋊Dic53—22no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—41no—
205C4⋊Dic53—42no—
206C4⋊Dic53—42no—
207C4⋊Dic53—81no—
208C4⋊Dic53—81no—
209C4⋊Dic53C2×D412yes—
210C4×Dic53—21no—
211C4×Dic53—21no—
212C4×Dic53—21no—
213C4×Dic53—21no—
214C4×Dic53—42no—
215C4×Dic53—42no—
216C4×Dic53—42no—
217C2×#88322—21no—
218C2×#88322—21no—
219C2×#88322—21no—
220C2×#88322—21no—
221C2×#88322—21no—
222C2×#88322C2×D412yes—
223C23×C53—22no—
224C23×C53—22no—
225C23×C53—22no—
226C23×C53—41no—
227C23×C53—41no—
228C23×C53—41no—
229C23×C53—41no—
230C23×C53C22×D411yescharacteristic
231Q8×C53—22no—
232Q8×C53—22no—
233Q8×C53—22no—
234Q8×C53—22no—
235Q8×C53—41no—
236Q8×C53—41no—
237Q8×C53—41no—
238Q8×C53—41no—
239D4×C53—22no—
240D4×C53—22no—
241D4×C53—22no—
242D4×C53C4×D414yes—
243C2×C4×C53—21no—
244C2×C4×C53—21no—
245C2×C4×C53—21no—
246C2×C4×C53—21no—
247C2×C4×C53—22no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—41no—
251C2×C4×C53—41no—
252C2×C4×C53—41no—
253C2×C4×C53—41no—
254C2×C4×C53—41no—
255C2×C4×C53—41no—
256C2×C4×C53—84no—
257C2×C4×C53C4×D412yes—
258C2×C4×C53C4×D412yes—
259C2×C4×C53C23×C411yescharacteristic
260C2×C4×C53C22×D411yescharacteristic
261C2×C4×C53C2×#4111yescharacteristic
262C2×C4×C53C2×#4111yescharacteristic
263C2×C4×C53H(5,𝔽2)11yescharacteristic
264C2×C4×C532−1+411yescharacteristic
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—22no—
268C2×Dic53—21no—
269C2×Dic53—21no—
270C2×Dic53—21no—
271C2×Dic53—21no—
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—41no—
281C2×Dic53—41no—
282C2×Dic53—41no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—84no—
289C2×Dic53—84no—
290C2×Dic53#13712yes—
291C8×C53—22no—
292C8×C53—22no—
293#88322—22no—
294#88322—41no—
295#23012—531nomaximal, Sylow
296C22×C53—24no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—42no—
300C22×C53—42no—
301C22×C53—42no—
302C22×C53—42no—
303C22×C53C2×C4×D411yescharacteristic
304C22×C53D4211yescharacteristic
305C22×C53#54811yescharacteristic
306C22×C53C2×#13711yescharacteristic
307C22×C53#57511yescharacteristic
308C22×C53#57711yescharacteristic
309C22×C53#57711yescharacteristic
310C4×C53—22no—
311C4×C53—22no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—42no—
315C4×C53C2×C4×D411yescharacteristic
316C4×C53#52011yescharacteristic
317C4×C53#54711yescharacteristic
318C4×C53#55311yescharacteristic
319Dic53—22no—
320Dic53—41no—
321Dic53—41no—
322Dic53—41no—
323Dic53—42no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327C2×#461D5311yescharacteristic
328C2×#440—531no—
329C2×C4×SD16—531no—
330#2365—531no—
331#2349—531no—
332#1960—531no—
333#1949—531no—
334#1935—531no—
335#1933—531no—
336#1877—531no—
337#1877—531no—
338#1865—531no—
339#1863—531no—
340#1771—531no—
341#1762—531no—
342C2×C53—44no—
343C2×C53#236111yescharacteristic
344C2×C53C4×#13711yescharacteristic
345C2×C53#294911yescharacteristic
346C2×C53#326611yescharacteristic
347C2×C53#326711yescharacteristic
348C2×C53#337211yescharacteristic
349C2×C53#337611yescharacteristic
350C22×SD16C2×D5311yescharacteristic
351C22×SD16C2×D5311yescharacteristic
352C2×C4⋊Q8C2×D5311yescharacteristic
353C2×C4×Q8—531no—
354C2×C4×Q8—531no—
355C2×C4×D4—531no—
356C2×C4×D4—531no—
357C2×C4×D4C2×D5311yescharacteristic
358#461Dic5318yescomplemented
359#440—1064no—
360C4×SD16—1064no—
361C2×#107—531no—
362C2×#106—531no—
363C2×C4⋊C8C2×D5311yescharacteristic
364C2×#103—531no—
365C2×#103—531no—
366C2×#103—531no—
367C2×#103—531no—
368C2×#103C2×D5311yescharacteristic
369C2×#103C2×D5311yescharacteristic
370C2×#102—531no—
371C2×#102—531no—
372C2×#97—531no—
373C2×#97—531no—
374C2×C4×C8—531no—
375#391—531no—
376#391—531no—
377#389—531no—
378#389—531no—
379#389—1061no—
380#387—1061no—
381#387—1061no—
382#385—1061no—
383#385—1061no—
384#382—531no—
385#380—1061no—
386C4×C4⋊C4—531no—
387C4×C4⋊C4—531no—
388C4×C4⋊C4—1061no—
389C2×#95—1061no—
390#340—531no—
391#340—531no—
392#340—531no—
393#340—531no—
394#336—531no—
395#336—531no—
396C53#2301211yescharacteristic, complemented, Sylow
397C22×Q8C22×D5311yescharacteristic
398C22×Q8C22×D5311yescharacteristic
399C22×D4C22×D5311yescharacteristic
400C23×C4—21no—
401C23×C4—1061no—
402C23×C4—1061no—
403C2×SD16—22no—
404C2×SD16—22no—
405C2×SD16—22no—
406C2×SD16—22no—
407C2×SD16C2×Dic5314yes—
408C2×SD16C2×Dic5314yes—
409C22×C8—531no—
410C22×C8—531no—
411C22×C8C22×D5311yescharacteristic
412C22×C8C22×D5311yescharacteristic
413C4⋊Q8—22no—
414C4⋊Q8C2×Dic5314yescomplemented
415C4×Q8—1062no—
416C4×Q8—1062no—
417C4×Q8—2121no—
418C4×Q8—2121no—
419C4×D4—22no—
420C4×D4—1062no—
421C4×D4—1062no—
422C4×D4—2121no—
423C4×D4—2121no—
424C4×D4C2×Dic5314yes—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—531no—
428C2×C4⋊C4—531no—
429C2×C4⋊C4—531no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—1061no—
434C2×C4⋊C4—1061no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4C22×D5311yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C42—531no—
442C2×C42—531no—
443C2×C42—1061no—
444C2×C42—1061no—
445C2×C42—1061no—
446C2×C42C22×D5311yescharacteristic
447#107—1062no—
448#106—1062no—
449C4⋊C8C2×Dic5314yescomplemented
450#103—1062no—
451#103—1062no—
452#103—2121no—
453#103—2121no—
454#103C2×Dic5314yescomplemented
455#103C2×Dic5314yescomplemented
456#102—534no—
457#102—1062no—
458#97—1062no—
459#97—1062no—
460C4×C8—1062no—
461#95—531no—
462#95—531no—
463#95—1061no—
464#95—1061no—
465#95—1061no—
466#95—1061no—
467#95—1061no—
468#95—1061no—
469#95—1061no—
470#95—1061no—
471#95—1061no—
472C24—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8C22×Dic5312yes—
484C2×Q8C22×Dic5312yes—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4D4×D5312yes—
489C2×D4#1171994712yes—
490C2×D4C22×Dic5312yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—24no—
495C22×C4—81no—
496C22×C4—81no—
497C22×C4—531no—
498C22×C4—531no—
499C22×C4—531no—
500C22×C4—531no—
501C22×C4—1061no—
502C22×C4—1061no—
503C22×C4—1061no—
504C22×C4—1061no—
505C22×C4—4241no—
506C22×C4—4241no—
507C22×C4—2122no—
508C22×C4—2122no—
509C22×C4—2122no—
510C22×C4—2122no—
511C22×C4D4×D5311yescharacteristic
512C22×C4D4×D5311yescharacteristic
513C22×C4C23×D5311yescharacteristic
514SD16—28no—
515SD16—28no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—532no—
521C2×C8—1061no—
522C2×C8—1061no—
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8C22×Dic5312yes—
527C2×C8C22×Dic5312yes—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—21no—
531C4⋊C4—532no—
532C4⋊C4—1062no—
533C4⋊C4—1061no—
534C4⋊C4—1061no—
535C4⋊C4—1061no—
536C4⋊C4—1061no—
537C4⋊C4—1061no—
538C4⋊C4—1061no—
539C4⋊C4—1061no—
540C4⋊C4—4241no—
541C4⋊C4—4241no—
542C4⋊C4—2121no—
543C4⋊C4—2122no—
544C4⋊C4—2122no—
545C4⋊C4C22×Dic5312yes—
546C22⋊C4—24no—
547C22⋊C4—2122no—
548C22⋊C4—2122no—
549C42—1061no—
550C42—1061no—
551C42—1061no—
552C42—1061no—
553C42—2122no—
554C42—2122no—
555C42—2122no—
556C42#1171994712yes—
557C42C22×Dic5312yes—
558C23—22no—
559C23—22no—
560C23—22no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23C2×D4×D5311yescharacteristic
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573Q8—41no—
574D4—22no—
575D4—22no—
576D4—22no—
577D4D4×Dic5314yes—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—84no—
592C2×C4—532no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1061no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—1061no—
602C2×C4—1061no—
603C2×C4—1061no—
604C2×C4—1062no—
605C2×C4—4244no—
606C2×C4—4244no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4—2121no—
615C2×C4—2121no—
616C2×C4—2121no—
617C2×C4—2121no—
618C2×C4D4×Dic5312yes—
619C2×C4D4×Dic5312yes—
620C2×C4C23×Dic5311yescharacteristic
621C2×C4C2×D4×D5311yescharacteristic
622C2×C4C2×#1171994711yescharacteristic
623C2×C4C2×#1171994711yescharacteristic
624C2×C4#1313326311yescharacteristic
625C2×C4#1313326411yescharacteristic
626C8—22no—
627C8—22no—
628C8—1062no—
629C8—2121no—
630C22—24no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22C2×D4×Dic5311yescharacteristic
638C22#1681125711yescharacteristic
639C22#1681127911yescharacteristic
640C22#137×D5311yescharacteristic
641C22#1681142211yescharacteristic
642C22#1681142911yescharacteristic
643C22#1681142911yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—1062no—
650C4—4241no—
651C4—4241no—
652C4—4241no—
653C4—2121no—
654C4—2121no—
655C4—2121no—
656C4—2122no—
657C4C2×D4×Dic5311yescharacteristic
658C4#1681110611yescharacteristic
659C4#1681127311yescharacteristic
660C4#1681132511yescharacteristic
661C2—44no—
662C2#2027773511yescharacteristic
663C2#137×Dic5311yescharacteristic
664C2#2028357511yescharacteristic
665C2#2028623111yescharacteristic
666C2#2028624611yescharacteristic
667C2#2028749111yescharacteristic
668C2#2028753511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#461C2
socleC23×C53C22×D4
prime radicalC22×D4×C53C23
cosocleC22×C4×C53C24
hypercenterC2×#461D53
hypocenterC53#23012
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
3. #20277775C2semidirect
5. #20275536C2semidirect
6. #20275485C2semidirect
7. #20275407C2semidirect
8. #20275393C2semidirect
11. #20274985C2semidirect
12. #20274971C2semidirect
13. #20274592C2semidirect
58. C53×#461C4semidirect
126. C4⋊Q8×C53C2×C4semidirect
134. C4⋊C8×C53C2×C4semidirect
135. #103×C53C2×C4semidirect
136. #103×C53C2×C4semidirect
358. #461Dic53semidirect
396. C53#23012semidirect
414. C4⋊Q8C2×Dic53semidirect
449. C4⋊C8C2×Dic53semidirect
454. #103C2×Dic53semidirect
455. #103C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107242
C2×SD16×Dic532
#202777752
#202776252
#202755362
#202754852
#202754072
#202753932
#202750512
#202749852
#202749712
#202745922
#202745402
C2×C53×#4612
#2301253