Finite Group Databasebeta

← #27309420

Subgroups — #27309420

Order 13568.

1758 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic53C211yesmaximal, characteristic
2C2×#16810509C212yesmaximal
3C2×C4⋊C4×Dic53C211yesmaximal, characteristic
4#20272986C211yesmaximal, characteristic
5#20272972C211yesmaximal, characteristic
6#20272968C211yesmaximal, characteristic
7#20272949C212yesmaximal
8#20272949C212yesmaximal
9#20272947C212yesmaximal
10#20272940C211yesmaximal, characteristic
11C2×C53×#395C211yesmaximal, characteristic
12C2×Q8×Dic53—28no—
13C2×Q8×Dic53—24no—
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4⋊Dic53C2212yes—
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4×Dic53C2212yes—
18C22×C4×Dic53C2211yescharacteristic
19C22×C4×Dic53C2211yescharacteristic
20C22×C4×Dic53C2211yescharacteristic
21#16810509—28no—
22C4⋊C4×Dic53—24no—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2211yescharacteristic
26C2×#13133124C2211yescharacteristic
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C23×Q8×C53C2211yescharacteristic
35C22×C4⋊C4×C53C2212yes—
36C22×C4⋊C4×C53C2211yescharacteristic
37C53×#395C418yescomplemented
38C2×#95×C53C2211yescharacteristic
39C2×#95×C53C2212yes—
40C23×Dic53C2312yes—
41C23×Dic53C2312yes—
42C23×Dic53C2311yescharacteristic
43C23×Dic53C2311yescharacteristic
44C23×Dic53C2311yescharacteristic
45C23×Dic53C2311yescharacteristic
46Q8×Dic53—232no—
47C2×C4⋊Dic53—22no—
48C2×C4⋊Dic53—24no—
49C2×C4⋊Dic53—24no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53—22no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53D418yes—
54C2×C4⋊Dic53Q814yescomplemented
55C2×C4×Dic53—24no—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—24no—
58C2×C4×Dic53—24no—
59C2×C4×Dic53—24no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53—22no—
63C2×C4×Dic53—22no—
64C2×C4×Dic53D414yes—
65#13133124—24no—
66#13133124—22no—
67#13133124—24no—
68#13133124—24no—
69#13133124—22no—
70#13133124—24no—
71#13133124—24no—
72#13133124—24no—
73#13133124—22no—
74#13133124—24no—
75#13133124—24no—
76C22×Q8×C53—24no—
77C22×Q8×C53—22no—
78C22×Q8×C53—22no—
79C22×Q8×C53—24no—
80C22×Q8×C53C2×C414yes—
81C23×C4×C53C2311yescharacteristic
82C23×C4×C53C2312yes—
83C23×C4×C53C2312yes—
84C23×C4×C53C2311yescharacteristic
85C23×C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53—24no—
87C2×C4⋊C4×C53—22no—
88C2×C4⋊C4×C53—22no—
89C2×C4⋊C4×C53—24no—
90C2×C4⋊C4×C53C2×C418yes—
91C2×C4⋊C4×C53C2×C414yes—
92#95×C53C2×C414yes—
93#95×C53C2×C418yes—
94C24×C53C2411yescharacteristic
95C2×Q8×C53—28no—
96C2×Q8×C53—216no—
97C2×Q8×C53—28no—
98C2×Q8×C53—28no—
99C2×Q8×C53—216no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53—21no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—22no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53C22×C414yes—
121C22×C4×C53C22×C412yes—
122C22×C4×C53C22×C414yes—
123C22×C4×C53C22×C412yes—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C2×D414yes—
126C22×C4×C53C2×D412yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53#4112yes—
129C22×C4×C53#4114yes—
130C22×C4×C53#4112yes—
131C22×Dic53—22no—
132C22×Dic53—21no—
133C22×Dic53—21no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—22no—
138C22×Dic53—21no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—22no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53C2×D414yes—
158C22×Dic53C2×D414yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D412yes—
161C22×Dic53C2×Q812yes—
162C22×Dic53C2×Q812yes—
163C22×Dic53#4112yes—
164C22×Dic53#4112yes—
165C4⋊C4×C53—28no—
166C4⋊C4×C53—216no—
167C4⋊Dic53—28no—
168C4⋊Dic53—216no—
169C4⋊Dic53—28no—
170C4×Dic53—28no—
171C4×Dic53—216no—
172C4×Dic53—28no—
173C4×Dic53—44no—
174C23×C53C23×C411yescharacteristic
175C23×C53C22×D411yescharacteristic
176C23×C53C22×D412yes—
177C23×C53C22×Q811yescharacteristic
178C23×C53C2×#4111yescharacteristic
179C23×C53C2×#4111yescharacteristic
180C23×C53C2×#4111yescharacteristic
181C23×C53C2×#4112yes—
182C23×C53C2×#4111yescharacteristic
183C23×C532−1+412yes—
184C23×C532−1+412yes—
185Q8×C53—232no—
186C2×C4×C53—24no—
187C2×C4×C53—24no—
188C2×C4×C53—24no—
189C2×C4×C53—22no—
190C2×C4×C53—22no—
191C2×C4×C53—24no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—24no—
196C2×C4×C53—24no—
197C2×C4×C53—24no—
198C2×C4×C53—24no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53C4×D414yes—
215C2×C4×C53C4×D418yes—
216C2×C4×C53C4×Q814yescomplemented
217C2×Dic53—22no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—22no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—24no—
224C2×Dic53—24no—
225C2×Dic53—24no—
226C2×Dic53—22no—
227C2×Dic53—22no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—42no—
235C2×Dic53—41no—
236C2×Dic53—42no—
237C2×Dic53—41no—
238C2×Dic53—42no—
239C2×Dic53—42no—
240C2×Dic53—42no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53#12014yes—
250C2×Dic53C22⋊Q814yes—
251#14174—531nomaximal, Sylow
252C22×C53C2×C4×D411yescharacteristic
253C22×C53C2×C4×D412yes—
254C22×C53C2×C4×Q811yescharacteristic
255C22×C53C4×#4111yescharacteristic
256C22×C53#51912yes—
257C22×C53C2×#12011yescharacteristic
258C22×C53C2×C22⋊Q811yescharacteristic
259C22×C53#52511yescharacteristic
260C22×C53#52512yes—
261C22×C53#52511yescharacteristic
262C22×C53#52911yescharacteristic
263C22×C53#53311yescharacteristic
264C22×C53#53611yescharacteristic
265C22×C53#53611yescharacteristic
266C22×C53#54112yes—
267C22×C53#54712yes—
268C22×C53#54712yes—
269C22×C53#54812yes—
270C22×C53D4×Q812yes—
271C22×C53#55212yes—
272C22×C53#55612yes—
273C22×C53#55612yes—
274C22×C53#55712yes—
275C4×C53—28no—
276C4×C53—24no—
277C4×C53—24no—
278C4×C53—44no—
279C4×C53—42no—
280Dic53—24no—
281Dic53—24no—
282Dic53—42no—
283Dic53—44no—
284Dic53—42no—
285Dic53—82no—
286C22×C4×Q8—531no—
287C2×#395D5311yescharacteristic
288C2×#387—532no—
289C2×C4×C4⋊C4—531no—
290#1446—531no—
291#1442—531no—
292#1441—531no—
293#1436—532no—
294#1436—532no—
295#1436—532no—
296#1434—531no—
297C2×C53C4×#12011yescharacteristic
298C2×C53C4×C22⋊Q811yescharacteristic
299C2×C53#232711yescharacteristic
300C2×C53#236212yes—
301C2×C53#236512yes—
302C2×C53#240611yescharacteristic
303C2×C53#240911yescharacteristic
304C2×C53#246212yes—
305C2×C53#246912yes—
306C2×C53#258111yescharacteristic
307C2×C53#259911yescharacteristic
308C23×Q8C2×D5311yescharacteristic
309C2×C4×Q8—1064no—
310C2×C4×Q8—1068no—
311C22×C4⋊C4—532no—
312C22×C4⋊C4—531no—
313C22×C4⋊C4—531no—
314C22×C4⋊C4C2×D5312yes—
315C22×C4⋊C4C2×D5311yescharacteristic
316C22×C42—531no—
317C22×C42—532no—
318C22×C42—531no—
319C22×C42—531no—
320#395Dic5318yescomplemented
321#387—1068no—
322C4×C4⋊C4—1064no—
323C2×#95—532no—
324C2×#95—532no—
325C2×#95—531no—
326C2×#95—532no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—531no—
330C2×#95—532no—
331C2×#95—532no—
332C2×#95—532no—
333C2×#95—531no—
334C2×#95C2×D5311yescharacteristic
335C2×#95C2×D5312yes—
336C53#1417411yescharacteristic, complemented, Sylow
337C22×Q8—22no—
338C22×Q8—24no—
339C22×Q8—24no—
340C22×Q8—22no—
341C22×Q8C2×Dic5314yes—
342C23×C4—531no—
343C23×C4—531no—
344C23×C4—532no—
345C23×C4—531no—
346C23×C4—531no—
347C23×C4—532no—
348C23×C4C22×D5311yescharacteristic
349C23×C4C22×D5312yes—
350C23×C4C22×D5311yescharacteristic
351C23×C4C22×D5312yes—
352C23×C4C22×D5311yescharacteristic
353C4×Q8—10632no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—24no—
358C2×C4⋊C4—538no—
359C2×C4⋊C4—534no—
360C2×C4⋊C4—1064no—
361C2×C4⋊C4—1064no—
362C2×C4⋊C4—1064no—
363C2×C4⋊C4—1062no—
364C2×C4⋊C4—1062no—
365C2×C4⋊C4—1062no—
366C2×C4⋊C4C2×Dic5318yes—
367C2×C4⋊C4C2×Dic5314yes—
368C2×C42—534no—
369C2×C42—1064no—
370C2×C42—1064no—
371C2×C42—1064no—
372C2×C42—1062no—
373C2×C42—1062no—
374C2×C42—1064no—
375C2×C42—1062no—
376C2×C42—1062no—
377C2×C42—1064no—
378#95—1064no—
379#95—1064no—
380#95—1062no—
381#95—1064no—
382#95—1064no—
383#95—1064no—
384#95—1062no—
385#95—1064no—
386#95—1064no—
387#95—1062no—
388#95—1064no—
389#95C2×Dic5314yes—
390#95C2×Dic5318yes—
391C24C23×D5311yescharacteristic
392C2×Q8—28no—
393C2×Q8—216no—
394C2×Q8—28no—
395C2×Q8—28no—
396C2×Q8—216no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—532no—
418C22×C4—534no—
419C22×C4—532no—
420C22×C4—532no—
421C22×C4—532no—
422C22×C4—534no—
423C22×C4—532no—
424C22×C4—532no—
425C22×C4—1062no—
426C22×C4—1061no—
427C22×C4—1061no—
428C22×C4—1062no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1061no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1061no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—1062no—
446C22×C4—1062no—
447C22×C4—1062no—
448C22×C4—1062no—
449C22×C4—1062no—
450C22×C4—1062no—
451C22×C4D4×D5312yes—
452C22×C4D4×D5314yes—
453C22×C4#1171994714yes—
454C22×C4#1171994712yes—
455C22×C4Q8×D5312yes—
456C22×C4#1171994912yes—
457C22×C4C22×Dic5314yes—
458C22×C4C22×Dic5312yes—
459C22×C4C22×Dic5314yes—
460C22×C4C22×Dic5312yes—
461C22×C4C22×Dic5312yes—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—10616no—
465C4⋊C4—1068no—
466C4⋊C4—1068no—
467C42—1068no—
468C42—10616no—
469C42—1068no—
470C42—2124no—
471C23C23×Dic5311yescharacteristic
472C23C2×D4×D5312yes—
473C23C2×D4×D5311yescharacteristic
474C23C2×Q8×D5311yescharacteristic
475C23#41×D5311yescharacteristic
476C23#41×D5311yescharacteristic
477C23C2×#1171994712yes—
478C23C2×#1171994711yescharacteristic
479C23C2×#1171994911yescharacteristic
480C23#1313326512yes—
481C23#1313326512yes—
482Q8—232no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—534no—
512C2×C4—534no—
513C2×C4—1064no—
514C2×C4—1064no—
515C2×C4—1064no—
516C2×C4—1064no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1062no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1062no—
527C2×C4—1062no—
528C2×C4—2122no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4—2122no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2121no—
540C2×C4—2121no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2121no—
544C2×C4—2121no—
545C2×C4D4×Dic5314yes—
546C2×C4D4×Dic5318yes—
547C2×C4Q8×Dic5314yescomplemented
548C22C2×D4×Dic5311yescharacteristic
549C22C2×D4×Dic5312yes—
550C22C2×Q8×Dic5311yescharacteristic
551C22#41×Dic5311yescharacteristic
552C22#1681110112yes—
553C22#120×D5311yescharacteristic
554C22C22⋊Q8×D5311yescharacteristic
555C22#1681112912yes—
556C22#1681112911yescharacteristic
557C22#1681113611yescharacteristic
558C22#1681115911yescharacteristic
559C22#1681117511yescharacteristic
560C22#1681119811yescharacteristic
561C22#1681119811yescharacteristic
562C22#1681123412yes—
563C22#1681127512yes—
564C22#1681127612yes—
565C22#1681128312yes—
566C22#1681128812yes—
567C22#1681130712yes—
568C22#1681134412yes—
569C22#1681134712yes—
570C22#1681134812yes—
571C4—24no—
572C4—28no—
573C4—24no—
574C4—42no—
575C4—44no—
576C4—1064no—
577C4—1064no—
578C4—4242no—
579C4—2124no—
580C4—2122no—
581C4—2122no—
582C2#120×Dic5311yescharacteristic
583C2C22⋊Q8×Dic5311yescharacteristic
584C2#2027743511yescharacteristic
585C2#2027774612yes—
586C2#2027777512yes—
587C2#2027805111yescharacteristic
588C2#2027807511yescharacteristic
589C2#2027869012yes—
590C2#2027878912yes—
591C2#2028007611yescharacteristic
592C2#2028028711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#395C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×#395D53
hypocenterC53#14174
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
37. C53×#395C4semidirect
54. C2×C4⋊Dic53Q8semidirect
216. C2×C4×C53C4×Q8semidirect
320. #395Dic53semidirect
336. C53#14174semidirect
547. C2×C4Q8×Dic53semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic532
C2×#168105092
C2×C4⋊C4×Dic532
#202729862
#202729722
#202729682
#202729492
#202729472
#202729402
C2×C53×#3952
#1417453