Finite Group Databasebeta

← #27312602

Subgroups — #27312602

Order 13568.

1614 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133215C211yesmaximal, characteristic
2C2×#16810530C211yesmaximal, characteristic
3#20272985C212yesmaximal
4#20272969C212yesmaximal
5#20272965C212yesmaximal
6#20272950C212yesmaximal
7#20272948C212yesmaximal
8#20272939C212yesmaximal
9C2×C53×#391C211yesmaximal, characteristic
10C23×Dic106C2211yescharacteristic
11C2×#13133215—24no—
12C2×#13133215—24no—
13C2×#13133215C418yescomplemented
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919C2212yes—
16C22×#11719919C2212yes—
17C22×C4×Dic53C2211yescharacteristic
18C22×C4×Dic53C2211yescharacteristic
19#16810530—24no—
20C2×#13133124C2212yes—
21C2×#13133124C2212yes—
22C2×#13133124C2212yes—
23C2×#13133124C2212yes—
24C2×#13133124C2212yes—
25C2×#13133124C2212yes—
26C2×#13133124C2212yes—
27C2×#13133124C2212yes—
28C2×#13133124C2212yes—
29C2×#13133124C2212yes—
30C23×Q8×C53C2211yescharacteristic
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×C42×C53C2211yescharacteristic
33C53×#391—24no—
34C2×#95×C53C2212yes—
35C2×#95×C53C2212yes—
36C22×Dic106—22no—
37C22×Dic106—22no—
38C22×Dic106—41no—
39C22×Dic106—41no—
40C22×Dic106—41no—
41C22×Dic106—41no—
42C22×Dic106C2×C414yes—
43C23×Dic53C2312yes—
44C23×Dic53C2312yes—
45C23×Dic53C2312yes—
46C23×Dic53C2312yes—
47#13133215—232no—
48C2×C4⋊Dic53—22no—
49C2×C4⋊Dic53—41no—
50C2×C4⋊Dic53—41no—
51C2×#11719919—24no—
52C2×#11719919—24no—
53C2×#11719919—24no—
54C2×#11719919—24no—
55C2×#11719919C2×C418yes—
56C2×#11719919C2×C418yes—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—22no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53—41no—
61C2×C4×Dic53—41no—
62C2×C4×Dic53C2×C414yes—
63#13133124—24no—
64#13133124—24no—
65#13133124—24no—
66#13133124—24no—
67#13133124—24no—
68#13133124—42no—
69#13133124—42no—
70#13133124—42no—
71#13133124—42no—
72#13133124Q818yescomplemented
73C22×Q8×C53—41no—
74C22×Q8×C53—41no—
75C22×Q8×C53—41no—
76C22×Q8×C53—41no—
77C22×Q8×C53C2×C414yes—
78C22×Q8×C53D414yes—
79C22×Q8×C53D414yes—
80C23×C4×C53C2312yes—
81C23×C4×C53C2312yes—
82C23×C4×C53C2311yescharacteristic
83C23×C4×C53C2311yescharacteristic
84C23×C4×C53C2311yescharacteristic
85C2×C4⋊C4×C53—22no—
86C2×C4⋊C4×C53—41no—
87C2×C4⋊C4×C53—41no—
88C2×C42×C53—22no—
89C2×C42×C53—41no—
90C2×C42×C53—41no—
91#95×C53—24no—
92#95×C53—24no—
93C24×C53C2411yescharacteristic
94C2×Q8×C53—44no—
95C2×Q8×C53—44no—
96C2×Q8×C53—44no—
97C2×Q8×C53—44no—
98C2×Q8×C53—44no—
99C2×Q8×C53—44no—
100C2×Q8×C53C22⋊C4116yes—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—21no—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
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×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53C22×C412yes—
127C22×C4×C53C22×C412yes—
128C22×C4×C53C22×C412yes—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D412yes—
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2×D412yes—
134C22×C4×C53C2×D412yes—
135C22×C4×C53C2×Q814yes—
136C22×Dic53—22no—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—22no—
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×Dic53—22no—
158C22×Dic53—22no—
159C22×Dic53C22×C414yes—
160C22×Dic53C22×C414yes—
161C22×Dic53C2×Q814yes—
162C22×Dic53C2×Q814yes—
163C22×Dic53#4114yes—
164C2×Dic106—28no—
165C2×Dic106—44no—
166C2×Dic106—44no—
167C2×Dic106—44no—
168C2×Dic106—44no—
169C2×Dic106—44no—
170C2×Dic106—44no—
171C4⋊C4×C53—44no—
172C42×C53—44no—
173C4⋊Dic53—44no—
174#11719919—216no—
175#11719919—216no—
176C4×Dic53—28no—
177C4×Dic53—44no—
178C23×C53C23×C411yescharacteristic
179C23×C53C22×D411yescharacteristic
180C23×C53C22×D411yescharacteristic
181C23×C53C22×Q812yes—
182C23×C53C2×#4112yes—
183C23×C53H(5,𝔽2)11yescharacteristic
184C23×C53H(5,𝔽2)11yescharacteristic
185C23×C53H(5,𝔽2)11yescharacteristic
186C23×C53H(5,𝔽2)11yescharacteristic
187C23×C532−1+412yes—
188C23×C532−1+412yes—
189Q8×C53—416no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53C2×C22⋊C414yes—
223C2×C4×C53C2×C22⋊C414yes—
224C2×C4×C53C2×C22⋊C414yes—
225C2×Dic53—24no—
226C2×Dic53—24no—
227C2×Dic53—24no—
228C2×Dic53—24no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—42no—
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—42no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53C4×Q818yes—
253Dic106—416no—
254#14450—531nomaximal, Sylow
255C22×C53C22×C22⋊C411yescharacteristic
256C22×C53C2×C4×Q812yes—
257C22×C53#51811yescharacteristic
258C22×C53#51811yescharacteristic
259C22×C53#51912yes—
260C22×C53#53411yescharacteristic
261C22×C53#53411yescharacteristic
262C22×C53#53411yescharacteristic
263C22×C53#53411yescharacteristic
264C22×C53#54212yes—
265C22×C53#54212yes—
266C22×C53#54412yes—
267C22×C53#54412yes—
268C22×C53#54812yes—
269C22×C53#54812yes—
270C22×C53D4×Q812yes—
271C22×C53D4×Q812yes—
272C22×C53#55412yes—
273C22×C53#55511yescharacteristic
274C22×C53Q8211yescharacteristic
275C22×C53#56311yescharacteristic
276C22×C53#56311yescharacteristic
277C22×C53#56312yes—
278C4×C53—24no—
279C4×C53—24no—
280C4×C53—24no—
281C4×C53—44no—
282C4×C53—82no—
283Dic53—28no—
284Dic53—44no—
285Dic53—44no—
286Dic53—82no—
287C22×C4⋊Q8—531no—
288C2×#391—531no—
289C2×#391D5311yescharacteristic
290#1446—532no—
291#1441—532no—
292#1441—532no—
293#1436—532no—
294#1436—532no—
295#1434—532no—
296C2×C53#230811yescharacteristic
297C2×C53#233112yes—
298C2×C53Q8×C22⋊C412yes—
299C2×C53#236911yescharacteristic
300C2×C53#238111yescharacteristic
301C2×C53#272912yes—
302C2×C53#272912yes—
303C2×C53#273111yescharacteristic
304C2×C53#273111yescharacteristic
305C2×C53#273411yescharacteristic
306C2×C53#273411yescharacteristic
307C23×Q8—531no—
308C23×Q8C2×D5311yescharacteristic
309C2×C4⋊Q8—538no—
310C2×C4⋊Q8—1064no—
311C2×C4⋊Q8—1064no—
312C22×C4⋊C4—532no—
313C22×C4⋊C4—532no—
314C22×C4⋊C4—531no—
315C22×C4⋊C4C2×D5311yescharacteristic
316C22×C42—531no—
317C22×C42—531no—
318C22×C42C2×D5311yescharacteristic
319#391—24no—
320#391—1064no—
321C2×#95—532no—
322C2×#95—532no—
323C2×#95—532no—
324C2×#95—532no—
325C2×#95—532no—
326C2×#95—532no—
327C2×#95—532no—
328C2×#95—532no—
329C2×#95—532no—
330C2×#95—532no—
331C2×#95C2×D5312yes—
332C2×#95C2×D5312yes—
333C53#1445011yescharacteristic, complemented, Sylow
334C22×Q8—41no—
335C22×Q8—41no—
336C22×Q8—41no—
337C22×Q8—41no—
338C22×Q8—534no—
339C22×Q8—1062no—
340C22×Q8—1062no—
341C22×Q8—2121no—
342C22×Q8—2121no—
343C22×Q8—2121no—
344C22×Q8—2121no—
345C22×Q8C4×D5314yes—
346C22×Q8D21214yes—
347C22×Q8C22⋊D5314yes—
348C23×C4—532no—
349C23×C4—532no—
350C23×C4—532no—
351C23×C4—532no—
352C23×C4C22×D5312yes—
353C23×C4C22×D5312yes—
354C23×C4C22×D5311yescharacteristic
355C23×C4C22×D5311yescharacteristic
356C23×C4C22×D5311yescharacteristic
357C4⋊Q8—10632no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—41no—
360C2×C4⋊C4—41no—
361C2×C4⋊C4—538no—
362C2×C4⋊C4—538no—
363C2×C4⋊C4—1064no—
364C2×C4⋊C4—1064no—
365C2×C4⋊C4—1064no—
366C2×C4⋊C4—1064no—
367C2×C4⋊C4—1062no—
368C2×C4⋊C4—2121no—
369C2×C4⋊C4—2121no—
370C2×C42—22no—
371C2×C42—41no—
372C2×C42—41no—
373C2×C42—534no—
374C2×C42—1062no—
375C2×C42—1062no—
376C2×C42—1062no—
377C2×C42—2121no—
378C2×C42—2121no—
379#95—24no—
380#95—24no—
381#95—538no—
382#95—1064no—
383#95—1064no—
384#95—1064no—
385#95—1064no—
386#95—1064no—
387#95—2122no—
388#95—2122no—
389#95—2122no—
390#95—2122no—
391C24C23×D5311yescharacteristic
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—1068no—
399C2×Q8—2124no—
400C2×Q8—2124no—
401C2×Q8—2124no—
402C2×Q8—2124no—
403C2×Q8—2124no—
404C2×Q8—2124no—
405C2×Q8#11719921116yes—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—534no—
432C22×C4—534no—
433C22×C4—534no—
434C22×C4—534no—
435C22×C4—534no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
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×C4—1062no—
452C22×C4—1062no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1062no—
459C22×C4C2×C4×D5312yes—
460C22×C4C2×C4×D5312yes—
461C22×C4C2×C4×D5312yes—
462C22×C4C2×D21212yes—
463C22×C4C2×D21212yes—
464C22×C4C2×D21212yes—
465C22×C4Q8×D5314yes—
466C22×C4C2×C22⋊D5312yes—
467C22×C4C2×C22⋊D5312yes—
468C22×C4C2×C22⋊D5312yes—
469C4⋊C4—44no—
470C4⋊C4—10616no—
471C4⋊C4—10616no—
472C4⋊C4—2124no—
473C42—44no—
474C42—1068no—
475C42—2124no—
476C23C22×C4×D5311yescharacteristic
477C23C22×C22⋊D5311yescharacteristic
478C23C22×D21211yescharacteristic
479C23C2×Q8×D5312yes—
480C23#41×D5312yes—
481C23#1313326211yescharacteristic
482C23#1313326211yescharacteristic
483C23#1313326211yescharacteristic
484C23#1313326211yescharacteristic
485C23#1313326412yes—
486C23#1313326512yes—
487Q8—416no—
488Q8—21216no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—538no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—1064no—
527C2×C4—1064no—
528C2×C4—1064no—
529C2×C4—1064no—
530C2×C4—1064no—
531C2×C4—1064no—
532C2×C4—1064no—
533C2×C4—1064no—
534C2×C4—2122no—
535C2×C4—2122no—
536C2×C4—2122no—
537C2×C4—2122no—
538C2×C4—2122no—
539C2×C4—2122no—
540C2×C4—2122no—
541C2×C4—2122no—
542C2×C4—2122no—
543C2×C4—2122no—
544C2×C4—2122no—
545C2×C4—2122no—
546C2×C4—2122no—
547C2×C4—2122no—
548C2×C4—2122no—
549C2×C4C2×#1171992114yes—
550C2×C4C2×#1171992114yes—
551C2×C4C2×#1171992114yes—
552C22C22×#1171992111yescharacteristic
553C22C4×Q8×D5312yes—
554C22#1681109611yescharacteristic
555C22#1681109611yescharacteristic
556C22#1681109912yes—
557C22#1681118211yescharacteristic
558C22#1681118211yescharacteristic
559C22#1681118511yescharacteristic
560C22#1681118511yescharacteristic
561C22#1681123712yes—
562C22#1681124312yes—
563C22#1681125012yes—
564C22#1681125212yes—
565C22#1681128012yes—
566C22#1681128512yes—
567C22Q8×C22⋊D5312yes—
568C22Q8×D21212yes—
569C22#1681132712yes—
570C22#1681133511yescharacteristic
571C22#1681135311yescharacteristic
572C22#1681137211yescharacteristic
573C22#1681137311yescharacteristic
574C22#1681137412yes—
575C4—24no—
576C4—24no—
577C4—24no—
578C4—44no—
579C4—82no—
580C4—1068no—
581C4—4242no—
582C4—2124no—
583C4—2124no—
584C2#2027729011yescharacteristic
585C2#2027745812yes—
586C2Q8×#1171992112yes—
587C2#2027780311yescharacteristic
588C2#2027789411yescharacteristic
589C2#2028169312yes—
590C2#2028169512yes—
591C2#2028172311yescharacteristic
592C2#2028172411yescharacteristic
593C2#2028174511yescharacteristic
594C2#2028174711yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutatorC23×C53C23×C4
FittingC2×C53×#391C2
socle, prime radical, cosocleC24×C53C24
hypercenterC2×#391D53
hypocenterC53#14450
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
13. C2×#13133215C4semidirect
72. #13133124Q8semidirect
333. C53#14450semidirect

Maximal subgroups

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

subgroupindex
C22×#131332152
C2×#168105302
#202729852
#202729692
#202729652
#202729502
#202729482
#202729392
C2×C53×#3912
#1445053