Finite Group Databasebeta

← #27290708

Subgroups — #27290708

Order 13568.

1658 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133132C211yesmaximal, characteristic
2#20277293C211yesmaximal, characteristic
3C2×#16810486C212yesmaximal, complemented
4C2×#16810389C212yesmaximal, complemented
5C22×#123×C53C211yesmaximal, characteristic
6C22×C22⋊Dic53C2212yes—
7C22×C22⋊Dic53C2212yes—
8C2×#13133132—24no—
9C2×#13133132—22no—
10#16810503—24no—
11#16810486—24no—
12#16810486—21no—
13#16810486—24no—
14#16810479—21no—
15C2×#13133124—24no—
16#16810389—24no—
17C23×D4×C53C2211yescharacteristic
18C24×C4×C53C2211yescharacteristic
19C2×#123×C53—22no—
20C2×#123×C53—24no—
21C2×#123×C53—24no—
22C2×#123×C53C414yescomplemented
23C22×C4⋊C4×C53—21no—
24C22×C22⋊C4×C53—21no—
25C22×C22⋊C4×C53C2211yescharacteristic
26C23×Dic53—22no—
27C23×Dic53D414yes—
28C2×C22⋊Dic53—24no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—22no—
36C2×C22⋊Dic53—22no—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—22no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53D414yes—
41#13133132—28no—
42#13133124—24no—
43#13133124—216no—
44C25×C53D411yescharacteristic
45C25×C53C2311yescharacteristic
46C22×D4×C53—22no—
47C22×D4×C53—22no—
48C22×D4×C53—22no—
49C22×D4×C53—22no—
50C22×D4×C53—22no—
51C22×D4×C53—22no—
52C22×D4×C53C2×C412yes—
53C22×D4×C53D412yes—
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53—21no—
57C23×C4×C53—22no—
58C23×C4×C53—22no—
59C23×C4×C53—22no—
60C23×C4×C53—22no—
61C23×C4×C53—22no—
62C23×C4×C53C2×C412yes—
63C23×C4×C53D411yescharacteristic
64C23×C4×C53D414yes—
65#123×C53—216no—
66#123×C53—28no—
67#123×C53—28no—
68C2×C4⋊C4×C53—28no—
69C2×C4⋊C4×C53—24no—
70C2×C22⋊C4×C53—28no—
71C2×C22⋊C4×C53—24no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—22no—
74C2×C22⋊C4×C53C2×C412yes—
75C2×C22⋊C4×C53D414yes—
76C24×C53—22no—
77C24×C53—21no—
78C24×C53—22no—
79C24×C53—21no—
80C24×C53—44no—
81C24×C53—44no—
82C24×C53C22⋊C412yes—
83C24×C53C22×C411yescharacteristic
84C24×C53C2×D411yescharacteristic
85C24×C53C2×D412yes—
86C24×C53C2×D411yescharacteristic
87C24×C53C2×D412yes—
88C24×C53C2×D412yes—
89C24×C53C2×D412yes—
90C24×C53#4112yes—
91C24×C53#4112yes—
92C2×D4×C53—22no—
93C2×D4×C53—22no—
94C2×D4×C53—24no—
95C2×D4×C53—24no—
96C2×D4×C53—24no—
97C2×D4×C53—22no—
98C2×D4×C53—24no—
99C2×D4×C53—22no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—42no—
103C2×D4×C53—42no—
104C2×D4×C53—42no—
105C2×D4×C53—42no—
106C2×D4×C53—42no—
107C2×D4×C53—42no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—24no—
112C22×C4×C53—24no—
113C22×C4×C53—24no—
114C22×C4×C53—24no—
115C22×C4×C53—22no—
116C22×C4×C53—24no—
117C22×C4×C53—24no—
118C22×C4×C53—21no—
119C22×C4×C53—24no—
120C22×C4×C53—24no—
121C22×C4×C53—22no—
122C22×C4×C53—24no—
123C22×C4×C53—21no—
124C22×C4×C53—44no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53—42no—
130C22×C4×C53—42no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53—42no—
134C22×C4×C53—42no—
135C22×C4×C53C22⋊C412yes—
136C22×C4×C53C22⋊C414yescomplemented
137C22×Dic53—24no—
138C22×Dic53—24no—
139C22×Dic53—24no—
140C22×Dic53—24no—
141C22×Dic53—48no—
142C22×Dic53—48no—
143C22×Dic53—42no—
144C22×Dic53—42no—
145C4⋊C4×C53—216no—
146C22⋊C4×C53—216no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—24no—
149C22⋊C4×C53C22⋊C414yescomplemented
150C22⋊Dic53—24no—
151C22⋊Dic53—24no—
152C22⋊Dic53—42no—
153C22⋊Dic53—42no—
154C22⋊Dic53—42no—
155C22⋊Dic53—42no—
156C22⋊Dic53—42no—
157C22⋊Dic53—42no—
158C22⋊Dic53—84no—
159C23×C53—28no—
160C23×C53—24no—
161C23×C53—22no—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—42no—
165C23×C53—44no—
166C23×C53—42no—
167C23×C53—44no—
168C23×C53—44no—
169C23×C53—44no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—48no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—44no—
176C23×C53—82no—
177C23×C53—84no—
178C23×C53C2×C22⋊C411yescharacteristic
179C23×C53C2×C22⋊C411yescharacteristic
180C23×C53C2×C22⋊C411yescharacteristic
181C23×C53C4×D412yes—
182C23×C53C4×D412yes—
183C23×C53#12012yes—
184C23×C53#12011yescharacteristic
185C23×C53#12011yescharacteristic
186C23×C53#12012yes—
187C23×C53#12012yes—
188C23×C53#12112yes—
189C23×C53#12112yes—
190C23×C53#12112yes—
191C23×C53#12112yes—
192C23×C53#12112yes—
193C23×C53#12112yes—
194C23×C53#12311yescharacteristic
195C23×C53#12312yes—
196C23×C53#12311yescharacteristic
197C23×C53#12412yes—
198C23×C53#12712yes—
199D4×C53—44no—
200D4×C53—44no—
201D4×C53—44no—
202D4×C53—44no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—22no—
206C2×C4×C53—28no—
207C2×C4×C53—22no—
208C2×C4×C53—22no—
209C2×C4×C53—44no—
210C2×C4×C53—48no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—44no—
214C2×C4×C53—44no—
215C2×C4×C53—44no—
216C2×C4×C53—42no—
217C2×C4×C53—44no—
218C2×C4×C53—44no—
219C2×C4×C53—42no—
220C2×Dic53—48no—
221C2×Dic53—48no—
222C2×Dic53—44no—
223C2×Dic53—44no—
224C2×Dic53—48no—
225C2×Dic53—84no—
226C2×Dic53—84no—
227C2×Dic53—84no—
228#10992—531nomaximal, Sylow
229C22×C53—24no—
230C22×C53—22no—
231C22×C53—22no—
232C22×C53—24no—
233C22×C53—24no—
234C22×C53—24no—
235C22×C53—24no—
236C22×C53—24no—
237C22×C53—24no—
238C22×C53—24no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—42no—
242C22×C53—41no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—41no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—44no—
250C22×C53—44no—
251C22×C53—42no—
252C22×C53—82no—
253C22×C53—84no—
254C22×C53—84no—
255C22×C53#37911yescharacteristic
256C22×C53#38612yes—
257C22×C53#38611yescharacteristic
258C22×C53#38611yescharacteristic
259C22×C53#39012yes—
260C22×C53#39212yes—
261C22×C53#39212yes—
262C22×C53#39412yes—
263C22×C53#39412yes—
264C22×C53#45814yes—
265C4×C53—44no—
266C4×C53—44no—
267C4×C53—42no—
268Dic53—84no—
269Dic53—84no—
270C22×#123D5311yescharacteristic
271C22×#99—531no—
272#2308—531no—
273C2×#381—532no—
274C2×#342—532no—
275C2×C53—22no—
276C2×C53—21no—
277C2×C53—21no—
278C2×C53—22no—
279C2×C53—22no—
280C2×C53—22no—
281C2×C53—22no—
282C2×C53—82no—
283C2×C53#143811yescharacteristic
284C2×C53#189012yes—
285C2×C53#202214yes—
286C23×D4C2×D5311yescharacteristic
287C24×C4C2×D5311yescharacteristic
288C2×#123—22no—
289C2×#123—24no—
290C2×#123—24no—
291C2×#123Dic5314yescomplemented
292C22×C4⋊C4—21no—
293C22×C22⋊C4—21no—
294C22×C22⋊C4—532no—
295C22×C22⋊C4—532no—
296C22×C22⋊C4C2×D5311yescharacteristic
297C2×#99—1064no—
298C2×#99—1062no—
299#386—1064no—
300#381—1064no—
301#381—1061no—
302#381—1064no—
303#379—1061no—
304C2×#95—1064no—
305#342—1064no—
306C53#1099211yescharacteristic, complemented, Sylow
307C25C22⋊D5311yescharacteristic
308C25C22×D5311yescharacteristic
309C22×D4—22no—
310C22×D4—22no—
311C22×D4—22no—
312C22×D4—22no—
313C22×D4—22no—
314C22×D4—22no—
315C22×D4C2×Dic5312yes—
316C22×D4C22⋊D5312yes—
317C23×C4—22no—
318C23×C4—22no—
319C23×C4—22no—
320C23×C4—22no—
321C23×C4—21no—
322C23×C4—21no—
323C23×C4—21no—
324C23×C4—22no—
325C23×C4—534no—
326C23×C4—1062no—
327C23×C4C2×Dic5312yes—
328C23×C4C22⋊D5311yescharacteristic
329C23×C4C22⋊D5314yes—
330#123—216no—
331#123—28no—
332#123—28no—
333C2×C4⋊C4—28no—
334C2×C4⋊C4—24no—
335C2×C22⋊C4—28no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—534no—
340C2×C22⋊C4—1062no—
341C2×C22⋊C4—1062no—
342C2×C22⋊C4—1062no—
343C2×C22⋊C4—1062no—
344C2×C22⋊C4—1064no—
345C2×C22⋊C4—1062no—
346C2×C22⋊C4—1062no—
347C2×C22⋊C4—1062no—
348C2×C22⋊C4—1062no—
349C2×C22⋊C4—1062no—
350C2×C22⋊C4—1062no—
351C2×C22⋊C4—1062no—
352C2×C22⋊C4C2×Dic5312yes—
353C2×C22⋊C4C22⋊D5314yes—
354#99—1068no—
355#95—10616no—
356#95—1064no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—44no—
362C24—44no—
363C24C22⋊Dic5312yes—
364C24D4×D5312yes—
365C24D4×D5312yes—
366C24#1171994712yes—
367C24#1171994712yes—
368C24C22×Dic5311yescharacteristic
369C24C2×C22⋊D5311yescharacteristic
370C24C2×C22⋊D5311yescharacteristic
371C24C2×C22⋊D5312yes—
372C24C2×C22⋊D5312yes—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—24no—
376C2×D4—24no—
377C2×D4—22no—
378C2×D4—24no—
379C2×D4—24no—
380C2×D4—22no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—42no—
386C2×D4—42no—
387C2×D4—42no—
388C2×D4—42no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—44no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—42no—
415C22×C4—42no—
416C22×C4—1064no—
417C22×C4—1064no—
418C22×C4—1064no—
419C22×C4—1064no—
420C22×C4—2128no—
421C22×C4—2122no—
422C22×C4—2122no—
423C22×C4—2128no—
424C22×C4C22⋊Dic5312yes—
425C22×C4C22⋊Dic5314yescomplemented
426C4⋊C4—216no—
427C22⋊C4—216no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—1064no—
431C22⋊C4—1064no—
432C22⋊C4—4244no—
433C22⋊C4—2122no—
434C22⋊C4—2122no—
435C22⋊C4—2122no—
436C22⋊C4—2122no—
437C22⋊C4—2122no—
438C22⋊C4—2122no—
439C22⋊C4C22⋊Dic5314yescomplemented
440C23—24no—
441C23—28no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—42no—
452C23—44no—
453C23—48no—
454C23—42no—
455C23—42no—
456C23—44no—
457C23—82no—
458C23—84no—
459C23C2×C22⋊Dic5311yescharacteristic
460C23C2×C22⋊Dic5311yescharacteristic
461C23C2×C22⋊Dic5311yescharacteristic
462C23D4×Dic5312yes—
463C23D4×Dic5312yes—
464C23#1313318512yes—
465C23#1313318511yescharacteristic
466C23#1313318511yescharacteristic
467C23#1313318612yes—
468C23#1313318612yes—
469C23#1313318812yes—
470C23#1313318812yes—
471C23#1313318912yes—
472C23#1313318912yes—
473C23#1313318912yes—
474C23#1313318912yes—
475C23#1313319811yescharacteristic
476C23#1313319812yes—
477C23#1313319811yescharacteristic
478C23#1313320412yes—
479C23#1313321312yes—
480D4—44no—
481D4—44no—
482D4—44no—
483D4—44no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—4244no—
502C2×C4—4244no—
503C2×C4—4244no—
504C2×C4—2128no—
505C2×C4—2128no—
506C2×C4—2124no—
507C2×C4—2124no—
508C2×C4—2128no—
509C22—24no—
510C22—24no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—22no—
516C22—24no—
517C22—22no—
518C22—24no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—41no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—41no—
527C22—42no—
528C22—42no—
529C22—44no—
530C22—42no—
531C22—44no—
532C22—82no—
533C22—84no—
534C22—84no—
535C22#1681047911yescharacteristic
536C22#1681050312yes—
537C22#1681050311yescharacteristic
538C22#1681050311yescharacteristic
539C22#1681052612yes—
540C22#1681053112yes—
541C22#1681053112yes—
542C22#1681053712yes—
543C22#1681053712yes—
544C22#1681081214yes—
545C4—44no—
546C4—44no—
547C4—42no—
548C4—4244no—
549C4—4244no—
550C2—21no—
551C2—22no—
552C2—21no—
553C2—22no—
554C2—22no—
555C2—22no—
556C2—22no—
557C2—82no—
558C2#2027295611yescharacteristic
559C2#2027512912yes—
560C2#2027584314yes—
subgroupG/H
centerC23#13133185
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC22×#123×C53C2
socleC23×C53#120
prime radicalC23×D4×C53C22
cosocleC25×C53C23
hypercenterC22×#123D53
hypocenterC53#10992
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
3. C2×#16810486C2semidirect
4. C2×#16810389C2semidirect
22. C2×#123×C53C4semidirect
136. C22×C4×C53C22⋊C4semidirect
149. C22⋊C4×C53C22⋊C4semidirect
291. C2×#123Dic53semidirect
306. C53#10992semidirect
425. C22×C4C22⋊Dic53semidirect
439. C22⋊C4C22⋊Dic53semidirect

Maximal subgroups

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

subgroupindex
C22×#131331322
#202772932
C2×#168104862
C2×#168103892
C22×#123×C532
#1099253