Finite Group Databasebeta

← #27301622

Subgroups — #27301622

Order 13568.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133132C211yesmaximal, characteristic
2C2×#16810418C212yesmaximal, complemented
3#20274630C211yesmaximal, characteristic
4#20272840C212yesmaximal, complemented
5C53×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×#13133132—24no—
8C2×#13133132—21no—
9C2×#13133132—21no—
10C2×#13133132C2212yes—
11C2×#13133132C2212yes—
12#16810479—21no—
13#16810418—24no—
14C23×D4×C53C2211yescharacteristic
15C2×#124×C53C411yescharacteristic, complemented
16C2×#124×C53C2211yescharacteristic
17C2×C22⋊Q8×C53C411yescharacteristic, complemented
18C53×#394—21no—
19C53×#394—21no—
20C53×#390C411yescharacteristic, complemented
21C53×#390C2211yescharacteristic
22C53×#379—21no—
23C53×#379—21no—
24C4×C22⋊C4×C53C411yescharacteristic, complemented
25C23×Dic53—41no—
26C2×C22⋊Dic53—24no—
27C2×C22⋊Dic53—22no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—42no—
30C2×C22⋊Dic53—41no—
31C2×C22⋊Dic53—41no—
32C2×C22⋊Dic53—41no—
33#13133132—24no—
34#13133132—22no—
35#13133132—22no—
36#13133132—22no—
37#13133132—42no—
38#13133132D414yes—
39C25×C53D411yescharacteristic
40C25×C53D411yescharacteristic
41C22×Q8×C53C2×C411yescharacteristic, complemented
42C22×D4×C53—22no—
43C22×D4×C53—22no—
44C22×D4×C53—41no—
45C22×D4×C53—41no—
46C22×D4×C53—42no—
47C22×D4×C53C2×C411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C22×D4×C53D411yescharacteristic
50C22×D4×C53C2311yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52#124×C53—22no—
53#124×C53—22no—
54#124×C53—42no—
55C22⋊Q8×C53—41no—
56C22⋊Q8×C53—41no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—21no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C42×C53C2×C411yescharacteristic, complemented
70C2×C42×C53C2×C411yescharacteristic, complemented
71#95×C53—21no—
72C24×C53—21no—
73C24×C53—22no—
74C24×C53—21no—
75C24×C53—21no—
76C24×C53—21no—
77C24×C53—21no—
78C24×C53—21no—
79C24×C53—22no—
80C24×C53—41no—
81C24×C53—42no—
82C24×C53—42no—
83C24×C53—81no—
84C24×C53—81no—
85C24×C53C22⋊C411yescharacteristic
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22⋊C411yescharacteristic
88C24×C53C2×D411yescharacteristic
89C24×C53C2×D411yescharacteristic
90C2×Q8×C53—22no—
91C2×Q8×C53—41no—
92C2×Q8×C53—41no—
93C2×D4×C53—22no—
94C2×D4×C53—22no—
95C2×D4×C53—21no—
96C2×D4×C53—22no—
97C2×D4×C53—21no—
98C2×D4×C53—22no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C2×D4×C53—41no—
103C2×D4×C53—41no—
104C2×D4×C53—41no—
105C2×D4×C53—44no—
106C2×D4×C53—44no—
107C2×D4×C53—44no—
108C2×D4×C53—44no—
109C2×D4×C53C2×D412yes—
110C2×D4×C53#4112yes—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—41no—
119C22×C4×C53C22⋊C411yescharacteristic
120C22×C4×C53C22×C411yescharacteristic
121C22×Dic53—44no—
122C22×Dic53—42no—
123C22×Dic53—42no—
124C22×Dic53—82no—
125C22×Dic53—82no—
126C4⋊C4×C53—42no—
127C4⋊C4×C53—41no—
128C22⋊C4×C53—42no—
129C22⋊C4×C53—42no—
130C22⋊C4×C53—41no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—42no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—42no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—81no—
137C22⋊C4×C53—81no—
138C22⋊C4×C53—81no—
139C22⋊C4×C53—81no—
140C42×C53—22no—
141C42×C53—41no—
142C22⋊Dic53—24no—
143C22⋊Dic53—44no—
144C22⋊Dic53—42no—
145C22⋊Dic53—42no—
146C22⋊Dic53—42no—
147C22⋊Dic53—82no—
148C22⋊Dic53—82no—
149C23×C53—22no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—22no—
153C23×C53—22no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53—21no—
165C23×C53—41no—
166C23×C53—42no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—41no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—81no—
181C23×C53—81no—
182C23×C53—81no—
183C23×C53—81no—
184C23×C53—84no—
185C23×C53—82no—
186C23×C53—82no—
187C23×C53—84no—
188C23×C53—84no—
189C23×C53#9911yescharacteristic
190C23×C53#9911yescharacteristic
191C23×C53#9912yes—
192C23×C53C2×C22⋊C411yescharacteristic
193C23×C53#12011yescharacteristic
194C23×C53#12112yes—
195C23×C53#12311yescharacteristic
196Q8×C53—82no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—44no—
200D4×C53—44no—
201D4×C53—42no—
202D4×C53—42no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53C4×D412yes—
223C2×Dic53—44no—
224C2×Dic53—88no—
225C2×Dic53—82no—
226C2×Dic53—82no—
227C2×Dic53—82no—
228C2×Dic53—82no—
229C2×Dic53—82no—
230#13090—531nomaximal, Sylow
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—22no—
239C22×C53—21no—
240C22×C53—22no—
241C22×C53—21no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—41no—
248C22×C53—44no—
249C22×C53—44no—
250C22×C53—42no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—81no—
263C22×C53—84no—
264C22×C53—81no—
265C22×C53—82no—
266C22×C53—82no—
267C22×C53—84no—
268C22×C53#35412yescomplemented
269C22×C53#38611yescharacteristic
270C22×C53C2×#9911yescharacteristic
271C22×C53C2×#9911yescharacteristic
272C4×C53—22no—
273C4×C53—41no—
274C4×C53—81no—
275C4×C53—81no—
276C4×C53—81no—
277C4×C53—81no—
278Dic53—84no—
279Dic53—162no—
280C22×#99—531no—
281#2435D5311yescharacteristic
282C2×#354—532no—
283#1781—531no—
284#1393—532no—
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—22no—
288C2×C53—41no—
289C2×C53—42no—
290C2×C53—41no—
291C2×C53—42no—
292C2×C53—41no—
293C2×C53—81no—
294C2×C53#189811yescharacteristic
295C2×C53C2×#35411yescharacteristic
296C2×C53#212711yescharacteristic
297C23×D4C2×D5311yescharacteristic
298C2×#124Dic5311yescharacteristic, complemented
299C2×#124C2×D5311yescharacteristic
300C2×C22⋊Q8Dic5311yescharacteristic, complemented
301C22×C22⋊C4—1061no—
302C2×#99—532no—
303C2×#99—532no—
304C2×#99—1064no—
305C2×#99—1061no—
306C2×#99—1061no—
307#394—21no—
308#394—21no—
309#390Dic5311yescharacteristic, complemented
310#390C2×D5311yescharacteristic
311#379—21no—
312#379—21no—
313#379—1061no—
314C4×C22⋊C4Dic5311yescharacteristic, complemented
315#354—1064no—
316C53#1309011yescharacteristic, complemented, Sylow
317C25C22⋊D5311yescharacteristic
318C25C22⋊D5311yescharacteristic
319C22×Q8C2×Dic5311yescharacteristic, complemented
320C22×D4—22no—
321C22×D4—22no—
322C22×D4—41no—
323C22×D4—41no—
324C22×D4—42no—
325C22×D4C2×Dic5311yescharacteristic
326C22×D4C22⋊D5311yescharacteristic
327C22×D4C22⋊D5311yescharacteristic
328C22×D4C22×D5311yescharacteristic
329C23×C4—2121no—
330C23×C4C2×Dic5311yescharacteristic
331#124—22no—
332#124—22no—
333#124—42no—
334C22⋊Q8—41no—
335C22⋊Q8—41no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—1064no—
349C2×C22⋊C4—1062no—
350C2×C22⋊C4—1062no—
351C2×C22⋊C4—2121no—
352C2×C22⋊C4—2121no—
353C2×C22⋊C4—2121no—
354C2×C22⋊C4—2122no—
355C2×C42C2×Dic5311yescharacteristic, complemented
356C2×C42C2×Dic5311yescharacteristic, complemented
357#99—534no—
358#99—1064no—
359#99—1062no—
360#99—1062no—
361#99—1062no—
362#99—2122no—
363#95—21no—
364C24—21no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—41no—
373C24—42no—
374C24—42no—
375C24—81no—
376C24—81no—
377C24C22⋊Dic5311yescharacteristic
378C24C22⋊Dic5311yescharacteristic
379C24C22⋊Dic5311yescharacteristic
380C24C2×C22⋊D5311yescharacteristic
381C24C2×C22⋊D5311yescharacteristic
382C2×Q8—22no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—22no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4—41no—
394C2×D4—41no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4D4×D5312yes—
402C2×D4#1171994712yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—4242no—
412C22×C4—4242no—
413C22×C4—2122no—
414C22×C4—2124no—
415C22×C4—2122no—
416C22×C4C22⋊Dic5311yescharacteristic
417C22×C4C22×Dic5311yescharacteristic
418C4⋊C4—41no—
419C4⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—81no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—1064no—
433C22⋊C4—4242no—
434C22⋊C4—4242no—
435C22⋊C4—2122no—
436C22⋊C4—2122no—
437C22⋊C4—2124no—
438C22⋊C4—2122no—
439C42—22no—
440C42—41no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—21no—
452C23—21no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—81no—
473C23—81no—
474C23—81no—
475C23—81no—
476C23—84no—
477C23—82no—
478C23—82no—
479C23—84no—
480C23—84no—
481C23#1313313211yescharacteristic
482C23#1313313211yescharacteristic
483C23#1313313212yes—
484C23C2×C22⋊Dic5311yescharacteristic
485C23#1313318511yescharacteristic
486C23#1313318912yes—
487C23#1313319811yescharacteristic
488Q8—82no—
489D4—42no—
490D4—42no—
491D4—44no—
492D4—44no—
493D4—42no—
494D4—42no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—4242no—
515C2×C4—4242no—
516C2×C4—4242no—
517C2×C4—4248no—
518C2×C4—4242no—
519C2×C4—4242no—
520C2×C4—2124no—
521C2×C4D4×Dic5312yes—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—21no—
531C22—22no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—41no—
537C22—42no—
538C22—41no—
539C22—44no—
540C22—44no—
541C22—42no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—81no—
554C22—84no—
555C22—81no—
556C22—82no—
557C22—82no—
558C22—84no—
559C22#1681041812yescomplemented
560C22#1681050311yescharacteristic
561C22C2×#1313313211yescharacteristic
562C22C2×#1313313211yescharacteristic
563C4—22no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C4—8482no—
570C4—4244no—
571C2—21no—
572C2—21no—
573C2—22no—
574C2—42no—
575C2—41no—
576C2—41no—
577C2—42no—
578C2—41no—
579C2—81no—
580C2#2027516911yescharacteristic
581C2C2×#1681041811yescharacteristic
582C2#2027638411yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2435C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenter#2435D53
hypocenterC53#13090
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2435

Complements

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

NH ≅ G/Ntype
2. C2×#16810418C2semidirect
4. #20272840C2semidirect
15. C2×#124×C53C4semidirect
17. C2×C22⋊Q8×C53C4semidirect
20. C53×#390C4semidirect
24. C4×C22⋊C4×C53C4semidirect
41. C22×Q8×C53C2×C4semidirect
58. C2×C4⋊C4×C53C2×C4semidirect
69. C2×C42×C53C2×C4semidirect
70. C2×C42×C53C2×C4semidirect
268. C22×C53#354semidirect
298. C2×#124Dic53semidirect
300. C2×C22⋊Q8Dic53semidirect
309. #390Dic53semidirect
314. C4×C22⋊C4Dic53semidirect
316. C53#13090semidirect
319. C22×Q8C2×Dic53semidirect
337. C2×C4⋊C4C2×Dic53semidirect
355. C2×C42C2×Dic53semidirect
356. C2×C42C2×Dic53semidirect
559. C22#16810418semidirect

Maximal subgroups

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

subgroupindex
C22×#131331322
C2×#168104182
#202746302
#202728402
C53×#24352
#1309053