Finite Group Databasebeta

← #27301592

Subgroups — #27301592

Order 13568.

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

normal subgroups only

This group has 894 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×#16810413C212yesmaximal, complemented
3#20274636C211yesmaximal, characteristic
4#20272833C212yesmaximal, complemented
5C53×#2405C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7C2×#13133134—21no—
8C2×#13133134C2212yes—
9C2×#13133132—24no—
10C2×#13133132—21no—
11C2×#13133132C2212yes—
12#16810587—21no—
13#16810413—24no—
14C23×D4×C53C2211yescharacteristic
15C2×#123×C53C411yescharacteristic, complemented
16C2×#123×C53C2211yescharacteristic
17C22×C22⋊C4×C53C411yescharacteristic, complemented
18C53×#394—21no—
19C53×#394—21no—
20C53×#386C411yescharacteristic, complemented
21C53×#386C2211yescharacteristic
22C53×#385—21no—
23C53×#379—21no—
24C53×#379C411yescharacteristic, complemented
25C23×Dic53—41no—
26C2×#11719917—22no—
27C2×C22⋊Dic53—24no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—42no—
30C2×C22⋊Dic53—41no—
31#13133134—22no—
32#13133134—22no—
33#13133134—42no—
34#13133132—24no—
35#13133132—22no—
36#13133132D414yes—
37C22⋊#88322—41no—
38C22⋊#88322—41no—
39C25×C53C2×C411yescharacteristic
40C25×C53D411yescharacteristic
41C22×D4×C53—22no—
42C22×D4×C53—22no—
43C22×D4×C53—41no—
44C22×D4×C53—41no—
45C22×D4×C53—42no—
46C22×D4×C53C2×C411yescharacteristic
47C22×D4×C53D411yescharacteristic
48C22×D4×C53D411yescharacteristic
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53C2×C411yescharacteristic, complemented
51C23×C4×C53C2×C411yescharacteristic, complemented
52C23×C4×C53D411yescharacteristic
53#123×C53—22no—
54#123×C53—22no—
55#123×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53—21no—
58C2×C22⋊C4×C53—21no—
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—41no—
67C2×C22⋊C4×C53—41no—
68C2×C22⋊C4×C53C2×C411yescharacteristic, complemented
69C2×C22⋊C4×C53C2×C411yescharacteristic, complemented
70#95×C53—21no—
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×C53C22×C411yescharacteristic
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—22no—
91C2×D4×C53—22no—
92C2×D4×C53—22no—
93C2×D4×C53—21no—
94C2×D4×C53—21no—
95C2×D4×C53—22no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—41no—
101C2×D4×C53—41no—
102C2×D4×C53—44no—
103C2×D4×C53—44no—
104C2×D4×C53—44no—
105C2×D4×C53—44no—
106C2×D4×C53C2×D412yes—
107C2×D4×C53#4112yes—
108C22×C4×C53—21no—
109C22×C4×C53—21no—
110C22×C4×C53—22no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—21no—
115C22×C4×C53—22no—
116C22×C4×C53—41no—
117C22×C4×C53—41no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53—81no—
121C22×C4×C53C22⋊C411yescharacteristic
122C22×C4×C53C2×D411yescharacteristic
123C22×Dic53—44no—
124C22×Dic53—42no—
125C22×Dic53—82no—
126C22×Dic53—82no—
127C4⋊C4×C53—42no—
128C4⋊C4×C53—42no—
129C22⋊C4×C53—22no—
130C22⋊C4×C53—42no—
131C22⋊C4×C53—42no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—42no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C22⋊C4×C53—81no—
139C22⋊C4×C53—81no—
140C22⋊C4×C53—81no—
141C22⋊Dic53—24no—
142C22⋊Dic53—44no—
143C22⋊Dic53—42no—
144#11719917—42no—
145#11719917—42no—
146C2×#88322—42no—
147C23×C53—22no—
148C23×C53—22no—
149C23×C53—22no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—21no—
153C23×C53—21no—
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—41no—
164C23×C53—42no—
165C23×C53—42no—
166C23×C53—41no—
167C23×C53—41no—
168C23×C53—42no—
169C23×C53—41no—
170C23×C53—41no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—42no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—42no—
177C23×C53—42no—
178C23×C53—81no—
179C23×C53—81no—
180C23×C53—81no—
181C23×C53—81no—
182C23×C53—84no—
183C23×C53—82no—
184C23×C53—82no—
185C23×C53—84no—
186C23×C53—84no—
187C23×C53#9911yescharacteristic
188C23×C53#9911yescharacteristic
189C23×C53#9912yes—
190C23×C53C2×C22⋊C411yescharacteristic
191C23×C53C4×D412yes—
192C23×C53#12011yescharacteristic
193C23×C53#12311yescharacteristic
194D4×C53—42no—
195D4×C53—42no—
196D4×C53—44no—
197D4×C53—44no—
198D4×C53—42no—
199D4×C53—42no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—42no—
206C2×C4×C53—42no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
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—82no—
220C2×C4×C53—82no—
221C2×C4×C53#12112yes—
222C2×Dic53—44no—
223C2×Dic53—88no—
224C2×Dic53—82no—
225C2×Dic53—82no—
226#88322—82no—
227#13084—531nomaximal, Sylow
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—22no—
237C22×C53—22no—
238C22×C53—21no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—44no—
246C22×C53—42no—
247C22×C53—44no—
248C22×C53—41no—
249C22×C53—41no—
250C22×C53—41no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—84no—
260C22×C53—81no—
261C22×C53—81no—
262C22×C53—82no—
263C22×C53—82no—
264C22×C53—84no—
265C22×C53#35212yescomplemented
266C22×C53#38611yescharacteristic
267C22×C53C2×#9911yescharacteristic
268C22×C53C2×#9911yescharacteristic
269C4×C53—22no—
270C4×C53—41no—
271C4×C53—81no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53—81no—
275Dic53—84no—
276C22×#99—531no—
277#2405D5311yescharacteristic
278C2×#352—532no—
279#1783—531no—
280#1390—532no—
281C2×C53—21no—
282C2×C53—21no—
283C2×C53—22no—
284C2×C53—41no—
285C2×C53—41no—
286C2×C53—42no—
287C2×C53—41no—
288C2×C53—42no—
289C2×C53—81no—
290C2×C53#189811yescharacteristic
291C2×C53C2×#35211yescharacteristic
292C2×C53#212111yescharacteristic
293C23×D4C2×D5311yescharacteristic
294C2×#123Dic5311yescharacteristic, complemented
295C2×#123C2×D5311yescharacteristic
296C22×C22⋊C4—1061no—
297C22×C22⋊C4Dic5311yescharacteristic, complemented
298C2×#100—532no—
299C2×#100—1061no—
300C2×#99—532no—
301C2×#99—1064no—
302C2×#99—1061no—
303#407—1061no—
304#394—21no—
305#394—21no—
306#386Dic5311yescharacteristic, complemented
307#386C2×D5311yescharacteristic
308#385—21no—
309#379—21no—
310#379Dic5311yescharacteristic, complemented
311#352—1064no—
312C53#1308411yescharacteristic, complemented, Sylow
313C25C2×Dic5311yescharacteristic
314C25C22⋊D5311yescharacteristic
315C22×D4—22no—
316C22×D4—22no—
317C22×D4—41no—
318C22×D4—41no—
319C22×D4—42no—
320C22×D4C2×Dic5311yescharacteristic
321C22×D4C22⋊D5311yescharacteristic
322C22×D4C22⋊D5311yescharacteristic
323C22×D4C22×D5311yescharacteristic
324C23×C4—2121no—
325C23×C4C2×Dic5311yescharacteristic, complemented
326C23×C4C2×Dic5311yescharacteristic, complemented
327C23×C4C22⋊D5311yescharacteristic
328C2×M16—1062no—
329#123—22no—
330#123—22no—
331#123—42no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—41no—
343C2×C22⋊C4—41no—
344C2×C22⋊C4—1064no—
345C2×C22⋊C4—1062no—
346C2×C22⋊C4—2121no—
347C2×C22⋊C4—2122no—
348C2×C22⋊C4C2×Dic5311yescharacteristic, complemented
349C2×C22⋊C4C2×Dic5311yescharacteristic, complemented
350#100—1062no—
351#100—1062no—
352#100—2122no—
353#99—534no—
354#99—1064no—
355#99—1062no—
356C22⋊C8—2121no—
357C22⋊C8—2121no—
358#95—21no—
359#95—21no—
360C24—21no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—41no—
369C24—42no—
370C24—42no—
371C24—81no—
372C24—81no—
373C24C22⋊Dic5311yescharacteristic
374C24C22⋊Dic5311yescharacteristic
375C24C22⋊Dic5311yescharacteristic
376C24C22×Dic5311yescharacteristic
377C24C2×C22⋊D5311yescharacteristic
378C2×D4—22no—
379C2×D4—22no—
380C2×D4—22no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—41no—
385C2×D4—41no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—41no—
389C2×D4—41no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4D4×D5312yes—
395C2×D4#1171994712yes—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—81no—
409C22×C4—4242no—
410C22×C4—4242no—
411C22×C4—2124no—
412C22×C4—2122no—
413C22×C4C22⋊Dic5311yescharacteristic
414C22×C4C2×C22⋊D5311yescharacteristic
415M16—2122no—
416M16—2122no—
417C2×C8—2122no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C22⋊C4—22no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—81no—
430C22⋊C4—81no—
431C22⋊C4—81no—
432C22⋊C4—1064no—
433C22⋊C4—2124no—
434C22⋊C4—2122no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—41no—
452C23—42no—
453C23—42no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—42no—
466C23—81no—
467C23—81no—
468C23—81no—
469C23—81no—
470C23—84no—
471C23—82no—
472C23—82no—
473C23—84no—
474C23—84no—
475C23#1313313211yescharacteristic
476C23#1313313211yescharacteristic
477C23#1313313212yes—
478C23C2×C22⋊Dic5311yescharacteristic
479C23D4×Dic5312yes—
480C23#1313318511yescharacteristic
481C23#1313319811yescharacteristic
482D4—42no—
483D4—42no—
484D4—44no—
485D4—44no—
486D4—42no—
487D4—42no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—4248no—
510C2×C4—4242no—
511C2×C4—4242no—
512C2×C4—2124no—
513C2×C4#1313318912yes—
514C8—4242no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—22no—
523C22—21no—
524C22—22no—
525C22—21no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—44no—
533C22—42no—
534C22—44no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—81no—
545C22—81no—
546C22—84no—
547C22—81no—
548C22—81no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22#1681041312yescomplemented
553C22#1681050311yescharacteristic
554C22C2×#1313313211yescharacteristic
555C22C2×#1313313211yescharacteristic
556C4—22no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—4244no—
563C2—21no—
564C2—21no—
565C2—22no—
566C2—41no—
567C2—41no—
568C2—42no—
569C2—41no—
570C2—42no—
571C2—81no—
572C2#2027516911yescharacteristic
573C2C2×#1681041311yescharacteristic
574C2#2027636111yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC24×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2405C2
socleC22×C53C2×#99
prime radicalC23×D4×C53C22
cosocleC22×D4×C53C23
hypercenter#2405D53
hypocenterC53#13084
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. C2×#16810413C2semidirect
4. #20272833C2semidirect
15. C2×#123×C53C4semidirect
17. C22×C22⋊C4×C53C4semidirect
20. C53×#386C4semidirect
24. C53×#379C4semidirect
50. C23×C4×C53C2×C4semidirect
51. C23×C4×C53C2×C4semidirect
68. C2×C22⋊C4×C53C2×C4semidirect
69. C2×C22⋊C4×C53C2×C4semidirect
265. C22×C53#352semidirect
294. C2×#123Dic53semidirect
297. C22×C22⋊C4Dic53semidirect
306. #386Dic53semidirect
310. #379Dic53semidirect
312. C53#13084semidirect
325. C23×C4C2×Dic53semidirect
326. C23×C4C2×Dic53semidirect
348. C2×C22⋊C4C2×Dic53semidirect
349. C2×C22⋊C4C2×Dic53semidirect
552. C22#16810413semidirect

Maximal subgroups

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

subgroupindex
C22×#131331322
C2×#168104132
#202746362
#202728332
C53×#24052
#1308453