Finite Group Databasebeta

← #27276764

Subgroups — #27276764

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810512C211yesmaximal, characteristic
2#20274693C211yesmaximal, characteristic, complemented
3#20274693C211yesmaximal, characteristic, complemented
4C2×#16810346C211yesmaximal, characteristic
5C2×#16810346C211yesmaximal, characteristic
6#20272558C211yesmaximal, characteristic, complemented
7C53×#1280C211yesmaximal, characteristic, complemented
8C22×#11719919C411yescharacteristic
9C22×#11719921C411yescharacteristic
10C22×#11719921C411yescharacteristic
11C22×#11719921C2211yescharacteristic
12C22×C4×Dic53C2211yescharacteristic
13C2×#13133146—21no—
14C2×#13133146—21no—
15C2×C22⋊#88322C2211yescharacteristic
16C2×C22⋊#88322C2211yescharacteristic
17#16810512—21no—
18#16810512—21no—
19#16810512—21no—
20#16810512—21no—
21C2×#13133124C411yescharacteristic
22#16810377—21no—
23#16810377—21no—
24#16810346—22no—
25#16810346—22no—
26C2×C22⋊C8×C53C2211yescharacteristic
27C2×C22⋊C8×C53C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C24×D53C2×C411yescharacteristic
30C23×Dic53C2×C411yescharacteristic
31C23×Dic53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C22×#88322—21no—
34C22×#88322—21no—
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—42no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×#11719921—42no—
47C2×#11719921—42no—
48C2×#11719921D412yes—
49C2×#11719921Q812yes—
50C2×C4×Dic53—42no—
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C2×C4×Dic53D411yescharacteristic
54C2×C4×Dic53D411yescharacteristic
55#13133146—42no—
56#13133146—42no—
57C22⋊#88322—21no—
58C22⋊#88322—21no—
59C22⋊#88322—21no—
60C22⋊#88322—21no—
61#13133124—21no—
62#13133124—21no—
63C23×C4×C53C2×C411yescharacteristic
64C23×C4×C53C2×C411yescharacteristic
65C23×C4×C53C2311yescharacteristic
66C22×C8×C53—21no—
67C22×C8×C53—21no—
68C22⋊C8×C53—21no—
69C22⋊C8×C53—21no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72#95×C53—21no—
73#95×C53—21no—
74C24×C53C22×C411yescharacteristic
75C23×D53—21no—
76C23×D53—21no—
77C23×D53—21no—
78C23×D53—21no—
79C23×D53—21no—
80C23×D53—21no—
81C23×D53—41no—
82C23×D53—41no—
83C23×D53—41no—
84C23×D53—41no—
85C23×D53C4⋊C412yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×Q811yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—21no—
107C22×Dic53—21no—
108C22×Dic53—21no—
109C22×Dic53—21no—
110C22×Dic53—22no—
111C22×Dic53—22no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114C22×Dic53—41no—
115C22×Dic53—41no—
116C22×Dic53C22⋊C411yescharacteristic
117C22×Dic53C22⋊C411yescharacteristic
118C22×Dic53C22⋊C411yescharacteristic
119C22×Dic53C22⋊C411yescharacteristic
120C22×Dic53C4⋊C412yes—
121C2×C8×C53—41no—
122C2×C8×C53—41no—
123C2×C8×C53—41no—
124C2×C8×C53—41no—
125C2×C8×C53—41no—
126C2×C8×C53—41no—
127#11719921—22no—
128#11719921—22no—
129#11719921—22no—
130#11719921—22no—
131#11719921—42no—
132#11719921—42no—
133#11719921—42no—
134#11719921—42no—
135#11719919—42no—
136#11719919—42no—
137C4×Dic53—22no—
138C4×Dic53—22no—
139C4×Dic53—41no—
140C4×Dic53—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53C2×C22⋊C411yescharacteristic
152C23×C53C2×C4⋊C411yescharacteristic
153C23×C53#11711yescharacteristic
154C23×C53#13111yescharacteristic
155C23×C53#13111yescharacteristic
156C23×C53#13111yescharacteristic
157C23×C53#13111yescharacteristic
158C22×D53—22no—
159C22×D53—22no—
160C22×D53—22no—
161C22×D53—42no—
162C22×D53—42no—
163C22×D53—41no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53—41no—
167C22×D53—41no—
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—42no—
172C22×D53—42no—
173C22×D53—81no—
174C22×D53—81no—
175C22×D53—81no—
176C22×D53—81no—
177C22×D53—81no—
178C22×D53—81no—
179C22×D53—81no—
180C22×D53—81no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53—81no—
199C2×C4×C53#12111yescharacteristic
200C2×C4×C53#12111yescharacteristic
201C2×C4×C53C22⋊Q811yescharacteristic
202C2×C4×C53C22⋊Q811yescharacteristic
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—22no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—21no—
211C2×Dic53—21no—
212C2×Dic53—21no—
213C2×Dic53—21no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—42no—
222C2×Dic53#9912yes—
223C2×Dic53#10012yes—
224C8×C53—81no—
225C8×C53—81no—
226#88322—81no—
227#88322—81no—
228#8561—531nomaximal, Sylow
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53#38011yescharacteristic
242C22×C53#40811yescharacteristic
243C22×C53#40811yescharacteristic
244C22×C53C2×#9911yescharacteristic
245C22×C53#41011yescharacteristic
246C22×C53C2×#10011yescharacteristic
247C22×C53#41311yescharacteristic
248C22×C53#42411yescharacteristic
249C22×C53#42411yescharacteristic
250C22×C53#43311yescharacteristic
251C22×C53#43311yescharacteristic
252C2×D53—42no—
253C2×D53—42no—
254C2×D53—42no—
255C2×D53—82no—
256C2×D53—82no—
257C2×D53—81no—
258C2×D53—81no—
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—82no—
262C2×D53—82no—
263C4×C53—21no—
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—81no—
268C4×C53—81no—
269Dic53—22no—
270Dic53—22no—
271Dic53—42no—
272Dic53—41no—
273Dic53—41no—
274C2×#388—531no—
275#1797—531no—
276#1797—531no—
277C2×#323—531no—
278C2×#323—531no—
279#1280—531no—
280#1280D5311yescharacteristic, complemented
281C2×C53—21no—
282C2×C53—21no—
283C2×C53—21no—
284C2×C53—21no—
285C2×C53#146811yescharacteristic
286C2×C53#146911yescharacteristic
287C2×C53#147111yescharacteristic
288C2×C53#147311yescharacteristic
289C2×C53#179811yescharacteristic
290C2×C53#179911yescharacteristic
291C2×C53#180011yescharacteristic
292D53—82no—
293C22×C4⋊C4—531no—
294C22×C22⋊C4—531no—
295C22×C22⋊C4—531no—
296C22×C22⋊C4—531no—
297C22×C42—531no—
298C2×C4⋊C8—1061no—
299C2×C4⋊C8—1061no—
300C2×C22⋊C8—531no—
301C2×C22⋊C8—531no—
302C2×C22⋊C8C2×D5311yescharacteristic
303C2×C22⋊C8C2×D5311yescharacteristic
304#388—1061no—
305#388—1061no—
306#388—1061no—
307#388—1061no—
308C2×#95—531no—
309C2×#95C2×D5311yescharacteristic
310#336—1061no—
311#336—1061no—
312#323—1062no—
313#323—1062no—
314C53#856111yescharacteristic, complemented, Sylow
315C25—531no—
316C23×C4—531no—
317C23×C4—531no—
318C23×C4—531no—
319C23×C4C4×D5311yescharacteristic
320C23×C4C4×D5311yescharacteristic
321C23×C4C22×D5311yescharacteristic
322C22×C8—21no—
323C22×C8—21no—
324C22×C8—1061no—
325C22×C8—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4—1061no—
328C2×C4⋊C4—2122no—
329C2×C22⋊C4—532no—
330C2×C22⋊C4—532no—
331C2×C22⋊C4—1061no—
332C2×C22⋊C4—1061no—
333C2×C22⋊C4—1061no—
334C2×C22⋊C4—1061no—
335C2×C22⋊C4—1061no—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C22⋊C4—1061no—
339C2×C22⋊C4—2122no—
340C2×C22⋊C4—2122no—
341C2×C42—531no—
342C2×C42—531no—
343C2×C42—531no—
344C2×C42—531no—
345C2×C42—2122no—
346C4⋊C8—2122no—
347C4⋊C8—2122no—
348C22⋊C8—21no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—1061no—
353C22⋊C8—1061no—
354C22⋊C8—1061no—
355C22⋊C8—1061no—
356#95—21no—
357#95—21no—
358#95—1061no—
359#95—1061no—
360C24—532no—
361C24—1061no—
362C24—1061no—
363C24—1061no—
364C24—1061no—
365C24—1061no—
366C24—1061no—
367C24—2121no—
368C24—2121no—
369C24—2121no—
370C24—2121no—
371C24C2×C4×D5311yescharacteristic
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—531no—
385C22×C4—531no—
386C22×C4—531no—
387C22×C4—531no—
388C22×C4—532no—
389C22×C4—1062no—
390C22×C4—1062no—
391C22×C4—1061no—
392C22×C4—1061no—
393C22×C4—1061no—
394C22×C4—1061no—
395C22×C4—1061no—
396C22×C4—1061no—
397C22×C4—2121no—
398C22×C4—2121no—
399C22×C4—2121no—
400C22×C4—2121no—
401C22×C4#1171994511yescharacteristic
402C22×C4#1171994511yescharacteristic
403C22×C4D4×D5311yescharacteristic
404C22×C4D4×D5311yescharacteristic
405C22×C4D4×D5311yescharacteristic
406C22×C4Q8×D5311yescharacteristic
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—2121no—
414C2×C8—2121no—
415C2×C8—2121no—
416C2×C8—2121no—
417C2×C8—2121no—
418C2×C8—2121no—
419C4⋊C4—2122no—
420C4⋊C4—2122no—
421C22⋊C4—1062no—
422C22⋊C4—1062no—
423C22⋊C4—1062no—
424C22⋊C4—1062no—
425C22⋊C4—2122no—
426C22⋊C4—2122no—
427C22⋊C4—2122no—
428C22⋊C4—2122no—
429C42—1062no—
430C42—1062no—
431C42—2121no—
432C42—2121no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—1062no—
438C23—1062no—
439C23—1062no—
440C23—4241no—
441C23—4241no—
442C23—4241no—
443C23—4241no—
444C23—4241no—
445C23—4241no—
446C23—4241no—
447C23—4241no—
448C23—2122no—
449C23—2122no—
450C23—2122no—
451C23—2122no—
452C23—2121no—
453C23—2121no—
454C23—2121no—
455C23—2121no—
456C23—2121no—
457C23—2121no—
458C23—2121no—
459C23—2121no—
460C23C22⋊C4×D5311yescharacteristic
461C23C4⋊C4×D5311yescharacteristic
462C23#1313317411yescharacteristic
463C23#1313322311yescharacteristic
464C23#1313322311yescharacteristic
465C23#1313322511yescharacteristic
466C23#1313322511yescharacteristic
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—532no—
486C2×C4—532no—
487C2×C4—1061no—
488C2×C4—1061no—
489C2×C4—1061no—
490C2×C4—1061no—
491C2×C4—1061no—
492C2×C4—1061no—
493C2×C4—1061no—
494C2×C4—1061no—
495C2×C4—1061no—
496C2×C4—1061no—
497C2×C4—1062no—
498C2×C4—2122no—
499C2×C4—2122no—
500C2×C4—2122no—
501C2×C4—2122no—
502C2×C4—2122no—
503C2×C4—2121no—
504C2×C4—2121no—
505C2×C4—2122no—
506C2×C4#1313319011yescharacteristic
507C2×C4#1313319011yescharacteristic
508C2×C4#1313319511yescharacteristic
509C2×C4#1313319511yescharacteristic
510C8—81no—
511C8—81no—
512C8—4241no—
513C8—4241no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—4242no—
527C22—4242no—
528C22—4242no—
529C22—4242no—
530C22—4241no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—2122no—
535C22—2122no—
536C22—2122no—
537C22#1681048211yescharacteristic
538C22#1681058811yescharacteristic
539C22#1681058811yescharacteristic
540C22#99×D5311yescharacteristic
541C22#1681059511yescharacteristic
542C22#100×D5311yescharacteristic
543C22#1681060611yescharacteristic
544C22#1681064811yescharacteristic
545C22#1681064811yescharacteristic
546C22#1681068211yescharacteristic
547C22#1681068511yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—81no—
553C4—81no—
554C4—1062no—
555C4—1062no—
556C4—2122no—
557C4—2121no—
558C4—2121no—
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—4242no—
564C2#2027306211yescharacteristic
565C2#2027306511yescharacteristic
566C2#2027307311yescharacteristic
567C2#2027308311yescharacteristic
568C2#2027469811yescharacteristic
569C2#2027470211yescharacteristic
570C2#2027470611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1280C2
socleC23×C53C2×C22⋊C4
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1280D53
hypocenterC53#8561
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
2. #20274693C2semidirect
3. #20274693C2semidirect
6. #20272558C2semidirect
7. C53×#1280C2semidirect
280. #1280D53semidirect
314. C53#8561semidirect

Maximal subgroups

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

subgroupindex
C2×#168105122
#202746932
C2×#168103462
#202725582
C53×#12802
#856153