Finite Group Databasebeta

← #27469815

Subgroups — #27469815

Order 13568.

836 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283018C211yesmaximal, characteristic
2C2×#16810488C211yesmaximal, characteristic, complemented
3#20276136C212yesmaximal, complemented
4#20276132C212yesmaximal, complemented
5#20275523C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275030C211yesmaximal, characteristic
8#20275023C211yesmaximal, characteristic, complemented
9#20274844C212yesmaximal, complemented
10C22⋊C8×Dic53C211yesmaximal, characteristic, complemented
11C53×#1872C211yesmaximal, characteristic, complemented
12C2×#13133206C2211yescharacteristic
13C2×#13133191C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16C2×#13133141C2212yes—
17C2×#13133139C2212yes—
18C2×#13133137C2212yes—
19C2×C22⋊#88322C2211yescharacteristic
20C2×C8×Dic53C2212yes—
21#16810530C2211yescharacteristic
22#16810524C2211yescharacteristic
23#16810513—21no—
24#16810513—21no—
25#16810500—21no—
26#16810500—21no—
27#16810488—21no—
28#16810488—21no—
29#16810488C2212yes—
30#16810488C2212yes—
31#16810488C2211yescharacteristic
32C2×#13133124—21no—
33C2×#13133124—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×C22⋊Q8×C53C2211yescharacteristic
39C22×C4⋊C4×C53C2211yescharacteristic
40C2×#103×C53C2212yes—
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#340C2212yes—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×#88322C2312yes—
49#13133206—24no—
50#13133191—22no—
51#13133191—22no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53—42no—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×#11719921—21no—
57C2×#11719921—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×C4×Dic53—24no—
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53D411yescharacteristic
64C2×C4×Dic53C2311yescharacteristic
65C2×C4×Dic53C2311yescharacteristic
66#13133141—42no—
67#13133139—42no—
68#13133137—42no—
69C22⋊#88322—22no—
70C8×Dic53—24no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—22no—
76#13133124—22no—
77#13133124—22no—
78#13133124—22no—
79C22×Q8×C53C2311yescharacteristic
80C23×C4×C53—21no—
81C23×C4×C53C2311yescharacteristic
82C22×C8×C53C2312yes—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—42no—
88C2×C4⋊C4×C53C2312yes—
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53—21no—
91#103×C53—42no—
92C22⋊C8×C53—22no—
93C24×C53C2×D411yescharacteristic
94C23×D53—21no—
95C2×Q8×C53—21no—
96C2×Q8×C53—21no—
97C2×Q8×C53—22no—
98C2×Q8×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—22no—
104C22×C4×C53—42no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4112yes—
109C22×C4×C53#4112yes—
110C22×C4×C53C2411yescharacteristic
111C2×C22⋊D53—42no—
112C2×C22⋊D53—42no—
113C2×C22⋊D53—42no—
114C22×Dic53—22no—
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—21no—
119C22×Dic53—22no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—42no—
123C22×Dic53—42no—
124C22×Dic53—42no—
125C22×Dic53C2×D411yescharacteristic
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C2×D212—21no—
130C2×D212—21no—
131C2×D212—22no—
132C2×D212—22no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—21no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—42no—
141C4⋊C4×C53—42no—
142C22⋊C4×C53—42no—
143#11719921—42no—
144#11719921—42no—
145#11719921—42no—
146#11719921—42no—
147#11719921—42no—
148C4⋊Dic53—21no—
149C4⋊Dic53—21no—
150C4⋊Dic53—22no—
151C4⋊Dic53—22no—
152C4⋊Dic53—42no—
153C4×Dic53—22no—
154C4×Dic53—22no—
155C4×Dic53—22no—
156C4×Dic53#4114yes—
157C2×#88322—22no—
158C2×#88322—22no—
159C2×#88322—22no—
160C23×C53—21no—
161C23×C53—21no—
162C23×C53—21no—
163C23×C53—21no—
164C23×C53#12412yes—
165C23×C53#13512yes—
166C23×C53#13512yes—
167C23×C53C22×D411yescharacteristic
168C22×D53—41no—
169C22×D53—41no—
170C22×D53—41no—
171C22×D53—41no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—41no—
175Q8×C53—42no—
176Q8×C53—42no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—24no—
180C2×C4×C53—24no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—84no—
191C2×C4×C53#12412yes—
192C2×C4×C53C2×#4112yes—
193C2×C4×C53C2×#4111yescharacteristic
194C2×C4×C53C2×#4111yescharacteristic
195C2×C4×C53H(5,𝔽2)12yes—
196C22⋊D53—84no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—44no—
208C2×Dic53—44no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—42no—
215C2×Dic53—84no—
216C2×Dic53#12312yes—
217C2×Dic53#12312yes—
218D212—42no—
219D212—42no—
220C8×C53—42no—
221#88322—42no—
222#27977—531nomaximal, Sylow
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53#48814yes—
230C22×C53C2×#12311yescharacteristic
231C22×C53C2×#12411yescharacteristic
232C22×C53#53511yescharacteristic
233C22×C53C2×#13511yescharacteristic
234C22×C53C2×#13511yescharacteristic
235C22×C53#57511yescharacteristic
236C22×C53#57511yescharacteristic
237C2×D53—81no—
238C2×D53—81no—
239C2×D53—81no—
240C2×D53—81no—
241C2×D53—81no—
242C2×D53—81no—
243C2×D53—81no—
244C4×C53—22no—
245C4×C53—81no—
246C4×C53—82no—
247C4×C53—81no—
248C4×C53#55312yes—
249C4×C53#55312yes—
250Dic53—22no—
251Dic53—22no—
252Dic53—41no—
253Dic53—41no—
254Dic53—81no—
255Dic53—82no—
256#2850—531no—
257C2×#382—531no—
258#2074—532no—
259#2073—532no—
260#1958—532no—
261#1874—531no—
262#1874—531no—
263#1872—531no—
264#1872D5311yescharacteristic, complemented
265#1835—532no—
266C4×C22⋊C8—531no—
267C2×C53—24no—
268C2×C53#250411yescharacteristic
269C2×C53#309111yescharacteristic
270C2×C53#309111yescharacteristic
271C2×C53C2×#48811yescharacteristic
272C2×C53#314211yescharacteristic
273C2×C53#321111yescharacteristic
274C2×C53#321111yescharacteristic
275D53—161no—
276C2×#124—531no—
277C2×C22⋊Q8C2×D5311yescharacteristic
278C2×#121—531no—
279C22×C4⋊C4—531no—
280C22×C4⋊C4C2×D5311yescharacteristic
281C22×C42—531no—
282C2×#103—532no—
283C2×#103C2×D5312yes—
284C2×#102—532no—
285C2×#102—532no—
286C2×C22⋊C8—531no—
287C2×C22⋊C8C2×D5311yescharacteristic
288C2×C4×C8—532no—
289#391—531no—
290#390—531no—
291#388—1061no—
292#388—1061no—
293#386—1061no—
294#386—1061no—
295#382—532no—
296#382—531no—
297#382—532no—
298#382—1061no—
299#382—1061no—
300C2×#95—1061no—
301C2×#95—1061no—
302#340—532no—
303#340—532no—
304#340—532no—
305#340C2×D5312yes—
306#336—532no—
307C53#2797711yescharacteristic, complemented, Sylow
308C22×Q8C22×D5311yescharacteristic
309C22×D4—531no—
310C22×D4—1061no—
311C23×C4—21no—
312C23×C4—531no—
313C23×C4—531no—
314C23×C4—1061no—
315C23×C4C22×D5311yescharacteristic
316C22×C8—532no—
317C22×C8C22×D5312yes—
318#124—1064no—
319C22⋊Q8—22no—
320C22⋊Q8—22no—
321#121—1062no—
322#121—1062no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—532no—
327C2×C4⋊C4—531no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—2122no—
330C2×C4⋊C4C22×D5312yes—
331C2×C4⋊C4C22×D5311yescharacteristic
332C2×C22⋊C4—21no—
333C2×C22⋊C4—1061no—
334C2×C22⋊C4—1061no—
335C2×C22⋊C4—1061no—
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C42—531no—
339C2×C42—531no—
340C2×C42—531no—
341C2×C42—531no—
342C2×C42—1064no—
343#103—42no—
344#103—2122no—
345#102—2122no—
346#102—2122no—
347C22⋊C8—22no—
348C22⋊C8—1062no—
349C4×C8—1064no—
350#95—1062no—
351#95—1062no—
352#95—1062no—
353#95—1062no—
354#95—1061no—
355#95—1061no—
356#95—1061no—
357#95—1061no—
358C24—1061no—
359C24D4×D5311yescharacteristic
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—22no—
364C2×D4—1062no—
365C2×D4—1061no—
366C2×D4—1061no—
367C2×D4—1062no—
368C2×D4—2122no—
369C2×D4—2122no—
370C2×D4—2122no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—531no—
380C22×C4—531no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—1062no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1061no—
388C22×C4—1062no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4—2122no—
394C22×C4#1171994512yes—
395C22×C4D4×D5311yescharacteristic
396C22×C4#1171994712yes—
397C22×C4C23×D5311yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—1062no—
402C2×C8—1062no—
403C2×C8—1062no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—1061no—
411C4⋊C4—1061no—
412C4⋊C4—1062no—
413C4⋊C4—1062no—
414C4⋊C4—2122no—
415C22⋊C4—42no—
416C22⋊C4—2122no—
417C22⋊C4—2122no—
418C22⋊C4—2122no—
419C22⋊C4—2122no—
420C22⋊C4—2122no—
421C42—534no—
422C42—1062no—
423C42—1062no—
424C42—1062no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—2121no—
430C23—2121no—
431C23—2121no—
432C23—2121no—
433C23—2121no—
434C23—2121no—
435C23—2121no—
436C23#1313320312yes—
437C23#1313323812yes—
438C23#1313324012yes—
439C23C2×D4×D5311yescharacteristic
440Q8—42no—
441Q8—42no—
442D4—4244no—
443D4—2122no—
444D4—2122no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—84no—
459C2×C4—532no—
460C2×C4—532no—
461C2×C4—1061no—
462C2×C4—1061no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—1061no—
468C2×C4—1061no—
469C2×C4—1061no—
470C2×C4—1061no—
471C2×C4—4244no—
472C2×C4—2124no—
473C2×C4—2124no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2122no—
477C2×C4—2121no—
478C2×C4—2122no—
479C2×C4—2122no—
480C2×C4#1313320312yes—
481C2×C4#41×D5312yes—
482C2×C4C2×#1171994711yescharacteristic
483C2×C4C2×#1171994511yescharacteristic
484C2×C4#1313326212yes—
485C8—42no—
486C8—2122no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—4241no—
494C22—4241no—
495C22—4241no—
496C22—4241no—
497C22—4241no—
498C22—4241no—
499C22—4241no—
500C22#1681097914yes—
501C22#123×D5311yescharacteristic
502C22C2×#1313320311yescharacteristic
503C22#1681118811yescharacteristic
504C22C2×#1313323811yescharacteristic
505C22C2×#1313324011yescharacteristic
506C22#1681141911yescharacteristic
507C22#1681142211yescharacteristic
508C4—22no—
509C4—81no—
510C4—81no—
511C4—82no—
512C4—1062no—
513C4—1062no—
514C4—4241no—
515C4—4242no—
516C4—2121no—
517C4—2121no—
518C4#1681131612yes—
519C4#1681132212yes—
520C2—24no—
521C2—8481no—
522C2#2027920511yescharacteristic
523C2#2028462911yescharacteristic
524C2#2028463111yescharacteristic
525C2C2×#1681097911yescharacteristic
526C2#2028511311yescharacteristic
527C2#2028568311yescharacteristic
528C2#2028568511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1872C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1872D53
hypocenterC53#27977
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
2. C2×#16810488C2semidirect
3. #20276136C2semidirect
4. #20276132C2semidirect
5. #20275523C2semidirect
8. #20275023C2semidirect
9. #20274844C2semidirect
10. C22⋊C8×Dic53C2semidirect
11. C53×#1872C2semidirect
264. #1872D53semidirect
307. C53#27977semidirect

Maximal subgroups

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

subgroupindex
#202830182
C2×#168104882
#202761362
#202761322
#202755232
#202750322
#202750302
#202750232
#202748442
C22⋊C8×Dic532
C53×#18722
#2797753