Finite Group Databasebeta

← #27466099

Subgroups — #27466099

Order 13568.

970 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277221C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic, complemented
3#20276275C212yesmaximal, complemented
4#20276240C212yesmaximal, complemented
5#20275470C212yesmaximal, complemented
6#20275032C211yesmaximal, characteristic
7#20275030C211yesmaximal, characteristic
8#20274902C211yesmaximal, characteristic, complemented
9#20274714C212yesmaximal, complemented
10#20274693C211yesmaximal, characteristic, complemented
11C53×#1852C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×#13133179C2211yescharacteristic
14C2×#13133177—21no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919—21no—
17C22×#11719919—21no—
18C22×C4×Dic53C2211yescharacteristic
19C2×#13133150C2212yes—
20C2×#13133146C2212yes—
21C2×#13133139C2212yes—
22C2×#13133137C2212yes—
23C2×C22⋊#88322C2211yescharacteristic
24#16810524C2211yescharacteristic
25#16810519C2211yescharacteristic
26#16810512—21no—
27#16810500—21no—
28#16810492—21no—
29#16810492—21no—
30#16810492C2212yes—
31#16810492C2212yes—
32C4⋊C4×Dic53C2211yescharacteristic
33#16810470—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×#107×C53C2212yes—
40C2×C22⋊C8×C53C2211yescharacteristic
41C53×#380C2211yescharacteristic
42C53×#340C2212yes—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C23×Dic53—21no—
46C23×Dic53D411yescharacteristic
47C23×Dic53D411yescharacteristic
48C22×C4×D53—21no—
49C22×#88322C2312yes—
50#13133191—22no—
51#13133191Q814yes—
52#13133179—24no—
53#13133177—44no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—42no—
56C2×C4⋊Dic53C2312yes—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×#11719919—21no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C2×#11719919—44no—
64C2×#11719919—44no—
65C2×#11719921—21no—
66C2×#11719921—21no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—24no—
70C2×C4×Dic53—21no—
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53D411yescharacteristic
73C2×C4×Dic53C2311yescharacteristic
74C2×C4×Dic53C2311yescharacteristic
75#13133150—42no—
76#13133146—24no—
77#13133139—42no—
78#13133137—42no—
79C22⋊#88322—22no—
80#13133124—21no—
81#13133124—21no—
82C23×C4×C53—21no—
83C23×C4×C53C2311yescharacteristic
84C22×C8×C53C2312yes—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—42no—
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2312yes—
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C22⋊C4×C53—21no—
91#107×C53—42no—
92C22⋊C8×C53—22no—
93#95×C53—21no—
94C24×C53C2×D411yescharacteristic
95C23×D53—21no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—42no—
102C22×C4×C53—42no—
103C22×C4×C53—42no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×Q812yes—
106C22×C4×C53#4112yes—
107C22×C4×C53C2411yescharacteristic
108C2×C22⋊D53—24no—
109C2×C22⋊D53—42no—
110C2×C22⋊D53—42no—
111C22×Dic53—22no—
112C22×Dic53—22no—
113C22×Dic53—22no—
114C22×Dic53—22no—
115C22×Dic53—22no—
116C22×Dic53—21no—
117C22×Dic53—22no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C22×Dic53D814yes—
122C22×Dic53C2×D411yescharacteristic
123C22×Dic53C2×D411yescharacteristic
124C22×Dic53C2×D411yescharacteristic
125C22×Dic53C2×D411yescharacteristic
126C2×D212—22no—
127C2×D212—22no—
128C2×D212—21no—
129C2×D212C2×Q812yes—
130C2×C4×D53—42no—
131C2×C4×D53—42no—
132C2×C4×D53—42no—
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—42no—
141C22⋊C4×C53—42no—
142#11719921—24no—
143#11719921—42no—
144#11719921—42no—
145#11719921—42no—
146C4⋊Dic53—21no—
147C4⋊Dic53—22no—
148C4⋊Dic53—22no—
149C4⋊Dic53—42no—
150C4⋊Dic53C2×Q812yes—
151#11719919—44no—
152#11719919—44no—
153#11719919—44no—
154#11719919—42no—
155#11719919—44no—
156C4×Dic53—22no—
157C4×Dic53—22no—
158C4×Dic53—22no—
159C4×Dic53—42no—
160C4×Dic53#4114yes—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—22no—
164C23×C53—21no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53C22⋊Q812yes—
169C23×C53C2×D812yes—
170C23×C53#13712yes—
171C23×C53C22×D411yescharacteristic
172C22×D53—22no—
173C22×D53—41no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—41no—
178C22×D53—41no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—24no—
182C2×C4×C53—24no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—41no—
186C2×C4×C53—41no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—42no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—84no—
193C2×C4×C53C22⋊Q812yes—
194C2×C4×C53C22×Q811yescharacteristic
195C2×C4×C53C2×#4112yes—
196C2×C4×C53C2×#4111yescharacteristic
197C2×C4×C53H(5,𝔽2)12yes—
198C22⋊D53—48no—
199C2×Dic53—28no—
200C2×Dic53—21no—
201C2×Dic53—21no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—28no—
207C2×Dic53—22no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—84no—
214C2×Dic53#12312yes—
215C2×Dic53#12312yes—
216C2×Dic53C2×D812yes—
217C2×Dic53C2×D812yes—
218C2×Dic53AGL(1,ℤ/8)12yes—
219C2×Dic53#13712yes—
220D212—24no—
221D212—42no—
222C4×D53—84no—
223C8×C53—42no—
224#88322—42no—
225#27677—531nomaximal, Sylow
226C22×C53—22no—
227C22×C53—22no—
228C22×C53—22no—
229C22×C53—22no—
230C22×C53—22no—
231C22×C53—22no—
232C22×C53#47414yes—
233C22×C53C2×C22⋊Q811yescharacteristic
234C22×C53C2×#12311yescharacteristic
235C22×C53#53511yescharacteristic
236C22×C53C22×D811yescharacteristic
237C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C53C2×#13711yescharacteristic
239C22×C53C2×#13711yescharacteristic
240C2×D53—42no—
241C2×D53—42no—
242C2×D53—42no—
243C2×D53—81no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C4×C53—22no—
248C4×C53—81no—
249C4×C53—81no—
250C4×C53—82no—
251C4×C53#55312yes—
252C4×C53#55412yes—
253Dic53—22no—
254Dic53—22no—
255Dic53—22no—
256Dic53—42no—
257Dic53—82no—
258Dic53#48014yes—
259C4×#121—531no—
260C2×#384—531no—
261#2099—532no—
262#2093—532no—
263#1947—532no—
264#1874—531no—
265#1874—531no—
266#1852—531no—
267#1852D5311yescharacteristic, complemented
268#1802—532no—
269#1797—531no—
270C2×C53—24no—
271C2×C53#248711yescharacteristic
272C2×C53C2×#47411yescharacteristic
273C2×C53#307911yescharacteristic
274C2×C53C2×#48011yescharacteristic
275C2×C53#309211yescharacteristic
276C2×C53#320511yescharacteristic
277C2×C53#321811yescharacteristic
278D53—82no—
279C2×#121—531no—
280C2×C4×D4—531no—
281C2×C4×D4—1061no—
282C22×C4⋊C4—531no—
283C22×C4⋊C4—1061no—
284C22×C4⋊C4—1061no—
285C22×C4⋊C4C2×D5311yescharacteristic
286C22×C42—531no—
287C2×#107—532no—
288C2×#107C2×D5312yes—
289C2×C4⋊C8—532no—
290C2×#102—532no—
291C2×#102—532no—
292C2×C22⋊C8—531no—
293C2×C22⋊C8C2×D5311yescharacteristic
294#390—531no—
295#389—531no—
296#388—1061no—
297#386—1061no—
298#384—532no—
299#384—532no—
300#384—1061no—
301#384—1061no—
302#380C2×D5311yescharacteristic
303C4×C4⋊C4—531no—
304C4×C22⋊C4—1061no—
305#340—532no—
306#340—532no—
307#340—532no—
308#340C2×D5312yes—
309#336—532no—
310C53#2767711yescharacteristic, complemented, Sylow
311C22×D4—531no—
312C22×D4—1061no—
313C23×C4—21no—
314C23×C4—531no—
315C23×C4—531no—
316C23×C4—1061no—
317C23×C4—1061no—
318C23×C4C22×D5311yescharacteristic
319C22×C8—532no—
320C22×C8C22×D5312yes—
321#121—534no—
322#121—1062no—
323C4×D4—1064no—
324C4×D4—2124no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—531no—
328C2×C4⋊C4—532no—
329C2×C4⋊C4—1062no—
330C2×C4⋊C4—1062no—
331C2×C4⋊C4—1062no—
332C2×C4⋊C4—1062no—
333C2×C4⋊C4—1061no—
334C2×C4⋊C4—1061no—
335C2×C4⋊C4—2122no—
336C2×C4⋊C4—2124no—
337C2×C4⋊C4—2124no—
338C2×C4⋊C4C22×D5312yes—
339C2×C4⋊C4C22×D5311yescharacteristic
340C2×C4⋊C4C22×D5311yescharacteristic
341C2×C22⋊C4—21no—
342C2×C22⋊C4—1061no—
343C2×C22⋊C4—1061no—
344C2×C22⋊C4—1061no—
345C2×C22⋊C4—1061no—
346C2×C42—531no—
347C2×C42—531no—
348C2×C42—531no—
349C2×C42—531no—
350C2×C42—1064no—
351C2×C42—1061no—
352#107—42no—
353#107—2122no—
354C4⋊C8—1064no—
355#102—2122no—
356#102—2122no—
357C22⋊C8—22no—
358C22⋊C8—1062no—
359#95—21no—
360#95—1061no—
361#95—1061no—
362C24—1061no—
363C24D4×D5311yescharacteristic
364C2×D4—532no—
365C2×D4—1064no—
366C2×D4—1062no—
367C2×D4—1061no—
368C2×D4—1062no—
369C2×D4—2122no—
370C2×D4—2122no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—531no—
380C22×C4—534no—
381C22×C4—531no—
382C22×C4—531no—
383C22×C4—531no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1062no—
388C22×C4—1062no—
389C22×C4—1061no—
390C22×C4—1062no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4—2122no—
394C22×C4—2122no—
395C22×C4—2122no—
396C22×C4—2122no—
397C22×C4#1171994512yes—
398C22×C4D4×D5311yescharacteristic
399C22×C4Q8×D5312yes—
400C22×C4C23×D5311yescharacteristic
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—1062no—
405C2×C8—1062no—
406C2×C8—1062no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—42no—
412C4⋊C4—532no—
413C4⋊C4—1061no—
414C4⋊C4—1062no—
415C4⋊C4—1062no—
416C4⋊C4—2122no—
417C4⋊C4—2124no—
418C4⋊C4—2124no—
419C4⋊C4—2124no—
420C4⋊C4—2122no—
421C4⋊C4—2124no—
422C22⋊C4—42no—
423C22⋊C4—1064no—
424C22⋊C4—2122no—
425C22⋊C4—2122no—
426C22⋊C4—2122no—
427C42—534no—
428C42—1062no—
429C42—1062no—
430C42—1062no—
431C42—2122no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—1062no—
437C23—2121no—
438C23—2121no—
439C23—2121no—
440C23—2121no—
441C23—2121no—
442C23—2121no—
443C23#1313319512yes—
444C23D8×D5312yes—
445C23#1313324912yes—
446C23C2×D4×D5311yescharacteristic
447D4—1064no—
448D4—2128no—
449D4—2122no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—84no—
464C2×C4—532no—
465C2×C4—532no—
466C2×C4—532no—
467C2×C4—532no—
468C2×C4—532no—
469C2×C4—532no—
470C2×C4—1068no—
471C2×C4—1068no—
472C2×C4—1061no—
473C2×C4—1061no—
474C2×C4—1061no—
475C2×C4—1061no—
476C2×C4—1061no—
477C2×C4—1061no—
478C2×C4—1062no—
479C2×C4—4244no—
480C2×C4—4244no—
481C2×C4—2122no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4#1313319512yes—
487C2×C4C2×Q8×D5311yescharacteristic
488C2×C4#41×D5312yes—
489C2×C4C2×#1171994511yescharacteristic
490C2×C4#1313326212yes—
491C8—42no—
492C8—2122no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—4241no—
500C22—4241no—
501C22—4241no—
502C22—4241no—
503C22—2122no—
504C22—2122no—
505C22—2122no—
506C22#1681090214yes—
507C22C2×#1313319511yescharacteristic
508C22#123×D5311yescharacteristic
509C22#1681118811yescharacteristic
510C22C2×D8×D5311yescharacteristic
511C22AGL(1,ℤ/8)×D5311yescharacteristic
512C22#137×D5311yescharacteristic
513C22C2×#1313324911yescharacteristic
514C4—22no—
515C4—81no—
516C4—82no—
517C4—81no—
518C4—534no—
519C4—1062no—
520C4—1062no—
521C4—1062no—
522C4—4242no—
523C4—2122no—
524C4#1681132212yes—
525C4#1681132712yes—
526C2—24no—
527C2—4242no—
528C2#2027903111yescharacteristic
529C2C2×#1681090211yescharacteristic
530C2#2028452811yescharacteristic
531C2#480×D5311yescharacteristic
532C2#2028464211yescharacteristic
533C2#2028563111yescharacteristic
534C2#2028574511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1852C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1852D53
hypocenterC53#27677
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
2. C2×#16810492C2semidirect
3. #20276275C2semidirect
4. #20276240C2semidirect
5. #20275470C2semidirect
8. #20274902C2semidirect
9. #20274714C2semidirect
10. #20274693C2semidirect
11. C53×#1852C2semidirect
267. #1852D53semidirect
310. C53#27677semidirect

Maximal subgroups

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

subgroupindex
#202772212
C2×#168104922
#202762752
#202762402
#202754702
#202750322
#202750302
#202749022
#202747142
#202746932
C53×#18522
#2767753