Finite Group Databasebeta

← #27465451

Subgroups — #27465451

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133191C211yesmaximal, characteristic
2#20276141C212yesmaximal, complemented
3#20275897C212yesmaximal
4#20275471C211yesmaximal, characteristic, complemented
5#20275471C211yesmaximal, characteristic, complemented
6#20275335C212yesmaximal, complemented
7#20275286C212yesmaximal, complemented
8#20274712C211yesmaximal, characteristic, complemented
9#20274712C211yesmaximal, characteristic, complemented
10#20274695C211yesmaximal, characteristic, complemented
11C53×#2285C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×C4×D212C2211yescharacteristic
14C2×C4×C22⋊D53—21no—
15C2×#13133150C2212yes—
16C2×#13133150C2212yes—
17C2×C4⋊#88322C2211yescharacteristic
18C2×C4⋊#88322C2211yescharacteristic
19C2×#13133137C2212yes—
20C2×#13133137C2212yes—
21C2×C22⋊#88322C2212yes—
22#16810525C2211yescharacteristic
23#16810521C2211yescharacteristic
24#16810514—21no—
25#16810502—21no—
26#16810483C2211yescharacteristic
27C4×C4⋊Dic53C2211yescharacteristic
28C4×#11719921—21no—
29#16810385C2212yes—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810385C2212yes—
33#16810378C2212yes—
34C2×#117×C53—21no—
35C22×C42×C53C2211yescharacteristic
36C53×#384C2211yescharacteristic
37C53×#384C2211yescharacteristic
38C53×#382—21no—
39C53×#380—21no—
40C53×#380C2212yes—
41C4×C4⋊C4×C53—21no—
42C4×C4⋊C4×C53C2212yes—
43C22×D212C2311yescharacteristic
44C22×C22⋊D53—21no—
45C22×C4×D53—21no—
46C22×#88322C2312yes—
47C22×#88322C2312yes—
48#13133191—22no—
49#13133191—22no—
50C4×D212—22no—
51C4×D212—22no—
52C4×C22⋊D53—44no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2311yescharacteristic
56C2×#11719919—21no—
57C2×C22⋊Dic53—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—21no—
61C2×#11719921—21no—
62C2×C4×Dic53—21no—
63#13133150—24no—
64#13133150—42no—
65C4⋊#88322—22no—
66C4⋊#88322—22no—
67#13133137—24no—
68#13133137—42no—
69C22⋊#88322—42no—
70#13133124—21no—
71#13133124—21no—
72#13133124—21no—
73C23×C4×C53D411yescharacteristic
74C23×C4×C53D411yescharacteristic
75C23×C4×C53C2311yescharacteristic
76#117×C53—44no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53C2312yes—
82C2×C4⋊C4×C53C2312yes—
83C2×C22⋊C4×C53—22no—
84C2×C42×C53—22no—
85C2×C42×C53—22no—
86C2×C42×C53—22no—
87C2×C42×C53D411yescharacteristic
88C2×C42×C53D411yescharacteristic
89C2×C42×C53C2311yescharacteristic
90C2×C42×C53C2311yescharacteristic
91#95×C53—22no—
92#95×C53—22no—
93C24×C53C2×D411yescharacteristic
94C23×D53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—41no—
107C22×C4×C53—41no—
108C22×C4×C53D814yes—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2411yescharacteristic
115C2×C22⋊D53—24no—
116C2×C22⋊D53—42no—
117C2×C22⋊D53—42no—
118C22×Dic53—21no—
119C22×Dic53—21no—
120C22×Dic53—21no—
121C2×D212—21no—
122C2×D212—21no—
123C2×D212—21no—
124C2×D212—21no—
125C2×D212—21no—
126C2×D212#4112yes—
127C2×C4×D53—42no—
128C2×C4×D53—42no—
129C2×C4×D53—42no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—22no—
134C4⋊C4×C53—42no—
135C4⋊C4×C53—42no—
136C4⋊C4×C53—42no—
137C4⋊C4×C53—42no—
138C22⋊C4×C53—44no—
139C42×C53—22no—
140C42×C53—22no—
141C42×C53—21no—
142C42×C53—21no—
143C42×C53—44no—
144C42×C53#4112yes—
145C42×C53#4112yes—
146C22⋊Dic53—42no—
147#11719921—24no—
148#11719921—42no—
149#11719921—42no—
150#11719921—42no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—21no—
155C4⋊Dic53—21no—
156C4⋊Dic53#4112yes—
157#11719919—42no—
158C4×Dic53—42no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—22no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—22no—
165C23×C53—21no—
166C23×C53—21no—
167C23×C53—21no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53C2×D812yes—
172C23×C53C22×D411yescharacteristic
173C22×D53—22no—
174C22×D53—41no—
175C22×D53—41no—
176C22×D53—41no—
177C22×D53—41no—
178C22×D53—41no—
179C22×D53—41no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—28no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—44no—
192C2×C4×C53—41no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53#12312yes—
203C2×C4×C53#12312yes—
204C2×C4×C53C2×D812yes—
205C2×C4×C53C2×D812yes—
206C2×C4×C53#13712yes—
207C2×C4×C53C2×#4111yescharacteristic
208C2×C4×C53C2×#4111yescharacteristic
209C2×C4×C53C2×#4111yescharacteristic
210C2×C4×C53C2×#4111yescharacteristic
211C2×C4×C53H(5,𝔽2)11yescharacteristic
212C2×C4×C532−1+411yescharacteristic
213C22⋊D53—48no—
214C2×Dic53—22no—
215C2×Dic53—41no—
216C2×Dic53—41no—
217C2×Dic53—41no—
218C2×Dic53—41no—
219C2×Dic53—41no—
220C2×Dic53—41no—
221C2×Dic53—41no—
222C2×Dic53—41no—
223D212—22no—
224D212—22no—
225D212—41no—
226D212—41no—
227C4×D53—84no—
228#88322—42no—
229#88322—42no—
230#27624—531nomaximal, Sylow
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—41no—
235C22×C53—41no—
236C22×C53—41no—
237C22×C53—41no—
238C22×C53C2×#12311yescharacteristic
239C22×C53#52511yescharacteristic
240C22×C53#53711yescharacteristic
241C22×C53C22×D811yescharacteristic
242C22×C53C2×#13711yescharacteristic
243C22×C53#57511yescharacteristic
244C22×C53#57511yescharacteristic
245C2×D53—42no—
246C2×D53—42no—
247C2×D53—42no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C2×D53—81no—
252C4×C53—22no—
253C4×C53—22no—
254C4×C53—41no—
255C4×C53—41no—
256C4×C53—82no—
257C4×C53—82no—
258C4×C53#48014yes—
259C4×C53#55211yescharacteristic
260C4×C53#55311yescharacteristic
261C4×C53#55511yescharacteristic
262C4×C53#55611yescharacteristic
263Dic53—42no—
264Dic53—81no—
265Dic53—81no—
266C4×#121—531no—
267#2285D5311yescharacteristic, complemented
268#2075—532no—
269#2033—532no—
270#1947—531no—
271#1947—531no—
272#1925—532no—
273#1918—532no—
274#1802—531no—
275#1802—531no—
276#1797—531no—
277C2×C53—42no—
278C2×C53#248511yescharacteristic
279C2×C53C2×#48011yescharacteristic
280C2×C53#309311yescharacteristic
281C2×C53#310011yescharacteristic
282C2×C53#311611yescharacteristic
283C2×C53#324511yescharacteristic
284C2×C53#325211yescharacteristic
285D53—82no—
286C2×#121—531no—
287C2×C4×D4—531no—
288C2×C4×D4—1061no—
289C2×#117—21no—
290C22×C42C2×D5311yescharacteristic
291C2×#107—532no—
292C2×#107—532no—
293C2×C4⋊C8—531no—
294C2×C4⋊C8—531no—
295C2×#102—532no—
296C2×#102—532no—
297C2×C22⋊C8—532no—
298#390—531no—
299#389—531no—
300#388—1061no—
301#386—1061no—
302#384C2×D5311yescharacteristic
303#384C2×D5311yescharacteristic
304#382—21no—
305#380—21no—
306#380—531no—
307#380C2×D5312yes—
308C4×C4⋊C4—21no—
309C4×C4⋊C4—531no—
310C4×C4⋊C4C2×D5312yes—
311C4×C22⋊C4—1061no—
312#340—532no—
313#340—532no—
314#340—532no—
315#340—532no—
316#336—532no—
317C53#2762411yescharacteristic, complemented, Sylow
318C22×D4—531no—
319C22×D4—1061no—
320C23×C4—1061no—
321C23×C4C22⋊D5311yescharacteristic
322C23×C4C22⋊D5311yescharacteristic
323C23×C4C22×D5311yescharacteristic
324C22×C8—532no—
325C22×C8—532no—
326#121—1062no—
327#121—1062no—
328C4×D4—1062no—
329C4×D4—1062no—
330C4×D4—2124no—
331#117—44no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—531no—
337C2×C4⋊C4—531no—
338C2×C4⋊C4—531no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4C22×D5312yes—
341C2×C4⋊C4C22×D5312yes—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—1061no—
344C2×C22⋊C4—1061no—
345C2×C22⋊C4—1061no—
346C2×C22⋊C4—1061no—
347C2×C22⋊C4—1061no—
348C2×C42—22no—
349C2×C42—22no—
350C2×C42—22no—
351C2×C42—1061no—
352C2×C42C22⋊D5311yescharacteristic
353C2×C42C22⋊D5311yescharacteristic
354C2×C42C22×D5311yescharacteristic
355C2×C42C22×D5311yescharacteristic
356#107—1064no—
357#107—2122no—
358C4⋊C8—1062no—
359C4⋊C8—1062no—
360#102—1064no—
361#102—2122no—
362C22⋊C8—2122no—
363#95—22no—
364#95—22no—
365#95—1061no—
366#95—1061no—
367#95—1061no—
368C24—1061no—
369C24C2×C22⋊D5311yescharacteristic
370C2×D4—532no—
371C2×D4—1064no—
372C2×D4—1061no—
373C2×D4—1061no—
374C2×D4—1061no—
375C2×D4—1061no—
376C2×D4—1061no—
377C2×D4—2122no—
378C2×D4—2122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—1061no—
393C22×C4—1061no—
394C22×C4—1061no—
395C22×C4—2122no—
396C22×C4—2122no—
397C22×C4—2122no—
398C22×C4#1171992214yes—
399C22×C4C2×C22⋊D5311yescharacteristic
400C22×C4C2×C22⋊D5311yescharacteristic
401C22×C4C2×C22⋊D5311yescharacteristic
402C22×C4C2×C22⋊D5311yescharacteristic
403C22×C4C2×C22⋊D5311yescharacteristic
404C22×C4C23×D5311yescharacteristic
405C2×C8—1062no—
406C2×C8—1062no—
407C2×C8—1062no—
408C2×C8—1062no—
409C2×C8—1062no—
410C2×C8—1062no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—532no—
420C4⋊C4—1061no—
421C4⋊C4—1061no—
422C4⋊C4—1061no—
423C4⋊C4—1061no—
424C4⋊C4—1061no—
425C4⋊C4—2122no—
426C22⋊C4—44no—
427C22⋊C4—1064no—
428C22⋊C4—2122no—
429C22⋊C4—2122no—
430C22⋊C4—2122no—
431C22⋊C4—2122no—
432C42—22no—
433C42—22no—
434C42—21no—
435C42—21no—
436C42—44no—
437C42—2122no—
438C42#1171994512yes—
439C42#1171994512yes—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—1062no—
447C23—2121no—
448C23—2121no—
449C23—2121no—
450C23—2121no—
451C23—2121no—
452C23—2121no—
453C23C2×#1171992212yes—
454C23C22×C22⋊D5311yescharacteristic
455D4—1062no—
456D4—1062no—
457D4—2128no—
458D4—2121no—
459D4—2121no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—28no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—1062no—
483C2×C4—4244no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4—2121no—
489C2×C4—2121no—
490C2×C4—2121no—
491C2×C4—2121no—
492C2×C4#1313320012yes—
493C2×C4#1313320012yes—
494C2×C4C2×#1171992212yes—
495C2×C4C2×#1171992212yes—
496C2×C4#1313324812yes—
497C2×C4#41×D5311yescharacteristic
498C2×C4#41×D5311yescharacteristic
499C2×C4C2×#1171994511yescharacteristic
500C2×C4C2×#1171994511yescharacteristic
501C2×C4#1313326311yescharacteristic
502C2×C4#1313326411yescharacteristic
503C8—2122no—
504C8—2122no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—4241no—
513C22—4241no—
514C22—4241no—
515C22—4241no—
516C22—2122no—
517C22—2122no—
518C22—2122no—
519C22C2×#1313320011yescharacteristic
520C22#1681113011yescharacteristic
521C22#1681120111yescharacteristic
522C22C22×#1171992211yescharacteristic
523C22C2×#1313324811yescharacteristic
524C22#1681142111yescharacteristic
525C22#1681142111yescharacteristic
526C4—22no—
527C4—22no—
528C4—41no—
529C4—41no—
530C4—82no—
531C4—82no—
532C4—4241no—
533C4—4241no—
534C4—2122no—
535C4#1681093614yes—
536C4#1681131211yescharacteristic
537C4#1681131711yescharacteristic
538C4#1681133611yescharacteristic
539C4#1681134111yescharacteristic
540C2—42no—
541C2—4242no—
542C2#2027900611yescharacteristic
543C2C2×#1681093611yescharacteristic
544C2#2028464611yescharacteristic
545C2#2028471111yescharacteristic
546C2#2028487111yescharacteristic
547C2#2028602211yescharacteristic
548C2#2028608211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2285C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#2285D53
hypocenterC53#27624
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
2. #20276141C2semidirect
4. #20275471C2semidirect
5. #20275471C2semidirect
6. #20275335C2semidirect
7. #20275286C2semidirect
8. #20274712C2semidirect
9. #20274712C2semidirect
10. #20274695C2semidirect
11. C53×#2285C2semidirect
267. #2285D53semidirect
317. C53#27624semidirect

Maximal subgroups

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

subgroupindex
C4×#131331912
#202761412
#202758972
#202754712
#202753352
#202752862
#202747122
#202746952
C53×#22852
#2762453