Finite Group Databasebeta

← #27465213

Subgroups — #27465213

Order 13568.

1124 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810483C211yesmaximal, characteristic
2#20276141C214yesmaximal, complemented
3#20274903C212yesmaximal, complemented
4#20274902C212yesmaximal, complemented
5#20274801C212yesmaximal, complemented
6#20274799C212yesmaximal, complemented
7#20274625C211yesmaximal, characteristic
8C2×C53×#380C211yesmaximal, characteristic
9C22×C4⋊Dic53C2211yescharacteristic
10C22×C4⋊Dic53C2211yescharacteristic
11C22×C22⋊Dic53—21no—
12C2×#13133150C2214yescomplemented
13C2×C22⋊#88322C2212yes—
14C2×C22⋊#88322C2212yes—
15C2×C22⋊#88322C2212yes—
16#16810483—21no—
17#16810483—21no—
18#16810483C2212yes—
19#16810483C2212yes—
20C2×#13133124—21no—
21#16810385C2214yescomplemented
22#16810385C2214yescomplemented
23#16810385C2214yescomplemented
24C24×C4×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C22×C4⋊C4×C53C2211yescharacteristic
27C22×C22⋊C4×C53—21no—
28C53×#380—21no—
29C53×#380—21no—
30C53×#380C2212yes—
31C53×#380C2212yes—
32C2×#95×C53—21no—
33C23×Dic53—21no—
34C23×Dic53—21no—
35C22×#88322C2314yes—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—24no—
39C2×C4⋊Dic53—42no—
40C2×C4⋊Dic53C2312yes—
41C2×C4⋊Dic53C2312yes—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—22no—
44C2×C22⋊Dic53—44no—
45#13133150—44no—
46C22⋊#88322—24no—
47C22⋊#88322—24no—
48C22⋊#88322—42no—
49#13133124—22no—
50#13133124—22no—
51C25×C53D411yescharacteristic
52C23×C4×C53—24no—
53C23×C4×C53—24no—
54C23×C4×C53—21no—
55C23×C4×C53—21no—
56C23×C4×C53—42no—
57C23×C4×C53D411yescharacteristic
58C23×C4×C53D411yescharacteristic
59C23×C4×C53D411yescharacteristic
60C23×C4×C53C2311yescharacteristic
61C23×C4×C53C2311yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—24no—
66C2×C4⋊C4×C53—42no—
67C2×C4⋊C4×C53C2312yes—
68C2×C4⋊C4×C53C2312yes—
69C2×C22⋊C4×C53—22no—
70C2×C22⋊C4×C53—22no—
71C2×C22⋊C4×C53—44no—
72#95×C53—22no—
73#95×C53—22no—
74C24×C53—22no—
75C24×C53—22no—
76C24×C53—22no—
77C24×C53—22no—
78C24×C53—41no—
79C24×C53—41no—
80C24×C53D814yes—
81C24×C53C2×D411yescharacteristic
82C24×C53C2×D411yescharacteristic
83C24×C53C2×D411yescharacteristic
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—22no—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53—416no—
103C22×C4×C53—42no—
104C22×C4×C53—42no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53#4114yes—
113C22×C4×C53#4114yes—
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C22×Dic53—42no—
120C22×Dic53—42no—
121C22×Dic53—42no—
122C22×Dic53—42no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53—42no—
126C4⋊C4×C53—42no—
127C22⋊C4×C53—44no—
128C22⋊C4×C53—44no—
129C22⋊Dic53—44no—
130C22⋊Dic53—44no—
131C4⋊Dic53—24no—
132C4⋊Dic53—24no—
133C4⋊Dic53—42no—
134C4⋊Dic53—42no—
135C2×#88322—24no—
136C2×#88322—24no—
137C2×#88322—24no—
138C23×C53—21no—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—28no—
153C23×C53—28no—
154C23×C53—44no—
155C23×C53—44no—
156C23×C53—44no—
157C23×C53—44no—
158C23×C53—44no—
159C23×C53#12312yes—
160C23×C53#12312yes—
161C23×C53C2×D812yes—
162C23×C53C2×D812yes—
163C23×C53C2×D812yes—
164C23×C53#13712yes—
165C23×C53#13712yes—
166C23×C53C22×D411yescharacteristic
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—28no—
172C2×C4×C53—28no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—44no—
177C2×C4×C53—44no—
178C2×C4×C53—44no—
179C2×C4×C53—44no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—84no—
187C2×C4×C53—84no—
188C2×C4×C53#12312yes—
189C2×C4×C53#12312yes—
190C2×C4×C53C2×#4112yes—
191C2×C4×C53C2×#4112yes—
192C2×C4×C53H(5,𝔽2)12yes—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—84no—
200C2×Dic53—84no—
201#88322—44no—
202#27600—531nomaximal, Sylow
203C22×C53—22no—
204C22×C53—22no—
205C22×C53—22no—
206C22×C53—22no—
207C22×C53—22no—
208C22×C53—22no—
209C22×C53—22no—
210C22×C53—22no—
211C22×C53—22no—
212C22×C53—22no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—416no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53#48014yes—
224C22×C53#48014yes—
225C22×C53C2×#12311yescharacteristic
226C22×C53C2×#12311yescharacteristic
227C22×C53#53411yescharacteristic
228C22×C53C22×D811yescharacteristic
229C22×C53C2×#13711yescharacteristic
230C22×C53C2×#13711yescharacteristic
231C22×C53#57511yescharacteristic
232C4×C53—22no—
233C4×C53—22no—
234C4×C53—41no—
235C4×C53—82no—
236C4×C53—82no—
237C4×C53#55114yes—
238Dic53—82no—
239Dic53—82no—
240C2×#380—531no—
241C2×#380D5311yescharacteristic
242#2075—534no—
243#1852—532no—
244#1852—532no—
245#1823—532no—
246#1823—532no—
247#1779—531no—
248C2×C53—24no—
249C2×C53—24no—
250C2×C53—42no—
251C2×C53#248311yescharacteristic
252C2×C53C2×#48011yescharacteristic
253C2×C53C2×#48011yescharacteristic
254C2×C53#309511yescharacteristic
255C2×C53#309511yescharacteristic
256C2×C53#318611yescharacteristic
257C2×C53#319511yescharacteristic
258C24×C4C2×D5311yescharacteristic
259C22×C4⋊C4—531no—
260C22×C4⋊C4—531no—
261C22×C4⋊C4C2×D5311yescharacteristic
262C22×C4⋊C4C2×D5311yescharacteristic
263C22×C22⋊C4—21no—
264C22×C22⋊C4—1061no—
265C2×#107—534no—
266C2×C22⋊C8—532no—
267C2×C22⋊C8—532no—
268C2×C22⋊C8—532no—
269#380—21no—
270#380—21no—
271#380—532no—
272#380—532no—
273#380—1061no—
274#380—1061no—
275#380C2×D5312yes—
276#380C2×D5312yes—
277C2×#95—21no—
278C2×#95—1061no—
279#340—534no—
280#340—534no—
281#340—534no—
282C53#2760011yescharacteristic, complemented, Sylow
283C25C22⋊D5311yescharacteristic
284C23×C4—24no—
285C23×C4—24no—
286C23×C4—21no—
287C23×C4—21no—
288C23×C4—42no—
289C23×C4—1061no—
290C23×C4—1061no—
291C23×C4C22⋊D5311yescharacteristic
292C23×C4C22⋊D5311yescharacteristic
293C23×C4C22⋊D5311yescharacteristic
294C23×C4C22×D5311yescharacteristic
295C23×C4C22×D5311yescharacteristic
296C23×C4C22×D5311yescharacteristic
297C22×C8—534no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—532no—
303C2×C4⋊C4—532no—
304C2×C4⋊C4—1064no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—2122no—
308C2×C4⋊C4C22×D5312yes—
309C2×C4⋊C4C22×D5312yes—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—1062no—
314C2×C22⋊C4—1062no—
315C2×C22⋊C4—2124no—
316#107—2124no—
317C22⋊C8—1064no—
318C22⋊C8—1064no—
319C22⋊C8—2122no—
320#95—22no—
321#95—22no—
322#95—1062no—
323#95—1062no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—41no—
329C24—41no—
330C24#1171992214yes—
331C24C2×C22⋊D5311yescharacteristic
332C24C2×C22⋊D5311yescharacteristic
333C24C2×C22⋊D5311yescharacteristic
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—416no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—1062no—
360C22×C4—1062no—
361C22×C4—2122no—
362C22×C4—2122no—
363C22×C4—2122no—
364C22×C4—2122no—
365C22×C4—2122no—
366C22×C4—2122no—
367C22×C4#1171994514yes—
368C22×C4#1171994714yes—
369C22×C4C2×C22⋊D5311yescharacteristic
370C22×C4C2×C22⋊D5311yescharacteristic
371C22×C4C2×C22⋊D5311yescharacteristic
372C22×C4C23×D5311yescharacteristic
373C2×C8—1064no—
374C2×C8—1064no—
375C2×C8—1064no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—1064no—
381C4⋊C4—1064no—
382C4⋊C4—2122no—
383C4⋊C4—2122no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—2124no—
387C22⋊C4—2124no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—28no—
403C23—28no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23#1313319812yes—
410C23#1313320012yes—
411C23C2×#1171992212yes—
412C23C2×#1171992212yes—
413C23C2×#1171992212yes—
414C23#1313324712yes—
415C23#1313324812yes—
416C23C22×C22⋊D5311yescharacteristic
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—28no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—4244no—
439C2×C4—4244no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4—2122no—
446C2×C4#1313319812yes—
447C2×C4#1313320012yes—
448C2×C4C2×#1171994712yes—
449C2×C4C2×#1171994512yes—
450C2×C4#1313326212yes—
451C8—2124no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—416no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22#1681093414yes—
473C22#1681093614yes—
474C22C2×#1313319811yescharacteristic
475C22C2×#1313320011yescharacteristic
476C22#1681118211yescharacteristic
477C22C22×#1171992211yescharacteristic
478C22C2×#1313324711yescharacteristic
479C22C2×#1313324811yescharacteristic
480C22#1681142111yescharacteristic
481C4—22no—
482C4—22no—
483C4—41no—
484C4—82no—
485C4—82no—
486C4—4242no—
487C4—4242no—
488C4#1681129914yes—
489C2—24no—
490C2—24no—
491C2—42no—
492C2#2027899311yescharacteristic
493C2C2×#1681093411yescharacteristic
494C2C2×#1681093611yescharacteristic
495C2#2028465911yescharacteristic
496C2#2028466311yescharacteristic
497C2#2028546311yescharacteristic
498C2#2028553611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#380C2
socleC23×C53C22×D4
prime radicalC25×C53D4
hypercenterC2×#380D53
hypocenterC53#27600
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. #20276141C2semidirect
3. #20274903C2semidirect
4. #20274902C2semidirect
5. #20274801C2semidirect
6. #20274799C2semidirect
12. C2×#13133150C22semidirect
21. #16810385C22semidirect
22. #16810385C22semidirect
23. #16810385C22semidirect
282. C53#27600semidirect

Maximal subgroups

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

subgroupindex
C2×#168104832
#202761412
#202749032
#202749022
#202748012
#202747992
#202746252
C2×C53×#3802
#2760053