Finite Group Databasebeta

← #27393606

Subgroups — #27393606

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D53C211yesmaximal, characteristic
2C2×#16810679C211yesmaximal, characteristic
3#20277119C211yesmaximal, characteristic
4#20275682C212yesmaximal, complemented
5#20275682C212yesmaximal, complemented
6#20275673C211yesmaximal, characteristic, complemented
7#20275672C211yesmaximal, characteristic, complemented
8#20275670C211yesmaximal, characteristic, complemented
9#20274804C212yesmaximal
10#20274800C212yesmaximal
11C53×#1989C211yesmaximal, characteristic, complemented
12C2×#13133209C2211yescharacteristic
13C2×#13133208C2211yescharacteristic
14C2×#13133208C2211yescharacteristic
15C2×#13133207—21no—
16C2×#13133207—21no—
17#125×D53—24no—
18#125×D53C418yescomplemented
19C2×#13133174—21no—
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C42×D53C2211yescharacteristic
25#16810679—24no—
26C2×C4⋊#88322C2211yescharacteristic
27C2×#13133129C2212yes—
28C2×#13133127C2212yes—
29#16810492C2212yes—
30#16810487—21no—
31#16810482—21no—
32#16810480C2212yes—
33C4×#11719919—21no—
34C4×C4⋊Dic53C2211yescharacteristic
35#16810386C2212yes—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810385C2212yes—
40#16810385C2212yes—
41C2×#125×C53C2211yescharacteristic
42C2×C4⋊C8×C53C2211yescharacteristic
43C4×C4⋊C4×C53C2211yescharacteristic
44C53×#340C2212yes—
45C53×#340C2212yes—
46C22×C4×D53—21no—
47C22×C4×D53—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C22×#88322C2312yes—
52#13133209—22no—
53#13133209C2×C414yes—
54#13133208—22no—
55#13133208—22no—
56#13133208C2×C414yes—
57#13133208C2×C414yes—
58#13133207—28no—
59#13133207—28no—
60#13133174—44no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53—21no—
63C2×C4⋊Dic53C2312yes—
64C2×C4⋊Dic53C2311yescharacteristic
65C2×C4⋊Dic53C2311yescharacteristic
66C2×#11719919—21no—
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—22no—
73C2×#11719919—22no—
74C4⋊C4×D53—28no—
75C4⋊C4×D53—28no—
76C4⋊C4×D53—22no—
77C4⋊C4×D53—22no—
78C4⋊C4×D53C2×C414yes—
79C4⋊C4×D53C2×C414yes—
80C2×#11719921—22no—
81C2×C4×Dic53—21no—
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53D411yescharacteristic
84C2×C4×Dic53C2311yescharacteristic
85C42×D53C2×C414yes—
86C42×D53D414yes—
87C4⋊#88322—22no—
88#13133129—24no—
89#13133127—24no—
90#13133124—21no—
91#13133124—22no—
92C22×C8×C53C2312yes—
93#125×C53—22no—
94#125×C53C2×C414yes—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53C2312yes—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C4⋊C4×C53C2311yescharacteristic
100C2×C42×C53—21no—
101C2×C42×C53C2311yescharacteristic
102C4⋊C8×C53—22no—
103#95×C53—21no—
104C23×D53C2×D411yescharacteristic
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2411yescharacteristic
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—22no—
114C22×Dic53C2×D411yescharacteristic
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C2×C4×D53—22no—
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53—22no—
121C2×C4×D53—24no—
122C2×C4×D53—24no—
123C2×C4×D53—42no—
124C2×C4×D53—44no—
125C2×C4×D53—42no—
126C2×C4×D53—42no—
127C2×C4×D53C22⋊C414yes—
128C2×C4×D53C22⋊C414yes—
129C2×C4×D53C22×C412yes—
130C2×C4×D53C2×D412yes—
131C2×C8×C53—22no—
132C2×C8×C53—24no—
133C4⋊C4×C53—24no—
134C4⋊C4×C53—24no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53C22×C412yes—
140C4⋊C4×C53C22×C412yes—
141C42×C53—42no—
142C42×C53C22×C412yes—
143C42×C53C2×D412yes—
144#11719921—44no—
145C4⋊Dic53—21no—
146C4⋊Dic53—21no—
147C4⋊Dic53—24no—
148C4⋊Dic53—24no—
149C4⋊Dic53—22no—
150C4⋊Dic53—22no—
151C4⋊Dic53C22×C412yes—
152C4⋊Dic53C22×C412yes—
153#11719919—24no—
154#11719919—24no—
155#11719919—24no—
156#11719919—24no—
157#11719919—24no—
158#11719919—24no—
159#11719919—44no—
160#11719919—44no—
161C4×Dic53—42no—
162C4×Dic53C22⋊C414yes—
163C4×Dic53C22⋊C414yes—
164C4×Dic53C22×C412yes—
165C4×Dic53C2×D412yes—
166C2×#88322—22no—
167C2×#88322—24no—
168C23×C53C22×D411yescharacteristic
169C22×D53—22no—
170C22×D53—22no—
171C22×D53C2×C22⋊C412yes—
172C22×D53AGL(1,ℤ/8)12yes—
173C22×D53#13712yes—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53C2×C22⋊C412yes—
186C2×C4×C53C2×C22⋊C412yes—
187C2×C4×C53C23×C411yescharacteristic
188C2×C4×C53C22×D411yescharacteristic
189C2×C4×C532−1+412yes—
190C2×C4×C532−1+412yes—
191C2×Dic53—22no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—22no—
195C2×Dic53—22no—
196C2×Dic53—22no—
197C2×Dic53—41no—
198C2×Dic53—41no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—42no—
202C2×Dic53—42no—
203C2×Dic53C2×C22⋊C412yes—
204C2×Dic53C2×C22⋊C412yes—
205C2×Dic53C2×C22⋊C412yes—
206C4×D53—28no—
207C4×D53—24no—
208C4×D53—22no—
209C4×D53—22no—
210C4×D53—44no—
211C4×D53—48no—
212C4×D53—48no—
213C8×C53—42no—
214#88322—42no—
215#21838—531nomaximal, Sylow
216C22×C53C22×C22⋊C411yescharacteristic
217C22×C53#53612yes—
218C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
219C22×C53C2×#13711yescharacteristic
220C22×C53#57512yes—
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—24no—
224C2×D53—24no—
225C2×D53#41714yes—
226C4×C53—22no—
227C4×C53—22no—
228C4×C53—42no—
229C4×C53—42no—
230C4×C53—82no—
231C4×C53#51912yes—
232C4×C53#53611yescharacteristic
233C4×C53#53611yescharacteristic
234Dic53—22no—
235Dic53—22no—
236Dic53—22no—
237Dic53—42no—
238Dic53—42no—
239Dic53—82no—
240C22×#125—531no—
241C2×#432—531no—
242#2285—531no—
243#1992—532no—
244#1992—532no—
245#1989—531no—
246#1989—531no—
247#1989—531no—
248#1989D5311yescharacteristic, complemented
249#1824—532no—
250#1823—532no—
251C2×C53#231011yescharacteristic
252C2×C53C2×#41711yescharacteristic
253C2×C53#289611yescharacteristic
254C2×C53#323412yes—
255C2×C53#323612yes—
256D53—24no—
257C2×#125—538no—
258C2×#125—531no—
259C2×#125—531no—
260C2×#125—531no—
261C2×#125—1061no—
262C2×#125—1061no—
263C2×#125—1064no—
264C2×#125C2×D5311yescharacteristic
265C2×#117—1061no—
266C22×C4⋊C4—531no—
267C22×C4⋊C4—531no—
268C22×C4⋊C4—1061no—
269C22×C4⋊C4—1061no—
270C22×C42—531no—
271#432—1064no—
272C2×C4⋊C8—531no—
273C2×C4⋊C8C2×D5311yescharacteristic
274C2×C22⋊C8—532no—
275C2×#97—532no—
276#384—532no—
277#382—1061no—
278#380—532no—
279#380—1061no—
280C4×C4⋊C4—531no—
281C4×C4⋊C4—1061no—
282C4×C4⋊C4C2×D5311yescharacteristic
283#340—532no—
284#340—532no—
285#340—532no—
286#340—532no—
287#340—532no—
288#340—532no—
289#340C2×D5312yes—
290#340C2×D5312yes—
291C53#2183811yescharacteristic, complemented, Sylow
292C23×C4—531no—
293C23×C4—531no—
294C23×C4—531no—
295C23×C4—1061no—
296C23×C4—1061no—
297C22×C8—532no—
298C22×C8C22×D5312yes—
299#125—22no—
300#125—534no—
301#125—534no—
302#125—534no—
303#125—1068no—
304#125—1068no—
305#125—1062no—
306#125—1062no—
307#125—1062no—
308#125C4×D5314yes—
309#117—2124no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—531no—
313C2×C4⋊C4—531no—
314C2×C4⋊C4—534no—
315C2×C4⋊C4—534no—
316C2×C4⋊C4—532no—
317C2×C4⋊C4—1062no—
318C2×C4⋊C4—1062no—
319C2×C4⋊C4—1068no—
320C2×C4⋊C4—1068no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4—1061no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—1062no—
330C2×C4⋊C4—1062no—
331C2×C4⋊C4C22×D5312yes—
332C2×C4⋊C4C22×D5311yescharacteristic
333C2×C4⋊C4C22×D5311yescharacteristic
334C2×C22⋊C4—1062no—
335C2×C42—21no—
336C2×C42—531no—
337C2×C42—531no—
338C2×C42—531no—
339C2×C42—534no—
340C2×C42—534no—
341C2×C42—1061no—
342C2×C42C22×D5311yescharacteristic
343C4⋊C8—22no—
344C4⋊C8—1062no—
345C22⋊C8—1064no—
346#97—1064no—
347#95—21no—
348#95—1062no—
349#95—1061no—
350C24—531no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—531no—
355C22×C4—531no—
356C22×C4—531no—
357C22×C4—534no—
358C22×C4—532no—
359C22×C4—532no—
360C22×C4—534no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4—1064no—
364C22×C4—1064no—
365C22×C4—1062no—
366C22×C4—1062no—
367C22×C4—1061no—
368C22×C4—1061no—
369C22×C4—1062no—
370C22×C4—2122no—
371C22×C4—2122no—
372C22×C4—2122no—
373C22×C4—2124no—
374C22×C4D4×D5311yescharacteristic
375C22×C4D4×D5311yescharacteristic
376C22×C4C23×D5311yescharacteristic
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—1062no—
380C2×C8—1064no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—532no—
388C4⋊C4—532no—
389C4⋊C4—1064no—
390C4⋊C4—1064no—
391C4⋊C4—1064no—
392C4⋊C4—1064no—
393C4⋊C4—1064no—
394C4⋊C4—1064no—
395C4⋊C4—1064no—
396C4⋊C4—1064no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—1062no—
400C4⋊C4—1062no—
401C4⋊C4—2124no—
402C4⋊C4—2124no—
403C4⋊C4C2×C4×D5312yes—
404C4⋊C4C2×C4×D5312yes—
405C22⋊C4—2124no—
406C42—42no—
407C42—534no—
408C42—534no—
409C42—532no—
410C42—532no—
411C42—2122no—
412C42C2×C4×D5312yes—
413C42D4×D5312yes—
414C23—532no—
415C23—532no—
416C23—532no—
417C23—1062no—
418C23—1062no—
419C23C2×D4×D5311yescharacteristic
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—532no—
432C2×C4—532no—
433C2×C4—532no—
434C2×C4—1068no—
435C2×C4—1061no—
436C2×C4—1061no—
437C2×C4—1062no—
438C2×C4—1062no—
439C2×C4—1064no—
440C2×C4—1062no—
441C2×C4—1062no—
442C2×C4—1062no—
443C2×C4—1062no—
444C2×C4—2124no—
445C2×C4—2128no—
446C2×C4—2128no—
447C2×C4—2122no—
448C2×C4—2122no—
449C2×C4—2122no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4—2122no—
453C2×C4C22⋊C4×D5312yes—
454C2×C4C22⋊C4×D5312yes—
455C2×C4C22×C4×D5311yescharacteristic
456C2×C4C2×D4×D5311yescharacteristic
457C2×C4#1313326412yes—
458C2×C4#1313326512yes—
459C8—42no—
460C8—2122no—
461C22—534no—
462C22—1062no—
463C22—1062no—
464C22—1064no—
465C22—1064no—
466C22C2×C22⋊C4×D5311yescharacteristic
467C22#1681119612yes—
468C22AGL(1,ℤ/8)×D5311yescharacteristic
469C22#137×D5311yescharacteristic
470C22#1681142012yes—
471C4—22no—
472C4—22no—
473C4—42no—
474C4—42no—
475C4—82no—
476C4—1062no—
477C4—1062no—
478C4—1062no—
479C4—4242no—
480C4—2122no—
481C4—2122no—
482C4#1681109912yes—
483C4#1681119311yescharacteristic
484C4#1681119811yescharacteristic
485C2—1064no—
486C2#2027729811yescharacteristic
487C2#417×D5311yescharacteristic
488C2#2028323011yescharacteristic
489C2#2028590312yes—
490C2#2028591912yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1989C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1989D53
hypocenterC53#21838
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #20275682C2semidirect
5. #20275682C2semidirect
6. #20275673C2semidirect
7. #20275672C2semidirect
8. #20275670C2semidirect
11. C53×#1989C2semidirect
18. #125×D53C4semidirect
248. #1989D53semidirect
291. C53#21838semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D532
C2×#168106792
#202771192
#202756822
#202756732
#202756722
#202756702
#202748042
#202748002
C53×#19892
#2183853