Finite Group Databasebeta

← #27393592

Subgroups — #27393592

Order 13568.

1042 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D53C211yesmaximal, characteristic
2C2×#16810675C211yesmaximal, characteristic
3#20277111C211yesmaximal, characteristic
4#20275678C212yesmaximal, complemented
5#20275678C212yesmaximal, 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×C4⋊C4×D53—21no—
20C2×C4⋊C4×D53—21no—
21C2×C4⋊C4×D53C2211yescharacteristic
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C42×D53C2211yescharacteristic
24#16810675—24no—
25C2×C4⋊#88322C2211yescharacteristic
26C2×#13133129C2212yes—
27C2×C8×Dic53C2212yes—
28#16810490—21no—
29#16810488C2212yes—
30#16810485—21no—
31#16810480C2212yes—
32C4×C4⋊Dic53C2211yescharacteristic
33C4×#11719921—21no—
34#16810469—21no—
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—
60C2×C4⋊Dic53—21no—
61C2×C4⋊Dic53—21no—
62C2×C4⋊Dic53C2312yes—
63C2×C4⋊Dic53C2311yescharacteristic
64C2×C4⋊Dic53C2311yescharacteristic
65C2×#11719919—21no—
66C2×#11719919—21no—
67C2×#11719919—21no—
68C2×#11719919—21no—
69C2×#11719919—21no—
70C2×#11719919—21no—
71C4⋊C4×D53—28no—
72C4⋊C4×D53—28no—
73C4⋊C4×D53—22no—
74C4⋊C4×D53—22no—
75C4⋊C4×D53C2×C414yes—
76C4⋊C4×D53C2×C414yes—
77C2×#11719921—22no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53D411yescharacteristic
80C2×C4×Dic53D411yescharacteristic
81C2×C4×Dic53C2311yescharacteristic
82C42×D53C2×C414yes—
83C42×D53D414yes—
84C4⋊#88322—22no—
85#13133129—24no—
86C8×Dic53—24no—
87#13133124—21no—
88#13133124—22no—
89#13133124—22no—
90#13133124—22no—
91C22×C8×C53C2312yes—
92#125×C53—22no—
93#125×C53C2×C414yes—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2311yescharacteristic
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C42×C53—21no—
100C2×C42×C53C2311yescharacteristic
101C4⋊C8×C53—22no—
102#95×C53—21no—
103C23×D53C2×D411yescharacteristic
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53—22no—
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2411yescharacteristic
110C22×Dic53—21no—
111C22×Dic53—21no—
112C22×Dic53—22no—
113C22×Dic53C2×D411yescharacteristic
114C22×Dic53C2×D411yescharacteristic
115C22×Dic53C2×D411yescharacteristic
116C2×C4×D53—24no—
117C2×C4×D53—22no—
118C2×C4×D53—22no—
119C2×C4×D53—24no—
120C2×C4×D53—24no—
121C2×C4×D53—41no—
122C2×C4×D53—41no—
123C2×C4×D53—42no—
124C2×C4×D53—42no—
125C2×C4×D53—44no—
126C2×C4×D53C22⋊C414yes—
127C2×C4×D53C22⋊C414yes—
128C2×C4×D53C22×C412yes—
129C2×C4×D53C2×D412yes—
130C2×C8×C53—22no—
131C2×C8×C53—24no—
132C4⋊C4×C53—24no—
133C4⋊C4×C53—24no—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—22no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53C22×C412yes—
139C4⋊C4×C53C22×C412yes—
140C42×C53—42no—
141C42×C53C22×C412yes—
142C42×C53C2×D412yes—
143#11719921—44no—
144C4⋊Dic53—21no—
145C4⋊Dic53—21no—
146C4⋊Dic53—24no—
147C4⋊Dic53—24no—
148C4⋊Dic53—22no—
149C4⋊Dic53—22no—
150C4⋊Dic53C22×C412yes—
151C4⋊Dic53C22×C412yes—
152#11719919—24no—
153#11719919—24no—
154#11719919—24no—
155#11719919—24no—
156#11719919—24no—
157#11719919—24no—
158C4×Dic53—42no—
159C4×Dic53C22⋊C414yes—
160C4×Dic53C22⋊C414yes—
161C4×Dic53C22×C412yes—
162C4×Dic53C2×D412yes—
163C2×#88322—22no—
164C2×#88322—24no—
165C23×C53C22×D411yescharacteristic
166C22×D53—22no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53C2×C22⋊C412yes—
170C22×D53#13514yes—
171C2×C4×C53—22no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53—42no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53C2×C22⋊C412yes—
183C2×C4×C53C2×C22⋊C412yes—
184C2×C4×C53C23×C411yescharacteristic
185C2×C4×C53C22×D411yescharacteristic
186C2×C4×C532−1+412yes—
187C2×C4×C532−1+412yes—
188C2×Dic53—22no—
189C2×Dic53—22no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—22no—
193C2×Dic53—22no—
194C2×Dic53—42no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—42no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53C2×C22⋊C412yes—
201C2×Dic53C2×C22⋊C412yes—
202C2×Dic53C2×C22⋊C412yes—
203C4×D53—28no—
204C4×D53—24no—
205C4×D53—22no—
206C4×D53—22no—
207C4×D53—44no—
208C4×D53—48no—
209C4×D53—48no—
210C8×C53—42no—
211#88322—42no—
212#21836—531nomaximal, Sylow
213C22×C53C22×C22⋊C411yescharacteristic
214C22×C53#53612yes—
215C22×C53C2×#13512yes—
216C22×C53#57511yescharacteristic
217C22×C53#57511yescharacteristic
218C2×D53—22no—
219C2×D53—22no—
220C2×D53—24no—
221C2×D53—24no—
222C2×D53#41614yes—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—42no—
226C4×C53—42no—
227C4×C53—82no—
228C4×C53#51912yes—
229C4×C53#53611yescharacteristic
230C4×C53#53611yescharacteristic
231Dic53—22no—
232Dic53—22no—
233Dic53—22no—
234Dic53—42no—
235Dic53—42no—
236Dic53—82no—
237C22×#125—531no—
238C2×#431—531no—
239#2283—531no—
240#1991—532no—
241#1991—532no—
242#1989—531no—
243#1989—531no—
244#1989—531no—
245#1989D5311yescharacteristic, complemented
246#1824—532no—
247#1823—532no—
248C2×C53#231011yescharacteristic
249C2×C53C2×#41611yescharacteristic
250C2×C53#289611yescharacteristic
251C2×C53#323212yes—
252C2×C53#323212yes—
253D53—24no—
254C2×#125—538no—
255C2×#125—531no—
256C2×#125—531no—
257C2×#125—531no—
258C2×#125—1061no—
259C2×#125—1061no—
260C2×#125—1064no—
261C2×#125C2×D5311yescharacteristic
262C22×C4⋊C4—531no—
263C22×C4⋊C4—531no—
264C22×C4⋊C4—1061no—
265C22×C4⋊C4—1061no—
266C22×C42—531no—
267#431—1064no—
268C2×C4⋊C8—531no—
269C2×C4⋊C8C2×D5311yescharacteristic
270C2×C22⋊C8—532no—
271C2×C4×C8—532no—
272#383—1061no—
273#382—532no—
274#381—1061no—
275#380—532no—
276C4×C4⋊C4—531no—
277C4×C4⋊C4C2×D5311yescharacteristic
278C4×C22⋊C4—1061no—
279#376—1061no—
280#340—532no—
281#340—532no—
282#340—532no—
283#340—532no—
284#340—532no—
285#340—532no—
286#340C2×D5312yes—
287#340C2×D5312yes—
288C53#2183611yescharacteristic, complemented, Sylow
289C23×C4—531no—
290C23×C4—531no—
291C23×C4—531no—
292C23×C4—1061no—
293C23×C4—1061no—
294C22×C8—532no—
295C22×C8C22×D5312yes—
296#125—22no—
297#125—534no—
298#125—534no—
299#125—534no—
300#125—1068no—
301#125—1068no—
302#125—1062no—
303#125—1062no—
304#125—1062no—
305#125C4×D5314yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—531no—
309C2×C4⋊C4—531no—
310C2×C4⋊C4—534no—
311C2×C4⋊C4—534no—
312C2×C4⋊C4—532no—
313C2×C4⋊C4—1062no—
314C2×C4⋊C4—1062no—
315C2×C4⋊C4—1068no—
316C2×C4⋊C4—1068no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4—1061no—
319C2×C4⋊C4—1061no—
320C2×C4⋊C4—1061no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4C22×D5312yes—
326C2×C4⋊C4C22×D5311yescharacteristic
327C2×C4⋊C4C22×D5311yescharacteristic
328C2×C22⋊C4—1062no—
329C2×C42—21no—
330C2×C42—531no—
331C2×C42—531no—
332C2×C42—531no—
333C2×C42—534no—
334C2×C42—534no—
335C2×C42—1061no—
336C2×C42C22×D5311yescharacteristic
337C4⋊C8—22no—
338C4⋊C8—1062no—
339C22⋊C8—1064no—
340C4×C8—1064no—
341#95—21no—
342#95—1062no—
343#95—1062no—
344#95—1062no—
345#95—1061no—
346C24—531no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—531no—
351C22×C4—531no—
352C22×C4—531no—
353C22×C4—534no—
354C22×C4—532no—
355C22×C4—532no—
356C22×C4—534no—
357C22×C4—1064no—
358C22×C4—1064no—
359C22×C4—1064no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—1061no—
363C22×C4—1061no—
364C22×C4—1062no—
365C22×C4—2121no—
366C22×C4—2121no—
367C22×C4—2124no—
368C22×C4—2122no—
369C22×C4—2122no—
370C22×C4D4×D5311yescharacteristic
371C22×C4D4×D5311yescharacteristic
372C22×C4C23×D5311yescharacteristic
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—1062no—
376C2×C8—1064no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—532no—
384C4⋊C4—532no—
385C4⋊C4—1064no—
386C4⋊C4—1064no—
387C4⋊C4—1064no—
388C4⋊C4—1064no—
389C4⋊C4—1064no—
390C4⋊C4—1064no—
391C4⋊C4—1064no—
392C4⋊C4—1064no—
393C4⋊C4—1061no—
394C4⋊C4—1061no—
395C4⋊C4—1062no—
396C4⋊C4—1062no—
397C4⋊C4C2×C4×D5312yes—
398C4⋊C4C2×C4×D5312yes—
399C22⋊C4—2124no—
400C42—42no—
401C42—534no—
402C42—534no—
403C42—532no—
404C42—532no—
405C42—2122no—
406C42C2×C4×D5312yes—
407C42D4×D5312yes—
408C23—532no—
409C23—534no—
410C23—1061no—
411C23—1061no—
412C23—1062no—
413C23C2×D4×D5311yescharacteristic
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—532no—
426C2×C4—532no—
427C2×C4—532no—
428C2×C4—1068no—
429C2×C4—1062no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1062no—
433C2×C4—1064no—
434C2×C4—1062no—
435C2×C4—1062no—
436C2×C4—1062no—
437C2×C4—1062no—
438C2×C4—2124no—
439C2×C4—2128no—
440C2×C4—2128no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4C22⋊C4×D5312yes—
448C2×C4C22⋊C4×D5312yes—
449C2×C4C22×C4×D5311yescharacteristic
450C2×C4C2×D4×D5311yescharacteristic
451C2×C4#1313326412yes—
452C2×C4#1313326512yes—
453C8—42no—
454C8—2122no—
455C22—534no—
456C22—1062no—
457C22—1062no—
458C22—1064no—
459C22—1064no—
460C22C2×C22⋊C4×D5311yescharacteristic
461C22#1681119612yes—
462C22#135×D5312yes—
463C22#1681141911yescharacteristic
464C22#1681142211yescharacteristic
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—1062no—
471C4—1062no—
472C4—1062no—
473C4—4242no—
474C4—2122no—
475C4—2122no—
476C4#1681109912yes—
477C4#1681119311yescharacteristic
478C4#1681119811yescharacteristic
479C2—1064no—
480C2#2027729811yescharacteristic
481C2#416×D5311yescharacteristic
482C2#2028323111yescharacteristic
483C2#2028588112yes—
484C2#2028588712yes—
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#21836
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. #20275678C2semidirect
5. #20275678C2semidirect
6. #20275673C2semidirect
7. #20275672C2semidirect
8. #20275670C2semidirect
11. C53×#1989C2semidirect
18. #125×D53C4semidirect
245. #1989D53semidirect
288. C53#21836semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D532
C2×#168106752
#202771112
#202756782
#202756732
#202756722
#202756702
#202748042
#202748002
C53×#19892
#2183653