Finite Group Databasebeta

← #27417466

Subgroups — #27417466

Order 13568.

1002 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810956C211yesmaximal, characteristic
2C2×#16810951C211yesmaximal, characteristic
3#20277335C211yesmaximal, characteristic
4#20276050C212yesmaximal, complemented
5#20275958C212yesmaximal, complemented
6#20275028C211yesmaximal, characteristic
7#20275028C211yesmaximal, characteristic
8#20274983C212yesmaximal, complemented
9#20274969C212yesmaximal, complemented
10#20274786C211yesmaximal, characteristic
11C2×C53×#541C211yesmaximal, characteristic
12C2×#13133176C2211yescharacteristic
13C22×C4⋊Dic53C2211yescharacteristic
14#16810956—24no—
15#16810951—24no—
16C2×#13133150C2212yes—
17C2×#13133148C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24#16810521C2212yes—
25#16810521C2212yes—
26#16810495—21no—
27#16810495—21no—
28#16810492—21no—
29#16810488—21no—
30#16810385C2212yes—
31#16810385C2212yes—
32#16810378C2212yes—
33#16810378C2212yes—
34C53×#541—24no—
35C53×#541—24no—
36C2×C4⋊Q8×C53C2211yescharacteristic
37C2×#126×C53—21no—
38C2×#126×C53—21no—
39C2×#125×C53—21no—
40C2×#125×C53—21no—
41C2×#125×C53C2211yescharacteristic
42C2×C22⋊Q8×C53C2211yescharacteristic
43C2×C22⋊Q8×C53C2211yescharacteristic
44C2×C4×Q8×C53C2211yescharacteristic
45C2×C4×Q8×C53C2211yescharacteristic
46C2×#117×C53C2211yescharacteristic
47C23×Dic53—21no—
48C22×#88322C2312yes—
49C22×#88322C2312yes—
50#13133176—24no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×C22⋊Dic53—21no—
59C2×C4×Dic53—21no—
60#13133150—24no—
61#13133148—24no—
62#13133141—24no—
63#13133141—24no—
64#13133141—24no—
65#13133141—24no—
66C22⋊#88322—22no—
67C22⋊#88322—22no—
68#13133124—22no—
69#13133124—22no—
70C22×Q8×C53C2311yescharacteristic
71C22×Q8×C53C2311yescharacteristic
72C23×C4×C53C2311yescharacteristic
73C4⋊Q8×C53—22no—
74C4⋊Q8×C53—22no—
75#126×C53—28no—
76#126×C53—28no—
77#125×C53—28no—
78#125×C53—28no—
79#125×C53—22no—
80#125×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53D414yes—
84C22⋊Q8×C53D414yes—
85C4×Q8×C53—22no—
86C4×Q8×C53—22no—
87C4×Q8×C53—22no—
88C4×Q8×C53—22no—
89#117×C53—22no—
90#117×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53D411yescharacteristic
100C2×C4⋊C4×C53D411yescharacteristic
101C2×C4⋊C4×C53C2311yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—21no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53D411yescharacteristic
106C2×C22⋊C4×C53D411yescharacteristic
107C2×C42×C53—21no—
108C2×C42×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C2×C42×C53C2311yescharacteristic
111C24×C53C2×D411yescharacteristic
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53C2×D412yes—
119C2×Q8×C53C2×D412yes—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D412yes—
132C22×C4×C53C2×D412yes—
133C22×C4×C53C2411yescharacteristic
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—42no—
137C22×Dic53—82no—
138C4⋊C4×C53—24no—
139C4⋊C4×C53—24no—
140C4⋊C4×C53—24no—
141C4⋊C4×C53—24no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—24no—
144C4⋊C4×C53—24no—
145C4⋊C4×C53—24no—
146C4⋊C4×C53—22no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53C2×D412yes—
151C4⋊C4×C53C2×D412yes—
152C22⋊C4×C53—24no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C42×C53—24no—
157C42×C53—24no—
158C42×C53—21no—
159C42×C53—21no—
160C42×C53—21no—
161C42×C53—21no—
162C22⋊Dic53—24no—
163C4⋊Dic53—22no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53—42no—
167C4⋊Dic53#4114yes—
168#11719919—82no—
169#11719919—82no—
170C4×Dic53—24no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53#12012yes—
182C23×C53C22×D411yescharacteristic
183Q8×C53—24no—
184Q8×C53—24no—
185Q8×C53—42no—
186Q8×C53—42no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—24no—
190C2×C4×C53—24no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—22no—
201C2×C4×C53—22no—
202C2×C4×C53—22no—
203C2×C4×C53—41no—
204C2×C4×C53—41no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—42no—
208C2×C4×C53—42no—
209C2×C4×C53#12012yes—
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C2×#4112yes—
213C2×C4×C532−1+412yes—
214C2×Dic53—24no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—42no—
218C2×Dic53—42no—
219C2×Dic53—42no—
220C2×Dic53—84no—
221#88322—42no—
222#88322—42no—
223#24071—531nomaximal, Sylow
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—22no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53C2×#12011yescharacteristic
230C22×C53#52511yescharacteristic
231C22×C53#53611yescharacteristic
232C22×C53#57712yes—
233C22×C53#57812yes—
234C4×C53—22no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—41no—
238C4×C53—41no—
239C4×C53—42no—
240C4×C53—41no—
241C4×C53—41no—
242C4×C53—42no—
243C4×C53#54712yes—
244C4×C53#54712yes—
245Dic53—44no—
246Dic53—82no—
247C2×#541D5311yescharacteristic
248C2×#484—531no—
249C2×#483—531no—
250#2317—531no—
251#2059—532no—
252#2043—532no—
253#1873—531no—
254#1873—531no—
255#1865—532no—
256#1863—532no—
257#1818—531no—
258C2×C53—42no—
259C2×C53#240611yescharacteristic
260C2×C53#301011yescharacteristic
261C2×C53#301111yescharacteristic
262C2×C53#312012yes—
263C2×C53#324212yes—
264#541—24no—
265#541—24no—
266C2×C4⋊Q8C2×D5311yescharacteristic
267C2×#126—21no—
268C2×#126—21no—
269C2×#125—21no—
270C2×#125—21no—
271C2×#125C2×D5311yescharacteristic
272C2×C22⋊Q8C2×D5311yescharacteristic
273C2×C22⋊Q8C2×D5311yescharacteristic
274C2×C4×Q8C2×D5311yescharacteristic
275C2×C4×Q8C2×D5311yescharacteristic
276C2×#117—531no—
277C2×#117C2×D5311yescharacteristic
278C22×C4⋊C4—531no—
279#484—1064no—
280#483—1064no—
281C2×#107—532no—
282C2×#106—532no—
283C2×#103—532no—
284C2×#103—532no—
285C2×#103—532no—
286C2×#103—532no—
287C2×C22⋊C8—531no—
288C2×C22⋊C8—531no—
289#389—532no—
290#389—532no—
291#385—1061no—
292#385—1061no—
293#384—1061no—
294#382—1061no—
295#340—532no—
296#340—532no—
297#336—532no—
298#336—532no—
299C53#2407111yescharacteristic, complemented, Sylow
300C22×Q8C22×D5311yescharacteristic
301C22×Q8C22×D5311yescharacteristic
302C23×C4—1061no—
303C23×C4C22×D5311yescharacteristic
304C22×C8—532no—
305C22×C8—532no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308#126—28no—
309#126—28no—
310#125—28no—
311#125—28no—
312#125—22no—
313#125—22no—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8C22⋊D5314yes—
317C22⋊Q8C22⋊D5314yes—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322#117—22no—
323#117—22no—
324#117—1064no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—532no—
334C2×C4⋊C4—532no—
335C2×C4⋊C4—1062no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1062no—
338C2×C4⋊C4—1062no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4C22⋊D5311yescharacteristic
341C2×C4⋊C4C22⋊D5311yescharacteristic
342C2×C4⋊C4C22×D5311yescharacteristic
343C2×C4⋊C4C22×D5311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—1061no—
347C2×C22⋊C4C22⋊D5311yescharacteristic
348C2×C22⋊C4C22⋊D5311yescharacteristic
349C2×C42—21no—
350C2×C42—21no—
351C2×C42—1061no—
352C2×C42C22×D5311yescharacteristic
353C2×C42C22×D5311yescharacteristic
354#107—1064no—
355#106—1064no—
356#103—1064no—
357#103—1064no—
358#103—1064no—
359#103—1064no—
360C22⋊C8—1062no—
361C22⋊C8—1062no—
362#95—1062no—
363#95—1062no—
364C24C2×C22⋊D5311yescharacteristic
365C2×Q8—21no—
366C2×Q8—21no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8C2×C22⋊D5312yes—
372C2×Q8C2×C22⋊D5312yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—1062no—
380C22×C4—1062no—
381C22×C4—4242no—
382C22×C4—2122no—
383C22×C4C2×C22⋊D5311yescharacteristic
384C22×C4C2×C22⋊D5311yescharacteristic
385C22×C4C2×C22⋊D5311yescharacteristic
386C22×C4C2×C22⋊D5311yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388C22×C4C2×C22⋊D5312yes—
389C22×C4C2×C22⋊D5312yes—
390C22×C4C23×D5311yescharacteristic
391C2×C8—1062no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1062no—
395C2×C8—1062no—
396C2×C8—1062no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—534no—
410C4⋊C4—1062no—
411C4⋊C4—1062no—
412C4⋊C4—1062no—
413C4⋊C4—4242no—
414C4⋊C4—4242no—
415C4⋊C4—2122no—
416C4⋊C4C2×C22⋊D5312yes—
417C4⋊C4C2×C22⋊D5312yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—1064no—
423C42—24no—
424C42—24no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—1064no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—21no—
434C23#1313318612yes—
435C23C22×C22⋊D5311yescharacteristic
436Q8—24no—
437Q8—24no—
438Q8—42no—
439Q8—42no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—1064no—
463C2×C4—4244no—
464C2×C4—2122no—
465C2×C4—2122no—
466C2×C4—2122no—
467C2×C4—2122no—
468C2×C4—2122no—
469C2×C4#1313318612yes—
470C2×C4C22×C22⋊D5311yescharacteristic
471C2×C4C22×C22⋊D5311yescharacteristic
472C2×C4#41×D5312yes—
473C2×C4#1313326412yes—
474C8—2122no—
475C8—2122no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22C2×#1313318611yescharacteristic
482C22#1681113011yescharacteristic
483C22#1681119411yescharacteristic
484C22#1681143112yes—
485C22#1681143812yes—
486C4—22no—
487C4—42no—
488C4—42no—
489C4—41no—
490C4—41no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—42no—
495C4—4242no—
496C4—2124no—
497C4#1681127212yes—
498C4#1681127212yes—
499C2—42no—
500C2#2027804611yescharacteristic
501C2#2028400111yescharacteristic
502C2#2028401011yescharacteristic
503C2#2028492312yes—
504C2#2028597712yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#541C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#541D53
hypocenterC53#24071
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
4. #20276050C2semidirect
5. #20275958C2semidirect
8. #20274983C2semidirect
9. #20274969C2semidirect
299. C53#24071semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109512
#202773352
#202760502
#202759582
#202750282
#202749832
#202749692
#202747862
C2×C53×#5412
#2407153