Finite Group Databasebeta

← #27412156

Subgroups — #27412156

Order 13568.

1202 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811178C211yesmaximal, characteristic
2C2×#16810662C212yesmaximal
3#389×D53C211yesmaximal, characteristic
4#20276056C212yesmaximal, complemented
5#20276055C212yesmaximal, complemented
6#20276045C211yesmaximal, characteristic, complemented
7#20276044C211yesmaximal, characteristic, complemented
8#20276043C211yesmaximal, characteristic, complemented
9#20274433C212yesmaximal
10#20274407C211yesmaximal, characteristic
11C53×#2058C211yesmaximal, characteristic, complemented
12C22×#11719912C2212yes—
13#16811178—28no—
14C2×#13133217C2211yescharacteristic
15C2×#13133208C2212yes—
16C2×#13133197—21no—
17C2×#13133193—21no—
18C2×Q8×Dic53—21no—
19C2×#13133182—21no—
20C2×#13133175C2212yes—
21C2×C4⋊C4×D53—21no—
22C2×C4⋊C4×D53C2211yescharacteristic
23C2×C4⋊C4×D53C2211yescharacteristic
24C2×C42×D53C2211yescharacteristic
25#16810662—28no—
26C2×#13133151C2212yes—
27C2×#13133150C2212yes—
28C2×#13133148C2212yes—
29#16810522C2211yescharacteristic
30#16810521C2211yescharacteristic
31#16810520—21no—
32#16810519C2211yescharacteristic
33#16810518—21no—
34#95×D53—21no—
35#16810386C2212yes—
36#16810385C2212yes—
37#16810385C2212yes—
38#16810378C2211yescharacteristic
39#16810377C2212yes—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×#106×C53C2212yes—
42C53×#389C2211yescharacteristic
43C53×#340C2212yes—
44C53×#336C2211yescharacteristic
45C22×Dic106—21no—
46C22×C4×D53—21no—
47C22×C4×D53—21no—
48C22×C4×D53D411yescharacteristic
49C22×C4×D53D411yescharacteristic
50C22×C4×D53C2311yescharacteristic
51C2×#11719912—24no—
52C2×#11719912—24no—
53C2×#11719912Q818yescomplemented
54C22×#88322C2312yes—
55#13133217—24no—
56#13133208—24no—
57#13133208—24no—
58#13133197—28no—
59#13133193—28no—
60Q8×Dic53—28no—
61#13133182—28no—
62#13133175—24no—
63#13133175D418yes—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2311yescharacteristic
68C2×C4⋊Dic53C2312yes—
69C2×C4⋊Dic53C2311yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×#11719919—22no—
73C2×#11719919—21no—
74C2×#11719919—21no—
75C4⋊C4×D53—22no—
76C4⋊C4×D53—24no—
77C4⋊C4×D53—44no—
78C4⋊C4×D53Q814yescomplemented
79C2×#11719921—22no—
80C2×C4×Dic53—22no—
81C2×C4×Dic53D411yescharacteristic
82C2×C4×Dic53D411yescharacteristic
83C2×C4×Dic53C2311yescharacteristic
84C42×D53—24no—
85#13133151—24no—
86#13133150—24no—
87#13133148—24no—
88#13133124—21no—
89#13133124—21no—
90#13133124—21no—
91C22×Q8×C53—21no—
92C22×C8×C53C2312yes—
93C4⋊Q8×C53—24no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53—21no—
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2312yes—
98C2×C4⋊C4×C53C2311yescharacteristic
99C2×C42×C53C2311yescharacteristic
100#106×C53—24no—
101#95×C53—21no—
102C23×D53C2×D411yescharacteristic
103C2×Q8×C53—24no—
104C2×Q8×C53—42no—
105C2×Q8×C53—42no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2411yescharacteristic
112C22×Dic53—21no—
113C22×Dic53—22no—
114C22×Dic53—21no—
115C22×Dic53C2×D411yescharacteristic
116C22×Dic53C2×D411yescharacteristic
117C22×Dic53C2×D411yescharacteristic
118C2×C4×D53—22no—
119C2×C4×D53—22no—
120C2×C4×D53—24no—
121C2×C4×D53—22no—
122C2×C4×D53—22no—
123C2×C4×D53—41no—
124C2×C4×D53—41no—
125C2×C4×D53—42no—
126C2×C4×D53—42no—
127C2×C4×D53—42no—
128C2×C4×D53—42no—
129C2×C4×D53—42no—
130C2×C4×D53C2×D414yes—
131C2×C4×D53C2×Q814yes—
132C2×C4×D53C2×Q812yes—
133C2×C4×D53#4112yes—
134C2×Dic106—24no—
135C2×Dic106—42no—
136C2×Dic106—42no—
137C2×C8×C53—22no—
138C2×C8×C53—22no—
139C2×C8×C53C2×Q814yes—
140C4⋊C4×C53—24no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—42no—
145C4⋊C4×C53C2×D414yes—
146C4⋊C4×C53C2×Q812yes—
147C42×C53—22no—
148#11719921—28no—
149C4⋊Dic53—24no—
150C4⋊Dic53—28no—
151C4⋊Dic53—22no—
152C4⋊Dic53—21no—
153C4⋊Dic53—22no—
154C4⋊Dic53—42no—
155C4⋊Dic53C2×D414yes—
156C4⋊Dic53C2×Q812yes—
157#11719919—28no—
158#11719919—24no—
159#11719919—42no—
160#11719919—42no—
161#11719919—42no—
162C4×Dic53—22no—
163C4×Dic53—22no—
164C4×Dic53—22no—
165C4×Dic53—28no—
166C2×#88322—22no—
167C2×#88322—22no—
168C2×#88322C2×Q814yes—
169#11719912—216no—
170C23×C53C22×D411yescharacteristic
171C22×D53—21no—
172C22×D53—21no—
173C22×D53—21no—
174C22×D53—21no—
175C22×D53#12012yes—
176C22×D53C22⋊Q812yes—
177C22×D53C4⋊Q812yes—
178Q8×C53—48no—
179C2×C4×C53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—24no—
184C2×C4×C53—22no—
185C2×C4×C53—41no—
186C2×C4×C53—42no—
187C2×C4×C53—41no—
188C2×C4×C53—42no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53AGL(1,ℤ/8)12yes—
193C2×C4×C53#13712yes—
194C2×C4×C53C22×D412yes—
195C2×C4×C53C22×Q812yes—
196C2×C4×C53C22×Q811yescharacteristic
197C2×C4×C53C2×#4111yescharacteristic
198C2×Dic53—24no—
199C2×Dic53—21no—
200C2×Dic53—21no—
201C2×Dic53—22no—
202C2×Dic53—21no—
203C2×Dic53—21no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—42no—
207C2×Dic53—41no—
208C2×Dic53—42no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53#12012yes—
214C2×Dic53C22⋊Q812yes—
215C2×Dic53C4⋊Q812yes—
216C4×D53—44no—
217C4×D53—44no—
218C4×D53—48no—
219C4×D53—84no—
220C4×D53C22⋊Q818yes—
221C4×D53C4⋊Q818yes—
222Dic106—48no—
223C8×C53—24no—
224#88322—24no—
225#23554—531nomaximal, Sylow
226C22×C53C2×#12011yescharacteristic
227C22×C53C2×C22⋊Q811yescharacteristic
228C22×C53C2×C4⋊Q811yescharacteristic
229C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
230C22×C53C2×#13711yescharacteristic
231C22×C53#57511yescharacteristic
232C22×C53#57511yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53#39514yes—
240C4×C53—22no—
241C4×C53—22no—
242C4×C53—42no—
243C4×C53—44no—
244C4×C53—81no—
245C4×C53C2×C22⋊Q812yes—
246C4×C53C2×C4⋊Q812yes—
247Dic53—42no—
248Dic53—44no—
249Dic53—41no—
250Dic53—41no—
251Dic53—81no—
252Dic53#39514yes—
253C2×#533—531no—
254C2×#428—532no—
255C2×#389—531no—
256#2059—532no—
257#2059—532no—
258#2058—531no—
259#2058D5311yescharacteristic, complemented
260#2057—531no—
261#2057—531no—
262#1736—532no—
263#1728—531no—
264C2×C53C2×#39511yescharacteristic
265C2×C53#300611yescharacteristic
266C2×C53#300711yescharacteristic
267C2×C53#307711yescharacteristic
268C2×C53#307811yescharacteristic
269C2×C53#316511yescharacteristic
270C2×C53#316611yescharacteristic
271D53—24no—
272C22×M16—532no—
273#533—1068no—
274C2×C4⋊Q8—531no—
275C2×C4⋊Q8C2×D5311yescharacteristic
276C2×#125—532no—
277C2×C22⋊Q8—1061no—
278C2×C22⋊Q8—1061no—
279C2×C4×Q8—1061no—
280C2×C4×Q8—1061no—
281C2×#117—532no—
282C22×C4⋊C4—531no—
283C22×C4⋊C4—531no—
284C22×C4⋊C4—1061no—
285C22×C42—531no—
286#428—1068no—
287C2×#107—532no—
288C2×#107—532no—
289C2×#106—532no—
290C2×#106C2×D5312yes—
291#389—531no—
292#389—531no—
293#389—531no—
294#389—1061no—
295#389—1061no—
296#389C2×D5311yescharacteristic
297C2×#95—1061no—
298#340—532no—
299#340—532no—
300#340—532no—
301#340C2×D5312yes—
302#336—532no—
303#336—531no—
304#336C2×D5311yescharacteristic
305C53#2355411yescharacteristic, complemented, Sylow
306C22×Q8—21no—
307C22×Q8—1061no—
308C23×C4—531no—
309C23×C4—531no—
310C23×C4—531no—
311C23×C4—1061no—
312C23×C4—1061no—
313C2×M16—538no—
314C2×M16—1064no—
315C2×M16—1064no—
316C22×C8—532no—
317C22×C8C22×D5312yes—
318C4⋊Q8—24no—
319C4⋊Q8—1064no—
320#125—1064no—
321#125—1064no—
322C22⋊Q8—1068no—
323C22⋊Q8—1068no—
324C4×Q8—1068no—
325C4×Q8—1068no—
326#117—538no—
327#117—1064no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—531no—
331C2×C4⋊C4—532no—
332C2×C4⋊C4—534no—
333C2×C4⋊C4—531no—
334C2×C4⋊C4—1062no—
335C2×C4⋊C4—1064no—
336C2×C4⋊C4—1061no—
337C2×C4⋊C4—1062no—
338C2×C4⋊C4—1062no—
339C2×C4⋊C4—1061no—
340C2×C4⋊C4—1061no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4—1061no—
344C2×C4⋊C4—2124no—
345C2×C4⋊C4C22×D5311yescharacteristic
346C2×C4⋊C4C22×D5312yes—
347C2×C4⋊C4C22×D5311yescharacteristic
348C2×C22⋊C4—1062no—
349C2×C42—531no—
350C2×C42—531no—
351C2×C42—531no—
352C2×C42—1064no—
353C2×C42—1062no—
354C2×C42C22×D5311yescharacteristic
355#107—1064no—
356#107—1064no—
357#106—24no—
358#106—1064no—
359#95—21no—
360#95—1061no—
361#95—1061no—
362#95—1061no—
363C24—531no—
364C2×Q8—24no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—1064no—
368C2×Q8—2122no—
369C2×Q8—2122no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—531no—
374C22×C4—534no—
375C22×C4—532no—
376C22×C4—534no—
377C22×C4—532no—
378C22×C4—531no—
379C22×C4—531no—
380C22×C4—1062no—
381C22×C4—1062no—
382C22×C4—1064no—
383C22×C4—1061no—
384C22×C4—1062no—
385C22×C4—1062no—
386C22×C4—1062no—
387C22×C4—1061no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2122no—
391C22×C4—2122no—
392C22×C4—2122no—
393C22×C4—2122no—
394C22×C4—2122no—
395C22×C4D4×D5311yescharacteristic
396C22×C4D4×D5311yescharacteristic
397C22×C4C23×D5311yescharacteristic
398M16—10616no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—534no—
402C2×C8—1062no—
403C2×C8—1062no—
404C2×C8Q8×D5314yes—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—534no—
411C4⋊C4—532no—
412C4⋊C4—1064no—
413C4⋊C4—1068no—
414C4⋊C4—1068no—
415C4⋊C4—1064no—
416C4⋊C4—1062no—
417C4⋊C4—1061no—
418C4⋊C4—1062no—
419C4⋊C4—2122no—
420C4⋊C4—2122no—
421C4⋊C4—2122no—
422C4⋊C4—2122no—
423C4⋊C4D4×D5314yes—
424C4⋊C4Q8×D5312yes—
425C22⋊C4—1068no—
426C42—22no—
427C42—1062no—
428C42—1062no—
429C42—1068no—
430C42—1062no—
431C23—532no—
432C23—532no—
433C23—532no—
434C23—1061no—
435C23—1061no—
436C23—1061no—
437C23—1061no—
438C23C2×D4×D5311yescharacteristic
439Q8—48no—
440Q8—2128no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—532no—
455C2×C4—532no—
456C2×C4—532no—
457C2×C4—538no—
458C2×C4—538no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1064no—
462C2×C4—1062no—
463C2×C4—1061no—
464C2×C4—1061no—
465C2×C4—1061no—
466C2×C4—1061no—
467C2×C4—4244no—
468C2×C4—2124no—
469C2×C4—2124no—
470C2×C4—2128no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2121no—
474C2×C4—2121no—
475C2×C4—2122no—
476C2×C4—2121no—
477C2×C4—2122no—
478C2×C4#1313324412yes—
479C2×C4#1313324612yes—
480C2×C4C2×D4×D5312yes—
481C2×C4C2×Q8×D5312yes—
482C2×C4C2×Q8×D5311yescharacteristic
483C2×C4#41×D5311yescharacteristic
484C8—24no—
485C8—1064no—
486C22—534no—
487C22—1062no—
488C22—1062no—
489C22—1062no—
490C22—1062no—
491C22—1062no—
492C22—1062no—
493C22#120×D5311yescharacteristic
494C22C22⋊Q8×D5311yescharacteristic
495C22C4⋊Q8×D5311yescharacteristic
496C22C2×#1313324411yescharacteristic
497C22C2×#1313324611yescharacteristic
498C22#1681141911yescharacteristic
499C22#1681142211yescharacteristic
500C4—22no—
501C4—22no—
502C4—44no—
503C4—42no—
504C4—81no—
505C4—534no—
506C4—4241no—
507C4—2121no—
508C4—2124no—
509C4—2121no—
510C4—2122no—
511C4C22⋊Q8×D5312yes—
512C4C4⋊Q8×D5312yes—
513C2—1064no—
514C2#395×D5311yescharacteristic
515C2#2028396611yescharacteristic
516C2#2028397011yescharacteristic
517C2#2028451311yescharacteristic
518C2#2028451511yescharacteristic
519C2#2028525611yescharacteristic
520C2#2028526611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2058C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2058D53
hypocenterC53#23554
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2058

Complements

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

NH ≅ G/Ntype
4. #20276056C2semidirect
5. #20276055C2semidirect
6. #20276045C2semidirect
7. #20276044C2semidirect
8. #20276043C2semidirect
11. C53×#2058C2semidirect
53. C2×#11719912Q8semidirect
78. C4⋊C4×D53Q8semidirect
259. #2058D53semidirect
305. C53#23554semidirect

Maximal subgroups

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

subgroupindex
C2×#168111782
C2×#168106622
#389×D532
#202760562
#202760552
#202760452
#202760442
#202760432
#202744332
#202744072
C53×#20582
#2355453