Finite Group Databasebeta

← #27276196

Subgroups — #27276196

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811346C211yesmaximal, characteristic
2#20275675C211yesmaximal, characteristic, complemented
3#20274712C211yesmaximal, characteristic
4C2×#16810368C211yesmaximal, characteristic, complemented
5C2×Q8⋊#88322C211yesmaximal, characteristic, complemented
6C2×#16810350C211yesmaximal, characteristic
7C53×#1803C211yesmaximal, characteristic, complemented
8#16811346—22no—
9#16811346—22no—
10#16811346—22no—
11#16811346C414yescomplemented
12C2×#13133217C2211yescharacteristic
13C2×#13133212—22no—
14C2×#13133205—21no—
15C2×#13133193—21no—
16C2×C4×Dic106—21no—
17C2×C4×D212C2211yescharacteristic
18C2×#13133175—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133137—21no—
21C2×C4×#88322C2211yescharacteristic
22#16810385—21no—
23#16810385—22no—
24#16810368—22no—
25Q8⋊#88322—22no—
26#16810350—22no—
27C2×C4×Q8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×#103×C53—21no—
30C53×#384C2211yescharacteristic
31C53×#340—21no—
32C22×Dic106—21no—
33C22×D212D411yescharacteristic
34C22×C4×D53—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37#13133217—21no—
38#13133217—21no—
39#13133217C2×C412yes—
40#13133217D412yes—
41#13133212—28no—
42#13133212—28no—
43#13133205—24no—
44#13133205—24no—
45#13133193—24no—
46#13133193—24no—
47C4×Dic106—24no—
48C4×Dic106—24no—
49C4×D212—21no—
50C4×D212—21no—
51C4×D212C2×C412yes—
52C4×D212D412yes—
53#13133175—24no—
54#13133175—24no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53D411yescharacteristic
57C2×C4⋊Dic53D412yes—
58C2×#11719919—22no—
59C2×#11719921—22no—
60C2×#11719921—22no—
61C2×#11719921—21no—
62C2×C4×Dic53—22no—
63C4⋊#88322—21no—
64C4⋊#88322—21no—
65#13133137—42no—
66C4×#88322—21no—
67C4×#88322—21no—
68C22×Q8×C53D411yescharacteristic
69C22×C8×C53—21no—
70C4×Q8×C53—21no—
71C4×Q8×C53—21no—
72C4×Q8×C53C2×C412yes—
73C4×Q8×C53D412yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53D411yescharacteristic
79C2×C42×C53—21no—
80C2×C42×C53C2311yescharacteristic
81C4⋊C8×C53—21no—
82C4⋊C8×C53—21no—
83#103×C53—42no—
84C23×D53—21no—
85C2×Q8×C53—21no—
86C2×Q8×C53—21no—
87C2×Q8×C53—21no—
88C2×Q8×C53—21no—
89C2×Q8×C53—21no—
90C2×Q8×C53C22⋊C412yes—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×Dic53—21no—
98C22×Dic53—22no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×D212—21no—
103C2×D212—21no—
104C2×D212C22⋊C412yes—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—41no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53—41no—
111C2×Dic106—22no—
112C2×Dic106—22no—
113C2×Dic106—41no—
114C2×Dic106—41no—
115C2×Dic106—41no—
116C2×Dic106—41no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C2×C8×C53—41no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—42no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53C22⋊C412yes—
129C42×C53—22no—
130C42×C53—22no—
131C42×C53C22×C411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53C2×D411yescharacteristic
135#11719921—28no—
136#11719921—28no—
137#11719921—22no—
138#11719921—22no—
139C4⋊Dic53—22no—
140C4⋊Dic53—22no—
141C4⋊Dic53—21no—
142C4⋊Dic53—22no—
143C4⋊Dic53C22⋊C412yes—
144C4⋊Dic53C22⋊C414yes—
145#11719919—28no—
146C4×Dic53—28no—
147C2×#88322—22no—
148C2×#88322—41no—
149C2×#88322—41no—
150C2×#88322—41no—
151C2×#88322—41no—
152C2×#88322—41no—
153C23×C53#12011yescharacteristic
154C22×D53—22no—
155C22×D53—41no—
156C22×D53—41no—
157C22×D53—41no—
158C22×D53—41no—
159C22×D53—41no—
160C22×D53—41no—
161Q8×C53—22no—
162Q8×C53—22no—
163Q8×C53—41no—
164Q8×C53—41no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—22no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—81no—
174C2×C4×C53—81no—
175C2×C4×C53—81no—
176C2×C4×C53C2×C22⋊C411yescharacteristic
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53C2×C22⋊C411yescharacteristic
179C2×C4×C53#12011yescharacteristic
180C2×C4×C53#12011yescharacteristic
181C2×C4×C53#12011yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53#13511yescharacteristic
184C2×C4×C53#13511yescharacteristic
185C2×C4×C53#13511yescharacteristic
186C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
187C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
188C2×C4×C53#13711yescharacteristic
189C2×C4×C53#13711yescharacteristic
190C2×Dic53—24no—
191C2×Dic53—22no—
192C2×Dic53—42no—
193C2×Dic53—41no—
194C2×Dic53—42no—
195C2×Dic53—41no—
196D212—22no—
197D212—22no—
198D212—41no—
199D212—41no—
200C4×D53—42no—
201C4×D53—42no—
202C4×D53—42no—
203C4×D53—42no—
204Dic106—42no—
205Dic106—42no—
206Dic106—42no—
207Dic106—42no—
208C8×C53—81no—
209#88322—42no—
210#88322—81no—
211#8449—531nomaximal, Sylow
212C22×C53#37911yescharacteristic
213C22×C53#44911yescharacteristic
214C22×C53#44911yescharacteristic
215C22×C53#45211yescharacteristic
216C22×C53#45211yescharacteristic
217C22×C53#45411yescharacteristic
218C22×C53#45611yescharacteristic
219C2×D53—42no—
220C2×D53—42no—
221C2×D53—42no—
222C2×D53—81no—
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—81no—
226C4×C53—21no—
227C4×C53—21no—
228C4×C53—42no—
229C4×C53—42no—
230C4×C53—161no—
231C4×C53#41611yescharacteristic
232C4×C53#41611yescharacteristic
233C4×C53#41711yescharacteristic
234C4×C53#41711yescharacteristic
235C4×C53#44911yescharacteristic
236C4×C53#44911yescharacteristic
237C4×C53#45211yescharacteristic
238C4×C53#45211yescharacteristic
239Dic53—44no—
240Dic53—42no—
241C2×#556—531no—
242#1990—531no—
243#1803D5311yescharacteristic, complemented
244#1802—531no—
245C2×#332—531no—
246C2×Q8⋊C8—531no—
247C2×D4⋊C8—531no—
248C2×C53#162811yescharacteristic
249C2×C53#163811yescharacteristic
250C2×C53#165011yescharacteristic
251C2×C53#165211yescharacteristic
252C2×C53#178811yescharacteristic
253C2×C53#178811yescharacteristic
254C2×C53#179311yescharacteristic
255D53—82no—
256#556—534no—
257#556—1062no—
258#556—1062no—
259#556—1062no—
260C2×C4⋊Q8—531no—
261C2×#126—1062no—
262C2×#124—1061no—
263C2×C22⋊Q8—1061no—
264C2×C4×Q8—1061no—
265C2×C4×Q8C2×D5311yescharacteristic
266C2×C4×D4—531no—
267C2×#117—1061no—
268C2×C4⋊C8—531no—
269C2×C4⋊C8C2×D5311yescharacteristic
270C2×#103—21no—
271C2×#102—1061no—
272C2×C4×C8—531no—
273#384C2×D5311yescharacteristic
274#340—21no—
275#340—1061no—
276#340—1062no—
277#332—1062no—
278Q8⋊C8—1062no—
279D4⋊C8—1062no—
280C53#844911yescharacteristic, complemented, Sylow
281C22×Q8—1061no—
282C22×Q8C22⋊D5311yescharacteristic
283C22×D4—531no—
284C23×C4—1061no—
285C22×C8—21no—
286C22×C8—1061no—
287C22×C8—1061no—
288C4⋊Q8—532no—
289C4⋊Q8—532no—
290C4⋊Q8—1061no—
291C4⋊Q8—1061no—
292#126—1068no—
293#126—1068no—
294#124—1064no—
295#124—1064no—
296C22⋊Q8—1064no—
297C22⋊Q8—1064no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—1064no—
301C4×Q8—1064no—
302C4×Q8C4×D5312yes—
303C4×Q8D21212yes—
304C4×D4—532no—
305C4×D4—532no—
306C4×D4—1061no—
307C4×D4—1061no—
308#117—1064no—
309#117—1064no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—531no—
315C2×C4⋊C4—532no—
316C2×C4⋊C4—1062no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4C22⋊D5311yescharacteristic
319C2×C22⋊C4—1062no—
320C2×C22⋊C4—1062no—
321C2×C22⋊C4—1061no—
322C2×C42—21no—
323C2×C42—1062no—
324C2×C42C22×D5311yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—1061no—
328C4⋊C8—1061no—
329#103—42no—
330#102—2122no—
331C4×C8—1061no—
332C4×C8—1061no—
333C24—1061no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—21no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—1062no—
340C2×Q8—1062no—
341C2×Q8—2121no—
342C2×Q8—2121no—
343C2×Q8—2121no—
344C2×Q8—2121no—
345C2×Q8#1171992112yes—
346C2×D4—532no—
347C2×D4—1061no—
348C2×D4—1061no—
349C2×D4—1061no—
350C2×D4—1061no—
351C2×D4—1061no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—41no—
355C22×C4—1061no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—1062no—
359C22×C4—2121no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4—2121no—
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C22×C4C2×C22⋊D5311yescharacteristic
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—1062no—
370C2×C8—2121no—
371C2×C8—2121no—
372C2×C8—2121no—
373C2×C8—2121no—
374C2×C8—2121no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—532no—
384C4⋊C4—534no—
385C4⋊C4—1068no—
386C4⋊C4—1062no—
387C4⋊C4—1062no—
388C4⋊C4—1061no—
389C4⋊C4—1062no—
390C4⋊C4#1171992112yes—
391C22⋊C4—1068no—
392C22⋊C4—1068no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C42—22no—
396C42—22no—
397C42—1068no—
398C42C2×C4×D5311yescharacteristic
399C42C2×D21211yescharacteristic
400C42D4×D5311yescharacteristic
401C42D4×D5311yescharacteristic
402C23—1062no—
403C23—2121no—
404C23—2121no—
405C23—2121no—
406C23—2121no—
407C23—2121no—
408C23—2121no—
409C23#1313318511yescharacteristic
410Q8—22no—
411Q8—22no—
412Q8—41no—
413Q8—41no—
414Q8—2122no—
415Q8—2122no—
416Q8—2122no—
417Q8—2122no—
418D4—1062no—
419D4—1062no—
420D4—2121no—
421D4—2121no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—1062no—
434C2×C4—1064no—
435C2×C4—2121no—
436C2×C4—2122no—
437C2×C4—2121no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4—2122no—
441C2×C4—2122no—
442C2×C4—2122no—
443C2×C4C22⋊C4×D5311yescharacteristic
444C2×C4C22⋊C4×D5311yescharacteristic
445C2×C4C2×#1171992111yescharacteristic
446C2×C4#1313318511yescharacteristic
447C2×C4#1313318711yescharacteristic
448C2×C4#1313318711yescharacteristic
449C2×C4#1313323711yescharacteristic
450C2×C4#1313323711yescharacteristic
451C2×C4#1313323811yescharacteristic
452C2×C4#1313324011yescharacteristic
453C2×C4#1313324311yescharacteristic
454C2×C4#1313324411yescharacteristic
455C2×C4#1313324811yescharacteristic
456C2×C4#1313324911yescharacteristic
457C8—81no—
458C8—4241no—
459C8—2122no—
460C22—4241no—
461C22—4241no—
462C22—4241no—
463C22—4241no—
464C22—2122no—
465C22—2122no—
466C22—2122no—
467C22#1681047811yescharacteristic
468C22#1681076611yescharacteristic
469C22#1681076711yescharacteristic
470C22#1681078311yescharacteristic
471C22#1681078411yescharacteristic
472C22#1681079411yescharacteristic
473C22#1681080411yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—161no—
479C4—2122no—
480C4—2124no—
481C4#1681061711yescharacteristic
482C4#1681061811yescharacteristic
483C4#1681062211yescharacteristic
484C4#1681062311yescharacteristic
485C4#1681076611yescharacteristic
486C4#1681077111yescharacteristic
487C4#1681078311yescharacteristic
488C4#1681078811yescharacteristic
489C2—4242no—
490C2#2027382411yescharacteristic
491C2#2027388611yescharacteristic
492C2#2027397011yescharacteristic
493C2#2027398311yescharacteristic
494C2#2027465311yescharacteristic
495C2#2027465411yescharacteristic
496C2#2027467211yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1803C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenter#1803D53
hypocenterC53#8449
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. #20275675C2semidirect
4. C2×#16810368C2semidirect
5. C2×Q8⋊#88322C2semidirect
7. C53×#1803C2semidirect
11. #16811346C4semidirect
243. #1803D53semidirect
280. C53#8449semidirect

Maximal subgroups

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

subgroupindex
C2×#168113462
#202756752
#202747122
C2×#168103682
C2×Q8⋊#883222
C2×#168103502
C53×#18032
#844953