Finite Group Databasebeta

← #27276200

Subgroups — #27276200

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#20275676C211yesmaximal, characteristic, complemented
3#20274720C211yesmaximal, characteristic, complemented
4#20274716C211yesmaximal, characteristic
5C2×#16810370C211yesmaximal, characteristic, complemented
6C2×#16810352C211yesmaximal, characteristic
7C2×C53×Q8⋊C8C211yesmaximal, 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×#13133141—21no—
21C2×#13133139—21no—
22#16810493C2211yescharacteristic
23#16810386—21no—
24#16810386—22no—
25#16810385—21no—
26#16810370—22no—
27#16810352—22no—
28C2×C4×Q8×C53C2211yescharacteristic
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C53×Q8⋊C8—22no—
32C22×Dic106—21no—
33C22×D212D411yescharacteristic
34C22×C4×D53—21no—
35C22×#88322—21no—
36#13133217—21no—
37#13133217—21no—
38#13133217C2×C412yes—
39#13133217D412yes—
40#13133212—28no—
41#13133212—28no—
42#13133205—24no—
43#13133205—24no—
44#13133193—24no—
45#13133193—24no—
46C4×Dic106—24no—
47C4×Dic106—24no—
48C4×D212—21no—
49C4×D212—21no—
50C4×D212C2×C412yes—
51C4×D212D412yes—
52#13133175—24no—
53#13133175—24no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—21no—
58C2×C4⋊Dic53D411yescharacteristic
59C2×C4⋊Dic53D412yes—
60C2×#11719919—22no—
61C2×#11719921—22no—
62C2×#11719921—22no—
63C2×#11719921—21no—
64C2×C4×Dic53—22no—
65C4⋊#88322—21no—
66C4⋊#88322—21no—
67#13133141—42no—
68#13133139—42no—
69C22×Q8×C53D411yescharacteristic
70C22×C8×C53—21no—
71C22×C8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53—21no—
74C4×Q8×C53C2×C412yes—
75C4×Q8×C53D412yes—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C42×C53—21no—
79C2×C42×C53C2311yescharacteristic
80C4⋊C8×C53—21no—
81C4⋊C8×C53—21no—
82C4×C8×C53—21no—
83C4×C8×C53—21no—
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×C53C2×D411yescharacteristic
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×Dic53—21no—
97C22×Dic53—22no—
98C22×Dic53—41no—
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—22no—
118C2×C8×C53—41no—
119C2×C8×C53—41no—
120C2×C8×C53—41no—
121C2×C8×C53—41no—
122C2×C8×C53—41no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53C22⋊C412yes—
127C42×C53—22no—
128C42×C53—22no—
129C42×C53C22×C411yescharacteristic
130C42×C53C2×D411yescharacteristic
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133#11719921—28no—
134#11719921—28no—
135#11719921—22no—
136#11719921—22no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—21no—
140C4⋊Dic53—22no—
141C4⋊Dic53—42no—
142C4⋊Dic53—41no—
143C4⋊Dic53—41no—
144C4⋊Dic53—41no—
145C4⋊Dic53—41no—
146C4⋊Dic53C22⋊C412yes—
147C4⋊Dic53C22⋊C414yes—
148#11719919—28no—
149C4×Dic53—28no—
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×C53C2×C22⋊C411yescharacteristic
174C2×C4×C53C2×C22⋊C411yescharacteristic
175C2×C4×C53C2×C22⋊C411yescharacteristic
176C2×C4×C53#12011yescharacteristic
177C2×C4×C53#12011yescharacteristic
178C2×C4×C53#12011yescharacteristic
179C2×C4×C53#13511yescharacteristic
180C2×C4×C53#13511yescharacteristic
181C2×C4×C53#13511yescharacteristic
182C2×C4×C53#13511yescharacteristic
183C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
184C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C53#13711yescharacteristic
186C2×C4×C53#13711yescharacteristic
187C2×Dic53—24no—
188C2×Dic53—22no—
189C2×Dic53—42no—
190C2×Dic53—41no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—81no—
194C2×Dic53—81no—
195C2×Dic53—81no—
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—42no—
209C8×C53—81no—
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#41611yescharacteristic
231C4×C53#41611yescharacteristic
232C4×C53#41711yescharacteristic
233C4×C53#41711yescharacteristic
234C4×C53#44911yescharacteristic
235C4×C53#44911yescharacteristic
236C4×C53#45211yescharacteristic
237C4×C53#45211yescharacteristic
238Dic53—44no—
239Dic53—42no—
240Dic53—161no—
241C2×#556—531no—
242#1990—531no—
243#1803—531no—
244#1802—531no—
245C2×#332—531no—
246C2×Q8⋊C8D5311yescharacteristic, complemented
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—1061no—
271C2×#102—1061no—
272C2×C4×C8C2×D5311yescharacteristic
273#384—531no—
274#340—1061no—
275#340—1062no—
276#340—1061no—
277#332—1062no—
278Q8⋊C8—22no—
279D4⋊C8—1062no—
280C53#844911yescharacteristic, complemented, Sylow
281C22×Q8—1061no—
282C22×Q8D21211yescharacteristic
283C22×D4—531no—
284C23×C4—1061no—
285C22×C8—21no—
286C22×C8—21no—
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×Q8C22⋊D5312yes—
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—531no—
312C2×C4⋊C4—532no—
313C2×C4⋊C4—1062no—
314C2×C4⋊C4—1061no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—1061no—
317C2×C4⋊C4—1061no—
318C2×C4⋊C4D21211yescharacteristic
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—2122no—
330#102—2122no—
331C4×C8—21no—
332C4×C8—21no—
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—1061no—
355C22×C4—1062no—
356C22×C4—1062no—
357C22×C4—1062no—
358C22×C4—2121no—
359C22×C4—2121no—
360C22×C4—2121no—
361C22×C4—2121no—
362C22×C4—2121no—
363C22×C4C2×D21211yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C22×C4D4×D5311yescharacteristic
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—2121no—
373C2×C8—2121no—
374C2×C8—2121no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—532no—
379C4⋊C4—534no—
380C4⋊C4—1068no—
381C4⋊C4—1062no—
382C4⋊C4—1061no—
383C4⋊C4—1062no—
384C4⋊C4—1062no—
385C4⋊C4—2122no—
386C4⋊C4—2121no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C4⋊C4—2121no—
390C4⋊C4#1171992112yes—
391C22⋊C4—1068no—
392C22⋊C4—1068no—
393C22⋊C4—1062no—
394C22⋊C4—1062no—
395C42—22no—
396C42—22no—
397C42—1068no—
398C42C2×C4×D5311yescharacteristic
399C42D4×D5311yescharacteristic
400C42D4×D5311yescharacteristic
401C42C2×C22⋊D5311yescharacteristic
402C23—1062no—
403C23—2121no—
404C23—2121no—
405C23—2121no—
406C23—2121no—
407C23—2121no—
408C23—2121no—
409C23#1313318711yescharacteristic
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—1062no—
431C2×C4—1064no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—4241no—
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#1313318511yescharacteristic
448C2×C4#1313318711yescharacteristic
449C2×C4#1313323811yescharacteristic
450C2×C4#1313323911yescharacteristic
451C2×C4#1313323911yescharacteristic
452C2×C4#1313324011yescharacteristic
453C2×C4#1313324411yescharacteristic
454C2×C4#1313324511yescharacteristic
455C2×C4#1313324911yescharacteristic
456C2×C4#1313325011yescharacteristic
457C8—42no—
458C8—81no—
459C8—4241no—
460C22—4241no—
461C22—4241no—
462C22—4241no—
463C22—4241no—
464C22—2122no—
465C22—2122no—
466C22—2122no—
467C22#1681047811yescharacteristic
468C22#1681077011yescharacteristic
469C22#1681077111yescharacteristic
470C22#1681078711yescharacteristic
471C22#1681078811yescharacteristic
472C22#1681079711yescharacteristic
473C22#1681080711yescharacteristic
474C4—21no—
475C4—21no—
476C4—42no—
477C4—42no—
478C4—8481no—
479C4—2124no—
480C4—2122no—
481C4#1681061811yescharacteristic
482C4#1681062011yescharacteristic
483C4#1681062311yescharacteristic
484C4#1681062511yescharacteristic
485C4#1681076711yescharacteristic
486C4#1681077011yescharacteristic
487C4#1681078411yescharacteristic
488C4#1681078711yescharacteristic
489C2—4242no—
490C2#2027382811yescharacteristic
491C2#2027389011yescharacteristic
492C2#2027397411yescharacteristic
493C2#2027398711yescharacteristic
494C2#2027465411yescharacteristic
495C2#2027465611yescharacteristic
496C2#2027467511yescharacteristic
subgroupG/H
centerC23#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC2×C53×Q8⋊C8C2
socleC23×C53#120
prime radicalC22×D212D4
cosocleC2×C42×C53C23
hypercenterC2×Q8⋊C8D53
hypocenterC53#8449
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #20275676C2semidirect
3. #20274720C2semidirect
5. C2×#16810370C2semidirect
7. C2×C53×Q8⋊C8C2semidirect
11. #16811346C4semidirect
246. C2×Q8⋊C8D53semidirect
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
#202756762
#202747202
#202747162
C2×#168103702
C2×#168103522
C2×C53×Q8⋊C82
#844953