Finite Group Databasebeta

← #27355811

Subgroups — #27355811

Order 13568.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283844C211yesmaximal, characteristic, complemented
2#20283836C211yesmaximal, characteristic, complemented
3C4×#13133239C211yesmaximal, characteristic
4#20273619C212yesmaximal, complemented
5#20273567C212yesmaximal, complemented
6#20273546C212yesmaximal, complemented
7#20273529C212yesmaximal, complemented
8#20273474C211yesmaximal, characteristic, complemented
9#20273155C211yesmaximal, characteristic, complemented
10#20273155C211yesmaximal, characteristic, complemented
11C53×#1451C211yesmaximal, characteristic, complemented
12C2×#13133239C411yescharacteristic
13C4×#11719945C2211yescharacteristic
14C4×#11719945C2211yescharacteristic
15#16810753C2212yes—
16C8×Dic106C2212yes—
17C4×Dic212C411yescharacteristic
18C4×Dic212C2211yescharacteristic
19C4×#11719913C411yescharacteristic
20C4×#11719913C411yescharacteristic
21C4×#11719913C2211yescharacteristic
22C4×#11719913C2211yescharacteristic
23C4×D424C411yescharacteristic
24C4×D424C2211yescharacteristic
25#16810702C2212yes—
26#16810698—22no—
27#16810697—22no—
28C8×D212C2212yes—
29#16810679—21no—
30#16810675—21no—
31#16810661C411yescharacteristic
32#16810647C2211yescharacteristic
33#16810647C2211yescharacteristic
34#16810645—21no—
35C4⋊C8×D53—21no—
36#16810620C411yescharacteristic
37#16810620C411yescharacteristic
38#16810376C2212yes—
39#16810376C2212yes—
40#16810355C2212yes—
41#16810355C2212yes—
42#16810352C2212yes—
43#16810352C2212yes—
44C53×#431C2211yescharacteristic
45C4×M16×C53C2211yescharacteristic
46C2×C4×C8×C53C2211yescharacteristic
47C53×#322C2212yes—
48C53×#322C2212yes—
49C2×#11719945C2×C411yescharacteristic
50C2×#11719945C2×C411yescharacteristic
51#13133239—24no—
52C2×Dic212C2×C411yescharacteristic
53C2×#11719913C2×C411yescharacteristic
54C2×#11719913C2×C411yescharacteristic
55C2×D424C2×C411yescharacteristic
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719912—22no—
59C2×C8×D53—22no—
60C4×Dic106C2×C411yescharacteristic
61C4×Dic106C2×C411yescharacteristic
62C4×Dic106C2311yescharacteristic
63C4×Dic106C2311yescharacteristic
64C4×D212C2×C411yescharacteristic
65C4×D212C2×C411yescharacteristic
66C4×D212C2311yescharacteristic
67C4×D212C2311yescharacteristic
68C4×C22⋊D53—21no—
69C4×C22⋊D53—21no—
70C4×C22⋊D53—21no—
71C4×C22⋊D53—21no—
72#13133176C2×C411yescharacteristic
73#13133176C2×C411yescharacteristic
74#13133174—21no—
75#13133174—21no—
76C42×D53—21no—
77C42×D53—21no—
78#13133151C2×C411yescharacteristic
79#13133151C2×C411yescharacteristic
80#13133149C2×C411yescharacteristic
81#13133149C2×C411yescharacteristic
82C4⋊#88322—21no—
83C4⋊#88322—21no—
84C4⋊#88322C2312yes—
85C4⋊#88322C2312yes—
86#13133146—22no—
87#13133146—22no—
88#13133142C2×C411yescharacteristic
89#13133142C2×C411yescharacteristic
90#13133142C2×C411yescharacteristic
91#13133142C2×C411yescharacteristic
92#13133139C2×C411yescharacteristic
93#13133139C2×C411yescharacteristic
94#13133139C2×C411yescharacteristic
95#13133139C2×C411yescharacteristic
96#13133129—22no—
97#13133129—22no—
98#13133127—22no—
99C8×Dic53—22no—
100C2×M16×C53—21no—
101C22×C8×C53C2×C411yescharacteristic
102C22×C8×C53C2×C411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105C4⋊C8×C53D411yescharacteristic
106C22⋊C8×C53D412yes—
107#97×C53—21no—
108C4×C8×C53C2×C411yescharacteristic
109C4×C8×C53C2×C411yescharacteristic
110C4×C8×C53C2312yes—
111C4×C8×C53C2311yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C4×C8×C53C2312yes—
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—21no—
118C2×C22⋊D53—21no—
119#11719945—22no—
120#11719945—22no—
121#11719945—41no—
122#11719945—41no—
123C2×D212C22×C411yescharacteristic
124C2×D212C22×C411yescharacteristic
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×Dic106C22×C411yescharacteristic
132C2×Dic106C22×C411yescharacteristic
133M16×C53—42no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C2×C8×C53—41no—
137C2×C8×C53C22×C411yescharacteristic
138C2×C8×C53C22×C411yescharacteristic
139C2×C8×C53C22×C411yescharacteristic
140C2×C8×C53C22×C411yescharacteristic
141C2×C8×C53C2×D412yes—
142C2×C8×C53C2×D412yes—
143C42×C53C22×C411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C22⋊Dic53—21no—
148C22⋊Dic53—21no—
149#11719921—21no—
150#11719921—21no—
151#11719921—21no—
152#11719921—21no—
153C4⋊Dic53C22×C411yescharacteristic
154C4⋊Dic53C22×C411yescharacteristic
155C4⋊Dic53C22×C411yescharacteristic
156C4⋊Dic53C22×C411yescharacteristic
157#11719919—21no—
158#11719919—21no—
159#11719919—21no—
160#11719919—21no—
161C4×Dic53—21no—
162C4×Dic53—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165C4×Dic53—21no—
166C4×Dic53—21no—
167#11719917—81no—
168#11719917—81no—
169C2×#88322—22no—
170C2×#88322—22no—
171Dic212—22no—
172D424—22no—
173#11719913—22no—
174#11719913M1614yescomplemented
175#11719912—82no—
176C8×D53—82no—
177C23×C53C4×D411yescharacteristic
178C22×D53—21no—
179C22×D53—21no—
180C2×C4×C53—21no—
181C2×C4×C53—21no—
182C2×C4×C53—21no—
183C2×C4×C53—21no—
184C2×C4×C53—22no—
185C2×C4×C53C4×D411yescharacteristic
186C2×C4×C53C4×D411yescharacteristic
187C2×C4×C53C4×D411yescharacteristic
188C2×C4×C53C23×C411yescharacteristic
189C2×C4×C53C22×D411yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C22⋊D53—42no—
192C22⋊D53—42no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199D212—21no—
200D212—21no—
201D212—21no—
202D212C2×M1612yes—
203C4×D53—42no—
204C4×D53—42no—
205C4×D53—41no—
206C4×D53—41no—
207C4×D53—41no—
208C4×D53—41no—
209C4×D53—42no—
210C4×D53—42no—
211Dic106—21no—
212Dic106—21no—
213Dic106—21no—
214Dic106C2×M1612yes—
215C8×C53—21no—
216C8×C53—21no—
217C8×C53—21no—
218C8×C53—22no—
219C8×C53—22no—
220C8×C53—42no—
221C8×C53C2×M1612yes—
222#88322—42no—
223#88322—42no—
224#18061—531nomaximal, Sylow
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53C2×C4×D411yescharacteristic
229C2×D53—22no—
230C2×D53—41no—
231C2×D53—41no—
232C2×D53—41no—
233C2×D53—41no—
234C2×D53—41no—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53#44412yes—
239C4×C53C22×M1611yescharacteristic
240C4×C53#56811yescharacteristic
241C4×C53#57611yescharacteristic
242C4×C53#57811yescharacteristic
243Dic53—22no—
244Dic53—41no—
245Dic53—41no—
246Dic53—41no—
247Dic53—41no—
248Dic53—41no—
249#2991—531no—
250#2990—531no—
251C4×#135—531no—
252#1592—532no—
253#1584—532no—
254#1581—532no—
255SD16⋊C8—532no—
256#1567—531no—
257#1489—531no—
258#1489—531no—
259#1451D5311yescharacteristic, complemented
260C2×C53—41no—
261C2×C53D4×M1611yescharacteristic
262C2×C53#295111yescharacteristic
263C2×C53C2×#44411yescharacteristic
264D53—42no—
265D53—81no—
266C2×#135—531no—
267C4×#41—531no—
268C4×#41—531no—
269#446—532no—
270C8×Q8—532no—
271C4×Q16—531no—
272C4×Q16—531no—
273C4×SD16—531no—
274C4×SD16—531no—
275C4×SD16—531no—
276C4×SD16—531no—
277C4×D8—531no—
278C4×D8—531no—
279#436—532no—
280#436—1062no—
281#435—1062no—
282C8×D4—532no—
283#432—1061no—
284#431—1061no—
285#431C2×D5311yescharacteristic
286#427—531no—
287#423—531no—
288#423—531no—
289#423—1061no—
290C2×C4⋊C8—1061no—
291#416—531no—
292#416—531no—
293C4×M16C2×D5311yescharacteristic
294C2×C4×C8C2×D5311yescharacteristic
295#335—532no—
296#335—532no—
297Q8⋊C8—532no—
298Q8⋊C8—532no—
299D4⋊C8—532no—
300D4⋊C8—532no—
301#322C2×D5312yes—
302#322C2×D5312yes—
303C53#1806111yescharacteristic, complemented, Sylow
304C2×#41—531no—
305C2×#41—531no—
306#135—1064no—
307C2×Q16—531no—
308C2×SD16—531no—
309C2×SD16—531no—
310C2×D8—531no—
311C2×M16—21no—
312C2×M16—1061no—
313C2×M16—1061no—
314C2×M16—1062no—
315C22×C8—1062no—
316C22×C8C4×D5311yescharacteristic
317C22×C8C4×D5311yescharacteristic
318C4×Q8—531no—
319C4×Q8—531no—
320C4×Q8—531no—
321C4×Q8—531no—
322C4×D4—531no—
323C4×D4—531no—
324C4×D4—531no—
325C4×D4—531no—
326C4×D4—1061no—
327C4×D4—1061no—
328C4×D4—1061no—
329C4×D4—1061no—
330#117—531no—
331#117—531no—
332#117—1061no—
333#117—1061no—
334C2×C42—1061no—
335C2×C42—1061no—
336C2×C42C22×D5311yescharacteristic
337#107—531no—
338#107—531no—
339#106—531no—
340#106—531no—
341C4⋊C8—532no—
342C4⋊C8—532no—
343C4⋊C8—1062no—
344C4⋊C8—1062no—
345C4⋊C8—1061no—
346C4⋊C8—1061no—
347C4⋊C8D21211yescharacteristic
348C4⋊C8D21211yescharacteristic
349#103—531no—
350#103—531no—
351#103—531no—
352#103—531no—
353#102—531no—
354#102—531no—
355#102—531no—
356#102—531no—
357C22⋊C8—1062no—
358C22⋊C8—1062no—
359C22⋊C8D21212yes—
360#97—21no—
361#97—1062no—
362C4×C8—1062no—
363C4×C8C4×D5311yescharacteristic
364C4×C8C4×D5311yescharacteristic
365C4×C8C22×D5312yes—
366C4×C8C22×D5311yescharacteristic
367C4×C8C22×D5311yescharacteristic
368C4×C8C22×D5312yes—
369#41—1062no—
370#41—1062no—
371#41—2121no—
372#41—2121no—
373C2×Q8—531no—
374C2×Q8—531no—
375C2×D4—531no—
376C2×D4—531no—
377C2×D4—1061no—
378C2×D4—1061no—
379C22×C4—1061no—
380C22×C4—1061no—
381C22×C4—1061no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4C2×C4×D5311yescharacteristic
386C22×C4C2×D21211yescharacteristic
387C22×C4#1171994511yescharacteristic
388Q16—1062no—
389SD16—534no—
390SD16—1062no—
391D8—1062no—
392M16—42no—
393M16—4242no—
394M16—4241no—
395M16—4241no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—41no—
399C2×C8—1062no—
400C2×C8—1062no—
401C2×C8—4242no—
402C2×C8C2×C4×D5311yescharacteristic
403C2×C8C2×C4×D5311yescharacteristic
404C2×C8C2×C4×D5311yescharacteristic
405C2×C8C2×C4×D5311yescharacteristic
406C2×C8C2×D21212yes—
407C2×C8C2×D21212yes—
408C4⋊C4—531no—
409C4⋊C4—531no—
410C4⋊C4—531no—
411C4⋊C4—531no—
412C4⋊C4—1061no—
413C4⋊C4—1061no—
414C4⋊C4—1061no—
415C4⋊C4—1061no—
416C22⋊C4—1061no—
417C22⋊C4—1061no—
418C22⋊C4—1061no—
419C22⋊C4—1061no—
420C22⋊C4—1061no—
421C22⋊C4—1061no—
422C42—1061no—
423C42—1061no—
424C42—1061no—
425C42—1061no—
426C42—1061no—
427C42—1061no—
428C42C2×C4×D5311yescharacteristic
429C42C2×D21211yescharacteristic
430C42#1171994511yescharacteristic
431C42C23×D5311yescharacteristic
432C23—1061no—
433C23—1061no—
434C23C4×D21211yescharacteristic
435Q8—532no—
436Q8—1061no—
437Q8—1061no—
438Q8—1061no—
439D4—532no—
440D4—1061no—
441D4—1061no—
442D4—1061no—
443D4—2122no—
444D4—2122no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—1061no—
451C2×C4—1061no—
452C2×C4—1061no—
453C2×C4—1061no—
454C2×C4—1061no—
455C2×C4—1061no—
456C2×C4—2122no—
457C2×C4—2122no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4—2121no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4C4×D21211yescharacteristic
465C2×C4C4×D21211yescharacteristic
466C2×C4C4×D21211yescharacteristic
467C2×C4C22×C4×D5311yescharacteristic
468C2×C4C22×D21211yescharacteristic
469C2×C4C2×#1171994511yescharacteristic
470C8—21no—
471C8—21no—
472C8—21no—
473C8—22no—
474C8—22no—
475C8—42no—
476C8—2122no—
477C8—2122no—
478C8M16×D5312yes—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—1062no—
483C22—2121no—
484C22—2121no—
485C22—2121no—
486C22—2121no—
487C22—2121no—
488C22C2×C4×D21211yescharacteristic
489C4—21no—
490C4—21no—
491C4—21no—
492C4—1062no—
493C4—2121no—
494C4—2121no—
495C4—2121no—
496C4—2121no—
497C4—2121no—
498C4#1681074612yes—
499C4C2×M16×D5311yescharacteristic
500C4#1681138911yescharacteristic
501C4#1681142811yescharacteristic
502C4#1681144011yescharacteristic
503C2—41no—
504C2—4241no—
505C2—2122no—
506C2M16×D21211yescharacteristic
507C2#2028359211yescharacteristic
508C2C2×#1681074611yescharacteristic
subgroupG/H
centerC2×C4C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1451C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133239C4
cosocleC42×C53C24
hypercenter#1451D53
hypocenterC53#18061
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1451

Complements

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

NH ≅ G/Ntype
1. #20283844C2semidirect
2. #20283836C2semidirect
4. #20273619C2semidirect
5. #20273567C2semidirect
6. #20273546C2semidirect
7. #20273529C2semidirect
8. #20273474C2semidirect
9. #20273155C2semidirect
10. #20273155C2semidirect
11. C53×#1451C2semidirect
174. #11719913M16semidirect
259. #1451D53semidirect
303. C53#18061semidirect

Maximal subgroups

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

subgroupindex
#202838442
#202838362
C4×#131332392
#202736192
#202735672
#202735462
#202735292
#202734742
#202731552
C53×#14512
#1806153