Finite Group Databasebeta

← #27538569

Subgroups — #27538569

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811061C211yesmaximal, characteristic, complemented
2#20284296C211yesmaximal, characteristic, complemented
3C2×#16810859C211yesmaximal, characteristic, complemented
4#20276828C212yesmaximal, complemented
5#20276818C212yesmaximal, complemented
6#20276811C212yesmaximal, complemented
7#20276801C212yesmaximal, complemented
8#20276503C211yesmaximal, characteristic, complemented
9#20276432C211yesmaximal, characteristic, complemented
10#20276428C211yesmaximal, characteristic, complemented
11C22×M32×C53C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C22×D424C2211yescharacteristic
14#16811134—21no—
15C2×#13133191—21no—
16#16811061—24no—
17C2×#13133162C2212yescomplemented
18C2×D848C2212yescomplemented
19#16810864C2211yescharacteristic, complemented
20#16810864C2211yescharacteristic, complemented
21#16810859—21no—
22#16810859—21no—
23#16810859C2212yes—
24#16810859C2212yes—
25#16810859C2211yescharacteristic, complemented
26#16810859C2211yescharacteristic, complemented
27#16810852—21no—
28#16810852—21no—
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810620—21no—
32C2×#13133139—21no—
33#16810451C2212yescomplemented
34#16810449C2212yescomplemented
35#16810428C2212yescomplemented
36#16810425C2212yescomplemented
37#16810425C2212yescomplemented
38#16810425C2212yescomplemented
39C23×C8×C53C2211yescharacteristic
40C2×M32×C53—24no—
41C2×M32×C53C2211yescharacteristic
42C2×M32×C53C2211yescharacteristic
43C2×M32×C53C2212yescomplemented
44C22×C16×C53C2212yescomplemented
45C2×#11719945—21no—
46C22×D212—21no—
47C22×C22⋊D53—41no—
48#13133239—22no—
49#13133239—22no—
50C2×Dic212C2311yescharacteristic
51C2×#11719913—21no—
52C2×D424—21no—
53C2×D424—22no—
54C2×D424—22no—
55C2×D424C2311yescharacteristic
56C2×D424C2312yes—
57#13133200—41no—
58#13133196—21no—
59#13133196—21no—
60#13133191—21no—
61#13133191—21no—
62#13133191—22no—
63#13133191—22no—
64#13133191—41no—
65#13133191—41no—
66#13133186—41no—
67C4×C22⋊D53—41no—
68C4×C22⋊D53—41no—
69#13133176—21no—
70C2×C4⋊Dic53—21no—
71C2×#11719921—41no—
72#13133162—24no—
73D848—24no—
74#13133151—21no—
75#13133151C2312yes—
76#13133151C2311yescharacteristic
77#13133151C2311yescharacteristic
78#13133149—21no—
79#13133142—21no—
80#13133142—21no—
81#13133139—21no—
82#13133139—21no—
83#13133139—22no—
84#13133139—22no—
85C23×C4×C53D411yescharacteristic
86C22×C8×C53—22no—
87C22×C8×C53—22no—
88C22×C8×C53D411yescharacteristic
89C22×C8×C53D411yescharacteristic
90C22×C8×C53D411yescharacteristic
91C22×C8×C53C2311yescharacteristic
92C22×C8×C53C2311yescharacteristic
93C22×C8×C53C2311yescharacteristic
94M32×C53—22no—
95M32×C53—24no—
96M32×C53D414yescomplemented
97C2×C16×C53—24no—
98C2×C16×C53C2312yes—
99C2×C16×C53C2312yes—
100C24×C53D811yescharacteristic
101C23×D53—41no—
102C22×C4×C53—22no—
103C22×C4×C53—22no—
104C22×C4×C53D811yescharacteristic
105C22×C4×C53D811yescharacteristic
106C22×C4×C53D811yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—42no—
114C2×C22⋊D53—42no—
115C2×C22⋊D53—84no—
116C22×Dic53—41no—
117#11719945—42no—
118#11719945—42no—
119C2×D212—22no—
120C2×D212—21no—
121C2×D212—42no—
122C2×D212—42no—
123C2×D212—41no—
124C2×C4×D53—41no—
125C2×Dic106—21no—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—22no—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—21no—
132C2×C8×C53—21no—
133C2×C8×C53C2×D411yescharacteristic
134C2×C8×C53C2×D411yescharacteristic
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D412yes—
137C2×C8×C53#4112yes—
138C2×C8×C53C2411yescharacteristic
139C22⋊Dic53—41no—
140C22⋊Dic53—41no—
141C22⋊Dic53—41no—
142#11719921—42no—
143#11719921—42no—
144#11719921—41no—
145#11719921—41no—
146C4⋊Dic53—22no—
147C4⋊Dic53—21no—
148C4⋊Dic53—21no—
149C4⋊Dic53—41no—
150#11719919—41no—
151#11719919—41no—
152C4×Dic53—41no—
153Dic212—21no—
154Dic212—21no—
155D424—21no—
156D424—22no—
157D424—22no—
158D424—21no—
159D424—42no—
160#11719913—42no—
161C16×C53—22no—
162C16×C53C2×D414yescomplemented
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—41no—
166C23×C53C2×D811yescharacteristic
167C23×C53C2×D811yescharacteristic
168C23×C53C2×D811yescharacteristic
169C22×D53—41no—
170C22×D53—42no—
171C22×D53—82no—
172C22×D53—82no—
173C22×D53—82no—
174C2×C4×C53—22no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53#12112yes—
182C2×C4×C53C2×D811yescharacteristic
183C2×C4×C53C2×D811yescharacteristic
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53#13512yes—
186C2×C4×C53C22×D411yescharacteristic
187C22⋊D53—82no—
188C22⋊D53—84no—
189C22⋊D53—82no—
190C22⋊D53—82no—
191C22⋊D53—84no—
192C2×Dic53—41no—
193C2×Dic53—42no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—82no—
197D212—42no—
198D212—41no—
199D212—42no—
200D212—41no—
201D212—82no—
202C4×D53—82no—
203Dic106—41no—
204Dic106—41no—
205C8×C53—21no—
206C8×C53—21no—
207C8×C53#12112yes—
208C8×C53C22×D411yescharacteristic
209C8×C53C2×#4111yescharacteristic
210#33605—531nomaximal, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—44no—
219C22×C53#46612yes—
220C22×C53#50912yescomplemented
221C22×C53C22×D811yescharacteristic
222C2×D53—82no—
223C2×D53—81no—
224C2×D53—81no—
225C2×D53—82no—
226C2×D53—81no—
227C2×D53—82no—
228C2×D53—164no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53#46612yes—
232C4×C53C2×#12111yescharacteristic
233C4×C53C2×#13511yescharacteristic
234Dic53—82no—
235Dic53—81no—
236Dic53—81no—
237Dic53—81no—
238C2×#509—531no—
239C22×M32D5311yescharacteristic, complemented
240#3050—531no—
241C2×#466—531no—
242#2220—532no—
243#2218—532no—
244#2217—532no—
245#2215—532no—
246#2159—531no—
247#2140—531no—
248#2139—531no—
249C2×C53—22no—
250C2×C53—22no—
251C2×C53—41no—
252C2×C53C2×#46611yescharacteristic
253C2×C53C2×#50911yescharacteristic
254C2×C53#341411yescharacteristic
255D53—161no—
256D53—162no—
257C2×#135—531no—
258C22×D8—531no—
259C23×C8C2×D5311yescharacteristic
260#525—1061no—
261C2×#121—1061no—
262#509—1064no—
263C2×SD32—532no—
264C2×D16—532no—
265C2×M32—24no—
266C2×M32C2×D5311yescharacteristic
267C2×M32C2×D5311yescharacteristic
268C2×M32C2×D5312yescomplemented
269C22×C16C2×D5312yescomplemented
270#467—531no—
271#467—531no—
272#466—532no—
273#466—532no—
274#466—531no—
275#466—531no—
276#466—1061no—
277#466—1061no—
278#465—1061no—
279#465—1061no—
280#427—531no—
281C2×#107—531no—
282#416—1061no—
283C2×#102—1061no—
284#367—532no—
285#366—532no—
286#358—532no—
287#357—532no—
288#357—532no—
289#357—532no—
290C53#3360511yescharacteristic, complemented, Sylow
291C2×#41—1061no—
292C22×D4—1061no—
293C22×D4—2121no—
294C23×C4D21211yescharacteristic
295#135—1062no—
296#135—1062no—
297C2×Q16—531no—
298C2×SD16—1061no—
299C2×D8—532no—
300C2×D8—531no—
301C2×D8—1061no—
302C2×D8—1062no—
303C2×D8—1062no—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D21211yescharacteristic
307C22×C8D21211yescharacteristic
308C22×C8D21211yescharacteristic
309C22×C8C22×D5311yescharacteristic
310C22×C8C22×D5311yescharacteristic
311C22×C8C22×D5311yescharacteristic
312#123—2121no—
313C22⋊Q8—1061no—
314C22⋊Q8—1061no—
315#121—1062no—
316#121—1062no—
317#121—1061no—
318#121—1061no—
319#121—2121no—
320#121—2121no—
321#120—2121no—
322C4×D4—2121no—
323C4×D4—2121no—
324#117—1061no—
325C2×C4⋊C4—1061no—
326C2×C22⋊C4—2121no—
327SD32—1064no—
328D16—1064no—
329M32—22no—
330M32—24no—
331M32C22⋊D5314yescomplemented
332C2×C16—24no—
333C2×C16C22×D5312yes—
334C2×C16C22×D5312yes—
335#107—531no—
336#107—531no—
337#107—532no—
338#107—1061no—
339#106—1061no—
340#103—1061no—
341#103—1061no—
342#102—1062no—
343#102—1062no—
344#102—1061no—
345#102—1061no—
346C24—2121no—
347C24D42411yescharacteristic
348#41—2122no—
349#41—2122no—
350C2×Q8—1061no—
351C2×D4—1062no—
352C2×D4—1061no—
353C2×D4—4244no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2122no—
358C2×D4—2122no—
359C2×D4—2122no—
360C2×D4—2121no—
361C2×D4—2122no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—2121no—
365C22×C4—2121no—
366C22×C4D42411yescharacteristic
367C22×C4D42411yescharacteristic
368C22×C4D42411yescharacteristic
369C22×C4C2×D21211yescharacteristic
370C22×C4C2×D21211yescharacteristic
371C22×C4C2×D21211yescharacteristic
372Q16—1061no—
373Q16—1061no—
374SD16—2122no—
375D8—1061no—
376D8—1062no—
377D8—1062no—
378D8—1061no—
379D8—2122no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8C2×D21211yescharacteristic
388C2×C8C2×D21211yescharacteristic
389C2×C8C2×D21211yescharacteristic
390C2×C8#1171994512yes—
391C2×C8C2×C22⋊D5312yes—
392C2×C8C23×D5311yescharacteristic
393C4⋊C4—1062no—
394C4⋊C4—1061no—
395C4⋊C4—1061no—
396C4⋊C4—2121no—
397C4⋊C4—2121no—
398C4⋊C4—2121no—
399C22⋊C4—2121no—
400C22⋊C4—2121no—
401C22⋊C4—2121no—
402C22⋊C4—2122no—
403C22⋊C4—2122no—
404C22⋊C4—2121no—
405C22⋊C4—2121no—
406C42—2121no—
407C16—22no—
408C16C2×C22⋊D5314yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—4242no—
413C23—4242no—
414C23—4242no—
415C23—2121no—
416C23—2122no—
417C23C2×D42411yescharacteristic
418C23C2×D42411yescharacteristic
419C23C2×D42411yescharacteristic
420Q8—2121no—
421Q8—2121no—
422D4—4244no—
423D4—4242no—
424D4—4244no—
425D4—4242no—
426D4—4242no—
427D4—4242no—
428D4—2122no—
429D4—2121no—
430D4—2122no—
431D4—2121no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—4242no—
440C2×C4—4242no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2122no—
444C2×C4—2121no—
445C2×C4#1313319112yes—
446C2×C4C2×D42411yescharacteristic
447C2×C4C2×D42411yescharacteristic
448C2×C4C2×D42411yescharacteristic
449C2×C4#1313323912yes—
450C2×C4C22×D21211yescharacteristic
451C8—21no—
452C8—21no—
453C8#1313319112yes—
454C8C22×C22⋊D5311yescharacteristic
455C8C2×#1171994511yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—8484no—
465C22—4242no—
466C22—4241no—
467C22—4241no—
468C22—4242no—
469C22—4241no—
470C22—4242no—
471C22#1681085912yes—
472C22#1681106112yescomplemented
473C22C22×D42411yescharacteristic
474C4—21no—
475C4—21no—
476C4—4242no—
477C4—4241no—
478C4—4241no—
479C4—4241no—
480C4#1681085912yes—
481C4C2×#1313319111yescharacteristic
482C4C2×#1313323911yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—8482no—
487C2—8481no—
488C2C2×#1681085911yescharacteristic
489C2C2×#1681106111yescharacteristic
490C2#2028785111yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC22×M32×C53C2
socleC22×C53C22×D8
hypercenterC22×M32D53
hypocenterC53#33605
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
1. C2×#16811061C2semidirect
2. #20284296C2semidirect
3. C2×#16810859C2semidirect
4. #20276828C2semidirect
5. #20276818C2semidirect
6. #20276811C2semidirect
7. #20276801C2semidirect
8. #20276503C2semidirect
9. #20276432C2semidirect
10. #20276428C2semidirect
11. C22×M32×C53C2semidirect
17. C2×#13133162C22semidirect
18. C2×D848C22semidirect
19. #16810864C22semidirect
20. #16810864C22semidirect
25. #16810859C22semidirect
26. #16810859C22semidirect
33. #16810451C22semidirect
34. #16810449C22semidirect
35. #16810428C22semidirect
36. #16810425C22semidirect
37. #16810425C22semidirect
38. #16810425C22semidirect
43. C2×M32×C53C22semidirect
44. C22×C16×C53C22semidirect
96. M32×C53D4semidirect
162. C16×C53C2×D4semidirect
220. C22×C53#509semidirect
239. C22×M32D53semidirect
268. C2×M32C2×D53semidirect
269. C22×C16C2×D53semidirect
290. C53#33605semidirect
331. M32C22⋊D53semidirect
408. C16C2×C22⋊D53semidirect
472. C22#16811061semidirect

Maximal subgroups

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

subgroupindex
C2×#168110612
#202842962
C2×#168108592
#202768282
#202768182
#202768112
#202768012
#202765032
#202764322
#202764282
C22×M32×C532
#3360553