Finite Group Databasebeta

← #27538580

Subgroups — #27538580

Order 13568.

686 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811066C211yesmaximal, characteristic, complemented
2#20284296C211yesmaximal, characteristic
3C2×#16810864C211yesmaximal, characteristic, complemented
4#20276828C212yesmaximal, complemented
5#20276823C212yesmaximal, complemented
6#20276811C212yesmaximal, complemented
7#20276806C212yesmaximal, complemented
8#20276503C211yesmaximal, characteristic, complemented
9#20276436C211yesmaximal, characteristic, complemented
10#20276428C211yesmaximal, characteristic, complemented
11C22×M32×C53C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13C22×Dic212C2211yescharacteristic
14#16811134—21no—
15C2×#13133196—21no—
16#16811066—24no—
17C2×Dic424C2212yescomplemented
18C2×#13133162C2212yescomplemented
19#16810864—21no—
20#16810864—21no—
21#16810864C2211yescharacteristic
22#16810864C2212yes—
23#16810864C2211yescharacteristic
24#16810864C2212yes—
25#16810859C2211yescharacteristic
26#16810859C2211yescharacteristic
27#16810852—21no—
28#16810852—21no—
29#16810661C2211yescharacteristic
30C2×#13133151C2211yescharacteristic
31#16810620—21no—
32C2×#13133142—21no—
33#16810451C2212yescomplemented
34#16810449C2212yescomplemented
35#16810428C2212yescomplemented
36#16810428C2212yescomplemented
37#16810428C2212yescomplemented
38#16810425C2212yescomplemented
39C23×C8×C53C2211yescharacteristic
40C2×M32×C53—24no—
41C2×M32×C53C2212yes—
42C2×M32×C53C2211yescharacteristic
43C2×M32×C53C2211yescharacteristic
44C22×C16×C53C2212yes—
45C2×#11719945—21no—
46C22×Dic106—21no—
47#13133239—22no—
48#13133239—22no—
49C2×Dic212—21no—
50C2×Dic212—22no—
51C2×Dic212—22no—
52C2×Dic212C2312yes—
53C2×Dic212C2311yescharacteristic
54C2×#11719913—21no—
55C2×D424C2311yescharacteristic
56#13133200—41no—
57#13133196—21no—
58#13133196—21no—
59#13133196—22no—
60#13133196—22no—
61#13133196—41no—
62#13133196—41no—
63#13133191—21no—
64#13133191—21no—
65#13133186—41no—
66C4×C22⋊D53—41no—
67C4×C22⋊D53—41no—
68#13133176—21no—
69C2×C4⋊Dic53—21no—
70C2×#11719919—41no—
71C2×C22⋊Dic53—41no—
72Dic424—24no—
73#13133162—24no—
74#13133151—21no—
75#13133151C2311yescharacteristic
76#13133151C2311yescharacteristic
77#13133151C2312yes—
78#13133149—21no—
79#13133142—21no—
80#13133142—21no—
81#13133142—22no—
82#13133142—22no—
83#13133139—21no—
84#13133139—21no—
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
101C22×C4×C53—22no—
102C22×C4×C53—22no—
103C22×C4×C53D811yescharacteristic
104C22×C4×C53D811yescharacteristic
105C22×C4×C53D811yescharacteristic
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114#11719945—42no—
115#11719945—42no—
116C2×D212—21no—
117C2×C4×D53—41no—
118C2×Dic106—21no—
119C2×Dic106—22no—
120C2×Dic106—42no—
121C2×Dic106—42no—
122C2×Dic106—41no—
123C2×C8×C53—22no—
124C2×C8×C53—22no—
125C2×C8×C53—22no—
126C2×C8×C53—21no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53C2×D411yescharacteristic
131C2×C8×C53C2×D411yescharacteristic
132C2×C8×C53C2×D411yescharacteristic
133C2×C8×C53C2×D412yes—
134C2×C8×C53#4112yes—
135C2×C8×C53C2411yescharacteristic
136C22⋊Dic53—42no—
137C22⋊Dic53—42no—
138C22⋊Dic53—41no—
139C22⋊Dic53—41no—
140C22⋊Dic53—41no—
141#11719921—41no—
142#11719921—41no—
143C4⋊Dic53—21no—
144C4⋊Dic53—21no—
145C4⋊Dic53—22no—
146C4⋊Dic53—41no—
147#11719919—42no—
148#11719919—42no—
149#11719919—41no—
150#11719919—41no—
151C4×Dic53—41no—
152Dic212—22no—
153Dic212—21no—
154Dic212—22no—
155Dic212—21no—
156Dic212—42no—
157D424—21no—
158D424—21no—
159#11719913—42no—
160C16×C53—22no—
161C16×C53C2×D414yes—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—41no—
165C23×C53C2×D811yescharacteristic
166C23×C53C2×D811yescharacteristic
167C23×C53C2×D811yescharacteristic
168C22×D53—41no—
169C2×C4×C53—22no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—21no—
174C2×C4×C53—22no—
175C2×C4×C53—22no—
176C2×C4×C53#12112yes—
177C2×C4×C53C2×D811yescharacteristic
178C2×C4×C53C2×D811yescharacteristic
179C2×C4×C53C2×D811yescharacteristic
180C2×C4×C53#13512yes—
181C2×C4×C53C22×D411yescharacteristic
182C22⋊D53—82no—
183C22⋊D53—82no—
184C22⋊D53—82no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—82no—
191C2×Dic53—82no—
192D212—41no—
193D212—41no—
194C4×D53—82no—
195Dic106—42no—
196Dic106—41no—
197Dic106—42no—
198Dic106—41no—
199Dic106—82no—
200C8×C53—21no—
201C8×C53—21no—
202C8×C53#12112yes—
203C8×C53C22×D411yescharacteristic
204C8×C53C2×#4111yescharacteristic
205#33606—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—44no—
214C22×C53#46612yes—
215C22×C53#51012yescomplemented
216C22×C53C22×D811yescharacteristic
217C2×D53—81no—
218C2×D53—81no—
219C2×D53—81no—
220C4×C53—21no—
221C4×C53—21no—
222C4×C53#46612yes—
223C4×C53C2×#12111yescharacteristic
224C4×C53C2×#13511yescharacteristic
225Dic53—81no—
226Dic53—82no—
227Dic53—81no—
228Dic53—82no—
229Dic53—81no—
230C2×#510—531no—
231C22×M32D5311yescharacteristic, complemented
232#3050—531no—
233C2×#467—531no—
234#2220—532no—
235#2219—532no—
236#2217—532no—
237#2216—532no—
238#2159—531no—
239#2141—531no—
240#2139—531no—
241C2×C53—22no—
242C2×C53—22no—
243C2×C53—41no—
244C2×C53C2×#46611yescharacteristic
245C2×C53C2×#51011yescharacteristic
246C2×C53#341411yescharacteristic
247D53—161no—
248C2×#135—531no—
249C22×Q16—531no—
250C23×C8C2×D5311yescharacteristic
251#525—1061no—
252C2×C22⋊Q8—1061no—
253#510—1064no—
254C2×Q32—532no—
255C2×SD32—532no—
256C2×M32—24no—
257C2×M32C2×D5311yescharacteristic
258C2×M32C2×D5312yes—
259C2×M32C2×D5311yescharacteristic
260C22×C16C2×D5312yes—
261#467—531no—
262#467—532no—
263#467—531no—
264#467—532no—
265#467—1061no—
266#467—1061no—
267#466—531no—
268#466—531no—
269#465—1061no—
270#465—1061no—
271#427—531no—
272C2×#107—531no—
273#416—1061no—
274C2×#103—1061no—
275#367—532no—
276#366—532no—
277#358—532no—
278#358—532no—
279#358—532no—
280#357—532no—
281C53#3360611yescharacteristic, complemented, Sylow
282C2×#41—1061no—
283C22×Q8—1061no—
284C23×C4D21211yescharacteristic
285#135—1062no—
286#135—1062no—
287C2×Q16—532no—
288C2×Q16—531no—
289C2×Q16—1061no—
290C2×Q16—1062no—
291C2×Q16—1062no—
292C2×SD16—1061no—
293C2×D8—531no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8D21211yescharacteristic
297C22×C8D21211yescharacteristic
298C22×C8D21211yescharacteristic
299C22×C8C22×D5311yescharacteristic
300C22×C8C22×D5311yescharacteristic
301C22×C8C22×D5311yescharacteristic
302#123—2121no—
303C22⋊Q8—1062no—
304C22⋊Q8—1062no—
305C22⋊Q8—1061no—
306C22⋊Q8—1061no—
307C22⋊Q8—2121no—
308C22⋊Q8—2121no—
309#121—1061no—
310#121—1061no—
311#120—2121no—
312C4×D4—2121no—
313C4×D4—2121no—
314#117—1061no—
315C2×C4⋊C4—1061no—
316C2×C4⋊C4—2121no—
317C2×C22⋊C4—2121no—
318Q32—1064no—
319SD32—1064no—
320M32—22no—
321M32—24no—
322M32C22⋊D5314yescomplemented
323C2×C16—24no—
324C2×C16C22×D5312yes—
325C2×C16C22×D5312yes—
326#107—532no—
327#107—531no—
328#107—531no—
329#107—1061no—
330#106—1061no—
331#103—1062no—
332#103—1062no—
333#103—1061no—
334#103—1061no—
335#102—1061no—
336#102—1061no—
337C24D42411yescharacteristic
338#41—2122no—
339#41—2122no—
340C2×Q8—1061no—
341C2×Q8—1062no—
342C2×Q8—2122no—
343C2×Q8—2122no—
344C2×Q8—2121no—
345C2×D4—1061no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—2121no—
352C22×C4—2121no—
353C22×C4—2121no—
354C22×C4D42411yescharacteristic
355C22×C4D42411yescharacteristic
356C22×C4D42411yescharacteristic
357C22×C4C2×D21211yescharacteristic
358C22×C4C2×D21211yescharacteristic
359C22×C4C2×D21211yescharacteristic
360Q16—1062no—
361Q16—1061no—
362Q16—1062no—
363Q16—1061no—
364Q16—2122no—
365SD16—2122no—
366D8—1061no—
367D8—1061no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8C2×D21211yescharacteristic
376C2×C8C2×D21211yescharacteristic
377C2×C8C2×D21211yescharacteristic
378C2×C8#1171994512yes—
379C2×C8C2×C22⋊D5312yes—
380C2×C8C23×D5311yescharacteristic
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—1062no—
384C4⋊C4—2121no—
385C4⋊C4—2122no—
386C4⋊C4—2122no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C22⋊C4—2122no—
390C22⋊C4—2122no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C22⋊C4—2121no—
396C42—2121no—
397C16—22no—
398C16C2×C22⋊D5314yes—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2121no—
403C23C2×D42411yescharacteristic
404C23C2×D42411yescharacteristic
405C23C2×D42411yescharacteristic
406Q8—4242no—
407Q8—2121no—
408Q8—2122no—
409Q8—2122no—
410Q8—2121no—
411D4—4242no—
412D4—4242no—
413D4—4242no—
414D4—2121no—
415D4—2121no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—4242no—
424C2×C4—4242no—
425C2×C4—4242no—
426C2×C4—2121no—
427C2×C4—2121no—
428C2×C4—2122no—
429C2×C4—2121no—
430C2×C4—2122no—
431C2×C4#1313319112yes—
432C2×C4C2×D42411yescharacteristic
433C2×C4C2×D42411yescharacteristic
434C2×C4C2×D42411yescharacteristic
435C2×C4#1313323912yes—
436C2×C4C22×D21211yescharacteristic
437C8—21no—
438C8—21no—
439C8#1313319112yes—
440C8C22×C22⋊D5311yescharacteristic
441C8C2×#1171994511yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22#1681085912yes—
454C22#1681106612yescomplemented
455C22C22×D42411yescharacteristic
456C4—21no—
457C4—21no—
458C4—4241no—
459C4—4242no—
460C4—4241no—
461C4—4241no—
462C4—4242no—
463C4#1681085912yes—
464C4C2×#1313319111yescharacteristic
465C4C2×#1313323911yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—8481no—
470C2C2×#1681085911yescharacteristic
471C2C2×#1681106611yescharacteristic
472C2#2028785111yescharacteristic
subgroupG/H
centerC22C22×D424
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC22×M32×C53C2
socleC22×C53C22×D8
prime radical#20284296C2
hypercenterC22×M32D53
hypocenterC53#33606
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×#16811066C2semidirect
3. C2×#16810864C2semidirect
4. #20276828C2semidirect
5. #20276823C2semidirect
6. #20276811C2semidirect
7. #20276806C2semidirect
8. #20276503C2semidirect
9. #20276436C2semidirect
10. #20276428C2semidirect
11. C22×M32×C53C2semidirect
17. C2×Dic424C22semidirect
18. C2×#13133162C22semidirect
33. #16810451C22semidirect
34. #16810449C22semidirect
35. #16810428C22semidirect
36. #16810428C22semidirect
37. #16810428C22semidirect
38. #16810425C22semidirect
96. M32×C53D4semidirect
215. C22×C53#510semidirect
231. C22×M32D53semidirect
281. C53#33606semidirect
322. M32C22⋊D53semidirect
454. C22#16811066semidirect

Maximal subgroups

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

subgroupindex
C2×#168110662
#202842962
C2×#168108642
#202768282
#202768232
#202768112
#202768062
#202765032
#202764362
#202764282
C22×M32×C532
#3360653