Finite Group Databasebeta

← #27538576

Subgroups — #27538576

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×#16811064C211yesmaximal, characteristic, complemented
2C22×#13133157C211yesmaximal, characteristic, complemented
3#20284296C211yesmaximal, characteristic
4#20276830C212yesmaximal, complemented
5#20276821C212yesmaximal, complemented
6#20276813C212yesmaximal, complemented
7#20276804C212yesmaximal, complemented
8#20276501C211yesmaximal, characteristic, complemented
9#20276435C211yesmaximal, characteristic, complemented
10#20276425C211yesmaximal, characteristic, complemented
11C2×C53×#467C211yesmaximal, characteristic, complemented
12C2×#13133239C2211yescharacteristic
13#16811134—21no—
14#16811064—24no—
15C2×#13133163C2212yescomplemented
16C2×#13133160C2212yescomplemented
17C2×#13133157—24no—
18C2×#13133157C2212yes—
19C2×#13133157C2211yescharacteristic
20C2×#13133157C2211yescharacteristic
21C22×#11719908C2212yes—
22#16810864C2211yescharacteristic
23#16810864C2211yescharacteristic
24#16810859C2211yescharacteristic
25#16810859C2211yescharacteristic
26#16810852—21no—
27#16810852—21no—
28#16810661C2211yescharacteristic
29#16810620—21no—
30#16810450C2212yescomplemented
31#16810448C2212yescomplemented
32#16810427C2212yescomplemented
33#16810427C2212yescomplemented
34#16810426C2212yescomplemented
35#16810423C2212yescomplemented
36C22×Q16×C53C2211yescharacteristic
37C23×C8×C53C2211yescharacteristic
38C2×C22⋊Q8×C53—21no—
39C53×#467—21no—
40C53×#467—21no—
41C53×#467C2212yes—
42C53×#467C2212yes—
43C2×#107×C53C2211yescharacteristic
44C2×#103×C53—21no—
45C2×#11719945—21no—
46#13133239—22no—
47#13133239—22no—
48C2×Dic212C2311yescharacteristic
49C2×#11719913—21no—
50C2×D424C2311yescharacteristic
51#13133200—41no—
52#13133196—21no—
53#13133196—21no—
54#13133191—21no—
55#13133191—21no—
56#13133186—41no—
57C4×C22⋊D53—41no—
58C4×C22⋊D53—41no—
59#13133176—21no—
60#13133163—24no—
61#13133160—24no—
62#13133157—22no—
63#13133157—24no—
64#13133157D414yescomplemented
65C2×#11719908—24no—
66C2×#11719908C2312yes—
67C2×#11719908C2312yes—
68#13133151C2311yescharacteristic
69#13133151C2311yescharacteristic
70#13133149—21no—
71#13133142—21no—
72#13133142—21no—
73#13133139—21no—
74#13133139—21no—
75C22×Q8×C53—21no—
76C23×C4×C53D411yescharacteristic
77C2×Q16×C53—21no—
78C2×Q16×C53—22no—
79C2×Q16×C53—22no—
80C2×Q16×C53C2312yes—
81C22×C8×C53—22no—
82C22×C8×C53—22no—
83C22×C8×C53D411yescharacteristic
84C22×C8×C53D411yescharacteristic
85C22×C8×C53D411yescharacteristic
86C22×C8×C53C2311yescharacteristic
87C22×C8×C53C2311yescharacteristic
88C22×C8×C53C2311yescharacteristic
89C22⋊Q8×C53—22no—
90C22⋊Q8×C53—22no—
91C22⋊Q8×C53—41no—
92C22⋊Q8×C53—41no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53—41no—
95C2×C22⋊C4×C53—41no—
96#107×C53—21no—
97#107×C53C2312yes—
98#103×C53—22no—
99#103×C53—22no—
100C24×C53D811yescharacteristic
101C2×Q8×C53—22no—
102C2×Q8×C53—41no—
103C2×Q8×C53—42no—
104C2×Q8×C53—42no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—41no—
108C22×C4×C53—41no—
109C22×C4×C53D811yescharacteristic
110C22×C4×C53D811yescharacteristic
111C22×C4×C53D811yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C2×C22⋊D53—41no—
116C2×C22⋊D53—41no—
117C2×C22⋊D53—41no—
118#11719945—42no—
119#11719945—42no—
120C2×D212—21no—
121C2×C4×D53—41no—
122C2×Dic106—21no—
123Q16×C53—22no—
124Q16×C53—22no—
125Q16×C53—42no—
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
139C4⋊C4×C53—22no—
140C4⋊C4×C53—41no—
141C4⋊C4×C53—42no—
142C4⋊C4×C53—42no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊Dic53—41no—
146C22⋊Dic53—41no—
147C22⋊Dic53—41no—
148#11719921—41no—
149#11719921—41no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152#11719919—41no—
153#11719919—41no—
154C4×Dic53—41no—
155Dic212—21no—
156Dic212—21no—
157D424—21no—
158D424—21no—
159#11719913—42no—
160#11719908—22no—
161#11719908C2×D414yes—
162C23×C53—22no—
163C23×C53—22no—
164C23×C53—41no—
165C23×C53C2×D811yescharacteristic
166C23×C53C2×D811yescharacteristic
167C23×C53C2×D811yescharacteristic
168C22×D53—41no—
169Q8×C53—42no—
170Q8×C53—42no—
171Q8×C53—82no—
172C2×C4×C53—22no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—82no—
182C2×C4×C53—82no—
183C2×C4×C53#12112yes—
184C2×C4×C53C2×D811yescharacteristic
185C2×C4×C53C2×D811yescharacteristic
186C2×C4×C53C2×D811yescharacteristic
187C2×C4×C53#13512yes—
188C2×C4×C53C22×D411yescharacteristic
189C22⋊D53—82no—
190C22⋊D53—82no—
191C22⋊D53—82no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195D212—41no—
196D212—41no—
197C4×D53—82no—
198Dic106—41no—
199Dic106—41no—
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—82no—
223C4×C53—82no—
224C4×C53#46612yes—
225C4×C53C2×#12111yescharacteristic
226C4×C53C2×#13511yescharacteristic
227Dic53—81no—
228Dic53—81no—
229Dic53—81no—
230C2×#510—531no—
231C22×M32—531no—
232#3050—531no—
233C2×#467D5311yescharacteristic, complemented
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×Q16C2×D5311yescharacteristic
250C23×C8C2×D5311yescharacteristic
251#525—1061no—
252C2×C22⋊Q8—21no—
253#510—1064no—
254C2×Q32—532no—
255C2×SD32—532no—
256C2×M32—531no—
257C2×M32—531no—
258C2×M32—532no—
259C2×M32—1064no—
260C22×C16—532no—
261#467—21no—
262#467—21no—
263#467—531no—
264#467—531no—
265#467C2×D5312yes—
266#467C2×D5312yes—
267#466—531no—
268#466—531no—
269#465—1061no—
270#465—1061no—
271#427—531no—
272C2×#107C2×D5311yescharacteristic
273#416—1061no—
274C2×#103—21no—
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—21no—
284C23×C4C22⋊D5311yescharacteristic
285#135—1062no—
286#135—1062no—
287C2×Q16—22no—
288C2×Q16—22no—
289C2×Q16—21no—
290C2×Q16—531no—
291C2×Q16C22×D5312yes—
292C2×SD16—1061no—
293C2×D8—531no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C22⋊D5311yescharacteristic
297C22×C8C22⋊D5311yescharacteristic
298C22×C8C22⋊D5311yescharacteristic
299C22×C8C22×D5311yescharacteristic
300C22×C8C22×D5311yescharacteristic
301C22×C8C22×D5311yescharacteristic
302#123—2121no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305C22⋊Q8—41no—
306C22⋊Q8—41no—
307C22⋊Q8—1061no—
308C22⋊Q8—1061no—
309#121—1061no—
310#121—1061no—
311#120—2121no—
312C4×D4—2121no—
313C4×D4—2121no—
314#117—1061no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—41no—
317C2×C22⋊C4—41no—
318Q32—1064no—
319SD32—1064no—
320M32—534no—
321M32—1064no—
322M32—1062no—
323C2×C16—532no—
324C2×C16—532no—
325C2×C16—1064no—
326#107—21no—
327#107—531no—
328#107—531no—
329#107C22×D5312yes—
330#106—1061no—
331#103—22no—
332#103—22no—
333#103—1061no—
334#103—1061no—
335#102—1061no—
336#102—1061no—
337C24#1171992211yescharacteristic
338#41—2122no—
339#41—2122no—
340C2×Q8—22no—
341C2×Q8—41no—
342C2×Q8—42no—
343C2×Q8—42no—
344C2×Q8—1061no—
345C2×D4—1061no—
346C2×D4—2121no—
347C2×D4—2121no—
348C2×D4—2121no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—2121no—
354C22×C4#1171992211yescharacteristic
355C22×C4#1171992211yescharacteristic
356C22×C4#1171992211yescharacteristic
357C22×C4C2×C22⋊D5311yescharacteristic
358C22×C4C2×C22⋊D5311yescharacteristic
359C22×C4C2×C22⋊D5311yescharacteristic
360Q16—22no—
361Q16—22no—
362Q16—42no—
363Q16—1061no—
364Q16—1061no—
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×C8D4×D5312yes—
376C2×C8#1171994712yes—
377C2×C8C2×C22⋊D5311yescharacteristic
378C2×C8C2×C22⋊D5311yescharacteristic
379C2×C8C2×C22⋊D5311yescharacteristic
380C2×C8C23×D5311yescharacteristic
381C4⋊C4—22no—
382C4⋊C4—41no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—1061no—
386C4⋊C4—1061no—
387C4⋊C4—2121no—
388C4⋊C4—2121no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—2121no—
392C22⋊C4—2121no—
393C22⋊C4—2121no—
394C22⋊C4—2121no—
395C22⋊C4—2121no—
396C42—2121no—
397C16—534no—
398C16—1062no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—2121no—
403C23C2×#1171992211yescharacteristic
404C23C2×#1171992211yescharacteristic
405C23C2×#1171992211yescharacteristic
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—2121no—
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—42no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—4242no—
428C2×C4—2121no—
429C2×C4—2121no—
430C2×C4—2121no—
431C2×C4#1313318912yes—
432C2×C4C2×#1171992211yescharacteristic
433C2×C4C2×#1171992211yescharacteristic
434C2×C4C2×#1171992211yescharacteristic
435C2×C4#1313323712yes—
436C2×C4C22×C22⋊D5311yescharacteristic
437C8—21no—
438C8—21no—
439C8#1313318912yes—
440C8C2×D4×D5311yescharacteristic
441C8C2×#1171994711yescharacteristic
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—44no—
450C22—4241no—
451C22—4241no—
452C22—4241no—
453C22#1681085712yes—
454C22#1681106412yescomplemented
455C22C22×#1171992211yescharacteristic
456C4—21no—
457C4—21no—
458C4—82no—
459C4—82no—
460C4—4241no—
461C4—4241no—
462C4—4241no—
463C4#1681085712yes—
464C4C2×#1313318911yescharacteristic
465C4C2×#1313323711yescharacteristic
466C2—22no—
467C2—22no—
468C2—41no—
469C2—8481no—
470C2C2×#1681085711yescharacteristic
471C2C2×#1681106411yescharacteristic
472C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#467C2
socleC22×C53C22×D8
prime radical#20284296C2
hypercenterC2×#467D53
hypocenterC53#33606
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#467

Complements

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

NH ≅ G/Ntype
1. C2×#16811064C2semidirect
2. C22×#13133157C2semidirect
4. #20276830C2semidirect
5. #20276821C2semidirect
6. #20276813C2semidirect
7. #20276804C2semidirect
8. #20276501C2semidirect
9. #20276435C2semidirect
10. #20276425C2semidirect
11. C2×C53×#467C2semidirect
15. C2×#13133163C22semidirect
16. C2×#13133160C22semidirect
30. #16810450C22semidirect
31. #16810448C22semidirect
32. #16810427C22semidirect
33. #16810427C22semidirect
34. #16810426C22semidirect
35. #16810423C22semidirect
64. #13133157D4semidirect
215. C22×C53#510semidirect
233. C2×#467D53semidirect
281. C53#33606semidirect
454. C22#16811064semidirect

Maximal subgroups

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

subgroupindex
C2×#168110642
C22×#131331572
#202842962
#202768302
#202768212
#202768132
#202768042
#202765012
#202764352
#202764252
C2×C53×#4672
#3360653