Finite Group Databasebeta

← #27369587

Subgroups — #27369587

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20284497C211yesmaximal, characteristic
3#20283823C211yesmaximal, characteristic
4#20274170C212yesmaximal, complemented
5#20274163C212yesmaximal, complemented
6#20274116C212yesmaximal, complemented
7#20274111C212yesmaximal, complemented
8#20273490C211yesmaximal, characteristic, complemented
9#20273405C211yesmaximal, characteristic, complemented
10C2×#16810372C211yesmaximal, characteristic, complemented
11C53×#3230C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810927—21no—
14#16810920—21no—
15#16810915C2211yescharacteristic
16#16810915C2212yes—
17#16810910C2211yescharacteristic
18#16810910C2212yes—
19#16810905—21no—
20#16810905C2211yescharacteristic
21#16810900—21no—
22#16810900C2211yescharacteristic
23C8×Dic106C2212yes—
24C8×D212C2212yes—
25#16810686—22no—
26#16810675—21no—
27#16810663—21no—
28#16810659—21no—
29#16810647C2211yescharacteristic
30C2×C4⋊#88322C2211yescharacteristic
31C2×C4⋊#88322C2211yescharacteristic
32C4⋊C8×D53—21no—
33#16810622—21no—
34#16810617—21no—
35#16810376C2212yes—
36#16810376C2212yes—
37#16810372C2212yes—
38#16810372C2212yes—
39#16810372C2212yes—
40#16810369C2212yes—
41C53×#533C2211yescharacteristic
42C2×C4⋊Q8×C53C2211yescharacteristic
43C53×#500C2212yes—
44C53×#487C2212yes—
45C53×#484—22no—
46C53×#477—21no—
47C53×#477—21no—
48C53×#431C2211yescharacteristic
49C2×#11719945D411yescharacteristic
50C2×#11719917—21no—
51C22×#88322—21no—
52C22×#88322—21no—
53C2×C8×D53—22no—
54C4×Dic106D411yescharacteristic
55C4×Dic106C2311yescharacteristic
56C4×D212D411yescharacteristic
57C4×D212C2311yescharacteristic
58C4×C22⋊D53—21no—
59C4×C22⋊D53—21no—
60#13133176D411yescharacteristic
61#13133174—21no—
62C42×D53—21no—
63#13133150—22no—
64#13133150—21no—
65#13133150—21no—
66#13133148—22no—
67#13133148—21no—
68#13133148—21no—
69C4⋊#88322—21no—
70C4⋊#88322—21no—
71C4⋊#88322—21no—
72C4⋊#88322C2312yes—
73C4⋊#88322C2312yes—
74C4⋊#88322C2311yescharacteristic
75C4⋊#88322C2311yescharacteristic
76#13133146—22no—
77#13133140—21no—
78#13133140—21no—
79#13133140—22no—
80#13133137—21no—
81#13133137—21no—
82#13133137—22no—
83#13133129—22no—
84C8×Dic53—22no—
85C22×Q8×C53—21no—
86C4⋊Q8×C53—21no—
87C4⋊Q8×C53—21no—
88C4⋊Q8×C53—21no—
89C4⋊Q8×C53C2311yescharacteristic
90C4⋊Q8×C53C2312yes—
91C4⋊Q8×C53C2311yescharacteristic
92#125×C53—21no—
93C22⋊Q8×C53—21no—
94C22⋊Q8×C53—21no—
95C4×Q8×C53—21no—
96C4×Q8×C53—21no—
97#117×C53—21no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—21no—
100C2×C42×C53C2311yescharacteristic
101#107×C53—22no—
102#107×C53—22no—
103C4⋊C8×C53D411yescharacteristic
104C4⋊C8×C53D411yescharacteristic
105#103×C53—22no—
106#103×C53—22no—
107C22⋊C8×C53D412yes—
108C4×C8×C53C2312yes—
109C2×Q8×C53—21no—
110C2×Q8×C53—21no—
111C2×Q8×C53—22no—
112C2×Q8×C53—41no—
113C2×Q8×C53—82no—
114C22×C4×C53—41no—
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C2×C22⋊D53—21no—
119#11719945—22no—
120#11719945Q1614yes—
121C2×D212C2×D411yescharacteristic
122C2×C4×D53—21no—
123C2×C4×D53—21no—
124C2×C4×D53—21no—
125C2×Dic106C2×D411yescharacteristic
126C2×C8×C53C2×D412yes—
127C2×C8×C53C2×D412yes—
128C4⋊C4×C53—22no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53—41no—
137C4⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C42×C53—41no—
140C42×C53C2×D411yescharacteristic
141C42×C53C2×D411yescharacteristic
142C42×C53#4111yescharacteristic
143C42×C53C2411yescharacteristic
144C22⋊Dic53—21no—
145#11719921—21no—
146#11719921—21no—
147C4⋊Dic53C2×D411yescharacteristic
148C4⋊Dic53C2×D411yescharacteristic
149#11719919—21no—
150#11719919—21no—
151C4×Dic53—21no—
152C4×Dic53—21no—
153C4×Dic53—21no—
154#11719917—42no—
155C2×#88322—22no—
156C2×#88322—21no—
157C2×#88322—22no—
158C2×#88322—21no—
159C2×#88322—42no—
160C2×#88322—81no—
161C8×D53—82no—
162C23×C53#12111yescharacteristic
163C22×D53—21no—
164Q8×C53—41no—
165Q8×C53—41no—
166Q8×C53—82no—
167Q8×C53—81no—
168Q8×C53—82no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—82no—
176C2×C4×C53#12111yescharacteristic
177C2×C4×C53#12111yescharacteristic
178C2×C4×C53#12111yescharacteristic
179C2×C4×C53C2×Q1612yes—
180C2×C4×C53#13512yes—
181C2×C4×C53C22×D411yescharacteristic
182C2×C4×C53C22×D411yescharacteristic
183C2×C4×C53C2×#4111yescharacteristic
184C22⋊D53—24no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188D212—21no—
189D212C2×Q1612yes—
190C4×D53—24no—
191C4×D53—41no—
192C4×D53—41no—
193C4×D53—81no—
194Dic106—21no—
195Dic106C2×Q1612yes—
196C8×C53—22no—
197C8×C53—22no—
198#88322—41no—
199#88322—41no—
200#88322—42no—
201#88322—42no—
202#19417—531nomaximal, Sylow
203C22×C53—21no—
204C22×C53—21no—
205C22×C53#46712yes—
206C22×C53C2×#12111yescharacteristic
207C2×D53—22no—
208C2×D53—41no—
209C2×D53—41no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—21no—
213C4×C53—81no—
214C4×C53—82no—
215C4×C53—81no—
216C4×C53#46712yes—
217C4×C53C22×Q1611yescharacteristic
218C4×C53C2×#13511yescharacteristic
219C4×C53#57711yescharacteristic
220C4×C53#57811yescharacteristic
221Dic53—22no—
222Dic53—41no—
223Dic53—41no—
224#3230D5311yescharacteristic, complemented
225#3079—531no—
226#3076—531no—
227#2988—531no—
228#1684—532no—
229#1683—532no—
230#1676—532no—
231#1675—532no—
232#1571—531no—
233#1550—531no—
234C2×#333—531no—
235C2×C53—22no—
236C2×C53#304611yescharacteristic
237C2×C53C2×#46711yescharacteristic
238C2×C53#306811yescharacteristic
239D53—42no—
240#533C2×D5311yescharacteristic
241C2×C4⋊Q8C2×D5311yescharacteristic
242C4×#41—531no—
243#500C2×D5312yes—
244#487C2×D5312yes—
245#484—22no—
246#479—1061no—
247#478—1061no—
248#477—21no—
249#477—21no—
250#477—531no—
251#477—532no—
252#476—531no—
253#476—532no—
254#475—531no—
255#475—1061no—
256#474—531no—
257#474—1061no—
258C8×Q8—532no—
259C8×D4—532no—
260C8×D4—1062no—
261#431—1061no—
262#431C2×D5311yescharacteristic
263#428—1061no—
264#427—1061no—
265#423—531no—
266C2×C4⋊C8—531no—
267C2×C4⋊C8—531no—
268C2×C4⋊C8—1061no—
269#417—1061no—
270#416—1061no—
271#335—532no—
272#335—532no—
273#333—532no—
274#333—532no—
275#333—532no—
276#332—532no—
277C53#1941711yescharacteristic, complemented, Sylow
278C2×#41—531no—
279C22×Q8—21no—
280C2×M16—1061no—
281C22×C8—1061no—
282C22×C8—1061no—
283C22×C8—1062no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8C22×D5312yes—
288C4⋊Q8C22×D5311yescharacteristic
289C4⋊Q8C22×D5311yescharacteristic
290#125—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—531no—
296C4×Q8—531no—
297C4×D4—531no—
298C4×D4—531no—
299C4×D4—1061no—
300C4×D4—1061no—
301#117—21no—
302#117—531no—
303#117—1061no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42—1061no—
307C2×C42C22×D5311yescharacteristic
308#107—22no—
309#107—22no—
310#107—1061no—
311#107—1061no—
312#107—1062no—
313#106—1061no—
314#106—1061no—
315#106—1062no—
316C4⋊C8—532no—
317C4⋊C8—532no—
318C4⋊C8—531no—
319C4⋊C8—531no—
320C4⋊C8—1062no—
321C4⋊C8—1061no—
322C4⋊C8—1061no—
323C4⋊C8—1061no—
324C4⋊C8C22⋊D5311yescharacteristic
325C4⋊C8C22⋊D5311yescharacteristic
326#103—22no—
327#103—22no—
328#103—1061no—
329#103—1061no—
330#103—1062no—
331#102—1061no—
332#102—1061no—
333#102—1062no—
334C22⋊C8—1062no—
335C22⋊C8C22⋊D5312yes—
336C4×C8—1062no—
337C4×C8C22×D5312yes—
338#41—534no—
339#41—1062no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×Q8—531no—
346C2×D4—531no—
347C2×D4—1061no—
348C22×C4—41no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4D4×D5311yescharacteristic
353C22×C4#1171994711yescharacteristic
354C22×C4C2×C22⋊D5311yescharacteristic
355M16—2122no—
356C2×C8—1061no—
357C2×C8—1062no—
358C2×C8—1061no—
359C2×C8—1062no—
360C2×C8—4242no—
361C2×C8—4241no—
362C2×C8—2122no—
363C2×C8C2×C22⋊D5312yes—
364C2×C8C2×C22⋊D5312yes—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—531no—
376C4⋊C4—531no—
377C4⋊C4—1061no—
378C4⋊C4—1061no—
379C22⋊C4—41no—
380C22⋊C4—1061no—
381C22⋊C4—1061no—
382C22⋊C4—1061no—
383C42—41no—
384C42—1061no—
385C42—1061no—
386C42—1061no—
387C42D4×D5311yescharacteristic
388C42#1171994711yescharacteristic
389C42C2×C22⋊D5311yescharacteristic
390C42C23×D5311yescharacteristic
391C23—1061no—
392C23#1313318811yescharacteristic
393Q8—41no—
394Q8—41no—
395Q8—81no—
396Q8—82no—
397Q8—82no—
398Q8—532no—
399Q8—1061no—
400D4—532no—
401D4—1064no—
402D4—1061no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—1061no—
411C2×C4—1064no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—4241no—
415C2×C4—2121no—
416C2×C4—2121no—
417C2×C4#1313318811yescharacteristic
418C2×C4#1313318811yescharacteristic
419C2×C4#1313318811yescharacteristic
420C2×C4Q16×D5312yes—
421C2×C4#1313324012yes—
422C2×C4C2×D4×D5311yescharacteristic
423C2×C4C22×C22⋊D5311yescharacteristic
424C2×C4C2×#1171994711yescharacteristic
425C8—22no—
426C8—22no—
427C8—2121no—
428C8—2121no—
429C8—2122no—
430C8—2122no—
431C22—21no—
432C22—21no—
433C22—1062no—
434C22—2121no—
435C22—2121no—
436C22#1681086112yes—
437C22C2×#1313318811yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—81no—
443C4—81no—
444C4—1062no—
445C4—2121no—
446C4—2121no—
447C4#1681086112yes—
448C4C2×Q16×D5311yescharacteristic
449C4C2×#1313324011yescharacteristic
450C4#1681143111yescharacteristic
451C4#1681143811yescharacteristic
452C2—22no—
453C2—2122no—
454C2#2028426111yescharacteristic
455C2C2×#1681086111yescharacteristic
456C2#2028443911yescharacteristic
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3230C2
socleC22×C53C2×#121
prime radicalC2×#11719945D4
hypercenter#3230D53
hypocenterC53#19417
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #20274170C2semidirect
5. #20274163C2semidirect
6. #20274116C2semidirect
7. #20274111C2semidirect
8. #20273490C2semidirect
9. #20273405C2semidirect
10. C2×#16810372C2semidirect
11. C53×#3230C2semidirect
224. #3230D53semidirect
277. C53#19417semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202844972
#202838232
#202741702
#202741632
#202741162
#202741112
#202734902
#202734052
C2×#168103722
C53×#32302
#1941753