Finite Group Databasebeta

← #27366560

Subgroups — #27366560

Order 13568.

664 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284769C212yesmaximal, complemented
2C8×#11719945C211yesmaximal, characteristic, complemented
3#20274128C212yesmaximal, complemented
4#20274121C212yesmaximal, complemented
5#20274116C211yesmaximal, characteristic, complemented
6#20274111C211yesmaximal, characteristic, complemented
7#20274072C211yesmaximal, characteristic, complemented
8#20274067C211yesmaximal, characteristic, complemented
9#20273463C211yesmaximal, characteristic, complemented
10#20273377C212yesmaximal, complemented
11C53×#3169C211yesmaximal, characteristic, complemented
12C4×#11719945C2211yescharacteristic
13#16810944—22no—
14#16810915C2212yes—
15#16810910C2212yes—
16#16810820C2212yes—
17#16810815C2212yes—
18#16810766—22no—
19C8×Dic106C2211yescharacteristic
20C8×Dic106C2211yescharacteristic
21C4×#11719923—22no—
22C4×#11719922—22no—
23C8×D212C2211yescharacteristic
24C8×D212C2211yescharacteristic
25C8×C22⋊D53—21no—
26C8×C22⋊D53—21no—
27#16810676C2211yescharacteristic
28#16810676C2212yes—
29#16810674—21no—
30C4×C8×D53—21no—
31#16810376C2211yescharacteristic
32#16810376C2211yescharacteristic
33#16810374C2212yes—
34#16810372C2212yes—
35#16810372C2212yes—
36#16810367C2212yes—
37#16810367C2212yes—
38C53×#532C2212yes—
39C53×#500C2211yescharacteristic
40C53×#493C2211yescharacteristic
41C53×#486C2212yes—
42C53×#466—21no—
43C53×#466—21no—
44C4×D8×C53—21no—
45C4×D8×C53—21no—
46C2×C4×C8×C53C2211yescharacteristic
47C2×#11719945D411yescharacteristic
48C2×#11719923—22no—
49C2×#11719922—22no—
50C2×C8×D53—21no—
51C2×C8×D53—21no—
52C4×Dic106D411yescharacteristic
53C4×Dic106C2311yescharacteristic
54C4×D212D411yescharacteristic
55C4×D212C2311yescharacteristic
56C4×C22⋊D53—21no—
57C4×C22⋊D53—21no—
58#13133176D411yescharacteristic
59#13133174—21no—
60C42×D53—21no—
61#13133150—22no—
62#13133148—22no—
63C4⋊#88322C2312yes—
64C4⋊#88322C2312yes—
65C4⋊#88322C2311yescharacteristic
66C4⋊#88322C2311yescharacteristic
67#13133146—21no—
68#13133146—21no—
69#13133140—22no—
70#13133138—22no—
71#13133138—22no—
72#13133137—22no—
73C22⋊#88322—22no—
74C22⋊#88322—21no—
75#13133129—21no—
76#13133129—21no—
77C4×#88322—21no—
78C4×#88322—22no—
79C8×Dic53—21no—
80C8×Dic53—21no—
81C2×#41×C53—22no—
82C2×D8×C53—21no—
83C2×D8×C53—21no—
84C22×C8×C53D411yescharacteristic
85C22×C8×C53D411yescharacteristic
86C4⋊Q8×C53C2312yes—
87#127×C53C2312yes—
88#124×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91C4×D4×C53—22no—
92C4×D4×C53—22no—
93C2×C42×C53C2311yescharacteristic
94#107×C53—21no—
95#107×C53—21no—
96#102×C53—22no—
97#102×C53—22no—
98C4×C8×C53D411yescharacteristic
99C4×C8×C53D411yescharacteristic
100C4×C8×C53C2311yescharacteristic
101C4×C8×C53C2311yescharacteristic
102#41×C53—84no—
103C2×Q8×C53—22no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53#4111yescharacteristic
112C2×C22⋊D53—21no—
113#11719945—22no—
114#11719945—22no—
115C2×D212C2×D411yescharacteristic
116C2×C4×D53—21no—
117C2×C4×D53—21no—
118C2×C4×D53—21no—
119C2×Dic106C2×D411yescharacteristic
120D8×C53—42no—
121D8×C53—42no—
122C2×C8×C53—22no—
123C2×C8×C53—41no—
124C2×C8×C53C2×D411yescharacteristic
125C2×C8×C53C2×D411yescharacteristic
126C2×C8×C53C2×D411yescharacteristic
127C2×C8×C53C2×D411yescharacteristic
128C4⋊C4×C53—22no—
129C4⋊C4×C53—22no—
130C22⋊C4×C53—42no—
131C22⋊C4×C53—42no—
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C42×C53C2411yescharacteristic
136C22⋊Dic53—21no—
137#11719923—44no—
138#11719922—44no—
139#11719921—21no—
140#11719921—21no—
141C4⋊Dic53C2×D411yescharacteristic
142C4⋊Dic53C2×D411yescharacteristic
143#11719919—21no—
144#11719919—21no—
145C4×Dic53—21no—
146C4×Dic53—21no—
147C4×Dic53—21no—
148C2×#88322—21no—
149C2×#88322—21no—
150C2×#88322—22no—
151C2×#88322—22no—
152C8×D53—42no—
153C8×D53—42no—
154C23×C53—42no—
155C23×C53#12111yescharacteristic
156C22×D53—21no—
157Q8×C53—82no—
158D4×C53—42no—
159D4×C53—42no—
160D4×C53—42no—
161D4×C53—42no—
162D4×C53—82no—
163D4×C53—84no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—22no—
169C2×C4×C53—42no—
170C2×C4×C53—84no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C2×#4111yescharacteristic
177C22⋊D53—24no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181D212—21no—
182D212#13512yes—
183C4×D53—24no—
184C4×D53—41no—
185C4×D53—41no—
186C4×D53—42no—
187Dic106—21no—
188Dic106#13512yes—
189C8×C53—21no—
190C8×C53—21no—
191C8×C53—21no—
192C8×C53#13512yes—
193#88322—41no—
194#88322—42no—
195#88322—42no—
196#88322—41no—
197#19128—531nomaximal, Sylow
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—82no—
202C22×C53—82no—
203C22×C53—82no—
204C22×C53C2×#12111yescharacteristic
205C2×D53—22no—
206C2×D53—41no—
207C2×D53—41no—
208C4×C53—21no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—82no—
212C4×C53#47212yes—
213C4×C53C2×#13511yescharacteristic
214C4×C53C2×#13511yescharacteristic
215C4×C53C2×#13511yescharacteristic
216C4×C53#57511yescharacteristic
217Dic53—22no—
218Dic53—41no—
219Dic53—41no—
220#3169D5311yescharacteristic, complemented
221#3106—532no—
222C8×#41—531no—
223#1678—532no—
224#1677—532no—
225#1676—531no—
226#1675—531no—
227#1668—531no—
228#1667—531no—
229#1564—531no—
230#1543—532no—
231C2×C53—41no—
232C2×C53—162no—
233C2×C53#303511yescharacteristic
234C2×C53#305011yescharacteristic
235C2×C53C2×#47211yescharacteristic
236D53—42no—
237#532C2×D5312yes—
238C4×#41—531no—
239#500C2×D5311yescharacteristic
240#493C2×D5311yescharacteristic
241#486C2×D5312yes—
242#482—1062no—
243#477—532no—
244#476—532no—
245#466—21no—
246#466—21no—
247#460—532no—
248#459—532no—
249#449—1062no—
250C8×Q8—531no—
251C8×Q8—531no—
252C4×SD16—1062no—
253C4×D8—21no—
254C4×D8—21no—
255C4×D8—1062no—
256C8×D4—531no—
257C8×D4—531no—
258C8×D4—1061no—
259C8×D4—1061no—
260#431—531no—
261#431—532no—
262#431—1061no—
263C2×C4×C8—1061no—
264C2×C4×C8C2×D5311yescharacteristic
265#335—531no—
266#335—531no—
267#334—532no—
268#333—532no—
269#333—532no—
270#331—532no—
271#331—532no—
272C53#1912811yescharacteristic, complemented, Sylow
273C2×#41—22no—
274C2×#41—531no—
275C2×SD16—1062no—
276C2×D8—21no—
277C2×D8—21no—
278C2×D8—1062no—
279C22×C8—1061no—
280C22×C8—1061no—
281C22×C8C22⋊D5311yescharacteristic
282C22×C8C22⋊D5311yescharacteristic
283C4⋊Q8C22×D5312yes—
284#127C22×D5312yes—
285#124—22no—
286#121—22no—
287#121—22no—
288C4×Q8—531no—
289C4×Q8—531no—
290C4×D4—22no—
291C4×D4—22no—
292C4×D4—531no—
293C4×D4—531no—
294C4×D4—1061no—
295C4×D4—1061no—
296#117—531no—
297#117—1061no—
298C2×C42—1061no—
299C2×C42C22×D5311yescharacteristic
300#107—21no—
301#107—21no—
302#107—1062no—
303#106—1062no—
304C4⋊C8—531no—
305C4⋊C8—531no—
306C4⋊C8—532no—
307C4⋊C8—532no—
308C4⋊C8—1061no—
309C4⋊C8—1061no—
310#103—1062no—
311#102—22no—
312#102—22no—
313#102—1062no—
314#102—1062no—
315#102—1062no—
316C22⋊C8—1062no—
317C22⋊C8—1061no—
318C22⋊C8—1061no—
319C22⋊C8—1061no—
320C4×C8—1061no—
321C4×C8—1061no—
322C4×C8—1062no—
323C4×C8—1061no—
324C4×C8C22⋊D5311yescharacteristic
325C4×C8C22⋊D5311yescharacteristic
326C4×C8C22×D5311yescharacteristic
327C4×C8C22×D5311yescharacteristic
328#41—84no—
329#41—1062no—
330#41—1062no—
331C2×Q8—22no—
332C2×Q8—531no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—531no—
338C2×D4—1061no—
339C22×C4—42no—
340C22×C4—1061no—
341C22×C4—1061no—
342C22×C4—1061no—
343C22×C4D4×D5311yescharacteristic
344C22×C4#1171994711yescharacteristic
345C22×C4C2×C22⋊D5311yescharacteristic
346SD16—2124no—
347D8—42no—
348D8—42no—
349D8—2124no—
350C2×C8—22no—
351C2×C8—41no—
352C2×C8—1062no—
353C2×C8—1062no—
354C2×C8—1061no—
355C2×C8—1061no—
356C2×C8—2122no—
357C2×C8—2122no—
358C2×C8C2×C22⋊D5311yescharacteristic
359C2×C8C2×C22⋊D5311yescharacteristic
360C2×C8C2×C22⋊D5311yescharacteristic
361C2×C8C2×C22⋊D5311yescharacteristic
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—531no—
365C4⋊C4—531no—
366C4⋊C4—1061no—
367C4⋊C4—1061no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—1061no—
371C22⋊C4—1061no—
372C22⋊C4—1061no—
373C42—1061no—
374C42—1061no—
375C42—1061no—
376C42D4×D5311yescharacteristic
377C42#1171994711yescharacteristic
378C42C2×C22⋊D5311yescharacteristic
379C42C23×D5311yescharacteristic
380C23—42no—
381C23—1061no—
382C23#1313318811yescharacteristic
383Q8—82no—
384Q8—532no—
385Q8—1061no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—42no—
390D4—82no—
391D4—84no—
392D4—532no—
393D4—1064no—
394D4—1061no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—42no—
401C2×C4—84no—
402C2×C4—1064no—
403C2×C4—1061no—
404C2×C4—1061no—
405C2×C4—1061no—
406C2×C4—2122no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4#1313318811yescharacteristic
410C2×C4#1313318811yescharacteristic
411C2×C4#1313318811yescharacteristic
412C2×C4C2×D4×D5311yescharacteristic
413C2×C4C22×C22⋊D5311yescharacteristic
414C2×C4C2×#1171994711yescharacteristic
415C8—21no—
416C8—21no—
417C8—21no—
418C8—2122no—
419C8—2122no—
420C8—2121no—
421C8—2121no—
422C8#1313323712yes—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—82no—
427C22—82no—
428C22—82no—
429C22—1062no—
430C22—2121no—
431C22—2121no—
432C22C2×#1313318811yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—82no—
437C4—1062no—
438C4—2121no—
439C4—2121no—
440C4#1681089012yes—
441C4#135×D5311yescharacteristic
442C4#135×D5311yescharacteristic
443C4C2×#1313323711yescharacteristic
444C4#1681142111yescharacteristic
445C2—41no—
446C2—162no—
447C2—2122no—
448C2#2028416411yescharacteristic
449C2#2028429211yescharacteristic
450C2C2×#1681089011yescharacteristic
subgroupG/H
centerC2×C4#13133188
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3169C2
socleC22×C53C2×#121
hypercenter#3169D53
hypocenterC53#19128
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #20284769C2semidirect
2. C8×#11719945C2semidirect
3. #20274128C2semidirect
4. #20274121C2semidirect
5. #20274116C2semidirect
6. #20274111C2semidirect
7. #20274072C2semidirect
8. #20274067C2semidirect
9. #20273463C2semidirect
10. #20273377C2semidirect
11. C53×#3169C2semidirect
220. #3169D53semidirect
272. C53#19128semidirect

Maximal subgroups

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

subgroupindex
#202847692
C8×#117199452
#202741282
#202741212
#202741162
#202741112
#202740722
#202740672
#202734632
#202733772
C53×#31692
#1912853