Finite Group Databasebeta

← #27484052

Subgroups — #27484052

Order 13568.

608 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285238C211yesmaximal, characteristic, complemented
2C2×#16810743C211yesmaximal, characteristic
3#20275889C212yesmaximal, complemented
4#20275886C212yesmaximal, complemented
5#20275610C212yesmaximal
6#20274837C212yesmaximal, complemented
7#20274769C211yesmaximal, characteristic, complemented
8#20274766C211yesmaximal, characteristic, complemented
9#20274758C211yesmaximal, characteristic, complemented
10#20274601C211yesmaximal, characteristic, complemented
11C53×#1812C211yesmaximal, characteristic, complemented
12C2×#13133237C2211yescharacteristic
13C2×#13133223C2211yescharacteristic
14#16811176C2212yes—
15#16811031—22no—
16#16811030—22no—
17#16810743—22no—
18#16810743—22no—
19C2×#13133153C2211yescharacteristic
20C2×#13133152C2211yescharacteristic
21#16810661C2211yescharacteristic
22#16810659C2212yes—
23#16810641C2212yes—
24#16810640C2212yes—
25C2×#13133144C2211yescharacteristic
26C2×#13133143C2211yescharacteristic
27#16810596C2212yes—
28#16810579C2212yes—
29C2×#13133127C2211yescharacteristic
30#16810395C2212yes—
31#16810393C2212yes—
32#16810382C2212yes—
33#16810379C2212yes—
34C2×#131×C53C2211yescharacteristic
35C53×#427C2211yescharacteristic
36C53×#427C2211yescharacteristic
37C53×#410C2212yes—
38C53×H(3,ℤ/4)C2212yes—
39C2×#11719945C2311yescharacteristic
40#13133237—22no—
41#13133237—22no—
42C2×#11719925Q811yescharacteristic, complemented
43C2×#11719923Q811yescharacteristic, complemented
44C2×#11719924Q811yescharacteristic, complemented
45C2×#11719922Q811yescharacteristic, complemented
46#13133223—21no—
47#13133223—21no—
48#13133223—21no—
49#13133223—21no—
50C2×#11719917C2312yes—
51C2×#11719917C2311yescharacteristic
52C2×#11719912—21no—
53C22×#88322C2311yescharacteristic
54C2×C8×D53—21no—
55#13133216—22no—
56#13133208—22no—
57#13133194—42no—
58C4×Dic106—42no—
59#13133176C2312yes—
60C2×C4×Dic53C2311yescharacteristic
61#13133153—21no—
62#13133153—21no—
63#13133152—22no—
64#13133151—21no—
65#13133150—22no—
66#13133149—21no—
67#13133148—22no—
68#13133145—42no—
69#13133144—22no—
70#13133144—42no—
71#13133143—22no—
72#13133131—42no—
73#13133128—22no—
74#13133127D412yes—
75#13133127D412yes—
76C4×#88322—22no—
77C2×#41×C53C2311yescharacteristic
78#131×C53—21no—
79#131×C53—21no—
80#131×C53—21no—
81#131×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53C2311yescharacteristic
84C22×C8×C53—21no—
85C22×C8×C53C2311yescharacteristic
86#117×C53C2312yes—
87#117×C53C2312yes—
88#107×C53—21no—
89#107×C53—21no—
90#106×C53—21no—
91#106×C53—21no—
92#99×C53—42no—
93#41×C53—21no—
94#41×C53—21no—
95#41×C53—21no—
96#41×C53—21no—
97C2×Q8×C53C2×Q811yescharacteristic
98C2×D4×C53—21no—
99C2×D4×C53C2×Q811yescharacteristic
100C22×C4×C53—21no—
101C22×C4×C53C2411yescharacteristic
102C2×C22⋊D53—21no—
103C22×Dic53—21no—
104#11719945—21no—
105#11719945—21no—
106#11719945—21no—
107#11719945—21no—
108C2×D212C2×Q811yescharacteristic
109C2×C4×D53—21no—
110C2×Dic106—42no—
111C2×Dic106C2×Q811yescharacteristic
112M16×C53—21no—
113M16×C53—21no—
114M16×C53—41no—
115M16×C53—41no—
116C2×C8×C53—21no—
117C2×C8×C53—41no—
118C2×C8×C53—41no—
119C2×C8×C53C2×D411yescharacteristic
120C2×C8×C53C2×D411yescharacteristic
121C2×C8×C53#4111yescharacteristic
122C2×C8×C53#4111yescharacteristic
123C2×C8×C53#4111yescharacteristic
124C2×C8×C53#4111yescharacteristic
125C4⋊C4×C53—22no—
126C4⋊C4×C53—22no—
127C22⋊C4×C53—22no—
128C22⋊C4×C53—22no—
129C42×C53—22no—
130C42×C53—22no—
131C22⋊Dic53—22no—
132#11719925—22no—
133#11719924—22no—
134#11719923—22no—
135#11719922—22no—
136C4⋊Dic53—22no—
137C4⋊Dic53—42no—
138#11719919—42no—
139#11719919—42no—
140C4×Dic53—21no—
141C4×Dic53—22no—
142C4×Dic53C2×D412yes—
143#11719917—21no—
144#11719917—22no—
145#11719917—21no—
146#11719917—22no—
147C2×#88322—22no—
148C2×#88322—21no—
149C2×#88322C2×D412yes—
150C2×#88322C2×Q811yescharacteristic
151C2×#88322C2×Q811yescharacteristic
152C2×#88322#4112yes—
153#11719912—41no—
154#11719912—41no—
155C8×D53—41no—
156C8×D53—41no—
157C23×C53—21no—
158C23×C53H(5,𝔽2)11yescharacteristic
159C22×D53—21no—
160Q8×C53—21no—
161Q8×C53—21no—
162D4×C53—21no—
163D4×C53—21no—
164D4×C53—41no—
165D4×C53—41no—
166C2×C4×C53—21no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53C22×D411yescharacteristic
172C2×C4×C53C22×Q811yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×C4×C53C2×#4111yescharacteristic
175C2×C4×C53C2×#4111yescharacteristic
176C2×C4×C53H(5,𝔽2)11yescharacteristic
177C22⋊D53—41no—
178C22⋊D53—41no—
179C2×Dic53—22no—
180C2×Dic53—21no—
181C2×Dic53—42no—
182C2×Dic53—81no—
183D212—21no—
184D212—21no—
185C4×D53—41no—
186C4×D53—41no—
187Dic106—21no—
188Dic106—21no—
189Dic106—84no—
190C8×C53—41no—
191C8×C53#12411yescharacteristic
192C8×C53#12411yescharacteristic
193C8×C53#12411yescharacteristic
194C8×C53#12411yescharacteristic
195#88322—41no—
196#88322—42no—
197#88322C22⋊Q812yes—
198#88322C22⋊Q812yes—
199#29155—531nomaximal, Sylow
200C22×C53—41no—
201C22×C53—41no—
202C22×C53—81no—
203C22×C53#53511yescharacteristic
204C22×C53#55411yescharacteristic
205C22×C53#55511yescharacteristic
206C2×D53—41no—
207C2×D53—81no—
208C4×C53—41no—
209C4×C53—82no—
210C4×C53—82no—
211C4×C53C2×C22⋊Q811yescharacteristic
212C4×C53C2×#12411yescharacteristic
213C4×C53#55411yescharacteristic
214C4×C53#55511yescharacteristic
215Dic53—42no—
216Dic53—41no—
217Dic53—82no—
218#3162—531no—
219C2×#444—531no—
220#2031—532no—
221#2031—532no—
222#1976—532no—
223#1833—532no—
224#1814—531no—
225#1814—531no—
226#1812—531no—
227#1812D5311yescharacteristic, complemented
228#1773—531no—
229C2×C53—21no—
230C2×C53—21no—
231C2×C53—21no—
232C2×C53—81no—
233C2×C53#256311yescharacteristic
234D53—81no—
235C2×#135—531no—
236C2×#131—531no—
237C2×#131C2×D5311yescharacteristic
238#533—532no—
239#501—1062no—
240#501—1062no—
241#444—1062no—
242#444—1062no—
243C2×#108—531no—
244C2×#108—531no—
245#427—531no—
246#427—532no—
247#427C2×D5311yescharacteristic
248#427C2×D5311yescharacteristic
249#421—532no—
250#421—532no—
251C2×#104—531no—
252C2×#104—531no—
253#410—532no—
254#410C2×D5312yes—
255#405—532no—
256C2×#97—531no—
257#344—532no—
258#344—532no—
259#339—532no—
260H(3,ℤ/4)—532no—
261H(3,ℤ/4)C2×D5312yes—
262C53#2915511yescharacteristic, complemented, Sylow
263C2×#41—531no—
264C2×#41C22×D5311yescharacteristic
265#135—1062no—
266#135—1062no—
267C2×Q16—531no—
268C2×SD16—531no—
269C2×SD16—531no—
270C2×D8—531no—
271#131—21no—
272#131—21no—
273#131—21no—
274#131—21no—
275#131—1061no—
276#131—1061no—
277#131—1061no—
278#131—1061no—
279C2×M16—21no—
280C2×M16—531no—
281C2×M16—532no—
282C2×M16—1061no—
283C2×M16C22×D5311yescharacteristic
284C22×C8—21no—
285C22×C8—531no—
286C22×C8—1061no—
287C22×C8C22×D5311yescharacteristic
288C4⋊Q8—1062no—
289#125—1062no—
290C22⋊Q8—2122no—
291C4×Q8—2122no—
292#117—532no—
293#117C22×D5312yes—
294#117C22×D5312yes—
295C2×C42—531no—
296#108—1062no—
297#108—1061no—
298#108—1061no—
299#107—21no—
300#107—21no—
301#107—1062no—
302#107—1061no—
303#106—21no—
304#106—21no—
305#106—1062no—
306#106—1061no—
307#104—1062no—
308#104—1062no—
309#104—2122no—
310#104—2122no—
311#99—42no—
312#99—2122no—
313#97—532no—
314#97—532no—
315#97—1062no—
316C4×C8—1062no—
317#41—21no—
318#41—21no—
319#41—21no—
320#41—21no—
321#41—1061no—
322#41—1061no—
323#41—1061no—
324#41—1061no—
325C2×Q8—531no—
326C2×Q8—2122no—
327C2×Q8Q8×D5311yescharacteristic
328C2×D4—21no—
329C2×D4—531no—
330C2×D4—1061no—
331C2×D4Q8×D5311yescharacteristic
332C22×C4—21no—
333C22×C4—1061no—
334C22×C4—1061no—
335C22×C4C23×D5311yescharacteristic
336Q16—1062no—
337SD16—1062no—
338SD16—1062no—
339D8—1062no—
340M16—21no—
341M16—21no—
342M16—41no—
343M16—41no—
344M16—1061no—
345M16—1061no—
346M16—1062no—
347M16—1062no—
348M16—2121no—
349M16—2121no—
350C2×C8—21no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—532no—
354C2×C8—531no—
355C2×C8—531no—
356C2×C8—532no—
357C2×C8—1062no—
358C2×C8—1061no—
359C2×C8—2121no—
360C2×C8—2121no—
361C2×C8#1171994511yescharacteristic
362C2×C8#1171994511yescharacteristic
363C2×C8D4×D5311yescharacteristic
364C2×C8D4×D5311yescharacteristic
365C2×C8#1171994711yescharacteristic
366C2×C8#1171994711yescharacteristic
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—1062no—
370C4⋊C4—2122no—
371C4⋊C4—2122no—
372C4⋊C4—2122no—
373C22⋊C4—22no—
374C22⋊C4—22no—
375C22⋊C4—1062no—
376C42—22no—
377C42—22no—
378C42—532no—
379C42—1061no—
380C42—1062no—
381C23—21no—
382C23—1061no—
383C23#1313326211yescharacteristic
384Q8—21no—
385Q8—21no—
386Q8—1061no—
387Q8—1061no—
388Q8—4244no—
389D4—21no—
390D4—21no—
391D4—41no—
392D4—41no—
393D4—1061no—
394D4—1061no—
395D4—2121no—
396D4—2121no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—1061no—
403C2×C4—1062no—
404C2×C4—4241no—
405C2×C4—2122no—
406C2×C4—2121no—
407C2×C4—2121no—
408C2×C4C2×D4×D5311yescharacteristic
409C2×C4C2×Q8×D5311yescharacteristic
410C2×C4#41×D5311yescharacteristic
411C2×C4C2×#1171994711yescharacteristic
412C2×C4C2×#1171994511yescharacteristic
413C2×C4#1313326311yescharacteristic
414C8—41no—
415C8—532no—
416C8—532no—
417C8—2122no—
418C8—2121no—
419C8#1313320311yescharacteristic
420C8#1313320311yescharacteristic
421C8#1313320311yescharacteristic
422C8#1313320311yescharacteristic
423C22—41no—
424C22—41no—
425C22—81no—
426C22—4241no—
427C22—2121no—
428C22#1681118911yescharacteristic
429C22#1681132811yescharacteristic
430C22#1681133711yescharacteristic
431C4—41no—
432C4—82no—
433C4—82no—
434C4—4242no—
435C4—2122no—
436C4—2121no—
437C4C22⋊Q8×D5311yescharacteristic
438C4C2×#1313320311yescharacteristic
439C4#1681133111yescharacteristic
440C4#1681133611yescharacteristic
441C2—21no—
442C2—21no—
443C2—21no—
444C2—81no—
445C2—4241no—
446C2#2027988511yescharacteristic
subgroupG/H
centerC4#16811336
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1812C2
socleC2×C53#2563
prime radicalC2×#13133237C22
hypercenter#1812D53
hypocenterC53#29155
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1812

Complements

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

NH ≅ G/Ntype
1. #20285238C2semidirect
3. #20275889C2semidirect
4. #20275886C2semidirect
6. #20274837C2semidirect
7. #20274769C2semidirect
8. #20274766C2semidirect
9. #20274758C2semidirect
10. #20274601C2semidirect
11. C53×#1812C2semidirect
42. C2×#11719925Q8semidirect
43. C2×#11719923Q8semidirect
44. C2×#11719924Q8semidirect
45. C2×#11719922Q8semidirect
227. #1812D53semidirect
262. C53#29155semidirect

Maximal subgroups

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

subgroupindex
#202852382
C2×#168107432
#202758892
#202758862
#202756102
#202748372
#202747692
#202747662
#202747582
#202746012
C53×#18122
#2915553