Finite Group Databasebeta

← #27469583

Subgroups — #27469583

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283024C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20276192C211yesmaximal, characteristic, complemented
4#20276191C211yesmaximal, characteristic, complemented
5#20276188C211yesmaximal, characteristic, complemented
6#20275678C212yesmaximal, complemented
7#20274917C212yesmaximal
8#20274911C212yesmaximal
9#20274878C212yesmaximal, complemented
10#20274472C211yesmaximal, characteristic
11C53×#2083C211yesmaximal, characteristic, complemented
12C2×#13133208C2211yescharacteristic
13C2×#13133175C2212yes—
14C2×C42×D53C2211yescharacteristic
15C2×#13133151C2212yes—
16C2×#13133148C2212yes—
17C2×#13133148C2212yes—
18C2×#13133129C2212yes—
19C2×C8×Dic53C2212yes—
20#16810521C2211yescharacteristic
21#16810509—21no—
22#16810505—21no—
23#16810497—21no—
24#16810494—21no—
25#16810493C2211yescharacteristic
26#16810488C2212yes—
27#16810488C2211yescharacteristic
28#16810480—21no—
29#16810480—21no—
30#16810480C2211yescharacteristic
31#16810480C2211yescharacteristic
32C4⋊C4×Dic53—21no—
33C4⋊C4×Dic53—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810378C2211yescharacteristic
38C2×#107×C53C2212yes—
39C53×#389C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#340C2212yes—
42C53×#336C2211yescharacteristic
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C22×C4×D53C2311yescharacteristic
46C22×#88322C2312yes—
47#13133208—24no—
48#13133175—42no—
49#13133175—42no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2311yescharacteristic
54C2×C4⋊Dic53C2311yescharacteristic
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719921—21no—
59C2×#11719921—21no—
60C2×#11719921—22no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53D411yescharacteristic
64C2×C4×Dic53C2311yescharacteristic
65C42×D53—24no—
66#13133151—42no—
67#13133148—42no—
68#13133148—42no—
69#13133129Q818yes—
70C8×Dic53—24no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—22no—
77#13133124—22no—
78C22×C8×C53C2312yes—
79C2×C4⋊C4×C53—21no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53—21no—
82C2×C4⋊C4×C53C2312yes—
83C2×C4⋊C4×C53C2311yescharacteristic
84C2×C4⋊C4×C53C2311yescharacteristic
85C2×C42×C53C2311yescharacteristic
86#107×C53—42no—
87#95×C53—21no—
88C23×D53C2×D411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—22no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53C2411yescharacteristic
95C22×Dic53—21no—
96C22×Dic53—22no—
97C22×Dic53—21no—
98C22×Dic53C2×D411yescharacteristic
99C22×Dic53C2×D411yescharacteristic
100C22×Dic53C2×D411yescharacteristic
101C2×C4×D53—22no—
102C2×C4×D53—21no—
103C2×C4×D53—21no—
104C2×C4×D53—22no—
105C2×C4×D53—22no—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—41no—
109C2×C4×D53—41no—
110C2×C4×D53C2×Q814yes—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53C2×Q814yes—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—42no—
120C4⋊C4×C53—42no—
121C4⋊C4×C53—42no—
122C42×C53—22no—
123#11719921—42no—
124#11719921—42no—
125#11719921—44no—
126C4⋊Dic53—22no—
127C4⋊Dic53—21no—
128C4⋊Dic53—22no—
129C4⋊Dic53—21no—
130C4⋊Dic53—22no—
131C4⋊Dic53—42no—
132C4⋊Dic53—42no—
133C4⋊Dic53—42no—
134#11719919—42no—
135#11719919—42no—
136C4×Dic53—22no—
137C4×Dic53—22no—
138C4×Dic53—44no—
139C4×Dic53#4114yes—
140C2×#88322—22no—
141C2×#88322—22no—
142C2×#88322C2×Q814yes—
143C23×C53C22×D411yescharacteristic
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—21no—
150C22×D53C4⋊Q812yes—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—21no—
155C2×C4×C53—41no—
156C2×C4×C53—42no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—42no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53#13512yes—
165C2×C4×C53#13512yes—
166C2×C4×C53C22×Q812yes—
167C2×C4×C53C2×#4112yes—
168C2×C4×C53H(5,𝔽2)11yescharacteristic
169C2×C4×C53H(5,𝔽2)11yescharacteristic
170C2×Dic53—21no—
171C2×Dic53—21no—
172C2×Dic53—21no—
173C2×Dic53—21no—
174C2×Dic53—21no—
175C2×Dic53—21no—
176C2×Dic53—41no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—41no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—41no—
184C2×Dic53—41no—
185C2×Dic53#12312yes—
186C2×Dic53#12312yes—
187C2×Dic53C4⋊Q812yes—
188C4×D53—24no—
189C4×D53—24no—
190C4×D53—44no—
191C4×D53—44no—
192C8×C53—24no—
193#88322—24no—
194#27951—531nomaximal, Sylow
195C22×C53C2×#12311yescharacteristic
196C22×C53C2×C4⋊Q811yescharacteristic
197C22×C53#53511yescharacteristic
198C22×C53C2×#13511yescharacteristic
199C22×C53C2×#13511yescharacteristic
200C22×C53#57511yescharacteristic
201C22×C53#57511yescharacteristic
202C2×D53—22no—
203C2×D53—22no—
204C2×D53—22no—
205C2×D53—41no—
206C2×D53—41no—
207C2×D53—41no—
208C2×D53—41no—
209C4×C53—22no—
210C4×C53—22no—
211C4×C53—81no—
212C4×C53—82no—
213C4×C53—81no—
214C4×C53#55412yes—
215C4×C53#55412yes—
216Dic53—22no—
217Dic53—22no—
218Dic53—22no—
219Dic53—81no—
220Dic53—82no—
221Dic53—81no—
222#2851—531no—
223#2285—531no—
224#2084—531no—
225#2084—531no—
226#2083—531no—
227#2083D5311yescharacteristic, complemented
228#1991—532no—
229#1854—532no—
230#1854—532no—
231#1846—532no—
232C4×C22⋊C8—531no—
233C2×C53#250211yescharacteristic
234C2×C53#309111yescharacteristic
235C2×C53#309111yescharacteristic
236C2×C53#316011yescharacteristic
237C2×C53#316611yescharacteristic
238C2×C53#321211yescharacteristic
239C2×C53#321211yescharacteristic
240D53—42no—
241C2×#125—531no—
242C2×#117—532no—
243C22×C42—531no—
244C2×#107—532no—
245C2×#107C2×D5312yes—
246C2×#106—532no—
247C2×#106—532no—
248C2×C22⋊C8—532no—
249C2×C4×C8—532no—
250#389—531no—
251#389C2×D5311yescharacteristic
252#387—1061no—
253#387—1061no—
254#385—1061no—
255#385—1061no—
256#384—531no—
257#384C2×D5311yescharacteristic
258#382—531no—
259#382—532no—
260#380—531no—
261#380—531no—
262#380—1061no—
263#380—1061no—
264C4×C4⋊C4—1061no—
265C4×C4⋊C4—1061no—
266#340—532no—
267#340—532no—
268#340—532no—
269#340C2×D5312yes—
270#336—531no—
271#336C2×D5311yescharacteristic
272C53#2795111yescharacteristic, complemented, Sylow
273C23×C4—531no—
274C23×C4—531no—
275C23×C4—531no—
276C22×C8—532no—
277C22×C8C22×D5312yes—
278#125—1064no—
279#117—2122no—
280#117—2122no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—532no—
285C2×C4⋊C4—531no—
286C2×C4⋊C4—531no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4—1061no—
290C2×C4⋊C4—1061no—
291C2×C4⋊C4—1061no—
292C2×C4⋊C4C22×D5312yes—
293C2×C4⋊C4C22×D5311yescharacteristic
294C2×C4⋊C4C22×D5311yescharacteristic
295C2×C22⋊C4—1062no—
296C2×C22⋊C4—1061no—
297C2×C22⋊C4—1061no—
298C2×C42—531no—
299C2×C42—531no—
300C2×C42—531no—
301C2×C42—1064no—
302C2×C42—1062no—
303C2×C42C22×D5311yescharacteristic
304#107—42no—
305#107—2122no—
306#106—2122no—
307#106—2122no—
308C22⋊C8—538no—
309C4×C8—1064no—
310#95—21no—
311#95—1062no—
312#95—1062no—
313#95—1061no—
314#95—1061no—
315#95—1061no—
316#95—1061no—
317#95—1061no—
318C24—531no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—531no—
323C22×C4—531no—
324C22×C4—531no—
325C22×C4—534no—
326C22×C4—1062no—
327C22×C4—1062no—
328C22×C4—1062no—
329C22×C4—1061no—
330C22×C4—1061no—
331C22×C4—1062no—
332C22×C4—1062no—
333C22×C4—1061no—
334C22×C4—1062no—
335C22×C4—1061no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4D4×D5311yescharacteristic
339C22×C4D4×D5311yescharacteristic
340C22×C4C23×D5311yescharacteristic
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—534no—
344C2×C8—1062no—
345C2×C8—1062no—
346C2×C8Q8×D5314yes—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—1062no—
356C4⋊C4—1061no—
357C4⋊C4—1062no—
358C4⋊C4—1061no—
359C4⋊C4—1062no—
360C4⋊C4—2122no—
361C4⋊C4—2122no—
362C4⋊C4—2122no—
363C4⋊C4—2122no—
364C4⋊C4—2122no—
365C22⋊C4—2124no—
366C22⋊C4—2122no—
367C22⋊C4—2122no—
368C42—22no—
369C42—534no—
370C42—1062no—
371C42—1062no—
372C42—2124no—
373C23—532no—
374C23—1061no—
375C23—1061no—
376C23—1061no—
377C23—1061no—
378C23—1061no—
379C23—1061no—
380C23C2×D4×D5311yescharacteristic
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—532no—
395C2×C4—532no—
396C2×C4—532no—
397C2×C4—1061no—
398C2×C4—1061no—
399C2×C4—1061no—
400C2×C4—1061no—
401C2×C4—1061no—
402C2×C4—1061no—
403C2×C4—1064no—
404C2×C4—1064no—
405C2×C4—2124no—
406C2×C4—2124no—
407C2×C4—2121no—
408C2×C4—2121no—
409C2×C4—2122no—
410C2×C4—2121no—
411C2×C4—2121no—
412C2×C4—2121no—
413C2×C4—2122no—
414C2×C4—2122no—
415C2×C4—2121no—
416C2×C4#1313323612yes—
417C2×C4#1313324012yes—
418C2×C4C2×Q8×D5312yes—
419C2×C4#41×D5312yes—
420C2×C4#1313326211yescharacteristic
421C2×C4#1313326311yescharacteristic
422C8—24no—
423C8—1064no—
424C22—1062no—
425C22—1062no—
426C22—1062no—
427C22—2121no—
428C22—2121no—
429C22—2121no—
430C22—2121no—
431C22#123×D5311yescharacteristic
432C22C4⋊Q8×D5311yescharacteristic
433C22#1681118911yescharacteristic
434C22C2×#1313323611yescharacteristic
435C22C2×#1313324011yescharacteristic
436C22#1681141911yescharacteristic
437C22#1681142211yescharacteristic
438C4—22no—
439C4—22no—
440C4—81no—
441C4—81no—
442C4—82no—
443C4—1062no—
444C4—1062no—
445C4—1062no—
446C4—4241no—
447C4—4242no—
448C4—4241no—
449C4#1681132712yes—
450C4#1681132912yes—
451C2—2122no—
452C2#2027918811yescharacteristic
453C2#2028462311yescharacteristic
454C2#2028463111yescharacteristic
455C2#2028522611yescharacteristic
456C2#2028526611yescharacteristic
457C2#2028568911yescharacteristic
458C2#2028569611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2083C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#2083D53
hypocenterC53#27951
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2083

Complements

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

NH ≅ G/Ntype
3. #20276192C2semidirect
4. #20276191C2semidirect
5. #20276188C2semidirect
6. #20275678C2semidirect
9. #20274878C2semidirect
11. C53×#2083C2semidirect
227. #2083D53semidirect
272. C53#27951semidirect

Maximal subgroups

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

subgroupindex
#202830242
#202771202
#202761922
#202761912
#202761882
#202756782
#202749172
#202749112
#202748782
#202744722
C53×#20832
#2795153