Finite Group Databasebeta

← #27283630

Subgroups — #27283630

Order 13568.

552 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275701C211yesmaximal, characteristic, complemented
2#20275552C211yesmaximal, characteristic
3#20274744C211yesmaximal, characteristic, complemented
4#20274491C211yesmaximal, characteristic, complemented
5#20273287C211yesmaximal, characteristic, complemented
6#20273283C211yesmaximal, characteristic
7C2×C53×#465C211yesmaximal, characteristic
8C2×#13133151—21no—
9C2×#13133132—22no—
10C2×C22⋊#88322C2211yescharacteristic
11C2×#13133127—21no—
12#16810483C2211yescharacteristic
13#16810483C2211yescharacteristic
14C4×C22⋊Dic53C2211yescharacteristic
15#16810390—22no—
16#16810386—21no—
17#16810377—21no—
18#16810360—22no—
19#16810357—22no—
20C22×SD16×C53C411yescharacteristic, complemented
21C23×C8×C53C2211yescharacteristic
22C2×C22⋊Q8×C53C2211yescharacteristic
23C2×#121×C53C2211yescharacteristic
24C53×#465—21no—
25C53×#465—21no—
26C53×#465—21no—
27C53×#465—21no—
28C2×#106×C53C411yescharacteristic, complemented
29C2×#103×C53C411yescharacteristic, complemented
30C2×#102×C53C411yescharacteristic, complemented
31C22×#88322—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53D412yes—
37C2×C4×Dic53—21no—
38#13133151—42no—
39#13133132—44no—
40C22⋊#88322—21no—
41C22⋊#88322D412yes—
42#13133127—42no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C22×Q8×C53C2×C411yescharacteristic
47C22×D4×C53—21no—
48C22×D4×C53C2×C411yescharacteristic
49C23×C4×C53C2311yescharacteristic
50C2×SD16×C53—21no—
51C2×SD16×C53—21no—
52C2×SD16×C53—42no—
53C22×C8×C53—42no—
54C22×C8×C53C2×C411yescharacteristic
55C22×C8×C53C2×C411yescharacteristic
56C22×C8×C53D411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C22×C8×C53D411yescharacteristic
59C22×C8×C53D411yescharacteristic
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66#121×C53—21no—
67#121×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C4⋊C4×C53C2×C411yescharacteristic
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73#106×C53—21no—
74#106×C53—21no—
75#103×C53—21no—
76#103×C53—21no—
77#102×C53—21no—
78#102×C53—21no—
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53#4111yescharacteristic
103C22×C4×C53#4111yescharacteristic
104C22×Dic53—22no—
105C22×Dic53—41no—
106C22×Dic53—41no—
107SD16×C53—42no—
108SD16×C53—42no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—41no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53C22⋊C411yescharacteristic
116C2×C8×C53C22⋊C411yescharacteristic
117C2×C8×C53C22⋊C411yescharacteristic
118C2×C8×C53C22⋊C411yescharacteristic
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—41no—
124C4⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C22⋊C4×C53—41no—
127C22⋊C4×C53—41no—
128C22⋊C4×C53—41no—
129C22⋊Dic53—22no—
130C22⋊Dic53—22no—
131C22⋊Dic53—42no—
132C22⋊Dic53—42no—
133C4⋊Dic53—41no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C2×#88322—22no—
140C2×#88322—41no—
141C2×#88322—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—81no—
147C23×C53—81no—
148C23×C53C4×D411yescharacteristic
149C23×C53#12111yescharacteristic
150C23×C53#12711yescharacteristic
151C23×C53AGL(1,ℤ/8)12yes—
152C23×C53#13712yes—
153Q8×C53—41no—
154Q8×C53—41no—
155Q8×C53—81no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—81no—
159D4×C53—82no—
160D4×C53—81no—
161D4×C53—81no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—81no—
176C2×C4×C53—81no—
177C2×C4×C53C2×C22⋊C411yescharacteristic
178C2×C4×C53C4×D411yescharacteristic
179C2×C4×C53#12111yescharacteristic
180C2×C4×C53#12411yescharacteristic
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—42no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C2×Dic53—81no—
188C2×Dic53—81no—
189C2×Dic53—81no—
190C8×C53—21no—
191C8×C53—21no—
192C8×C53—21no—
193C8×C53#9912yescomplemented
194#88322—42no—
195#9786—531nomaximal, Sylow
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—162no—
208C22×C53#39011yescharacteristic
209C22×C53#44012yes—
210C22×C53#46812yes—
211C22×C53#49611yescharacteristic
212C22×C53#49711yescharacteristic
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—81no—
219C4×C53C2×#9911yescharacteristic
220C4×C53#41011yescharacteristic
221Dic53—82no—
222Dic53—161no—
223Dic53—161no—
224C2×#465D5311yescharacteristic
225#1996—531no—
226#1964—531no—
227#1808—531no—
228#1752—531no—
229#1520—531no—
230#1519—531no—
231C2×C53—22no—
232C2×C53—21no—
233C2×C53—22no—
234C2×C53—21no—
235C2×C53—161no—
236C2×C53#196411yescharacteristic
237C2×C53#197311yescharacteristic
238C2×C53#197611yescharacteristic
239C22×SD16Dic5311yescharacteristic, complemented
240C23×C8C2×D5311yescharacteristic
241C2×C22⋊Q8C2×D5311yescharacteristic
242C2×#121C2×D5311yescharacteristic
243#465—21no—
244#465—21no—
245#465—21no—
246#465—21no—
247C2×#107—1061no—
248C2×#106Dic5311yescharacteristic, complemented
249C2×#103Dic5311yescharacteristic, complemented
250C2×#102Dic5311yescharacteristic, complemented
251C2×#99—1062no—
252C2×C22⋊C8—531no—
253C2×#97—1061no—
254#380—531no—
255#380—531no—
256C4×C22⋊C4—531no—
257#342—1062no—
258#340—1061no—
259#336—1061no—
260#328—1062no—
261#327—1062no—
262C53#978611yescharacteristic, complemented, Sylow
263C22×Q8C2×Dic5311yescharacteristic
264C22×D4—21no—
265C22×D4C2×Dic5311yescharacteristic
266C23×C4C22×D5311yescharacteristic
267C2×SD16—21no—
268C2×SD16—21no—
269C2×SD16—42no—
270C22×C8—42no—
271C22×C8—1061no—
272C22×C8C2×Dic5311yescharacteristic
273C22×C8C2×Dic5311yescharacteristic
274C22×C8C22⋊D5311yescharacteristic
275C22×C8C22⋊D5311yescharacteristic
276C22×C8C22⋊D5311yescharacteristic
277C22×C8C22⋊D5311yescharacteristic
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—1061no—
288C2×C4⋊C4—1061no—
289C2×C4⋊C4C2×Dic5311yescharacteristic
290C2×C4⋊C4C2×Dic5311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—532no—
294C2×C22⋊C4—1061no—
295C2×C22⋊C4—1061no—
296C2×C42—1061no—
297#107—2122no—
298#106—21no—
299#106—21no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#99—2124no—
305C22⋊C8—532no—
306C22⋊C8—1061no—
307#97—2122no—
308#95—1061no—
309#95—1061no—
310#95—1061no—
311C24—21no—
312C24D4×D5311yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—1062no—
331C22×C4—2121no—
332C22×C4—2121no—
333C22×C4D4×D5311yescharacteristic
334C22×C4#1171994711yescharacteristic
335C22×C4#1171994711yescharacteristic
336C22×C4C22×Dic5311yescharacteristic
337C22×C4C2×C22⋊D5311yescharacteristic
338C22×C4C2×C22⋊D5311yescharacteristic
339SD16—42no—
340SD16—42no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—1062no—
348C2×C8—2121no—
349C2×C8—2121no—
350C2×C8C22⋊Dic5311yescharacteristic
351C2×C8C22⋊Dic5311yescharacteristic
352C2×C8C22⋊Dic5311yescharacteristic
353C2×C8C22⋊Dic5311yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—2121no—
361C4⋊C4—2121no—
362C4⋊C4—2121no—
363C4⋊C4—2121no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—1062no—
369C22⋊C4—1062no—
370C22⋊C4—2122no—
371C22⋊C4—2122no—
372C42—2121no—
373C42—2121no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23D4×Dic5311yescharacteristic
381C23#1313318811yescharacteristic
382C23#1313321311yescharacteristic
383C23#1313324412yes—
384C23#1313324612yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—4241no—
410C2×C4—4241no—
411C2×C4—4241no—
412C2×C4—4241no—
413C2×C4—4241no—
414C2×C4—4241no—
415C2×C4—2122no—
416C2×C4—2122no—
417C2×C4—2122no—
418C2×C4C2×C22⋊Dic5311yescharacteristic
419C2×C4D4×Dic5311yescharacteristic
420C2×C4#1313318811yescharacteristic
421C2×C4#1313320411yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—2122no—
426C8#1313313212yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#1681052611yescharacteristic
440C22#1681072412yes—
441C22#1681086612yes—
442C22#1681100911yescharacteristic
443C22#1681101411yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—81no—
448C4—81no—
449C4—81no—
450C4—8481no—
451C4—8481no—
452C4—4242no—
453C4C2×#1313313211yescharacteristic
454C4#1681059711yescharacteristic
455C2—21no—
456C2—22no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#2027555211yescharacteristic
461C2#2027559811yescharacteristic
462C2#2027561411yescharacteristic
subgroupG/H
centerC22#16810526
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#465C2
socleC22×C53#390
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenterC2×#465D53
hypocenterC53#9786
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#465

Complements

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

NH ≅ G/Ntype
1. #20275701C2semidirect
3. #20274744C2semidirect
4. #20274491C2semidirect
5. #20273287C2semidirect
20. C22×SD16×C53C4semidirect
28. C2×#106×C53C4semidirect
29. C2×#103×C53C4semidirect
30. C2×#102×C53C4semidirect
193. C8×C53#99semidirect
239. C22×SD16Dic53semidirect
248. C2×#106Dic53semidirect
249. C2×#103Dic53semidirect
250. C2×#102Dic53semidirect
262. C53#9786semidirect
426. C8#13133132semidirect

Maximal subgroups

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

subgroupindex
#202757012
#202755522
#202747442
#202744912
#202732872
#202732832
C2×C53×#4652
#978653