Finite Group Databasebeta

← #27280027

Subgroups — #27280027

Order 13568.

620 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284179C211yesmaximal, characteristic
2#20273274C211yesmaximal, characteristic, complemented
3#20273209C211yesmaximal, characteristic, complemented
4#20273106C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2349C211yesmaximal, characteristic, complemented
8C2×#13133243—21no—
9#16811173C2211yescharacteristic
10#16810838—22no—
11#16810828C412yescomplemented
12#16810818C412yescomplemented
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810622—21no—
17#16810356—21no—
18#16810356—21no—
19#16810356—22no—
20D4⋊#88322—22no—
21C2×C4×D4×C53—21no—
22C2×C4×D4×C53C2211yescharacteristic
23C53×#389—21no—
24C53×#385—22no—
25C53×#382C2211yescharacteristic
26C53×#380—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C2×#95×C53—21no—
29C2×#11719945—21no—
30#13133243—42no—
31#13133243—42no—
32C2×#11719923—22no—
33C2×#11719922—22no—
34C2×#11719917—21no—
35#13133217C2×C411yescharacteristic, complemented
36#13133214C2×C411yescharacteristic, complemented
37#13133205—21no—
38#13133191—21no—
39#13133191—21no—
40C4×D212—21no—
41C4×D212—21no—
42C4⋊#88322—21no—
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45C4⋊#88322C2×C412yescomplemented
46#13133140—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×D4×C53D411yescharacteristic
53C23×C4×C53—21no—
54C23×C4×C53—22no—
55C4×D4×C53—21no—
56C4×D4×C53—21no—
57C4×D4×C53—42no—
58C4×D4×C53—42no—
59C4×D4×C53C2×C412yes—
60C4×D4×C53D412yes—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—21no—
67C2×C42×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#95×C53—22no—
70#95×C53—21no—
71#95×C53D411yescharacteristic
72#95×C53D411yescharacteristic
73#95×C53D412yes—
74C24×C53—21no—
75C2×D4×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—21no—
78C2×D4×C53C22⋊C412yes—
79C2×D4×C53D812yes—
80C2×D4×C53SD1612yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—22no—
84C22×C4×C53—22no—
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—44no—
88C22×C4×C53—44no—
89C22×C4×C53—82no—
90C22×C4×C53—82no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C2×C22⋊D53—41no—
95#11719945—41no—
96#11719945—41no—
97#11719945—41no—
98#11719945—41no—
99C2×D212—21no—
100C2×D212—21no—
101C2×D212—21no—
102C2×C4×D53—41no—
103C2×Dic106—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—42no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—82no—
109C4⋊C4×C53C22⋊C412yes—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—44no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53#4111yescharacteristic
118C42×C53#4111yescharacteristic
119#11719923—44no—
120#11719922—44no—
121#11719921—41no—
122#11719921—41no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125#11719917—41no—
126#11719917—41no—
127C2×#88322—22no—
128C2×#88322—41no—
129C2×#88322—41no—
130C23×C53—22no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53#12011yescharacteristic
136C22×D53—41no—
137D4×C53—22no—
138D4×C53—22no—
139D4×C53—22no—
140D4×C53#10214yes—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—82no—
155C2×C4×C53—82no—
156C2×C4×C53—88no—
157C2×C4×C53C2×C22⋊C411yescharacteristic
158C2×C4×C53C4×D411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53#12111yescharacteristic
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12311yescharacteristic
163C2×C4×C53C2×D811yescharacteristic
164C2×C4×C53C2×SD1611yescharacteristic
165C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C53#13711yescharacteristic
167C22⋊D53—81no—
168C22⋊D53—81no—
169C2×Dic53—41no—
170D212—41no—
171D212—41no—
172D212—81no—
173D212—81no—
174C4×D53—81no—
175C4×D53—81no—
176Dic106—41no—
177Dic106—41no—
178#88322—42no—
179#88322—81no—
180#88322—81no—
181#9179—531nomaximal, Sylow
182C22×C53—21no—
183C22×C53—21no—
184C22×C53—24no—
185C22×C53—44no—
186C22×C53—44no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53#38611yescharacteristic
190C22×C53#44711yescharacteristic
191C22×C53#44811yescharacteristic
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C4×C53—21no—
196C4×C53—21no—
197C4×C53—21no—
198C4×C53—21no—
199C4×C53—42no—
200C4×C53—42no—
201C4×C53—81no—
202C4×C53—82no—
203C4×C53C2×#10211yescharacteristic
204C4×C53#41711yescharacteristic
205C4×C53#48011yescharacteristic
206C4×C53#48111yescharacteristic
207Dic53—81no—
208#3036—531no—
209#2349D5311yescharacteristic, complemented
210#1517—531no—
211#1502—531no—
212#1478—531no—
213#1303—531no—
214#1286—531no—
215C2×C53—21no—
216C2×C53—21no—
217C2×C53—44no—
218C2×C53#187411yescharacteristic
219C2×C53#190411yescharacteristic
220C2×C53#190611yescharacteristic
221D53—161no—
222C2×AGL(1,ℤ/8)—1061no—
223#532—531no—
224C2×C4×D4—21no—
225C2×C4×D4C2×D5311yescharacteristic
226#463—1062no—
227#461—532no—
228#459—532no—
229#431—531no—
230#431—531no—
231#423—531no—
232#417—1061no—
233#389—21no—
234#385—22no—
235#382C2×D5311yescharacteristic
236#380—21no—
237C4×C4⋊C4C2×D5311yescharacteristic
238C2×#95—21no—
239#327—1061no—
240#327—1061no—
241#327—1062no—
242D4⋊C8—1062no—
243C53#917911yescharacteristic, complemented, Sylow
244C2×#41—1061no—
245C22×D4C22⋊D5311yescharacteristic
246C23×C4—21no—
247C23×C4—22no—
248AGL(1,ℤ/8)—2122no—
249AGL(1,ℤ/8)—2122no—
250C2×SD16—1062no—
251C2×D8—1062no—
252C2×M16—1061no—
253C4⋊Q8—531no—
254#127—531no—
255#124—1061no—
256#121—1061no—
257#121—1061no—
258C4×D4—21no—
259C4×D4—21no—
260C4×D4—42no—
261C4×D4—42no—
262C4×D4—1061no—
263C4×D4—1061no—
264C4×D4C4×D5312yes—
265C4×D4D21212yes—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4C22⋊D5311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C42—21no—
273C2×C42C22×D5311yescharacteristic
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278#103—1062no—
279#102—1062no—
280C22⋊C8—1061no—
281C22⋊C8—1061no—
282C4×C8—1061no—
283C4×C8—1061no—
284#95—22no—
285#95—21no—
286#95C22⋊D5311yescharacteristic
287#95C22⋊D5311yescharacteristic
288#95C22⋊D5312yes—
289C24—21no—
290#41—2121no—
291#41—2121no—
292#41—2121no—
293#41—2121no—
294C2×Q8—1061no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—1061no—
299C2×D4—1061no—
300C2×D4—1061no—
301C2×D4—2121no—
302C2×D4#1171992112yes—
303C2×D4#1171992212yes—
304C2×D4#1171992412yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—2121no—
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5311yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319SD16—2124no—
320D8—2124no—
321M16—2121no—
322M16—2121no—
323C2×C8—1062no—
324C2×C8—2121no—
325C2×C8—2121no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—82no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4#1171992112yes—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—44no—
337C22⋊C4—2121no—
338C22⋊C4—2121no—
339C42—41no—
340C42—41no—
341C42C2×C4×D5311yescharacteristic
342C42C2×D21211yescharacteristic
343C42#1171994511yescharacteristic
344C42#1171994511yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—2121no—
351C23#1313318611yescharacteristic
352Q8—2121no—
353Q8—2121no—
354D4—22no—
355D4—22no—
356D4—22no—
357D4—4241no—
358D4—4241no—
359D4—4241no—
360D4—4241no—
361D4—2121no—
362D4—2121no—
363D4#1313313714yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—88no—
380C2×C4—4241no—
381C2×C4—4241no—
382C2×C4—2121no—
383C2×C4C2×#1171992111yescharacteristic
384C2×C4C4×C22⋊D5311yescharacteristic
385C2×C4C4×C22⋊D5311yescharacteristic
386C2×C4#1313319111yescharacteristic
387C2×C4#1313319111yescharacteristic
388C2×C4#1313320011yescharacteristic
389C2×C4C2×#1171992211yescharacteristic
390C2×C4C2×#1171992411yescharacteristic
391C2×C4#1313324311yescharacteristic
392C2×C4#1313324811yescharacteristic
393C8—4241no—
394C8—4241no—
395C8—2122no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—44no—
400C22—44no—
401C22—42no—
402C22—42no—
403C22—4241no—
404C22—4241no—
405C22—4241no—
406C22#1681050211yescharacteristic
407C22#1681075711yescharacteristic
408C22#1681076211yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—4241no—
418C4C2×#1313313711yescharacteristic
419C4#1681062211yescharacteristic
420C4#1681093611yescharacteristic
421C4#1681094111yescharacteristic
422C2—21no—
423C2—21no—
424C2—44no—
425C2—8481no—
426C2#2027503211yescharacteristic
427C2#2027520211yescharacteristic
428C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2349C2
socleC22×C53#386
prime radical#20284179C2
cosocleC2×C42×C53C23
hypercenter#2349D53
hypocenterC53#9179
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2349

Complements

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

NH ≅ G/Ntype
2. #20273274C2semidirect
3. #20273209C2semidirect
4. #20273106C2semidirect
5. #20272618C2semidirect
6. #20272573C2semidirect
7. C53×#2349C2semidirect
11. #16810828C4semidirect
12. #16810818C4semidirect
35. #13133217C2×C4semidirect
36. #13133214C2×C4semidirect
45. C4⋊#88322C2×C4semidirect
209. #2349D53semidirect
243. C53#9179semidirect

Maximal subgroups

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

subgroupindex
#202841792
#202732742
#202732092
#202731062
#202726182
#202725732
C53×#23492
#917953