Finite Group Databasebeta

← #27280392

Subgroups — #27280392

Order 13568.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284746C211yesmaximal, characteristic
2#20275543C211yesmaximal, characteristic
3#20273209C211yesmaximal, characteristic
4#20273144C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C53×#2295C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810941—21no—
10#16810908C412yescomplemented
11#16810823C412yescomplemented
12#16810762—21no—
13C4×#11719924—22no—
14#16810676C2211yescharacteristic
15#16810676C2211yescharacteristic
16#16810647C2211yescharacteristic
17C2×#13133145—21no—
18C2×#13133145—21no—
19#16810356—22no—
20Q8⋊#88322—22no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#391—21no—
23C53×#389—21no—
24C53×#387—22no—
25C4×C4⋊C4×C53—21no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C53×#376—21no—
28C43×C53C2211yescharacteristic
29C2×#11719945—21no—
30C2×#11719924—22no—
31C2×#11719917—21no—
32#13133217C2×C411yescharacteristic
33#13133214C2×C411yescharacteristic
34#13133205—21no—
35#13133191—21no—
36#13133191—21no—
37C4×D212—21no—
38C4×D212—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322C2×C412yes—
44#13133145—42no—
45#13133145—42no—
46#13133141—22no—
47#13133137—22no—
48C22⋊#88322—21no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C22×Q8×C53D411yescharacteristic
53C4×Q8×C53—21no—
54C4×Q8×C53—21no—
55C4×Q8×C53C2×C412yes—
56C4×Q8×C53D412yes—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C42×C53—21no—
62C2×C42×C53—21no—
63C2×C42×C53—21no—
64C2×C42×C53—21no—
65C2×C42×C53D411yescharacteristic
66C2×C42×C53D411yescharacteristic
67C2×C42×C53C2311yescharacteristic
68#95×C53—22no—
69#95×C53—22no—
70#95×C53—22no—
71#95×C53D412yes—
72C2×Q8×C53—21no—
73C2×Q8×C53—42no—
74C2×Q8×C53C22⋊C412yes—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—21no—
79C22×C4×C53—22no—
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C2×C22⋊D53—41no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87#11719945—41no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×C4×D53—41no—
92C2×Dic106—21no—
93C4⋊C4×C53—22no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—44no—
98C4⋊C4×C53C22⋊C412yes—
99C42×C53—22no—
100C42×C53—22no—
101C42×C53—21no—
102C42×C53—21no—
103C42×C53—21no—
104C42×C53—42no—
105C42×C53—42no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53SD1612yescomplemented
109C42×C53C22×C411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53#4111yescharacteristic
112C42×C53#4111yescharacteristic
113#11719924—44no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—22no—
121C2×#88322—41no—
122C2×#88322—41no—
123C23×C53#12011yescharacteristic
124C22×D53—41no—
125Q8×C53—22no—
126Q8×C53—22no—
127Q8×C53—42no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—42no—
143C2×C4×C53C2×C22⋊C411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C4×D411yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12311yescharacteristic
149C2×C4×C53C2×SD1611yescharacteristic
150C2×C4×C53#13511yescharacteristic
151C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C53#13711yescharacteristic
153C22⋊D53—81no—
154C22⋊D53—81no—
155C2×Dic53—41no—
156D212—42no—
157D212—41no—
158D212—41no—
159D212—81no—
160C4×D53—81no—
161C4×D53—81no—
162Dic106—41no—
163Dic106—41no—
164#88322—42no—
165#88322—81no—
166#88322—81no—
167#9238—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53#38611yescharacteristic
171C22×C53#44811yescharacteristic
172C22×C53#44911yescharacteristic
173C2×D53—81no—
174C2×D53—81no—
175C2×D53—81no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—42no—
180C4×C53—42no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53—82no—
184C4×C53#42112yes—
185C4×C53C4×SD1611yescharacteristic
186C4×C53#44011yescharacteristic
187C4×C53#46011yescharacteristic
188C4×C53#46411yescharacteristic
189Dic53—81no—
190#3103—531no—
191#2295D5311yescharacteristic, complemented
192#1962—531no—
193#1502—531no—
194#1487—531no—
195#1303—531no—
196#1276—531no—
197C2×C53—21no—
198C2×C53—21no—
199C2×C53#188011yescharacteristic
200C2×C53#189711yescharacteristic
201C2×C53#190611yescharacteristic
202D53—161no—
203#532—531no—
204C2×C4×Q8C2×D5311yescharacteristic
205#481—1061no—
206#475—532no—
207#460—532no—
208#448—1061no—
209C4×SD16—1062no—
210#431—531no—
211#431—531no—
212#423—531no—
213C2×#104—1061no—
214C2×#104—1061no—
215#391—21no—
216#389—21no—
217#387—22no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2×D5311yescharacteristic
220#376—21no—
221C43C2×D5311yescharacteristic
222#327—1062no—
223Q8⋊C8—1062no—
224C53#923811yescharacteristic, complemented, Sylow
225C2×#41—1061no—
226C22×Q8C22⋊D5311yescharacteristic
227C2×SD16—1062no—
228C2×M16—1061no—
229C4⋊Q8—531no—
230#127—531no—
231#124—1061no—
232#121—1061no—
233#121—1061no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8C4×D5312yes—
237C4×Q8D21212yes—
238C4×D4—1061no—
239C4×D4—1061no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4C22⋊D5311yescharacteristic
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42C22⋊D5311yescharacteristic
249C2×C42C22⋊D5311yescharacteristic
250C2×C42C22×D5311yescharacteristic
251#106—1062no—
252C4⋊C8—532no—
253C4⋊C8—1061no—
254C4⋊C8—1061no—
255C4⋊C8—1061no—
256#104—2122no—
257#104—2122no—
258#103—1062no—
259#102—1062no—
260C22⋊C8—1061no—
261C22⋊C8—1061no—
262C4×C8—1061no—
263C4×C8—1061no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊D5312yes—
268#41—2121no—
269#41—2121no—
270#41—2121no—
271#41—2121no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—1061no—
275C2×Q8#1171992112yes—
276C2×D4—1061no—
277C2×D4—1061no—
278C2×D4—1061no—
279C2×D4—2121no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—2121no—
286C22×C4C2×C22⋊D5311yescharacteristic
287C22×C4C2×C22⋊D5311yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289SD16—2124no—
290M16—2121no—
291M16—2121no—
292C2×C8—1062no—
293C2×C8—2121no—
294C2×C8—2121no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—1061no—
301C4⋊C4—1061no—
302C4⋊C4#1171992112yes—
303C22⋊C4—2121no—
304C22⋊C4—2121no—
305C42—22no—
306C42—22no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—42no—
311C42—42no—
312C42—41no—
313C42—41no—
314C42#1171992412yescomplemented
315C42C2×C4×D5311yescharacteristic
316C42C2×D21211yescharacteristic
317C42#1171994511yescharacteristic
318C42#1171994511yescharacteristic
319C23—2121no—
320C23#1313318611yescharacteristic
321Q8—22no—
322Q8—22no—
323Q8—42no—
324Q8—2121no—
325Q8—2121no—
326D4—4241no—
327D4—4241no—
328D4—4241no—
329D4—2122no—
330D4—2121no—
331D4—2121no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—4241no—
348C2×C4—4241no—
349C2×C4—2121no—
350C2×C4C2×#1171992111yescharacteristic
351C2×C4C4×C22⋊D5311yescharacteristic
352C2×C4C4×C22⋊D5311yescharacteristic
353C2×C4#1313319111yescharacteristic
354C2×C4#1313319111yescharacteristic
355C2×C4#1313320011yescharacteristic
356C2×C4C2×#1171992411yescharacteristic
357C2×C4#1313323711yescharacteristic
358C2×C4#1313324211yescharacteristic
359C2×C4#1313324811yescharacteristic
360C8—4241no—
361C8—4241no—
362C8—2122no—
363C22—21no—
364C22—21no—
365C22—4241no—
366C22—4241no—
367C22—4241no—
368C22#1681050211yescharacteristic
369C22#1681076211yescharacteristic
370C22#1681076811yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—4241no—
380C4#1681064112yes—
381C4C4×#1171992411yescharacteristic
382C4#1681072111yescharacteristic
383C4#1681082311yescharacteristic
384C4#1681084511yescharacteristic
385C2—21no—
386C2—21no—
387C2—8481no—
388C2#2027507011yescharacteristic
389C2#2027516311yescharacteristic
390C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2295C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2295D53
hypocenterC53#9238
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2295

Complements

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

NH ≅ G/Ntype
4. #20273144C2semidirect
5. #20272618C2semidirect
6. #20272548C2semidirect
7. C53×#2295C2semidirect
10. #16810908C4semidirect
11. #16810823C4semidirect
108. C42×C53SD16semidirect
191. #2295D53semidirect
224. C53#9238semidirect
314. C42#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847462
#202755432
#202732092
#202731442
#202726182
#202725482
C53×#22952
#923853