Finite Group Databasebeta

← #27280048

Subgroups — #27280048

Order 13568.

500 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284187C211yesmaximal, characteristic
2#20273274C211yesmaximal, characteristic
3#20273209C211yesmaximal, characteristic
4#20273110C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272573C211yesmaximal, characteristic, complemented
7C53×#2356C211yesmaximal, characteristic, complemented
8C2×#13133248—21no—
9#16811173C2211yescharacteristic
10#16810845—22no—
11#16810833C412yescomplemented
12#16810823C412yescomplemented
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15#16810647C2211yescharacteristic
16#16810622—21no—
17#16810356—21no—
18#16810356—21no—
19#16810356—22no—
20Q8⋊#88322—22no—
21C2×C4×Q8×C53C2211yescharacteristic
22C53×#391—21no—
23C53×#387—22no—
24C53×#383—21no—
25C53×#382C2211yescharacteristic
26C4×C4⋊C4×C53—21no—
27C4×C4⋊C4×C53C2211yescharacteristic
28C53×#376—21no—
29C2×#11719945—21no—
30#13133248—42no—
31#13133248—42no—
32C2×#11719925—22no—
33C2×#11719924—22no—
34C2×#11719917—21no—
35#13133217C2×C411yescharacteristic
36#13133214C2×C411yescharacteristic
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×C412yes—
46#13133140—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—21no—
58C2×C4⋊C4×C53—21no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C42×C53—22no—
62C2×C42×C53—21no—
63C2×C42×C53—21no—
64C2×C42×C53C2311yescharacteristic
65#95×C53—22no—
66#95×C53—22no—
67#95×C53—22no—
68#95×C53—21no—
69#95×C53D411yescharacteristic
70#95×C53D411yescharacteristic
71#95×C53D412yes—
72C2×Q8×C53—21no—
73C2×Q8×C53—21no—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C2×C22⋊D53—41no—
87#11719945—41no—
88#11719945—41no—
89#11719945—41no—
90#11719945—41no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212—21no—
94C2×C4×D53—41no—
95C2×Dic106—21no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—21no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—41no—
102C4⋊C4×C53C22⋊C412yes—
103C42×C53—22no—
104C42×C53—22no—
105C42×C53—44no—
106C42×C53—41no—
107C42×C53—41no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719925—44no—
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—41no—
128Q8×C53—41no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53C2×C22⋊C411yescharacteristic
146C2×C4×C53C4×D411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12311yescharacteristic
151C2×C4×C53#13511yescharacteristic
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
155C22⋊D53—81no—
156C22⋊D53—81no—
157C2×Dic53—41no—
158D212—41no—
159D212—41no—
160D212—81no—
161D212—81no—
162C4×D53—81no—
163C4×D53—81no—
164Dic106—41no—
165Dic106—41no—
166#88322—42no—
167#88322—81no—
168#88322—81no—
169#9182—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#38611yescharacteristic
173C22×C53#44911yescharacteristic
174C22×C53#44911yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—42no—
185C4×C53—81no—
186C4×C53—82no—
187C4×C53#41611yescharacteristic
188C4×C53#41811yescharacteristic
189C4×C53#48211yescharacteristic
190C4×C53#48211yescharacteristic
191Dic53—81no—
192#3037—531no—
193#2356D5311yescharacteristic, complemented
194#1517—531no—
195#1502—531no—
196#1479—531no—
197#1303—531no—
198#1286—531no—
199C2×C53—21no—
200C2×C53—21no—
201C2×C53#187511yescharacteristic
202C2×C53#190511yescharacteristic
203C2×C53#190611yescharacteristic
204D53—161no—
205C2×#137—1061no—
206#532—531no—
207C2×C4×Q8C2×D5311yescharacteristic
208#464—1062no—
209#462—532no—
210#460—532no—
211#431—531no—
212#431—531no—
213#423—531no—
214#417—1061no—
215#391—21no—
216#387—22no—
217#383—21no—
218#382C2×D5311yescharacteristic
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2×D5311yescharacteristic
221#376—21no—
222#327—1061no—
223#327—1061no—
224#327—1062no—
225Q8⋊C8—1062no—
226C53#918211yescharacteristic, complemented, Sylow
227C2×#41—1061no—
228C22×Q8C22⋊D5311yescharacteristic
229#137—2122no—
230#137—2122no—
231C2×Q16—1062no—
232C2×SD16—1062no—
233C2×M16—1061no—
234C4⋊Q8—531no—
235#127—531no—
236#124—1061no—
237#121—1061no—
238#121—1061no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8C4×D5312yes—
242C4×Q8D21212yes—
243C4×D4—1061no—
244C4×D4—1061no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4C22⋊D5311yescharacteristic
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42C22×D5311yescharacteristic
253C4⋊C8—532no—
254C4⋊C8—1061no—
255C4⋊C8—1061no—
256C4⋊C8—1061no—
257#103—1062no—
258#102—1062no—
259C22⋊C8—1061no—
260C22⋊C8—1061no—
261C4×C8—1061no—
262C4×C8—1061no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—21no—
267#95C22⋊D5311yescharacteristic
268#95C22⋊D5311yescharacteristic
269#95C22⋊D5312yes—
270#41—2121no—
271#41—2121no—
272#41—2121no—
273#41—2121no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—1061no—
280C2×Q8#1171992112yes—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—2121no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—2121no—
291C22×C4C2×C22⋊D5311yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293C22×C4C2×C22⋊D5311yescharacteristic
294Q16—2124no—
295SD16—2124no—
296M16—2121no—
297M16—2121no—
298C2×C8—1062no—
299C2×C8—2121no—
300C2×C8—2121no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—1061no—
308C4⋊C4—1061no—
309C4⋊C4#1171992112yes—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C42—22no—
313C42—22no—
314C42—44no—
315C42—41no—
316C42—41no—
317C42C2×C4×D5311yescharacteristic
318C42C2×D21211yescharacteristic
319C42#1171994511yescharacteristic
320C42#1171994511yescharacteristic
321C23—2121no—
322C23#1313318611yescharacteristic
323Q8—22no—
324Q8—22no—
325Q8—41no—
326Q8—41no—
327Q8—2121no—
328Q8—2121no—
329D4—4241no—
330D4—4241no—
331D4—4241no—
332D4—4241no—
333D4—2121no—
334D4—2121no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—4241no—
352C2×C4—4241no—
353C2×C4—2121no—
354C2×C4C2×#1171992111yescharacteristic
355C2×C4C4×C22⋊D5311yescharacteristic
356C2×C4C4×C22⋊D5311yescharacteristic
357C2×C4#1313319111yescharacteristic
358C2×C4#1313319111yescharacteristic
359C2×C4#1313320011yescharacteristic
360C2×C4#1313323711yescharacteristic
361C2×C4#1313323711yescharacteristic
362C2×C4#1313324211yescharacteristic
363C2×C4#1313324211yescharacteristic
364C8—4241no—
365C8—4241no—
366C8—2122no—
367C22—21no—
368C22—21no—
369C22—4241no—
370C22—4241no—
371C22—4241no—
372C22#1681050211yescharacteristic
373C22#1681076811yescharacteristic
374C22#1681076811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—81no—
383C4—82no—
384C4—4241no—
385C4#1681061711yescharacteristic
386C4#1681062711yescharacteristic
387C4#1681094711yescharacteristic
388C4#1681094711yescharacteristic
389C2—21no—
390C2—21no—
391C2—8481no—
392C2#2027503711yescharacteristic
393C2#2027520911yescharacteristic
394C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2356C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2356D53
hypocenterC53#9182
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
4. #20273110C2semidirect
5. #20272618C2semidirect
6. #20272573C2semidirect
7. C53×#2356C2semidirect
11. #16810833C4semidirect
12. #16810823C4semidirect
193. #2356D53semidirect
226. C53#9182semidirect

Maximal subgroups

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

subgroupindex
#202841872
#202732742
#202732092
#202731102
#202726182
#202725732
C53×#23562
#918253