Finite Group Databasebeta

← #27280034

Subgroups — #27280034

Order 13568.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 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×#2351C211yesmaximal, 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×C53—21no—
22C2×C4×Q8×C53C2211yescharacteristic
23C53×#389—21no—
24C53×#389—22no—
25C53×#382—21no—
26C53×#382C2211yescharacteristic
27C4×C4⋊C4×C53—21no—
28C4×C4⋊C4×C53C2211yescharacteristic
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×C53—42no—
56C4×Q8×C53—42no—
57C4×Q8×C53C2×C412yes—
58C4×Q8×C53D412yes—
59C2×C4⋊C4×C53—22no—
60C2×C4⋊C4×C53—22no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C42×C53—22no—
66C2×C42×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—
74C2×Q8×C53—21no—
75C2×Q8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C2×Q8×C53D812yes—
79C2×Q8×C53SD1612yes—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—22no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C22⋊D53—41no—
89#11719945—41no—
90#11719945—41no—
91#11719945—41no—
92#11719945—41no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×C4×D53—41no—
97C2×Dic106—21no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53—44no—
103C4⋊C4×C53—44no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53C22⋊C412yes—
107C42×C53—22no—
108C42×C53—22no—
109C42×C53—44no—
110C42×C53—41no—
111C42×C53—41no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116#11719925—44no—
117#11719924—44no—
118#11719921—41no—
119#11719921—41no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122#11719917—41no—
123#11719917—41no—
124C2×#88322—22no—
125C2×#88322—41no—
126C2×#88322—41no—
127C23×C53#12011yescharacteristic
128C22×D53—41no—
129Q8×C53—22no—
130Q8×C53—22no—
131Q8×C53—22no—
132Q8×C53#10214yes—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—22no—
137C2×C4×C53—22no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—42no—
145C2×C4×C53—41no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53—41no—
149C2×C4×C53C2×C22⋊C411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53#12111yescharacteristic
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12311yescharacteristic
155C2×C4×C53C2×D811yescharacteristic
156C2×C4×C53C2×SD1611yescharacteristic
157C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C53#13711yescharacteristic
159C22⋊D53—81no—
160C22⋊D53—81no—
161C2×Dic53—41no—
162D212—41no—
163D212—41no—
164D212—81no—
165D212—81no—
166C4×D53—81no—
167C4×D53—81no—
168Dic106—41no—
169Dic106—41no—
170#88322—42no—
171#88322—81no—
172#88322—81no—
173#9180—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53#38611yescharacteristic
177C22×C53#44711yescharacteristic
178C22×C53#44811yescharacteristic
179C2×D53—81no—
180C2×D53—81no—
181C2×D53—81no—
182C4×C53—24no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—42no—
188C4×C53—42no—
189C4×C53—82no—
190C4×C53—81no—
191C4×C53C2×#10211yescharacteristic
192C4×C53#41711yescharacteristic
193C4×C53#48011yescharacteristic
194C4×C53#48111yescharacteristic
195Dic53—81no—
196#3037—531no—
197#2351D5311yescharacteristic, complemented
198#1517—531no—
199#1502—531no—
200#1479—531no—
201#1303—531no—
202#1286—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53#187411yescharacteristic
206C2×C53#190511yescharacteristic
207C2×C53#190611yescharacteristic
208D53—161no—
209C2×#137—1061no—
210#532—531no—
211C2×C4×Q8—21no—
212C2×C4×Q8C2×D5311yescharacteristic
213#464—1062no—
214#462—532no—
215#460—532no—
216#431—531no—
217#431—531no—
218#423—531no—
219#417—1061no—
220#389—21no—
221#389—22no—
222#382—21no—
223#382C2×D5311yescharacteristic
224C4×C4⋊C4—21no—
225C4×C4⋊C4C2×D5311yescharacteristic
226#327—1061no—
227#327—1061no—
228#327—1062no—
229Q8⋊C8—1062no—
230C53#918011yescharacteristic, complemented, Sylow
231C2×#41—1061no—
232C22×Q8C22⋊D5311yescharacteristic
233#137—2122no—
234#137—2122no—
235C2×Q16—1062no—
236C2×SD16—1062no—
237C2×M16—1061no—
238C4⋊Q8—531no—
239#127—531no—
240#124—1061no—
241#121—1061no—
242#121—1061no—
243C4×Q8—21no—
244C4×Q8—21no—
245C4×Q8—42no—
246C4×Q8—42no—
247C4×Q8C4×D5312yes—
248C4×Q8D21212yes—
249C4×D4—1061no—
250C4×D4—1061no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4C22⋊D5311yescharacteristic
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—21no—
260C2×C42C22×D5311yescharacteristic
261C4⋊C8—532no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265#103—1062no—
266#102—1062no—
267C22⋊C8—1061no—
268C22⋊C8—1061no—
269C4×C8—1061no—
270C4×C8—1061no—
271#95—22no—
272#95—21no—
273#95C22⋊D5311yescharacteristic
274#95C22⋊D5311yescharacteristic
275#95C22⋊D5312yes—
276#41—2121no—
277#41—2121no—
278#41—2121no—
279#41—2121no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—1061no—
284C2×Q8#1171992112yes—
285C2×Q8#1171992212yes—
286C2×Q8#1171992412yes—
287C2×D4—1061no—
288C2×D4—1061no—
289C2×D4—1061no—
290C2×D4—2121no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—2121no—
297C22×C4C2×C22⋊D5311yescharacteristic
298C22×C4C2×C22⋊D5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300Q16—2124no—
301SD16—2124no—
302M16—2121no—
303M16—2121no—
304C2×C8—1062no—
305C2×C8—2121no—
306C2×C8—2121no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—1061no—
316C4⋊C4—1061no—
317C4⋊C4#1171992112yes—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C42—22no—
321C42—22no—
322C42—44no—
323C42—41no—
324C42—41no—
325C42C2×C4×D5311yescharacteristic
326C42C2×D21211yescharacteristic
327C42#1171994511yescharacteristic
328C42#1171994511yescharacteristic
329C23—2121no—
330C23#1313318611yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—22no—
334Q8—2121no—
335Q8—2121no—
336Q8#1313313714yes—
337D4—4241no—
338D4—4241no—
339D4—4241no—
340D4—4241no—
341D4—2121no—
342D4—2121no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—41no—
359C2×C4—4241no—
360C2×C4—4241no—
361C2×C4—2121no—
362C2×C4C2×#1171992111yescharacteristic
363C2×C4C4×C22⋊D5311yescharacteristic
364C2×C4C4×C22⋊D5311yescharacteristic
365C2×C4#1313319111yescharacteristic
366C2×C4#1313319111yescharacteristic
367C2×C4#1313320011yescharacteristic
368C2×C4C2×#1171992211yescharacteristic
369C2×C4C2×#1171992411yescharacteristic
370C2×C4#1313324311yescharacteristic
371C2×C4#1313324811yescharacteristic
372C8—4241no—
373C8—4241no—
374C8—2122no—
375C22—21no—
376C22—21no—
377C22—4241no—
378C22—4241no—
379C22—4241no—
380C22#1681050211yescharacteristic
381C22#1681075711yescharacteristic
382C22#1681076211yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—24no—
388C4—42no—
389C4—42no—
390C4—82no—
391C4—81no—
392C4—4241no—
393C4C2×#1313313711yescharacteristic
394C4#1681062211yescharacteristic
395C4#1681093611yescharacteristic
396C4#1681094111yescharacteristic
397C2—21no—
398C2—21no—
399C2—8481no—
400C2#2027503211yescharacteristic
401C2#2027520911yescharacteristic
402C2#2027521611yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2351C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2351D53
hypocenterC53#9180
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
4. #20273110C2semidirect
5. #20272618C2semidirect
6. #20272573C2semidirect
7. C53×#2351C2semidirect
11. #16810833C4semidirect
12. #16810823C4semidirect
197. #2351D53semidirect
230. C53#9180semidirect

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×#23512
#918053