Finite Group Databasebeta

← #27283252

Subgroups — #27283252

Order 13568.

506 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#20285290C211yesmaximal, characteristic
2#20275634C211yesmaximal, characteristic, complemented
3C4×#13133150C211yesmaximal, characteristic, complemented
4#20273210C211yesmaximal, characteristic, complemented
5#20273209C211yesmaximal, characteristic, complemented
6M16×#88322C211yesmaximal, characteristic, complemented
7C53×#1980C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811026C411yescharacteristic, complemented
10#16810999C411yescharacteristic, complemented
11#16810971C411yescharacteristic, complemented
12#16810969C411yescharacteristic, complemented
13#16810857—21no—
14#16810857—21no—
15C4×#11719922—21no—
16C4×#11719922—21no—
17#16810676C2211yescharacteristic
18C2×#13133150—21no—
19C2×#13133143—22no—
20C2×C4×#88322C2211yescharacteristic
21C4×C4⋊Dic53C2211yescharacteristic
22#16810385—21no—
23#16810357—22no—
24#16810356—22no—
25#16810344—21no—
26C8×#88322—21no—
27C53×#532C2211yescharacteristic
28C2×#104×C53—22no—
29C4×M16×C53C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C2×#11719945—21no—
32C2×#11719922—21no—
33C2×#11719922—21no—
34C22×#88322D411yescharacteristic
35C22×#88322D411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133214C2×C411yescharacteristic
38#13133205—21no—
39#13133191—21no—
40#13133191—21no—
41C4×D212—21no—
42C4×D212—21no—
43C2×C4⋊Dic53—21no—
44C2×C4×Dic53—21no—
45#13133150—24no—
46#13133150—21no—
47#13133150—21no—
48C4⋊#88322—21no—
49#13133143—44no—
50#13133138—21no—
51#13133138—21no—
52#13133137—21no—
53#13133137—21no—
54C22⋊#88322D412yes—
55C4×#88322—21no—
56C4×#88322C2×C411yescharacteristic
57C4×#88322C2×C411yescharacteristic
58C4×#88322D411yescharacteristic
59C4×#88322D411yescharacteristic
60#13133124—21no—
61C2×#41×C53—21no—
62C2×M16×C53D412yes—
63C4⋊Q8×C53C2×C411yescharacteristic
64#127×C53C2×C411yescharacteristic
65#124×C53—21no—
66#121×C53—21no—
67#121×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C2×C4⋊C4×C53—21no—
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#104×C53—44no—
74#97×C53—21no—
75C4×C8×C53—21no—
76#95×C53—21no—
77#41×C53—42no—
78#41×C53—42no—
79C2×Q8×C53—21no—
80C2×D4×C53—21no—
81C2×D4×C53—21no—
82C2×D4×C53—21no—
83C2×D4×C53—41no—
84C22×C4×C53—41no—
85C22×C4×C53—41no—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C2×C22⋊D53—41no—
90C22×Dic53—41no—
91#11719945—42no—
92#11719945—42no—
93C2×D212—21no—
94C2×D212—21no—
95C2×D212—21no—
96C2×C4×D53—41no—
97C2×Dic106—21no—
98M16×C53—22no—
99M16×C53D814yescomplemented
100C2×C8×C53—22no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C22⋊C4×C53—41no—
106C22⋊C4×C53—41no—
107C42×C53—42no—
108C42×C53C22×C411yescharacteristic
109C42×C53C2×D411yescharacteristic
110C42×C53#4111yescharacteristic
111C42×C53#4111yescharacteristic
112#11719922—42no—
113#11719922—42no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—22no—
117C4⋊Dic53—21no—
118C4⋊Dic53—21no—
119C4⋊Dic53—41no—
120C4×Dic53—42no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—41no—
124C2×#88322C22⋊C411yescharacteristic
125C2×#88322C22⋊C411yescharacteristic
126C2×#88322C22⋊C411yescharacteristic
127C2×#88322C22⋊C411yescharacteristic
128C23×C53—41no—
129C23×C53#12711yescharacteristic
130C22×D53—41no—
131Q8×C53—42no—
132D4×C53—41no—
133D4×C53—41no—
134D4×C53—42no—
135D4×C53—81no—
136D4×C53—81no—
137D4×C53—81no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—41no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53—81no—
144C2×C4×C53—81no—
145C2×C4×C53—82no—
146C2×C4×C53C2×C22⋊C411yescharacteristic
147C2×C4×C53C4×D411yescharacteristic
148C2×C4×C53C4×D411yescharacteristic
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12411yescharacteristic
152C2×C4×C53C2×D812yes—
153C2×C4×C53#13512yes—
154C22⋊D53—81no—
155C22⋊D53—81no—
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—81no—
159C2×Dic53—81no—
160D212—41no—
161D212—41no—
162D212—42no—
163D212—81no—
164C4×D53—82no—
165Dic106—42no—
166C8×C53—24no—
167#88322—21no—
168#88322—21no—
169#88322—21no—
170#88322—42no—
171#88322#10412yescomplemented
172#9719—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53#39011yescharacteristic
179C22×C53#49311yescharacteristic
180C22×C53#49511yescharacteristic
181C2×D53—81no—
182C2×D53—81no—
183C2×D53—81no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—82no—
188C4×C53—81no—
189C4×C53C2×#10411yescharacteristic
190C4×C53#42111yescharacteristic
191C4×C53C4×D812yes—
192C4×C53#46612yes—
193Dic53—82no—
194Dic53—81no—
195#3169—531no—
196#1980—531no—
197#1980D5311yescharacteristic, complemented
198C4×#107—531no—
199#1502—531no—
200#1502—531no—
201C8×M16—531no—
202C2×C53—21no—
203C2×C53—21no—
204C2×C53—161no—
205C2×C53#197011yescharacteristic
206C2×C53#197611yescharacteristic
207C2×C53#198011yescharacteristic
208D53—161no—
209#532—531no—
210#532C2×D5311yescharacteristic
211#500—531no—
212#493—531no—
213#486—531no—
214#486—531no—
215#466—1061no—
216#466—1061no—
217C4×D8—1061no—
218C4×D8—1061no—
219#431—531no—
220C2×#107—1061no—
221C2×#104—22no—
222C2×#104—1062no—
223C4×M16C2×D5311yescharacteristic
224C2×C4×C8—531no—
225C4×C4⋊C4—531no—
226C4×C4⋊C4C2×D5311yescharacteristic
227#340—1061no—
228#327—1062no—
229#327—1062no—
230#322—1061no—
231C82—1061no—
232C53#971911yescharacteristic, complemented, Sylow
233C2×#41—21no—
234C2×#41—1061no—
235C2×D8—1061no—
236C2×D8—1061no—
237C2×M16C22⋊D5312yes—
238C22×C8—531no—
239C22×C8—531no—
240C4⋊Q8—531no—
241C4⋊Q8C4×D5311yescharacteristic
242#127—531no—
243#127C4×D5311yescharacteristic
244#124—21no—
245#124—1061no—
246#121—21no—
247#121—21no—
248#121—1061no—
249#121—1061no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—1061no—
253C4×D4—1061no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1061no—
256C2×C42—21no—
257C2×C42—1061no—
258C2×C42C22×D5311yescharacteristic
259#107—1061no—
260#107—1061no—
261#107—1064no—
262C4⋊C8—1061no—
263#104—44no—
264#104—2124no—
265#102—1061no—
266#102—1061no—
267#102—1061no—
268#102—1061no—
269C22⋊C8—532no—
270#97—21no—
271C4×C8—21no—
272C4×C8—531no—
273C4×C8—531no—
274C4×C8—531no—
275C4×C8—531no—
276C4×C8—1061no—
277#95—21no—
278#95—1061no—
279#41—42no—
280#41—42no—
281#41—2122no—
282#41—2122no—
283C2×Q8—21no—
284C2×Q8—1061no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C2×D4—1061no—
290C2×D4—1061no—
291C2×D4—1061no—
292C2×D4—2121no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4D4×D5311yescharacteristic
298C22×C4D4×D5311yescharacteristic
299C22×C4C2×C22⋊D5311yescharacteristic
300D8—2122no—
301D8—2122no—
302M16—22no—
303M16#1171992214yescomplemented
304C2×C8—22no—
305C2×C8—531no—
306C2×C8—531no—
307C2×C8—531no—
308C2×C8—531no—
309C2×C8—1062no—
310C2×C8—1062no—
311C2×C8—2121no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—1062no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C4⋊C4—2121no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—2121no—
323C22⋊C4—2121no—
324C42—42no—
325C42—2122no—
326C42C2×C4×D5311yescharacteristic
327C42#1171994511yescharacteristic
328C42D4×D5311yescharacteristic
329C42#1171994711yescharacteristic
330C23—41no—
331C23—2121no—
332C23#1313321311yescharacteristic
333Q8—42no—
334Q8—2122no—
335D4—41no—
336D4—41no—
337D4—42no—
338D4—81no—
339D4—81no—
340D4—81no—
341D4—4241no—
342D4—4241no—
343D4—4241no—
344D4—2121no—
345D4—2121no—
346D4—2122no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—4241no—
356C2×C4—4241no—
357C2×C4—4242no—
358C2×C4—2122no—
359C2×C4—2121no—
360C2×C4C22⋊C4×D5311yescharacteristic
361C2×C4#1313317711yescharacteristic
362C2×C4C4×C22⋊D5311yescharacteristic
363C2×C4#1313318911yescharacteristic
364C2×C4#1313319011yescharacteristic
365C2×C4#1313320311yescharacteristic
366C2×C4C2×#1171992212yes—
367C2×C4#1313323712yes—
368C8—24no—
369C8—532no—
370C8—1061no—
371C8—1061no—
372C8—1061no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—4241no—
380C22—4241no—
381C22—4241no—
382C22#1681052311yescharacteristic
383C22#1681099911yescharacteristic
384C22#1681100511yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—82no—
389C4—81no—
390C4—4242no—
391C4—4241no—
392C4#104×D5311yescharacteristic
393C4#1681063811yescharacteristic
394C4C4×#1171992212yes—
395C4#1681085712yes—
396C2—21no—
397C2—21no—
398C2—161no—
399C2—8481no—
400C2#2027558011yescharacteristic
401C2#2027561211yescharacteristic
402C2#2027563211yescharacteristic
subgroupG/H
centerC2×C4#13133189
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1980C2
socleC22×C53#390
prime radical#20285290C2
cosocleC2×C42×C53C23
hypercenter#1980D53
hypocenterC53#9719
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #20275634C2semidirect
3. C4×#13133150C2semidirect
4. #20273210C2semidirect
5. #20273209C2semidirect
6. M16×#88322C2semidirect
7. C53×#1980C2semidirect
9. #16811026C4semidirect
10. #16810999C4semidirect
11. #16810971C4semidirect
12. #16810969C4semidirect
99. M16×C53D8semidirect
171. #88322#104semidirect
197. #1980D53semidirect
232. C53#9719semidirect
303. M16#11719922semidirect

Maximal subgroups

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

subgroupindex
#202852902
#202756342
C4×#131331502
#202732102
#202732092
M16×#883222
C53×#19802
#971953