Finite Group Databasebeta

← #27278809

Subgroups — #27278809

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284786C211yesmaximal, characteristic
2#20283840C211yesmaximal, characteristic
3#20275692C211yesmaximal, characteristic, complemented
4#20273137C211yesmaximal, characteristic, complemented
5#20272605C211yesmaximal, characteristic, complemented
6#20272548C211yesmaximal, characteristic, complemented
7C4×#104×C53C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810964—21no—
10#16810908C411yescharacteristic, complemented
11#16810903C411yescharacteristic, complemented
12#16810833C411yescharacteristic, complemented
13#16810828C411yescharacteristic, complemented
14#16810785—21no—
15#16810754—21no—
16C4×#11719925—21no—
17C4×#11719923—21no—
18#16810695—22no—
19D4×#88322—21no—
20#16810680—21no—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24C2×C4⋊#88322—21no—
25#16810383—22no—
26#16810355—21no—
27#16810355—21no—
28C4×#41×C53C2211yescharacteristic
29C2×#104×C53—21no—
30C4×M16×C53C2211yescharacteristic
31C43×C53C2211yescharacteristic
32C53×#339—21no—
33C2×#11719925—21no—
34C2×#11719923—21no—
35C2×#11719917D412yes—
36C22×#88322—22no—
37#13133217C2×C411yescharacteristic
38#13133217C2×C411yescharacteristic
39#13133209—21no—
40#13133196—21no—
41#13133196—21no—
42C4×Dic106—21no—
43C4×Dic106—21no—
44#13133176—21no—
45#13133150—21no—
46#13133148—21no—
47C4⋊#88322—22no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322C2×C411yescharacteristic
51C4⋊#88322C2×C411yescharacteristic
52C4⋊#88322D411yescharacteristic
53C4⋊#88322D411yescharacteristic
54#13133141—21no—
55#13133140—21no—
56#13133140—21no—
57#13133138—21no—
58C22⋊#88322—22no—
59C22⋊#88322—21no—
60C22⋊#88322—21no—
61#13133128—22no—
62C4×#88322—21no—
63C4×#88322—21no—
64C2×#41×C53D411yescharacteristic
65C2×M16×C53—21no—
66C4×Q8×C53C2×C411yescharacteristic
67C4×Q8×C53D411yescharacteristic
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53C2×C411yescharacteristic
71C4×D4×C53D411yescharacteristic
72#117×C53—21no—
73#117×C53D411yescharacteristic
74C2×C42×C53—21no—
75C2×C42×C53—21no—
76C2×C42×C53—21no—
77C2×C42×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#104×C53—24no—
80#97×C53—21no—
81C4×C8×C53—21no—
82#41×C53—22no—
83#41×C53—41no—
84C2×Q8×C53C22⋊C411yescharacteristic
85C2×D4×C53—21no—
86C2×D4×C53C22⋊C411yescharacteristic
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—41no—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53#4111yescharacteristic
94C2×Dic106—21no—
95C2×Dic106—21no—
96M16×C53—22no—
97M16×C53—41no—
98C2×C8×C53—41no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53C22⋊C411yescharacteristic
102C4⋊C4×C53C22⋊C411yescharacteristic
103C22⋊C4×C53—21no—
104C22⋊C4×C53—21no—
105C22⋊C4×C53—21no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—21no—
109C42×C53—22no—
110C42×C53—22no—
111C42×C53—22no—
112C42×C53—41no—
113C42×C53—41no—
114C42×C53—41no—
115C42×C53C22×C411yescharacteristic
116C42×C53C2×D411yescharacteristic
117C42×C53C2×D411yescharacteristic
118C42×C53#4111yescharacteristic
119C22⋊Dic53—41no—
120#11719925—42no—
121#11719923—42no—
122C4⋊Dic53—21no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126#11719919—41no—
127#11719919—41no—
128C4×Dic53—41no—
129#11719917—42no—
130C2×#88322—21no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—21no—
134C2×#88322—22no—
135C2×#88322—82no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138Q8×C53—21no—
139Q8×C53—21no—
140D4×C53—21no—
141D4×C53—42no—
142D4×C53#10412yes—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12311yescharacteristic
166C2×Dic53—41no—
167C2×Dic53—41no—
168Dic106—41no—
169Dic106—41no—
170Dic106—41no—
171Dic106—41no—
172C8×C53—42no—
173#88322—41no—
174#88322—41no—
175#88322—41no—
176#88322—42no—
177#88322—41no—
178#8971—531nomaximal, Sylow
179C22×C53—21no—
180C22×C53—22no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53#38611yescharacteristic
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—44no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53C2×#10411yescharacteristic
194C4×C53#42111yescharacteristic
195C4×C53#44311yescharacteristic
196C4×C53#44411yescharacteristic
197C4×C53#45511yescharacteristic
198C4×C53#45611yescharacteristic
199Dic53—81no—
200Dic53—81no—
201#3107—531no—
202#2990—531no—
203#1994—531no—
204C4×#104D5311yescharacteristic
205#1485—531no—
206#1299—531no—
207#1276—531no—
208C2×C53—41no—
209C2×C53—42no—
210C2×C53#187111yescharacteristic
211C2×C53#187611yescharacteristic
212C2×C53#188211yescharacteristic
213#533—531no—
214C4×#41C2×D5311yescharacteristic
215#485—1061no—
216#475—531no—
217#474—531no—
218#462—531no—
219#461—531no—
220#452—1061no—
221#446—1061no—
222C4×Q16—1061no—
223C4×SD16—1061no—
224#435—1062no—
225C8×D4—1061no—
226#432—1061no—
227#431—531no—
228#431—531no—
229#423—531no—
230C2×C4⋊C8—1061no—
231C2×#104—21no—
232C4×M16C2×D5311yescharacteristic
233C43C2×D5311yescharacteristic
234#339—21no—
235#339—1062no—
236Q8⋊C8—1061no—
237Q8⋊C8—1061no—
238C53#897111yescharacteristic, complemented, Sylow
239C2×#41C22⋊D5311yescharacteristic
240C2×Q16—1061no—
241C2×SD16—1061no—
242C2×M16—21no—
243C2×M16—532no—
244C22×C8—1062no—
245C4⋊Q8—531no—
246C4⋊Q8—531no—
247#125—1061no—
248C22⋊Q8—1061no—
249C22⋊Q8—1061no—
250C4×Q8—1061no—
251C4×Q8—1061no—
252C4×Q8C4×D5311yescharacteristic
253C4×Q8D21211yescharacteristic
254C4×D4—21no—
255C4×D4—21no—
256C4×D4C4×D5311yescharacteristic
257C4×D4D21211yescharacteristic
258#117—21no—
259#117—1061no—
260#117C22⋊D5311yescharacteristic
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42C22×D5311yescharacteristic
266#107—1061no—
267#106—1061no—
268C4⋊C8—531no—
269C4⋊C8—531no—
270C4⋊C8—531no—
271C4⋊C8—531no—
272C4⋊C8—1062no—
273C4⋊C8—1061no—
274C4⋊C8—1061no—
275#104—24no—
276#103—1061no—
277#103—1061no—
278#103—1061no—
279#102—1061no—
280C22⋊C8—1062no—
281C22⋊C8—1061no—
282C22⋊C8—1061no—
283#97—21no—
284#97—1062no—
285C4×C8—21no—
286C4×C8—1061no—
287C4×C8—1061no—
288#41—22no—
289#41—41no—
290C2×Q8—1061no—
291C2×Q8—1061no—
292C2×Q8#1171992111yescharacteristic
293C2×D4—21no—
294C2×D4#1171992111yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4D4×D5311yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—2122no—
303SD16—2122no—
304M16—22no—
305M16—41no—
306M16—2122no—
307C2×C8—41no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—1061no—
311C2×C8—1061no—
312C2×C8—1062no—
313C2×C8—4242no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C4⋊C4—1061no—
319C4⋊C4—1061no—
320C4⋊C4—2121no—
321C4⋊C4—2121no—
322C4⋊C4#1171992111yescharacteristic
323C4⋊C4#1171992111yescharacteristic
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—2121no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—2121no—
338C42C2×C4×D5311yescharacteristic
339C42C2×D21211yescharacteristic
340C42D4×D5311yescharacteristic
341C42#1171994711yescharacteristic
342C23—21no—
343C23#1313318911yescharacteristic
344Q8—21no—
345Q8—21no—
346Q8—2121no—
347Q8—2121no—
348Q8—2121no—
349Q8—2121no—
350D4—21no—
351D4—42no—
352D4#1313314512yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4C2×#1171992111yescharacteristic
373C2×C4#1313317711yescharacteristic
374C2×C4#1313317711yescharacteristic
375C2×C4#1313318711yescharacteristic
376C2×C4#1313318911yescharacteristic
377C2×C4#1313320111yescharacteristic
378C8—42no—
379C8—2121no—
380C8—2122no—
381C8—2121no—
382C8—2121no—
383C8—2121no—
384C22—21no—
385C22—22no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22#1681050011yescharacteristic
391C4—21no—
392C4—21no—
393C4—21no—
394C4—44no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—4241no—
399C4—4241no—
400C4C2×#1313314511yescharacteristic
401C4#1681064111yescharacteristic
402C4#1681073711yescharacteristic
403C4#1681074211yescharacteristic
404C4#1681080211yescharacteristic
405C4#1681080711yescharacteristic
406C2—41no—
407C2—42no—
408C2#2027501511yescharacteristic
409C2#2027504011yescharacteristic
410C2#2027508211yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC4×#104×C53C2
socleC22×C53#386
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenterC4×#104D53
hypocenterC53#8971
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#104

Complements

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

NH ≅ G/Ntype
3. #20275692C2semidirect
4. #20273137C2semidirect
5. #20272605C2semidirect
6. #20272548C2semidirect
10. #16810908C4semidirect
11. #16810903C4semidirect
12. #16810833C4semidirect
13. #16810828C4semidirect
238. C53#8971semidirect

Maximal subgroups

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

subgroupindex
#202847862
#202838402
#202756922
#202731372
#202726052
#202725482
C4×#104×C532
#897153