Finite Group Databasebeta

← #27280175

Subgroups — #27280175

Order 13568.

478 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284503C211yesmaximal, characteristic
2#20274735C211yesmaximal, characteristic
3#20273215C211yesmaximal, characteristic, complemented
4#20273171C211yesmaximal, characteristic, complemented
5#20272611C211yesmaximal, characteristic, complemented
6#20272555C211yesmaximal, characteristic, complemented
7C8×#41×C53C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810930—21no—
10#16810923—21no—
11#16810918C411yescharacteristic, complemented
12#16810913C411yescharacteristic, complemented
13#16810908C411yescharacteristic, complemented
14#16810903C411yescharacteristic, complemented
15#16810659—21no—
16#16810647C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133144—21no—
19#16810619—21no—
20C4×C4⋊Dic53C2211yescharacteristic
21#16810382—21no—
22#16810361—22no—
23#16810355—21no—
24#16810355—21no—
25C4×#41×C53C2211yescharacteristic
26C8×Q8×C53—21no—
27C8×D4×C53—21no—
28C8×D4×C53—22no—
29C53×#431—21no—
30C53×#431C2211yescharacteristic
31C2×C4×C8×C53—21no—
32C2×C4×C8×C53C2211yescharacteristic
33C2×#11719917—21no—
34C22×#88322—21no—
35#13133217C2×C411yescharacteristic
36#13133217C2×C411yescharacteristic
37#13133209—21no—
38#13133196—21no—
39#13133196—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42#13133176—21no—
43C2×C4⋊Dic53—21no—
44C2×C4×Dic53—21no—
45#13133150—21no—
46#13133150—21no—
47#13133148—21no—
48#13133148—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322—21no—
52C4⋊#88322C2×C411yescharacteristic
53C4⋊#88322C2×C411yescharacteristic
54#13133144—42no—
55#13133141—21no—
56#13133141—21no—
57#13133138—21no—
58#13133138—21no—
59#13133124—21no—
60C2×#41×C53D411yescharacteristic
61C22×C8×C53—22no—
62C22×C8×C53—21no—
63C4×Q8×C53C2×C411yescharacteristic
64C4×Q8×C53D411yescharacteristic
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69#117×C53—21no—
70#117×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73C4⋊C8×C53—22no—
74C4⋊C8×C53—21no—
75C22⋊C8×C53—22no—
76C22⋊C8×C53D412yes—
77C4×C8×C53—22no—
78C4×C8×C53—21no—
79C4×C8×C53—21no—
80C4×C8×C53D411yescharacteristic
81C4×C8×C53D411yescharacteristic
82#41×C53—21no—
83#41×C53—21no—
84#41×C53SD1612yes—
85#41×C53Q1612yes—
86C2×Q8×C53C22⋊C411yescharacteristic
87C2×D4×C53—21no—
88C2×D4×C53C22⋊C411yescharacteristic
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D411yescharacteristic
94C22×C4×C53#4111yescharacteristic
95C22×Dic53—41no—
96C2×Dic106—21no—
97C2×Dic106—21no—
98C2×C8×C53—21no—
99C2×C8×C53—22no—
100C2×C8×C53—21no—
101C2×C8×C53—42no—
102C2×C8×C53—44no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53C22⋊C411yescharacteristic
106C4⋊C4×C53C22⋊C411yescharacteristic
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53—21no—
110C42×C53—21no—
111C42×C53—21no—
112C42×C53—21no—
113C42×C53C22×C411yescharacteristic
114C42×C53C2×D411yescharacteristic
115C42×C53C2×D411yescharacteristic
116C42×C53#4111yescharacteristic
117C22⋊Dic53—41no—
118C4⋊Dic53—22no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121C4⋊Dic53—21no—
122C4⋊Dic53—21no—
123C4⋊Dic53—41no—
124#11719919—41no—
125#11719919—41no—
126C4×Dic53—41no—
127C4×Dic53—41no—
128C4×Dic53—41no—
129#11719917—41no—
130#11719917—41no—
131C2×#88322—21no—
132C2×#88322—21no—
133C2×#88322—41no—
134C2×#88322—41no—
135C2×#88322—41no—
136C23×C53—21no—
137C23×C53#12111yescharacteristic
138Q8×C53—21no—
139Q8×C53#10312yes—
140D4×C53—22no—
141D4×C53—22no—
142D4×C53—21no—
143D4×C53#10312yes—
144C2×C4×C53—21no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—22no—
148C2×C4×C53—22no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—41no—
154C2×C4×C53—41no—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C4×D411yescharacteristic
157C2×C4×C53C4×D411yescharacteristic
158C2×C4×C53#12011yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53#12311yescharacteristic
161C2×C4×C53C2×SD1611yescharacteristic
162C2×C4×C53C2×Q1611yescharacteristic
163C2×C4×C53#13511yescharacteristic
164C2×C4×C53#13511yescharacteristic
165C2×Dic53—42no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—81no—
169C2×Dic53—81no—
170Dic106—41no—
171Dic106—41no—
172Dic106—81no—
173C8×C53—22no—
174C8×C53—42no—
175C8×C53—41no—
176#88322—41no—
177#88322—41no—
178#88322—81no—
179#9207—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—22no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53#38611yescharacteristic
186C22×C53#46511yescharacteristic
187C22×C53#46711yescharacteristic
188C4×C53—21no—
189C4×C53—22no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—41no—
194C4×C53C2×#10311yescharacteristic
195C4×C53#41611yescharacteristic
196C4×C53#44311yescharacteristic
197C4×C53#44411yescharacteristic
198C4×C53#46511yescharacteristic
199C4×C53#46711yescharacteristic
200Dic53—82no—
201Dic53—81no—
202Dic53—81no—
203#3076—531no—
204C8×#41D5311yescharacteristic
205#1806—531no—
206#1503—531no—
207#1493—531no—
208#1301—531no—
209#1279—531no—
210C2×C53—21no—
211C2×C53—21no—
212C2×C53—42no—
213C2×C53#187611yescharacteristic
214C2×C53#189211yescharacteristic
215C2×C53#190611yescharacteristic
216#533—531no—
217C4×#41C2×D5311yescharacteristic
218#479—1061no—
219#478—1061no—
220#477—531no—
221#476—531no—
222#475—531no—
223#474—531no—
224C8×Q8—21no—
225C8×D4—21no—
226C8×D4—22no—
227#431—21no—
228#431C2×D5311yescharacteristic
229#427—1061no—
230#423—531no—
231C2×C4⋊C8—531no—
232C2×#104—1061no—
233#416—1061no—
234C2×C4×C8—21no—
235C2×C4×C8C2×D5311yescharacteristic
236C4×C4⋊C4—531no—
237#339—1061no—
238#328—1062no—
239Q8⋊C8—1061no—
240Q8⋊C8—1061no—
241C53#920711yescharacteristic, complemented, Sylow
242C2×#41D21211yescharacteristic
243C2×M16—1061no—
244C22×C8—22no—
245C22×C8—21no—
246C22×C8—1061no—
247C4⋊Q8—531no—
248C4⋊Q8—531no—
249#125—1061no—
250C22⋊Q8—1061no—
251C22⋊Q8—1061no—
252C4×Q8—1061no—
253C4×Q8—1061no—
254C4×Q8C4×D5311yescharacteristic
255C4×Q8C22⋊D5311yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C4×D5311yescharacteristic
259C4×D4C22⋊D5311yescharacteristic
260#117—21no—
261#117—1061no—
262#117D21211yescharacteristic
263C2×C4⋊C4—1061no—
264C2×C42—21no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267#107—1061no—
268#107—1061no—
269#106—1061no—
270#106—1061no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—531no—
274C4⋊C8—531no—
275C4⋊C8—1061no—
276C4⋊C8—1061no—
277C4⋊C8—1061no—
278#104—2122no—
279#103—1061no—
280#103—1061no—
281#102—1061no—
282#102—1061no—
283C22⋊C8—22no—
284C22⋊C8C22⋊D5312yes—
285C4×C8—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8C22⋊D5311yescharacteristic
289C4×C8C22⋊D5311yescharacteristic
290#95—1061no—
291#41—21no—
292#41—21no—
293#41#1171991312yes—
294#41Dic21212yes—
295C2×Q8—1061no—
296C2×Q8—1061no—
297C2×Q8#1171992111yescharacteristic
298C2×D4—21no—
299C2×D4#1171992111yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—2121no—
304C22×C4C2×D21211yescharacteristic
305C22×C4#1171994511yescharacteristic
306C22×C4C2×C22⋊D5311yescharacteristic
307M16—2121no—
308M16—2121no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—42no—
313C2×C8—44no—
314C2×C8—1061no—
315C2×C8—1061no—
316C2×C8—2121no—
317C2×C8—2121no—
318C2×C8—2121no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—1062no—
322C4⋊C4—1061no—
323C4⋊C4—1061no—
324C4⋊C4—1061no—
325C4⋊C4—1061no—
326C4⋊C4—2121no—
327C4⋊C4—2121no—
328C4⋊C4—2121no—
329C4⋊C4#1171992111yescharacteristic
330C4⋊C4#1171992111yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—2121no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—2121no—
339C42—2121no—
340C42—2121no—
341C42C2×C4×D5311yescharacteristic
342C42#1171994511yescharacteristic
343C42C2×C22⋊D5311yescharacteristic
344C42C2×C22⋊D5311yescharacteristic
345C23—21no—
346C23#1313319111yescharacteristic
347Q8—21no—
348Q8—4241no—
349Q8—2121no—
350Q8—2121no—
351Q8#1313314212yes—
352D4—22no—
353D4—22no—
354D4—21no—
355D4#1313314212yes—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—4241no—
368C2×C4—4241no—
369C2×C4—2122no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4C2×#1171992111yescharacteristic
373C2×C4C4×C22⋊D5311yescharacteristic
374C2×C4C4×C22⋊D5311yescharacteristic
375C2×C4#1313318611yescharacteristic
376C2×C4#1313319111yescharacteristic
377C2×C4#1313320011yescharacteristic
378C2×C4C2×#1171991311yescharacteristic
379C2×C4C2×Dic21211yescharacteristic
380C2×C4#1313323911yescharacteristic
381C2×C4#1313323911yescharacteristic
382C8—22no—
383C8—42no—
384C8—41no—
385C8—4241no—
386C8—2121no—
387C8—2121no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22#1681050211yescharacteristic
394C22#1681085211yescharacteristic
395C22#1681086411yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—4242no—
403C4—4241no—
404C4—4241no—
405C4C2×#1313314211yescharacteristic
406C4#1681062011yescharacteristic
407C4#1681073811yescharacteristic
408C4#1681074311yescharacteristic
409C4#1681085211yescharacteristic
410C4#1681086411yescharacteristic
411C2—21no—
412C2—21no—
413C2—42no—
414C2#2027504211yescharacteristic
415C2#2027513811yescharacteristic
416C2#2027521611yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC8×#41×C53C2
socleC22×C53#386
prime radicalC2×#41×C53D4
cosocleC2×C42×C53C23
hypercenterC8×#41D53
hypocenterC53#9207
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C8×#41

Complements

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

NH ≅ G/Ntype
3. #20273215C2semidirect
4. #20273171C2semidirect
5. #20272611C2semidirect
6. #20272555C2semidirect
11. #16810918C4semidirect
12. #16810913C4semidirect
13. #16810908C4semidirect
14. #16810903C4semidirect
241. C53#9207semidirect

Maximal subgroups

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

subgroupindex
#202845032
#202747352
#202732152
#202731712
#202726112
#202725552
C8×#41×C532
#920753