Finite Group Databasebeta

← #27280217

Subgroups — #27280217

Order 13568.

450 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#20284775C211yesmaximal, characteristic
2#20273306C211yesmaximal, characteristic, complemented
3#20273167C211yesmaximal, characteristic, complemented
4#20273154C211yesmaximal, characteristic, complemented
5#20272605C211yesmaximal, characteristic, complemented
6#20272552C211yesmaximal, characteristic, complemented
7C53×#2922C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16810947—21no—
10#16810918C411yescharacteristic
11#16810913C411yescharacteristic
12#16810823C411yescharacteristic
13#16810818C411yescharacteristic
14#16810768—21no—
15C4×#11719924—21no—
16C4×#11719922—21no—
17#16810676C2211yescharacteristic
18#16810647C2211yescharacteristic
19C4×#11719917C2211yescharacteristic
20#16810362—22no—
21Q8⋊#88322—21no—
22#16810352—22no—
23D4⋊#88322—21no—
24C4×#41×C53C2211yescharacteristic
25C53×#433—22no—
26C53×#431—22no—
27C53×#424—21no—
28C53×#408—21no—
29C4×M16×C53C2211yescharacteristic
30C2×C4×C8×C53C2211yescharacteristic
31C2×#11719945—21no—
32C2×#11719924—21no—
33C2×#11719922—21no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36#13133217C2×C411yescharacteristic
37#13133214C2×C411yescharacteristic
38#13133205—21no—
39#13133191—21no—
40#13133191—21no—
41C4×D212—21no—
42C4×D212—21no—
43#13133150—21no—
44#13133148—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322C2×C411yescharacteristic
48C4⋊#88322C2×C411yescharacteristic
49#13133141—21no—
50#13133138—21no—
51#13133137—21no—
52#13133137—21no—
53C22⋊#88322—21no—
54#13133128—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57C2×#41×C53D411yescharacteristic
58C2×M16×C53—21no—
59C22×C8×C53—21no—
60C4×Q8×C53C2×C411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53C2×C411yescharacteristic
65C4×D4×C53D411yescharacteristic
66#117×C53—21no—
67#117×C53D411yescharacteristic
68C2×C42×C53—21no—
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—24no—
71C22⋊C8×C53—24no—
72#97×C53D412yes—
73C4×C8×C53—21no—
74C4×C8×C53—21no—
75C4×C8×C53D412yes—
76#41×C53—41no—
77#41×C53—41no—
78C2×Q8×C53C22⋊C411yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53C22⋊C411yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53#4111yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C2×C22⋊D53—41no—
88#11719945—81no—
89#11719945—81no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×C4×D53—41no—
94C2×Dic106—21no—
95M16×C53—41no—
96M16×C53—41no—
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53—41no—
100C2×C8×C53—41no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53C22⋊C411yescharacteristic
104C4⋊C4×C53C22⋊C411yescharacteristic
105C22⋊C4×C53—21no—
106C22⋊C4×C53—21no—
107C22⋊C4×C53—21no—
108C42×C53—21no—
109C42×C53—21no—
110C42×C53—21no—
111C42×C53C22×C411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53C2×D411yescharacteristic
115#11719924—42no—
116#11719922—42no—
117#11719921—41no—
118#11719921—41no—
119C4⋊Dic53—21no—
120C4⋊Dic53—21no—
121#11719917—22no—
122#11719917—41no—
123#11719917—81no—
124C2×#88322—21no—
125C2×#88322—21no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53—21no—
129C23×C53#12311yescharacteristic
130C22×D53—41no—
131Q8×C53—21no—
132Q8×C53—21no—
133D4×C53—21no—
134D4×C53—21no—
135D4×C53—42no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—21no—
145C2×C4×C53—42no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53C2×C22⋊C411yescharacteristic
151C2×C4×C53C4×D411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53#12011yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C22⋊D53—82no—
157C2×Dic53—41no—
158D212—41no—
159D212—41no—
160D212—41no—
161D212—41no—
162D212—81no—
163C4×D53—81no—
164C4×D53—81no—
165Dic106—81no—
166C8×C53—42no—
167C8×C53—42no—
168#88322—42no—
169#88322—41no—
170#88322—41no—
171#88322—81no—
172#9213—531nomaximal, Sylow
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—41no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53#38611yescharacteristic
180C2×D53—81no—
181C2×D53—81no—
182C2×D53—81no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53#42111yescharacteristic
190C4×C53#42111yescharacteristic
191C4×C53#44311yescharacteristic
192C4×C53#44411yescharacteristic
193C4×C53#47111yescharacteristic
194C4×C53#47211yescharacteristic
195Dic53—81no—
196#3106—531no—
197#2922D5311yescharacteristic, complemented
198#1525—531no—
199#1492—531no—
200#1489—531no—
201#1299—531no—
202#1278—531no—
203C2×C53—41no—
204C2×C53—81no—
205C2×C53#187611yescharacteristic
206C2×C53#189711yescharacteristic
207C2×C53#190711yescharacteristic
208D53—161no—
209#532—531no—
210C4×#41C2×D5311yescharacteristic
211#482—1061no—
212#477—531no—
213#476—531no—
214#460—531no—
215#459—531no—
216#449—1061no—
217C4×SD16—1061no—
218C4×D8—1061no—
219#433—22no—
220#431—22no—
221#431—531no—
222#424—21no—
223#423—531no—
224#408—21no—
225C4×M16—531no—
226C4×M16C2×D5311yescharacteristic
227C2×C4×C8C2×D5311yescharacteristic
228#329—1062no—
229Q8⋊C8—1061no—
230D4⋊C8—1061no—
231D4⋊C8—1062no—
232C53#921311yescharacteristic, complemented, Sylow
233C2×#41—1061no—
234C2×#41C22⋊D5311yescharacteristic
235C2×SD16—1061no—
236C2×D8—1061no—
237C2×M16—21no—
238C2×M16—1061no—
239C2×M16—1061no—
240C22×C8—21no—
241C4⋊Q8—531no—
242#127—531no—
243#124—1061no—
244#121—1061no—
245#121—1061no—
246C4×Q8C4×D5311yescharacteristic
247C4×Q8D21211yescharacteristic
248C4×D4—21no—
249C4×D4—21no—
250C4×D4—1061no—
251C4×D4—1061no—
252C4×D4C4×D5311yescharacteristic
253C4×D4D21211yescharacteristic
254#117—21no—
255#117C22⋊D5311yescharacteristic
256C2×C42—21no—
257C2×C42C22×D5311yescharacteristic
258#107—1061no—
259#106—1061no—
260C4⋊C8—24no—
261C4⋊C8—531no—
262C4⋊C8—531no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265#103—1061no—
266#102—1061no—
267#102—1061no—
268#102—1061no—
269C22⋊C8—24no—
270C22⋊C8—1061no—
271#97—1061no—
272#97C22⋊D5312yes—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—1061no—
276C4×C8—1061no—
277C4×C8C22⋊D5312yes—
278#41—41no—
279#41—41no—
280#41—4241no—
281#41—4241no—
282C2×Q8—1061no—
283C2×Q8#1171992111yescharacteristic
284C2×D4—21no—
285C2×D4—1061no—
286C2×D4—1061no—
287C2×D4—1061no—
288C2×D4—2121no—
289C2×D4#1171992111yescharacteristic
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—2121no—
294C22×C4#1171994511yescharacteristic
295C22×C4#1171994511yescharacteristic
296C22×C4C2×C22⋊D5311yescharacteristic
297SD16—2122no—
298D8—2122no—
299M16—41no—
300M16—41no—
301M16—1062no—
302M16—4241no—
303M16—2121no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—1061no—
309C2×C8—1061no—
310C2×C8—2121no—
311C2×C8—2121no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—1061no—
315C4⋊C4—1061no—
316C4⋊C4#1171992111yescharacteristic
317C4⋊C4#1171992111yescharacteristic
318C22⋊C4—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—2121no—
322C22⋊C4—2121no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42C2×C4×D5311yescharacteristic
327C42C2×D21211yescharacteristic
328C42C2×C22⋊D5311yescharacteristic
329C42C2×C22⋊D5311yescharacteristic
330C23—21no—
331C23—2121no—
332C23#1313320011yescharacteristic
333Q8—21no—
334Q8—21no—
335Q8—4241no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—4241no—
340D4—4242no—
341D4—2121no—
342D4—2121no—
343D4—2121no—
344D4—2121no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
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#1313318611yescharacteristic
366C2×C4#1313319111yescharacteristic
367C2×C4#1313319111yescharacteristic
368C8—42no—
369C8—42no—
370C8—4241no—
371C8—2122no—
372C8—2121no—
373C8—2121no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—4241no—
381C22—4241no—
382C22—4241no—
383C22#1681050211yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—41no—
388C4—41no—
389C4—41no—
390C4—4241no—
391C4#1681064111yescharacteristic
392C4#1681064111yescharacteristic
393C4#1681073811yescharacteristic
394C4#1681074311yescharacteristic
395C4#1681088611yescharacteristic
396C4#1681089311yescharacteristic
397C2—41no—
398C2—81no—
399C2—8481no—
400C2#2027504211yescharacteristic
401C2#2027516311yescharacteristic
402C2#2027522311yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2922C2
socleC22×C53#386
prime radical#20284775C2
cosocleC2×C42×C53C23
hypercenter#2922D53
hypocenterC53#9213
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2922

Complements

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

NH ≅ G/Ntype
2. #20273306C2semidirect
3. #20273167C2semidirect
4. #20273154C2semidirect
5. #20272605C2semidirect
6. #20272552C2semidirect
7. C53×#2922C2semidirect
197. #2922D53semidirect
232. C53#9213semidirect

Maximal subgroups

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

subgroupindex
#202847752
#202733062
#202731672
#202731542
#202726052
#202725522
C53×#29222
#921353