Finite Group Databasebeta

← #27369258

Subgroups — #27369258

Order 13568.

708 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#20285640C211yesmaximal, characteristic, complemented
2#20284179C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic, complemented
4#20274136C214yesmaximal, complemented
5#20274058C214yesmaximal, complemented
6#20273485C211yesmaximal, characteristic, complemented
7#20273400C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic, complemented
9C2×C53×#474C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11#16811173C2211yescharacteristic
12#16810997C2212yescomplemented
13#16810973C2212yescomplemented
14#16810838—22no—
15#16810828C2212yes—
16#16810825—22no—
17#16810818C2212yes—
18#16810762—22no—
19D4×#88322—22no—
20D4×#88322C2214yes—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810622—21no—
25C2×C22⋊#88322—22no—
26#16810373C2214yescomplemented
27#16810370C2212yescomplemented
28#16810369C2214yescomplemented
29#16810366C2212yescomplemented
30C2×C4⋊Q8×C53C2211yescharacteristic
31C2×C4×D4×C53C2211yescharacteristic
32C53×#474—22no—
33C53×#474C2214yes—
34C2×#107×C53—21no—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#102×C53—21no—
37C2×#11719945—21no—
38#13133243—44no—
39C2×#11719923—22no—
40C2×#11719924—24no—
41C2×#11719922—22no—
42C2×#11719917—21no—
43C22×#88322—24no—
44#13133217C2311yescharacteristic
45#13133214C2311yescharacteristic
46#13133205—21no—
47#13133191—21no—
48#13133191—21no—
49C4×D212—21no—
50C4×D212—21no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133141—24no—
56#13133140—22no—
57#13133137—22no—
58C22⋊#88322—24no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61C4×#88322—21no—
62C4×#88322C2312yes—
63C22×Q8×C53—21no—
64C22×D4×C53D411yescharacteristic
65C23×C4×C53—21no—
66C22×C8×C53—21no—
67C4⋊Q8×C53—21no—
68C4⋊Q8×C53—21no—
69C4⋊Q8×C53—21no—
70C4⋊Q8×C53C2312yes—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53D412yes—
74C4×D4×C53C2312yes—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53—21no—
77C2×C4⋊C4×C53D411yescharacteristic
78C2×C22⋊C4×C53—21no—
79C2×C42×C53C2311yescharacteristic
80#107×C53—24no—
81C4⋊C8×C53—21no—
82C4⋊C8×C53C2312yes—
83#102×C53—24no—
84C24×C53—21no—
85C2×Q8×C53—22no—
86C2×Q8×C53—41no—
87C2×Q8×C53—82no—
88C2×D4×C53—21no—
89C2×D4×C53—22no—
90C2×D4×C53SD1614yescomplemented
91C2×D4×C53C2×D412yes—
92C22×C4×C53—21no—
93C22×C4×C53—22no—
94C22×C4×C53—22no—
95C22×C4×C53—41no—
96C22×C4×C53—82no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C2×C22⋊D53—41no—
101#11719945—42no—
102#11719945—42no—
103C2×D212—21no—
104C2×D212—21no—
105C2×D212—21no—
106C2×C4×D53—41no—
107C2×Dic106—21no—
108C2×C8×C53—22no—
109C2×C8×C53—42no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—41no—
114C4⋊C4×C53—41no—
115C4⋊C4×C53C2×D412yes—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122#11719924—84no—
123#11719923—44no—
124#11719922—44no—
125#11719921—41no—
126#11719921—41no—
127C4⋊Dic53—21no—
128C4⋊Dic53—21no—
129#11719917—42no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—84no—
133C2×#88322C2×D414yes—
134C23×C53—22no—
135C23×C53—24no—
136C23×C53—81no—
137C23×C53—81no—
138C23×C53#12111yescharacteristic
139C22×D53—41no—
140Q8×C53—44no—
141Q8×C53—82no—
142D4×C53—22no—
143D4×C53—24no—
144D4×C53C2×SD1614yes—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—44no—
149C2×C4×C53—42no—
150C2×C4×C53—82no—
151C2×C4×C53—81no—
152C2×C4×C53—82no—
153C2×C4×C53#12111yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53C2×SD1612yes—
157C2×C4×C53#13512yes—
158C2×C4×C53C22×D411yescharacteristic
159C2×C4×C53C22×D411yescharacteristic
160C2×C4×C53C2×#4111yescharacteristic
161C22⋊D53—82no—
162C2×Dic53—41no—
163D212—41no—
164D212—41no—
165D212—42no—
166D212—81no—
167C4×D53—82no—
168Dic106—41no—
169Dic106—41no—
170C8×C53—42no—
171#88322—24no—
172#88322—42no—
173#88322—42no—
174#19385—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—24no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—84no—
181C22×C53#46512yes—
182C22×C53C2×#12111yescharacteristic
183C2×D53—81no—
184C2×D53—81no—
185C2×D53—81no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—42no—
190C4×C53—82no—
191C4×C53#46512yes—
192C4×C53C22×SD1611yescharacteristic
193C4×C53C2×#13511yescharacteristic
194C4×C53#57612yes—
195Dic53—81no—
196#3206—531no—
197C2×#474D5311yescharacteristic, complemented
198#3036—531no—
199#2972—531no—
200#1679—534no—
201#1665—534no—
202#1570—531no—
203#1549—531no—
204#1540—531no—
205C2×C53—22no—
206C2×C53—44no—
207C2×C53#304111yescharacteristic
208C2×C53C2×#46511yescharacteristic
209C2×C53#306911yescharacteristic
210D53—161no—
211C2×AGL(1,ℤ/8)—1061no—
212#532—531no—
213C2×C4⋊Q8C2×D5311yescharacteristic
214C2×C4×D4C2×D5311yescharacteristic
215#492—532no—
216#487—532no—
217#474—22no—
218#474C2×D5314yes—
219#463—1062no—
220#461—532no—
221#461—1062no—
222#459—532no—
223#448—1062no—
224C8×D4—534no—
225C8×D4—1062no—
226#431—531no—
227#431—531no—
228C2×#107—21no—
229#423—531no—
230C2×C4⋊C8C2×D5311yescharacteristic
231#417—1061no—
232C2×#102—21no—
233C2×C22⋊C8—1062no—
234#334—534no—
235#332—534no—
236#332—532no—
237#331—532no—
238C53#1938511yescharacteristic, complemented, Sylow
239C2×#41—1061no—
240C22×Q8—21no—
241C22×D4C22⋊D5311yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—2124no—
244C2×SD16—1064no—
245C2×SD16—1062no—
246C2×D8—1062no—
247C2×M16—1061no—
248C22×C8—21no—
249C22×C8—1064no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—531no—
254C4⋊Q8C22×D5312yes—
255#127—531no—
256#124—1061no—
257#121—1061no—
258#121—1061no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—1061no—
262C4×D4—1061no—
263C4×D4C22⋊D5312yes—
264C4×D4C22×D5312yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4C22⋊D5311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C22×D5311yescharacteristic
270#107—24no—
271C4⋊C8—21no—
272C4⋊C8—532no—
273C4⋊C8—532no—
274C4⋊C8—532no—
275C4⋊C8—1061no—
276C4⋊C8C22×D5312yes—
277#103—1064no—
278#103—1062no—
279#102—24no—
280#102—1062no—
281C22⋊C8—532no—
282C22⋊C8—1064no—
283C22⋊C8—1061no—
284C4×C8—532no—
285C4×C8—1061no—
286C24—21no—
287#41—2122no—
288#41—2122no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×Q8—1061no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—1061no—
296C2×D4—1061no—
297C2×D4—1061no—
298C2×D4—2121no—
299C2×D4#1171992414yescomplemented
300C2×D4C2×C22⋊D5312yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—82no—
306C22×C4—2121no—
307C22×C4D4×D5311yescharacteristic
308C22×C4#1171994711yescharacteristic
309C22×C4C2×C22⋊D5311yescharacteristic
310SD16—4244no—
311SD16—2124no—
312D8—2124no—
313M16—2122no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—534no—
317C2×C8—1062no—
318C2×C8—1062no—
319C2×C8—4244no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—1061no—
326C4⋊C4—1061no—
327C4⋊C4C2×C22⋊D5312yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—2121no—
331C22⋊C4—2121no—
332C42D4×D5311yescharacteristic
333C42#1171994711yescharacteristic
334C42C2×C22⋊D5311yescharacteristic
335C42C23×D5311yescharacteristic
336C23—22no—
337C23—24no—
338C23—81no—
339C23—81no—
340C23—2121no—
341C23#1313318911yescharacteristic
342Q8—44no—
343Q8—82no—
344Q8—2121no—
345Q8—2121no—
346D4—24no—
347D4—22no—
348D4—4241no—
349D4—4242no—
350D4—2121no—
351D4—2121no—
352D4—2122no—
353D4C2×#1171992414yes—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4242no—
363C2×C4—2121no—
364C2×C4#1313318911yescharacteristic
365C2×C4#1313318911yescharacteristic
366C2×C4#1313318911yescharacteristic
367C2×C4C2×#1171992412yes—
368C2×C4#1313323712yes—
369C2×C4C2×D4×D5311yescharacteristic
370C2×C4C22×C22⋊D5311yescharacteristic
371C2×C4C2×#1171994711yescharacteristic
372C8—42no—
373C8—1064no—
374C8—2122no—
375C8—2122no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—4241no—
383C22—4241no—
384C22—4241no—
385C22#1681085412yes—
386C22C2×#1313318911yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—82no—
392C4—4241no—
393C4#1681085412yes—
394C4C22×#1171992411yescharacteristic
395C4C2×#1313323711yescharacteristic
396C4#1681142412yes—
397C2—22no—
398C2—44no—
399C2—8481no—
400C2#2028421511yescharacteristic
401C2C2×#1681085411yescharacteristic
402C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#474C2
socleC22×C53C2×#121
prime radical#20284179C2
hypercenterC2×#474D53
hypocenterC53#19385
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #20285640C2semidirect
3. #20283739C2semidirect
4. #20274136C2semidirect
5. #20274058C2semidirect
6. #20273485C2semidirect
7. #20273400C2semidirect
8. #20273365C2semidirect
9. C2×C53×#474C2semidirect
12. #16810997C22semidirect
13. #16810973C22semidirect
26. #16810373C22semidirect
27. #16810370C22semidirect
28. #16810369C22semidirect
29. #16810366C22semidirect
90. C2×D4×C53SD16semidirect
197. C2×#474D53semidirect
238. C53#19385semidirect
299. C2×D4#11719924semidirect

Maximal subgroups

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

subgroupindex
#202856402
#202841792
#202837392
#202741362
#202740582
#202734852
#202734002
#202733652
C2×C53×#4742
#1938553