Finite Group Databasebeta

← #27369325

Subgroups — #27369325

Order 13568.

728 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285836C211yesmaximal, characteristic, complemented
2#20284179C211yesmaximal, characteristic
3#20283739C211yesmaximal, characteristic, complemented
4#20274154C214yesmaximal, complemented
5#20274102C214yesmaximal, complemented
6#20273489C211yesmaximal, characteristic, complemented
7#20273387C211yesmaximal, characteristic, complemented
8#20273365C211yesmaximal, characteristic, complemented
9C2×C53×#461C211yesmaximal, characteristic, complemented
10C2×#13133243—21no—
11#16811173C2211yescharacteristic
12#16811026C2212yescomplemented
13#16810971C2212yescomplemented
14#16810936—22no—
15#16810900—22no—
16#16810838—22no—
17#16810828C2212yes—
18#16810818C2212yes—
19D4×#88322—22no—
20D4×#88322C2214yes—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23#16810647C2211yescharacteristic
24#16810622—21no—
25C2×C22⋊#88322—22no—
26#16810375C2214yescomplemented
27#16810370C2212yescomplemented
28#16810369C2214yescomplemented
29#16810366C2212yescomplemented
30C22×SD16×C53—21no—
31C2×C4⋊Q8×C53C2211yescharacteristic
32C2×C4×D4×C53C2211yescharacteristic
33C53×#461—22no—
34C53×#461C2214yes—
35C2×C4⋊C8×C53C2211yescharacteristic
36C2×#103×C53—21no—
37C2×#11719945—21no—
38#13133243—44no—
39C2×#11719923—22no—
40C2×#11719922—22no—
41C2×#11719917—21no—
42C22×#88322—24no—
43#13133217C2311yescharacteristic
44#13133214C2311yescharacteristic
45#13133205—21no—
46#13133191—21no—
47#13133191—21no—
48C4×D212—21no—
49C4×D212—21no—
50#13133150—24no—
51C4⋊#88322—21no—
52C4⋊#88322D412yes—
53C4⋊#88322C2312yes—
54C4⋊#88322C2312yes—
55#13133140—22no—
56#13133137—22no—
57#13133137—24no—
58C22⋊#88322—24no—
59C22⋊#88322—21no—
60C22⋊#88322D412yes—
61C4×#88322—21no—
62C4×#88322C2312yes—
63C22×Q8×C53—21no—
64C22×D4×C53D411yescharacteristic
65C23×C4×C53—21no—
66C2×SD16×C53—24no—
67C2×SD16×C53—44no—
68C22×C8×C53—21no—
69C4⋊Q8×C53—21no—
70C4⋊Q8×C53—21no—
71C4⋊Q8×C53—21no—
72C4⋊Q8×C53C2312yes—
73C4×D4×C53—21no—
74C4×D4×C53—21no—
75C4×D4×C53D412yes—
76C4×D4×C53C2312yes—
77C2×C4⋊C4×C53—21no—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53D411yescharacteristic
80C2×C22⋊C4×C53—21no—
81C2×C42×C53C2311yescharacteristic
82C4⋊C8×C53—21no—
83C4⋊C8×C53C2312yes—
84#103×C53—24no—
85C24×C53—21no—
86C2×Q8×C53—22no—
87C2×Q8×C53—41no—
88C2×Q8×C53—42no—
89C2×Q8×C53—42no—
90C2×D4×C53—21no—
91C2×D4×C53—22no—
92C2×D4×C53D814yes—
93C2×D4×C53C2×D412yes—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—41no—
98C22×C4×C53—82no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53#4111yescharacteristic
102C2×C22⋊D53—41no—
103#11719945—42no—
104#11719945—42no—
105C2×D212—21no—
106C2×D212—21no—
107C2×D212—21no—
108C2×C4×D53—41no—
109C2×Dic106—21no—
110SD16×C53—48no—
111C2×C8×C53—22no—
112C2×C8×C53—42no—
113C4⋊C4×C53—21no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C4⋊C4×C53C2×D412yes—
119C22⋊C4×C53—22no—
120C22⋊C4×C53—22no—
121C42×C53C2×D411yescharacteristic
122C42×C53C2×D411yescharacteristic
123C42×C53#4111yescharacteristic
124C42×C53C2411yescharacteristic
125#11719923—44no—
126#11719922—44no—
127#11719921—41no—
128#11719921—41no—
129C4⋊Dic53—21no—
130C4⋊Dic53—21no—
131#11719917—42no—
132C2×#88322—22no—
133C2×#88322—22no—
134C2×#88322—84no—
135C2×#88322C2×D414yes—
136C23×C53—22no—
137C23×C53—24no—
138C23×C53—81no—
139C23×C53—81no—
140C23×C53#12111yescharacteristic
141C22×D53—41no—
142Q8×C53—42no—
143Q8×C53—42no—
144Q8×C53—82no—
145D4×C53—22no—
146D4×C53—24no—
147D4×C53C2×D814yes—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—44no—
152C2×C4×C53—42no—
153C2×C4×C53—82no—
154C2×C4×C53—81no—
155C2×C4×C53—82no—
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53C2×D812yes—
160C2×C4×C53#13512yes—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C2×#4111yescharacteristic
164C22⋊D53—82no—
165C2×Dic53—41no—
166D212—41no—
167D212—41no—
168D212—42no—
169D212—81no—
170C4×D53—82no—
171Dic106—41no—
172Dic106—41no—
173C8×C53—42no—
174#88322—24no—
175#88322—42no—
176#88322—42no—
177#19392—531nomaximal, Sylow
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—24no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—84no—
184C22×C53#46612yes—
185C22×C53C2×#12111yescharacteristic
186C2×D53—81no—
187C2×D53—81no—
188C2×D53—81no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—82no—
194C4×C53#46612yes—
195C4×C53C22×D811yescharacteristic
196C4×C53C2×#13511yescharacteristic
197C4×C53#57712yes—
198Dic53—81no—
199#3226—531no—
200#3036—531no—
201C2×#461D5311yescharacteristic, complemented
202#2972—531no—
203#1681—534no—
204#1673—534no—
205#1571—531no—
206#1546—531no—
207#1540—531no—
208C2×C53—22no—
209C2×C53—44no—
210C2×C53#304211yescharacteristic
211C2×C53C2×#46611yescharacteristic
212C2×C53#306911yescharacteristic
213D53—161no—
214C2×AGL(1,ℤ/8)—1061no—
215C22×SD16—21no—
216#532—531no—
217C2×C4⋊Q8C2×D5311yescharacteristic
218C2×C4×D4C2×D5311yescharacteristic
219#500—532no—
220#486—532no—
221#480—1062no—
222#474—1062no—
223#463—1062no—
224#461—22no—
225#461—532no—
226#461C2×D5314yes—
227#459—532no—
228C8×D4—534no—
229C8×D4—1062no—
230#431—531no—
231#431—531no—
232#423—531no—
233C2×C4⋊C8C2×D5311yescharacteristic
234#417—1061no—
235C2×#103—21no—
236C2×C22⋊C8—1062no—
237#335—534no—
238#332—534no—
239#332—532no—
240#331—532no—
241C53#1939211yescharacteristic, complemented, Sylow
242C2×#41—1061no—
243C22×Q8—21no—
244C22×D4C22⋊D5311yescharacteristic
245C23×C4—21no—
246AGL(1,ℤ/8)—2124no—
247C2×SD16—24no—
248C2×SD16—44no—
249C2×SD16—1062no—
250C2×D8—1062no—
251C2×M16—1061no—
252C22×C8—21no—
253C22×C8—1064no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8—21no—
257C4⋊Q8—531no—
258C4⋊Q8C22×D5312yes—
259#127—531no—
260#124—1061no—
261#121—1061no—
262#121—1061no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—1061no—
266C4×D4—1061no—
267C4×D4C22⋊D5312yes—
268C4×D4C22×D5312yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4C22⋊D5311yescharacteristic
272C2×C22⋊C4—21no—
273C2×C42C22×D5311yescharacteristic
274#107—1064no—
275C4⋊C8—21no—
276C4⋊C8—532no—
277C4⋊C8—532no—
278C4⋊C8—532no—
279C4⋊C8—1061no—
280C4⋊C8C22×D5312yes—
281#103—24no—
282#103—1062no—
283#102—1064no—
284#102—1062no—
285C22⋊C8—532no—
286C22⋊C8—1064no—
287C22⋊C8—1061no—
288C4×C8—532no—
289C4×C8—1061no—
290C24—21no—
291#41—2122no—
292#41—2122no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—1061no—
298C2×D4—21no—
299C2×D4—22no—
300C2×D4—1061no—
301C2×D4—1061no—
302C2×D4—1061no—
303C2×D4—2121no—
304C2×D4#1171992214yes—
305C2×D4C2×C22⋊D5312yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4—2121no—
312C22×C4D4×D5311yescharacteristic
313C22×C4#1171994711yescharacteristic
314C22×C4C2×C22⋊D5311yescharacteristic
315SD16—48no—
316SD16—2124no—
317D8—2124no—
318M16—2122no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—534no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8—4244no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4C2×C22⋊D5312yes—
333C22⋊C4—22no—
334C22⋊C4—22no—
335C22⋊C4—2121no—
336C22⋊C4—2121no—
337C42D4×D5311yescharacteristic
338C42#1171994711yescharacteristic
339C42C2×C22⋊D5311yescharacteristic
340C42C23×D5311yescharacteristic
341C23—22no—
342C23—24no—
343C23—81no—
344C23—81no—
345C23—2121no—
346C23#1313318911yescharacteristic
347Q8—42no—
348Q8—42no—
349Q8—82no—
350Q8—2121no—
351Q8—2121no—
352D4—24no—
353D4—22no—
354D4—4241no—
355D4—4242no—
356D4—2121no—
357D4—2121no—
358D4—2122no—
359D4C2×#1171992214yes—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—4242no—
369C2×C4—2121no—
370C2×C4#1313318911yescharacteristic
371C2×C4#1313318911yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4C2×#1171992212yes—
374C2×C4#1313323712yes—
375C2×C4C2×D4×D5311yescharacteristic
376C2×C4C22×C22⋊D5311yescharacteristic
377C2×C4C2×#1171994711yescharacteristic
378C8—42no—
379C8—1064no—
380C8—2122no—
381C8—2122no—
382C22—21no—
383C22—21no—
384C22—24no—
385C22—42no—
386C22—42no—
387C22—84no—
388C22—4241no—
389C22—4241no—
390C22—4241no—
391C22#1681085712yes—
392C22C2×#1313318911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—82no—
398C4—4241no—
399C4#1681085712yes—
400C4C22×#1171992211yescharacteristic
401C4C2×#1313323711yescharacteristic
402C4#1681142912yes—
403C2—22no—
404C2—44no—
405C2—8481no—
406C2#2028422411yescharacteristic
407C2C2×#1681085711yescharacteristic
408C2#2028444911yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC2×C53×#461C2
socleC22×C53C2×#121
prime radical#20284179C2
hypercenterC2×#461D53
hypocenterC53#19392
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#461

Complements

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

NH ≅ G/Ntype
1. #20285836C2semidirect
3. #20283739C2semidirect
4. #20274154C2semidirect
5. #20274102C2semidirect
6. #20273489C2semidirect
7. #20273387C2semidirect
8. #20273365C2semidirect
9. C2×C53×#461C2semidirect
12. #16811026C22semidirect
13. #16810971C22semidirect
26. #16810375C22semidirect
27. #16810370C22semidirect
28. #16810369C22semidirect
29. #16810366C22semidirect
201. C2×#461D53semidirect
241. C53#19392semidirect

Maximal subgroups

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

subgroupindex
#202858362
#202841792
#202837392
#202741542
#202741022
#202734892
#202733872
#202733652
C2×C53×#4612
#1939253