Finite Group Databasebeta

← #27280934

Subgroups — #27280934

Order 13568.

480 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284524C211yesmaximal, characteristic
2#20283824C211yesmaximal, characteristic
3#20274736C211yesmaximal, characteristic
4#20273396C211yesmaximal, characteristic, complemented
5#20272618C211yesmaximal, characteristic, complemented
6#20272590C211yesmaximal, characteristic, complemented
7C53×#1987C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810915—21no—
10#16810910—21no—
11#16810905C412yescomplemented
12#16810900C412yescomplemented
13#16810747—21no—
14#16810688—21no—
15C8×C22⋊D53—22no—
16#16810676C2211yescharacteristic
17#16810675—21no—
18#16810663—21no—
19#16810647C2211yescharacteristic
20#16810647C2211yescharacteristic
21C4⋊C8×D53—21no—
22C2×#13133145—21no—
23#16810622—21no—
24#16810383—21no—
25#16810372—21no—
26#16810372—21no—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C2×C4⋊C8×C53C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#340—22no—
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C2×#11719917—21no—
34C2×C8×D53—22no—
35C4×Dic106C2×C411yescharacteristic
36C4×Dic106D411yescharacteristic
37C4×D212C2×C411yescharacteristic
38C4×D212D411yescharacteristic
39C4×C22⋊D53—21no—
40C4×C22⋊D53—21no—
41#13133176D411yescharacteristic
42#13133174—21no—
43C42×D53—21no—
44#13133150—22no—
45#13133148—22no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322—21no—
49C4⋊#88322C2×C412yes—
50C4⋊#88322D411yescharacteristic
51C4⋊#88322D411yescharacteristic
52#13133146—22no—
53#13133145—42no—
54#13133140—22no—
55#13133137—22no—
56C22⋊#88322—21no—
57#13133129—22no—
58C4×#88322—21no—
59C8×Dic53—22no—
60C22×Q8×C53—21no—
61C22×C8×C53D412yes—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53C2×C412yes—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C4⋊C4×C53—21no—
69C2×C42×C53—21no—
70C2×C42×C53C2311yescharacteristic
71C4⋊C8×C53—21no—
72C4⋊C8×C53—21no—
73#95×C53—21no—
74C2×Q8×C53—22no—
75C2×Q8×C53—41no—
76C2×Q8×C53—82no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C2×C22⋊D53—21no—
83#11719945—21no—
84#11719945—21no—
85#11719945SD1612yes—
86#11719945Q1612yes—
87C2×D212C22⋊C411yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—21no—
91C2×Dic106C22⋊C411yescharacteristic
92C2×C8×C53—22no—
93C2×C8×C53—42no—
94C4⋊C4×C53—22no—
95C4⋊C4×C53—22no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—41no—
98C4⋊C4×C53—41no—
99C4⋊C4×C53—41no—
100C42×C53—41no—
101C42×C53—41no—
102C42×C53C22×C411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53C2×D411yescharacteristic
105C42×C53#4111yescharacteristic
106C22⋊Dic53—21no—
107#11719921—21no—
108#11719921—21no—
109C4⋊Dic53C22⋊C411yescharacteristic
110C4⋊Dic53C22⋊C411yescharacteristic
111#11719919—21no—
112#11719919—21no—
113C4×Dic53—21no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116#11719917—41no—
117#11719917—41no—
118#11719917—41no—
119#11719917—41no—
120C2×#88322—22no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—41no—
124C8×D53—82no—
125C23×C53#12111yescharacteristic
126C22×D53—21no—
127Q8×C53—82no—
128Q8×C53—82no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—81no—
135C2×C4×C53—82no—
136C2×C4×C53—81no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12311yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53C2×Q1611yescharacteristic
145C2×C4×C53#13511yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C22⋊D53—22no—
148C22⋊D53—22no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152D212—21no—
153D212#10312yes—
154C4×D53—22no—
155C4×D53—22no—
156C4×D53—41no—
157C4×D53—41no—
158C4×D53—81no—
159Dic106—21no—
160Dic106#10312yes—
161C8×C53—42no—
162#88322—42no—
163#88322—41no—
164#88322—41no—
165#88322—81no—
166#9315—531nomaximal, Sylow
167C22×C53—21no—
168C22×C53—21no—
169C22×C53#38611yescharacteristic
170C22×C53#46511yescharacteristic
171C22×C53#46711yescharacteristic
172C2×D53—22no—
173C2×D53—41no—
174C2×D53—41no—
175C4×C53—21no—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—82no—
179C4×C53—82no—
180C4×C53C2×#10311yescharacteristic
181C4×C53#41611yescharacteristic
182C4×C53#45511yescharacteristic
183C4×C53#45611yescharacteristic
184C4×C53#46511yescharacteristic
185C4×C53#46711yescharacteristic
186Dic53—22no—
187Dic53—41no—
188Dic53—41no—
189#3079—531no—
190#2988—531no—
191#1987D5311yescharacteristic, complemented
192#1806—531no—
193#1548—531no—
194#1303—531no—
195#1292—531no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53#188211yescharacteristic
199C2×C53#189211yescharacteristic
200C2×C53#190311yescharacteristic
201D53—42no—
202C2×C4⋊Q8C2×D5311yescharacteristic
203C4×#41—531no—
204#477—1061no—
205#476—1061no—
206#475—532no—
207#474—532no—
208C8×Q8—1061no—
209C8×D4—1061no—
210C8×D4—1062no—
211#431—531no—
212#431—1061no—
213#428—1061no—
214#423—531no—
215#423—531no—
216C2×C4⋊C8—1061no—
217C2×C4⋊C8C2×D5311yescharacteristic
218C2×#104—1061no—
219#417—1061no—
220C4×C4⋊C4C2×D5311yescharacteristic
221#340—22no—
222#339—1061no—
223#333—1061no—
224#333—1061no—
225C53#931511yescharacteristic, complemented, Sylow
226C2×#41—531no—
227C22×Q8—21no—
228C2×M16—1061no—
229C2×M16—1061no—
230C22×C8—1062no—
231C22×C8C22⋊D5312yes—
232C4⋊Q8—21no—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8C4×D5312yes—
236C4×Q8—531no—
237C4×Q8—531no—
238C4×D4—531no—
239C4×D4—531no—
240C4×D4—1061no—
241C4×D4—1061no—
242#117—531no—
243#117—1061no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C42—21no—
248C2×C42—1061no—
249C2×C42C22×D5311yescharacteristic
250#107—1062no—
251#106—1062no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—532no—
255C4⋊C8—531no—
256C4⋊C8—531no—
257C4⋊C8—1062no—
258C4⋊C8—1061no—
259C4⋊C8—1061no—
260C4⋊C8—1061no—
261#104—2122no—
262#103—1062no—
263#102—1062no—
264C22⋊C8—1062no—
265C22⋊C8—1061no—
266C4×C8—1062no—
267C4×C8—1061no—
268#95—21no—
269#41—532no—
270#41—532no—
271#41—1061no—
272#41—1061no—
273C2×Q8—22no—
274C2×Q8—41no—
275C2×Q8—82no—
276C2×Q8—531no—
277C2×D4—531no—
278C2×D4—1061no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4D4×D5311yescharacteristic
285C22×C4#1171994711yescharacteristic
286C22×C4C2×C22⋊D5311yescharacteristic
287M16—2121no—
288M16—2121no—
289M16—2121no—
290M16—2121no—
291C2×C8—22no—
292C2×C8—42no—
293C2×C8—1062no—
294C2×C8—1061no—
295C2×C8—1061no—
296C2×C8—4242no—
297C2×C8—2121no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—531no—
305C4⋊C4—531no—
306C4⋊C4—1061no—
307C4⋊C4—1061no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C42—41no—
312C42—41no—
313C42—1061no—
314C42—1061no—
315C42—1061no—
316C42C2×C4×D5311yescharacteristic
317C42#1171994511yescharacteristic
318C42D4×D5311yescharacteristic
319C42D4×D5311yescharacteristic
320C23—1061no—
321C23#1313318811yescharacteristic
322Q8—82no—
323Q8—82no—
324Q8—532no—
325Q8—1061no—
326D4—532no—
327D4—1062no—
328D4—1062no—
329D4—1061no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—81no—
338C2×C4—1061no—
339C2×C4—1062no—
340C2×C4—1062no—
341C2×C4—1061no—
342C2×C4—1061no—
343C2×C4—4241no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4C22⋊C4×D5311yescharacteristic
347C2×C4#1313317711yescharacteristic
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318511yescharacteristic
350C2×C4#1313319011yescharacteristic
351C2×C4#1313319911yescharacteristic
352C2×C4SD16×D5311yescharacteristic
353C2×C4Q16×D5311yescharacteristic
354C2×C4#1313323811yescharacteristic
355C2×C4#1313324011yescharacteristic
356C8—42no—
357C8—4241no—
358C8—2121no—
359C8—2121no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—1062no—
364C22—2121no—
365C22—2121no—
366C22#1681049911yescharacteristic
367C22#1681084911yescharacteristic
368C22#1681086111yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—82no—
374C4—1062no—
375C4—2121no—
376C4—2121no—
377C4#103×D5311yescharacteristic
378C4#1681061811yescharacteristic
379C4#1681080011yescharacteristic
380C4#1681080511yescharacteristic
381C4#1681085511yescharacteristic
382C4#1681086311yescharacteristic
383C2—21no—
384C2—21no—
385C2—2122no—
386C2#2027508711yescharacteristic
387C2#2027513511yescharacteristic
388C2#2027519211yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1987C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1987D53
hypocenterC53#9315
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1987

Complements

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

NH ≅ G/Ntype
4. #20273396C2semidirect
5. #20272618C2semidirect
6. #20272590C2semidirect
7. C53×#1987C2semidirect
11. #16810905C4semidirect
12. #16810900C4semidirect
191. #1987D53semidirect
225. C53#9315semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202838242
#202747362
#202733962
#202726182
#202725902
C53×#19872
#931553