Finite Group Databasebeta

← #27280928

Subgroups — #27280928

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#20283823C211yesmaximal, characteristic
3#20275664C211yesmaximal, characteristic, complemented
4#20274732C211yesmaximal, characteristic
5#20272617C211yesmaximal, characteristic, complemented
6#20272589C211yesmaximal, characteristic, complemented
7C53×#1548C211yesmaximal, characteristic, complemented
8C4×#11719945C2211yescharacteristic
9#16810915—21no—
10#16810910—21no—
11#16810905C412yescomplemented
12#16810900C412yescomplemented
13C8×Dic106—21no—
14C8×D212—21no—
15#16810686—22no—
16#16810675—21no—
17#16810663—21no—
18#16810647C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C4⋊C8×D53—21no—
21C2×#13133143—21no—
22#16810622—21no—
23C4×C4⋊Dic53C2211yescharacteristic
24#16810385—22no—
25#16810384—21no—
26C2×C4⋊Q8×C53C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C53×#423C2211yescharacteristic
29C53×#333—21no—
30C53×#333—21no—
31C2×#11719945D411yescharacteristic
32C2×#11719917—21no—
33C22×#88322D412yes—
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—
43C2×C4⋊Dic53—21no—
44C2×C4×Dic53—21no—
45C42×D53—21no—
46#13133150—22no—
47#13133148—22no—
48C4⋊#88322—21no—
49C4⋊#88322—21no—
50C4⋊#88322—21no—
51C4⋊#88322C2×C412yes—
52#13133146—22no—
53#13133143—42no—
54#13133140—22no—
55#13133137—22no—
56#13133129—22no—
57C8×Dic53—22no—
58#13133124—21no—
59C22×Q8×C53—21no—
60C2×M16×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53C2×C412yes—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—21no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D411yescharacteristic
71C4⋊C8×C53D411yescharacteristic
72C22⋊C8×C53—21no—
73C4×C8×C53—21no—
74C2×Q8×C53—22no—
75C2×Q8×C53—41no—
76C2×Q8×C53—82no—
77C22×C4×C53—41no—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C2×C22⋊D53—21no—
82C22×Dic53—41no—
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
92M16×C53—41no—
93M16×C53—41no—
94C2×C8×C53—21no—
95C2×C8×C53—21no—
96C2×C8×C53—41no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C42×C53C22×C411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53C2×D411yescharacteristic
104C42×C53#4111yescharacteristic
105C22⋊Dic53—21no—
106#11719921—21no—
107#11719921—21no—
108C4⋊Dic53—22no—
109C4⋊Dic53—41no—
110C4⋊Dic53C22⋊C411yescharacteristic
111C4⋊Dic53C22⋊C411yescharacteristic
112#11719919—21no—
113#11719919—21no—
114C4×Dic53—21no—
115C4×Dic53—21no—
116C4×Dic53—21no—
117C4×Dic53—41no—
118C4×Dic53—41no—
119#11719917—41no—
120#11719917—41no—
121C2×#88322—22no—
122C2×#88322—22no—
123C2×#88322—42no—
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—82no—
134C2×C4×C53C2×C22⋊C411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12311yescharacteristic
140C2×C4×C53C2×SD1611yescharacteristic
141C2×C4×C53C2×Q1611yescharacteristic
142C2×C4×C53#13511yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C22⋊D53—22no—
145C22⋊D53—22no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—42no—
150C2×Dic53—81no—
151C2×Dic53—81no—
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—41no—
162C8×C53—41no—
163C8×C53—81no—
164#88322—42no—
165#88322—42no—
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×C53C2×#10311yescharacteristic
180C4×C53#41611yescharacteristic
181C4×C53#45511yescharacteristic
182C4×C53#45611yescharacteristic
183C4×C53#46511yescharacteristic
184C4×C53#46711yescharacteristic
185Dic53—22no—
186Dic53—41no—
187Dic53—41no—
188Dic53—82no—
189#3079—531no—
190#2988—531no—
191#1987—531no—
192#1806—531no—
193#1548D5311yescharacteristic, complemented
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—1061no—
212#431C2×D5311yescharacteristic
213#428—1061no—
214#423—531no—
215#423C2×D5311yescharacteristic
216C2×C4⋊C8—531no—
217C2×C4⋊C8—1061no—
218C2×#104—1061no—
219#417—1061no—
220C4×C4⋊C4—531no—
221#340—1062no—
222#339—1061no—
223#333—21no—
224#333—21no—
225C53#931511yescharacteristic, complemented, Sylow
226C2×#41—531no—
227C22×Q8—21no—
228C2×M16—21no—
229C2×M16—1061no—
230C22×C8—532no—
231C22×C8—1062no—
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—1061no—
247C2×C42—1061no—
248C2×C42—1061no—
249C2×C42C22×D5311yescharacteristic
250#107—1062no—
251#106—1062no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—532no—
255C4⋊C8—1062no—
256C4⋊C8—1061no—
257C4⋊C8—1061no—
258C4⋊C8—1061no—
259C4⋊C8C22⋊D5311yescharacteristic
260C4⋊C8C22⋊D5311yescharacteristic
261#104—2122no—
262#103—1062no—
263#102—1062no—
264C22⋊C8—21no—
265C22⋊C8—1062no—
266C4×C8—21no—
267C4×C8—1062no—
268#95—1061no—
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—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—2121no—
284C22×C4#1171994511yescharacteristic
285C22×C4D4×D5311yescharacteristic
286C22×C4D4×D5311yescharacteristic
287M16—41no—
288M16—41no—
289M16—2121no—
290M16—2121no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—41no—
294C2×C8—1062no—
295C2×C8—1062no—
296C2×C8—4242no—
297C2×C8—2122no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—531no—
303C4⋊C4—531no—
304C4⋊C4—1062no—
305C4⋊C4—1061no—
306C4⋊C4—1061no—
307C4⋊C4—2121no—
308C22⋊C4—1061no—
309C22⋊C4—1061no—
310C22⋊C4—1061no—
311C42—1061no—
312C42—1061no—
313C42—1061no—
314C42—2121no—
315C42—2121no—
316C42C2×C4×D5311yescharacteristic
317C42D4×D5311yescharacteristic
318C42#1171994711yescharacteristic
319C42C2×C22⋊D5311yescharacteristic
320C23—1061no—
321C23#1313319011yescharacteristic
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—82no—
335C2×C4—1061no—
336C2×C4—1062no—
337C2×C4—1062no—
338C2×C4—1061no—
339C2×C4—1061no—
340C2×C4—4241no—
341C2×C4—4241no—
342C2×C4—4241no—
343C2×C4—2122no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4C22⋊C4×D5311yescharacteristic
347C2×C4#1313317711yescharacteristic
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318511yescharacteristic
350C2×C4#1313318811yescharacteristic
351C2×C4#1313319911yescharacteristic
352C2×C4SD16×D5311yescharacteristic
353C2×C4Q16×D5311yescharacteristic
354C2×C4#1313323811yescharacteristic
355C2×C4#1313324011yescharacteristic
356C8—41no—
357C8—41no—
358C8—81no—
359C8—2122no—
360C8—2122no—
361C22—21no—
362C22—21no—
363C22—1062no—
364C22—2121no—
365C22—2121no—
366C22#1681049911yescharacteristic
367C22#1681085511yescharacteristic
368C22#1681086311yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—1062no—
374C4—4242no—
375C4—2121no—
376C4—2121no—
377C4#103×D5311yescharacteristic
378C4#1681061811yescharacteristic
379C4#1681079911yescharacteristic
380C4#1681080411yescharacteristic
381C4#1681084911yescharacteristic
382C4#1681086111yescharacteristic
383C2—21no—
384C2—21no—
385C2—2122no—
386C2#2027508111yescharacteristic
387C2#2027513511yescharacteristic
388C2#2027519811yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1548C2
socleC22×C53#386
prime radicalC2×#11719945D4
cosocleC2×C42×C53C23
hypercenter#1548D53
hypocenterC53#9315
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
3. #20275664C2semidirect
5. #20272617C2semidirect
6. #20272589C2semidirect
7. C53×#1548C2semidirect
11. #16810905C4semidirect
12. #16810900C4semidirect
193. #1548D53semidirect
225. C53#9315semidirect

Maximal subgroups

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

subgroupindex
#202845242
#202838232
#202756642
#202747322
#202726172
#202725892
C53×#15482
#931553