Finite Group Databasebeta

← #27351915

Subgroups — #27351915

Order 13568.

1254 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288479C211yesmaximal, characteristic
2#20277688C211yesmaximal, characteristic, complemented
3#20273276C212yesmaximal, complemented
4#20273245C214yesmaximal, complemented
5#20273232C214yesmaximal, complemented
6#20273211C212yesmaximal, complemented
7C53×#2975C211yesmaximal, characteristic, complemented
8C2×#13133262—21no—
9#16811277C414yescomplemented
10#16811268—24no—
11D4×D212C414yescomplemented
12#16811204—21no—
13#16811185—22no—
14#16811173C2212yes—
15C2×#13133191C2214yes—
16#16810526—21no—
17#16810516—22no—
18#16810489C2211yescharacteristic, complemented
19#16810486C2212yescomplemented
20C4⋊C4×Dic53—21no—
21C4×C4⋊Dic53C2211yescharacteristic, complemented
22C4×C22⋊Dic53C2212yescomplemented
23#16810358C2214yescomplemented
24#16810358C2214yescomplemented
25#16810358C2218yescomplemented
26C2×C4×D4×C53C2211yescharacteristic
27C53×#436—24no—
28C53×#431C2212yescomplemented
29C53×#407C2214yescomplemented
30#13133262—44no—
31#13133262—44no—
32C2×#11719945—22no—
33C2×#11719947—24no—
34C22×D212C2×C414yes—
35C22×C22⋊D53—24no—
36C2×D4×D53—24no—
37#13133217C2×C412yes—
38#13133214C2×C412yes—
39#13133205—22no—
40#13133201—28no—
41#13133197—24no—
42#13133192—24no—
43#13133191—24no—
44#13133191—24no—
45#13133191—22no—
46#13133191C2×C418yes—
47#13133191C2314yes—
48#13133187—28no—
49C4×D212—22no—
50C4×D212C2×C414yes—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—21no—
53C2×C4⋊Dic53C2×C414yes—
54C2×C22⋊Dic53—22no—
55C2×C22⋊Dic53—22no—
56C2×#11719921—24no—
57C2×C4×Dic53—21no—
58C2×C4×Dic53—22no—
59#13133124—21no—
60#13133124—22no—
61#13133124C2311yescharacteristic
62#13133124C2312yes—
63#13133124C2311yescharacteristic
64C22×D4×C53D411yescharacteristic
65C23×C4×C53C2312yes—
66C2×M16×C53—24no—
67C4×D4×C53C2×C414yes—
68C4×D4×C53D414yes—
69C2×C4⋊C4×C53D411yescharacteristic
70C2×C22⋊C4×C53D412yes—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—22no—
73C22⋊C8×C53—24no—
74C22⋊C8×C53C2314yes—
75C4×C8×C53—22no—
76C24×C53C2×D412yes—
77C23×D53—24no—
78C2×D4×C53—22no—
79C2×D4×C53—22no—
80C2×D4×C53C22⋊C414yes—
81C22×C4×C53—82no—
82C22×C4×C53C22×C414yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D412yes—
85C22×C4×C53C2×D414yes—
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2411yescharacteristic
88C2×C22⋊D53—28no—
89C2×C22⋊D53—24no—
90C2×C22⋊D53—48no—
91C2×C22⋊D53—48no—
92C22×Dic53—24no—
93C22×Dic53—21no—
94C22×Dic53—22no—
95C22×Dic53—21no—
96#11719947—416no—
97D4×D53—416no—
98#11719945—44no—
99#11719945—44no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—24no—
103C2×D212—84no—
104C2×D212C22×C414yes—
105C2×C4×D53—24no—
106C2×Dic106—22no—
107M16×C53—84no—
108C2×C8×C53—24no—
109C4⋊C4×C53C22⋊C414yes—
110C22⋊C4×C53C22⋊C418yes—
111C42×C53C22×C412yes—
112C42×C53C2×D412yes—
113C22⋊Dic53—44no—
114C22⋊Dic53—82no—
115#11719921—28no—
116#11719921—24no—
117#11719921—24no—
118C4⋊Dic53—24no—
119C4⋊Dic53—22no—
120C4⋊Dic53—42no—
121C4⋊Dic53—42no—
122C4⋊Dic53C22×C414yes—
123C4×Dic53—42no—
124C4×Dic53—82no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—42no—
128C23×C53—42no—
129C23×C53C2×C22⋊C414yes—
130C23×C53C22×D411yescharacteristic
131C22×D53—24no—
132C22×D53—48no—
133C22×D53—84no—
134C22×D53—84no—
135D4×C53—28no—
136C2×C4×C53—44no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53—44no—
140C2×C4×C53—41no—
141C2×C4×C53C2×C22⋊C414yes—
142C2×C4×C53C2×C22⋊C412yes—
143C2×C4×C53C2×C22⋊C412yes—
144C2×C4×C53AGL(1,ℤ/8)12yes—
145C2×C4×C53AGL(1,ℤ/8)12yes—
146C2×C4×C53C23×C411yescharacteristic
147C2×C4×C53C22×D411yescharacteristic
148C22⋊D53—416no—
149C22⋊D53—44no—
150C22⋊D53—44no—
151C2×Dic53—22no—
152C2×Dic53—24no—
153C2×Dic53—48no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—42no—
157C2×Dic53—41no—
158C2×Dic53—42no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—42no—
162D212—44no—
163D212—44no—
164D212—82no—
165D212—84no—
166C4×D53—48no—
167Dic106—44no—
168C8×C53—44no—
169#17626—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—24no—
173C22×C53—24no—
174C22×C53—24no—
175C22×C53—82no—
176C22×C53—82no—
177C22×C53C22×C22⋊C411yescharacteristic
178C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
179C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
180C2×D53—44no—
181C2×D53—44no—
182C2×D53—44no—
183C2×D53—168no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—24no—
187C4×C53—22no—
188C4×C53#41814yes—
189Dic53—42no—
190Dic53—44no—
191Dic53—81no—
192Dic53—82no—
193#3496—531no—
194#2975D5311yescharacteristic, complemented
195#2357—531no—
196#1517—532no—
197#1510—534no—
198#1507—534no—
199#1502—532no—
200C2×C53—21no—
201C2×C53—21no—
202C2×C53—44no—
203C2×C53#288311yescharacteristic
204C2×C53C2×#41811yescharacteristic
205C2×C53#290111yescharacteristic
206D53—84no—
207C2×H(5,𝔽2)—1061no—
208#547—534no—
209#546—1064no—
210D42—534no—
211#537—1061no—
212#534—1062no—
213#532—532no—
214C2×#121—534no—
215C2×C4×D4C2×D5311yescharacteristic
216#436—24no—
217#431C2×D5312yescomplemented
218#407C2×D5314yescomplemented
219#390—1061no—
220#388—1062no—
221#382—531no—
222#381—532no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4—1061no—
225C4×C22⋊C4—532no—
226#327—534no—
227#327—534no—
228#327—538no—
229C53#1762611yescharacteristic, complemented, Sylow
230H(5,𝔽2)—2124no—
231H(5,𝔽2)—2124no—
232C2×#41—1064no—
233C2×#41—1062no—
234C22×D4—534no—
235C22×D4—1064no—
236C22×D4—1064no—
237C22×D4D21211yescharacteristic
238C23×C4C22×D5312yes—
239C2×M16—24no—
240C4⋊Q8—532no—
241#127—532no—
242#124—1062no—
243#123—1068no—
244C22⋊Q8—1064no—
245#121—538no—
246#121—534no—
247#121—1064no—
248#121—1062no—
249#121—1064no—
250#121—1064no—
251#120—1068no—
252C4×D4—534no—
253C4×D4—1062no—
254C4×D4C4×D5314yes—
255C4×D4C22⋊D5314yes—
256C2×C4⋊C4—534no—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1061no—
259C2×C4⋊C4D21211yescharacteristic
260C2×C22⋊C4—1064no—
261C2×C22⋊C4—1062no—
262C2×C22⋊C4—1062no—
263C2×C22⋊C4D21212yes—
264C2×C42—1062no—
265C2×C42—1061no—
266C2×C42C22×D5311yescharacteristic
267C4⋊C8—22no—
268C22⋊C8—24no—
269C22⋊C8C22×D5314yes—
270C4×C8—22no—
271#95—531no—
272#95—532no—
273#95—531no—
274#95—1062no—
275#95—1061no—
276C24—1064no—
277C24C2×D21212yes—
278#41—21216no—
279#41—2124no—
280#41—2124no—
281C2×Q8—1062no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—534no—
285C2×D4—1068no—
286C2×D4—1064no—
287C2×D4—1062no—
288C2×D4—1064no—
289C2×D4—1062no—
290C2×D4—4244no—
291C2×D4—21216no—
292C2×D4—2128no—
293C2×D4—2128no—
294C2×D4#1171992114yes—
295C22×C4—82no—
296C22×C4—1064no—
297C22×C4—1061no—
298C22×C4—1064no—
299C22×C4—1062no—
300C22×C4—1061no—
301C22×C4C2×C4×D5314yes—
302C22×C4C2×D21212yes—
303C22×C4C2×D21211yescharacteristic
304C22×C4C2×D21211yescharacteristic
305C22×C4C2×C22⋊D5314yes—
306C22×C4C23×D5311yescharacteristic
307M16—84no—
308C2×C8—24no—
309C4⋊C4—534no—
310C4⋊C4—1064no—
311C4⋊C4—1062no—
312C4⋊C4—2122no—
313C4⋊C4—2122no—
314C4⋊C4#1171992114yes—
315C22⋊C4—1068no—
316C22⋊C4—1064no—
317C22⋊C4—1064no—
318C22⋊C4—4242no—
319C22⋊C4—2124no—
320C22⋊C4#1171992118yes—
321C42—4242no—
322C42—2122no—
323C42C2×C4×D5312yes—
324C42C2×C22⋊D5312yes—
325C23—22no—
326C23—22no—
327C23—42no—
328C23—42no—
329C23—1064no—
330C23—4244no—
331C23—4244no—
332C23—2128no—
333C23C2×#1171992114yes—
334C23C22×D21211yescharacteristic
335Q8—2124no—
336D4—28no—
337D4—4242no—
338D4—4244no—
339D4—2124no—
340D4—21216no—
341D4—2124no—
342D4—2124no—
343D4—2124no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—1062no—
350C2×C4—1064no—
351C2×C4—2122no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4—2121no—
358C2×C4—2128no—
359C2×C4—2121no—
360C2×C4—2128no—
361C2×C4C2×#1171992112yes—
362C2×C4C2×#1171992114yes—
363C2×C4C2×#1171992112yes—
364C2×C4#1313324512yes—
365C2×C4#1313324512yes—
366C2×C4C22×C4×D5311yescharacteristic
367C2×C4C22×C22⋊D5311yescharacteristic
368C8—44no—
369C22—21no—
370C22—21no—
371C22—24no—
372C22—24no—
373C22—24no—
374C22—82no—
375C22—82no—
376C22—8488no—
377C22—2124no—
378C22—2124no—
379C22—2124no—
380C22C22×#1171992111yescharacteristic
381C22C2×#1313324511yescharacteristic
382C22C2×#1313324511yescharacteristic
383C4—21no—
384C4—21no—
385C4—24no—
386C4—22no—
387C4—4242no—
388C4—4241no—
389C4—2122no—
390C4—2124no—
391C4#1681062914yes—
392C2—21no—
393C2—21no—
394C2—44no—
395C2—4244no—
396C2#2028317511yescharacteristic
397C2C2×#1681062911yescharacteristic
398C2#2028326011yescharacteristic
subgroupG/H
centerC22C22×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2975C2
socleC22×C53C22×C22⋊C4
prime radical#20288479C2
cosocleC22×C4×C53C24
hypercenter#2975D53
hypocenterC53#17626
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #20277688C2semidirect
3. #20273276C2semidirect
4. #20273245C2semidirect
5. #20273232C2semidirect
6. #20273211C2semidirect
7. C53×#2975C2semidirect
9. #16811277C4semidirect
11. D4×D212C4semidirect
18. #16810489C22semidirect
19. #16810486C22semidirect
21. C4×C4⋊Dic53C22semidirect
22. C4×C22⋊Dic53C22semidirect
23. #16810358C22semidirect
24. #16810358C22semidirect
25. #16810358C22semidirect
28. C53×#431C22semidirect
29. C53×#407C22semidirect
194. #2975D53semidirect
217. #431C2×D53semidirect
218. #407C2×D53semidirect
229. C53#17626semidirect

Maximal subgroups

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

subgroupindex
#202884792
#202776882
#202732762
#202732452
#202732322
#202732112
C53×#29752
#1762653