Finite Group Databasebeta

← #27440392

Subgroups — #27440392

Order 13568.

1370 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1370 conjugacy classes of subgroups — too many to draw. Only its 191 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20288479C211yesmaximal, characteristic
2#20285111C212yesmaximal, complemented
3#20275939C214yesmaximal, complemented
4#20275633C212yesmaximal, complemented
5#20274668C214yesmaximal, complemented
6C4×#13133134C211yesmaximal, characteristic, complemented
7C53×#2542C211yesmaximal, characteristic, complemented
8C2×#13133262C2212yes—
9#16811277—22no—
10#16811268—24no—
11D4×D212—22no—
12#16811204C2212yes—
13#16811185—22no—
14#16811173C2212yes—
15C2×#13133191—22no—
16#16810985—24no—
17#16810975—22no—
18#16810971—22no—
19C2×#13133145C2218yescomplemented
20#16810622C2218yescomplemented
21C2×#13133134C2212yes—
22C4×#11719917C2212yescomplemented
23#16810392C2212yescomplemented
24C2×C4⋊Q8×C53C2211yescharacteristic, complemented
25C2×C22⋊Q8×C53—22no—
26C2×C4×D4×C53C2211yescharacteristic
27C53×#399—22no—
28C53×#390C2212yes—
29C53×#384C2212yescomplemented
30C4×C4⋊C4×C53C2211yescharacteristic, complemented
31#13133262—28no—
32#13133262—44no—
33C2×#11719945C2314yes—
34C2×#11719947—24no—
35C22×D212—22no—
36C22×C22⋊D53—24no—
37C2×D4×D53—24no—
38C2×#11719917C2314yes—
39#13133217D412yes—
40#13133214D412yes—
41#13133205D414yes—
42#13133201—28no—
43#13133197—24no—
44#13133192—24no—
45#13133191—24no—
46#13133191—24no—
47#13133191—22no—
48#13133191—24no—
49#13133191—22no—
50#13133187—28no—
51C4×D212—22no—
52C4×D212—22no—
53C2×C4⋊Dic53—22no—
54C2×#11719921—24no—
55#13133145—48no—
56#13133140—28no—
57#13133137—28no—
58#13133134—24no—
59#13133128—22no—
60C4×#88322—22no—
61C22×Q8×C53—21no—
62C22×D4×C53C2311yescharacteristic
63C23×C4×C53D412yes—
64C4⋊Q8×C53—22no—
65C4⋊Q8×C53—22no—
66C22⋊Q8×C53—48no—
67C4×D4×C53—22no—
68C4×D4×C53—22no—
69C2×C4⋊C4×C53—21no—
70C2×C4⋊C4×C53—24no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53C2312yes—
73C2×C4⋊C4×C53C2311yescharacteristic
74C2×C22⋊C4×C53—24no—
75C2×C22⋊C4×C53D412yes—
76C2×C42×C53C2312yes—
77C2×C42×C53C2311yescharacteristic
78#95×C53—21no—
79C24×C53C2×D412yes—
80C23×D53—24no—
81C2×Q8×C53—44no—
82C2×Q8×C53—41no—
83C2×Q8×C53—41no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53#4114yes—
87C22×C4×C53—22no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—44no—
92C22×C4×C53C2×D411yescharacteristic
93C22×C4×C53C2×D412yes—
94C22×C4×C53C2×D411yescharacteristic
95C22×C4×C53C2411yescharacteristic
96C2×C22⋊D53—28no—
97C2×C22⋊D53—24no—
98C2×C22⋊D53—48no—
99C2×C22⋊D53—48no—
100C22×Dic53—24no—
101#11719947—416no—
102D4×D53—416no—
103#11719945—28no—
104#11719945—28no—
105C2×D212—22no—
106C2×D212—24no—
107C2×D212—22no—
108C2×D212—84no—
109C2×D212C2×D414yes—
110C2×C4×D53—24no—
111C2×Dic106C2×D414yes—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53—24no—
115C4⋊C4×C53—48no—
116C4⋊C4×C53—42no—
117C22⋊C4×C53—24no—
118C22⋊C4×C53—48no—
119C42×C53—24no—
120C42×C53C2×D412yes—
121C42×C53C2×D412yes—
122#11719921—28no—
123#11719921—24no—
124#11719921—24no—
125C4⋊Dic53—24no—
126C4⋊Dic53—22no—
127C4⋊Dic53—22no—
128#11719917—24no—
129#11719917—24no—
130C2×#88322—24no—
131C23×C53—22no—
132C23×C53—22no—
133C23×C53—42no—
134C23×C53—42no—
135C23×C53#12414yes—
136C23×C53C22×D411yescharacteristic
137C22×D53—24no—
138C22×D53—48no—
139C22×D53—84no—
140C22×D53—84no—
141Q8×C53—82no—
142Q8×C53—82no—
143D4×C53—28no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—22no—
147C2×C4×C53—41no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—44no—
151C2×C4×C53—42no—
152C2×C4×C53—44no—
153C2×C4×C53—42no—
154C2×C4×C53—82no—
155C2×C4×C53#12012yes—
156C2×C4×C53#12011yescharacteristic
157C2×C4×C53#12011yescharacteristic
158C2×C4×C53C22×D412yes—
159C2×C4×C53C2×#4112yes—
160C2×C4×C53H(5,𝔽2)12yes—
161C22⋊D53—416no—
162C22⋊D53—44no—
163C22⋊D53—44no—
164C2×Dic53—24no—
165C2×Dic53—48no—
166D212—28no—
167D212—44no—
168D212—82no—
169D212—84no—
170C4×D53—48no—
171Dic106—28no—
172#88322—44no—
173#25870—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—24no—
177C22×C53—44no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—82no—
181C22×C53C2×#12011yescharacteristic
182C22×C53C2×#12411yescharacteristic
183C22×C53#53511yescharacteristic
184C2×D53—44no—
185C2×D53—44no—
186C2×D53—44no—
187C2×D53—168no—
188C4×C53—21no—
189C4×C53—22no—
190C4×C53—44no—
191C4×C53—42no—
192C4×C53—82no—
193C4×C53#45312yes—
194C4×C53#54614yes—
195Dic53—44no—
196#3496—531no—
197#3141—532no—
198#2542D5311yescharacteristic, complemented
199#2039—534no—
200#1980—532no—
201#1792—534no—
202C4×#100—531no—
203C2×C53—21no—
204C2×C53—21no—
205C2×C53—44no—
206C2×C53#243511yescharacteristic
207C2×C53C2×#45311yescharacteristic
208C2×C53#302211yescharacteristic
209D53—84no—
210C2×H(5,𝔽2)—532no—
211#547—1062no—
212#546—1064no—
213D42—1062no—
214#537—532no—
215#534—1062no—
216#532—532no—
217C2×C4⋊Q8C2×D5311yescharacteristic, complemented
218C2×C22⋊Q8—22no—
219C2×#121—1062no—
220C2×C4×D4C2×D5311yescharacteristic
221#489—1064no—
222#487—1062no—
223#486—1062no—
224C2×#104—538no—
225#417—538no—
226C2×#100—532no—
227C4×M16—532no—
228#399—22no—
229#390C2×D5312yes—
230#384C2×D5312yescomplemented
231C4×C4⋊C4C2×D5311yescharacteristic, complemented
232#343—532no—
233C53#2587011yescharacteristic, complemented, Sylow
234H(5,𝔽2)—1068no—
235H(5,𝔽2)—2124no—
236C2×#41—534no—
237C2×#41—1064no—
238C22×Q8—21no—
239C22×D4—1062no—
240C22×D4—1064no—
241C22×D4—1064no—
242C22×D4C22×D5311yescharacteristic
243C23×C4C22⋊D5312yes—
244C2×M16—534no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—532no—
248#127—532no—
249#124—534no—
250#123—1068no—
251C22⋊Q8—48no—
252C22⋊Q8—1064no—
253#121—1064no—
254#121—1064no—
255#121—1064no—
256#121—1062no—
257#121—1064no—
258#121—1062no—
259#120—1068no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—1062no—
263C4×D4—1062no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—1062no—
268C2×C4⋊C4C22×D5312yes—
269C2×C4⋊C4C22×D5311yescharacteristic
270C2×C22⋊C4—24no—
271C2×C22⋊C4—1064no—
272C2×C22⋊C4C22⋊D5312yes—
273C2×C42C22×D5312yes—
274C2×C42C22×D5311yescharacteristic
275#104—2128no—
276#103—1068no—
277#102—1068no—
278#100—1064no—
279#97—1062no—
280C4×C8—1062no—
281#95—21no—
282C24—1064no—
283C24C2×C22⋊D5312yes—
284#41—1068no—
285#41—1068no—
286#41—21216no—
287C2×Q8—44no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—534no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—534no—
294C2×D4—1064no—
295C2×D4—1068no—
296C2×D4—1064no—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—4244no—
300C2×D4—21216no—
301C2×D4—2128no—
302C2×D4—2128no—
303C2×D4#1171994914yes—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—44no—
309C22×C4—1064no—
310C22×C4—1064no—
311C22×C4C2×C22⋊D5311yescharacteristic
312C22×C4C2×C22⋊D5312yes—
313C22×C4C2×C22⋊D5311yescharacteristic
314C22×C4C23×D5311yescharacteristic
315M16—1064no—
316M16—1064no—
317C2×C8—1064no—
318C4⋊C4—24no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—48no—
322C4⋊C4—42no—
323C4⋊C4—1064no—
324C4⋊C4—1062no—
325C4⋊C4—1062no—
326C22⋊C4—24no—
327C22⋊C4—48no—
328C22⋊C4—1068no—
329C22⋊C4—1064no—
330C22⋊C4—1064no—
331C42—24no—
332C42D4×D5312yes—
333C42D4×D5312yes—
334C23—22no—
335C23—22no—
336C23—42no—
337C23—42no—
338C23—1064no—
339C23—4244no—
340C23—4244no—
341C23—2128no—
342C23#1313320614yes—
343C23C22×C22⋊D5311yescharacteristic
344Q8—82no—
345Q8—82no—
346Q8—1068no—
347D4—28no—
348D4—1068no—
349D4—4242no—
350D4—4244no—
351D4—21216no—
352D4—2124no—
353D4—2124no—
354D4—2124no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—1064no—
367C2×C4—2128no—
368C2×C4—2128no—
369C2×C4#1313318512yes—
370C2×C4#1313318511yescharacteristic
371C2×C4#1313318511yescharacteristic
372C2×C4C2×D4×D5312yes—
373C2×C4C2×#1171994912yes—
374C2×C4#1313326312yes—
375C8—2124no—
376C22—21no—
377C22—21no—
378C22—24no—
379C22—44no—
380C22—42no—
381C22—42no—
382C22—82no—
383C22—8488no—
384C22—2124no—
385C22—2124no—
386C22—2124no—
387C22C2×#1313318511yescharacteristic
388C22C2×#1313320611yescharacteristic
389C22#1681118711yescharacteristic
390C4—21no—
391C4—22no—
392C4—44no—
393C4—42no—
394C4—82no—
395C4—2124no—
396C4#1681079012yes—
397C4#1681126414yes—
398C2—21no—
399C2—21no—
400C2—44no—
401C2—4244no—
402C2#2027836711yescharacteristic
403C2C2×#1681079011yescharacteristic
404C2#2028409311yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2542C2
socleC22×C53C2×#120
prime radical#20288479C2
hypercenter#2542D53
hypocenterC53#25870
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2542

Complements

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

NH ≅ G/Ntype
2. #20285111C2semidirect
3. #20275939C2semidirect
4. #20275633C2semidirect
5. #20274668C2semidirect
6. C4×#13133134C2semidirect
7. C53×#2542C2semidirect
19. C2×#13133145C22semidirect
20. #16810622C22semidirect
22. C4×#11719917C22semidirect
23. #16810392C22semidirect
24. C2×C4⋊Q8×C53C22semidirect
29. C53×#384C22semidirect
30. C4×C4⋊C4×C53C22semidirect
198. #2542D53semidirect
217. C2×C4⋊Q8C2×D53semidirect
230. #384C2×D53semidirect
231. C4×C4⋊C4C2×D53semidirect
233. C53#25870semidirect

Maximal subgroups

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

subgroupindex
#202884792
#202851112
#202759392
#202756332
#202746682
C4×#131331342
C53×#25422
#2587053