Finite Group Databasebeta

← #27489651

Subgroups — #27489651

Order 13568.

1316 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1316 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#20283975C212yesmaximal, complemented
2#20279196C211yesmaximal, characteristic, complemented
3#20275941C214yesmaximal, complemented
4#20275762C214yesmaximal, complemented
5#20274795C211yesmaximal, characteristic, complemented
6#20274659C212yesmaximal, complemented
7C2×C53×#537C211yesmaximal, characteristic
8C2×#13133247C2214yescomplemented
9C2×#13133196C2211yescharacteristic
10#16810785—24no—
11#16810779—22no—
12#16810774—22no—
13C2×#13133144C2218yescomplemented
14#16810624C2214yescomplemented
15#16810587C2212yes—
16#16810561—22no—
17#16810528C2212yescomplemented
18#16810488C2212yescomplemented
19#16810486—22no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810392C2214yescomplemented
22C22×#41×C53C2212yes—
23C53×#537—24no—
24C53×#537—22no—
25C53×#537C2214yes—
26C2×C22⋊Q8×C53—22no—
27C2×#121×C53—22no—
28C2×#121×C53—22no—
29C22×C4⋊C4×C53C2211yescharacteristic
30C22×Dic106C2311yescharacteristic
31#13133247—48no—
32C2×#11719925—24no—
33C2×#11719923—24no—
34C2×#11719917C2314yes—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53C2312yes—
42#13133144—48no—
43#13133141—24no—
44#13133138—24no—
45C22⋊#88322—22no—
46C22⋊#88322—22no—
47#13133124—24no—
48#13133124—21no—
49#13133124—24no—
50C2×#41×C53—22no—
51C2×#41×C53—22no—
52C2×#41×C53—48no—
53C2×#41×C53D414yes—
54C2×#41×C53C2314yes—
55C22×Q8×C53D412yes—
56C22×D4×C53—22no—
57C22×D4×C53—22no—
58C22×D4×C53D412yes—
59C23×C4×C53—22no—
60C23×C4×C53D412yes—
61C23×C4×C53C2311yescharacteristic
62C22⋊Q8×C53—28no—
63C22⋊Q8×C53—28no—
64#121×C53—28no—
65#121×C53—28no—
66#121×C53—28no—
67#121×C53—28no—
68C2×C4⋊C4×C53—22no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53D412yes—
72C2×C4⋊C4×C53C2312yes—
73C2×C22⋊C4×C53—24no—
74C24×C53—22no—
75C24×C53C2×D411yescharacteristic
76#41×C53—28no—
77#41×C53—28no—
78#41×C53—44no—
79#41×C53—44no—
80#41×C53—48no—
81#41×C53—48no—
82C2×Q8×C53—22no—
83C2×Q8×C53—44no—
84C2×Q8×C53C2×D414yes—
85C2×D4×C53—24no—
86C2×D4×C53—24no—
87C2×D4×C53—28no—
88C2×D4×C53—22no—
89C2×D4×C53—44no—
90C2×D4×C53—48no—
91C2×D4×C53—84no—
92C2×D4×C53C2×D414yes—
93C22×C4×C53—24no—
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—24no—
99C22×C4×C53—48no—
100C22×C4×C53—44no—
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D414yes—
105C22×C4×C53C2411yescharacteristic
106C22×Dic53—22no—
107C22×Dic53—21no—
108C22×Dic53—21no—
109C2×Dic106—22no—
110C2×Dic106—24no—
111C4⋊C4×C53—28no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C22⋊C4×C53—216no—
115C22⋊Dic53—42no—
116C22⋊Dic53—41no—
117C22⋊Dic53—41no—
118#11719925—48no—
119#11719923—48no—
120C4⋊Dic53—22no—
121#11719919—42no—
122C4×Dic53—24no—
123#11719917—28no—
124C2×#88322—24no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—24no—
128C23×C53—41no—
129C23×C53—44no—
130C23×C53—44no—
131C23×C53—44no—
132C23×C53#12011yescharacteristic
133C23×C53#12011yescharacteristic
134C23×C53C22×D411yescharacteristic
135Q8×C53—24no—
136Q8×C53—24no—
137Q8×C53—44no—
138D4×C53—24no—
139D4×C53—24no—
140D4×C53—44no—
141D4×C53—48no—
142D4×C53—48no—
143D4×C53—88no—
144C2×C4×C53—22no—
145C2×C4×C53—24no—
146C2×C4×C53—24no—
147C2×C4×C53—24no—
148C2×C4×C53—24no—
149C2×C4×C53—48no—
150C2×C4×C53—44no—
151C2×C4×C53—44no—
152C2×C4×C53—48no—
153C2×C4×C53—44no—
154C2×C4×C53—82no—
155C2×C4×C53#12012yes—
156C2×C4×C53C22×D412yes—
157C2×C4×C53H(5,𝔽2)14yes—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—42no—
164C2×Dic53—42no—
165Dic106—44no—
166#88322—44no—
167#29618—531nomaximal, Sylow
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—44no—
176C22×C53—44no—
177C22×C53—44no—
178C22×C53—44no—
179C22×C53—88no—
180C22×C53#45512yes—
181C22×C53C2×#12011yescharacteristic
182C22×C53#53512yes—
183C4×C53—22no—
184C4×C53—44no—
185C4×C53—44no—
186C4×C53#53414yes—
187Dic53—82no—
188Dic53—81no—
189Dic53—81no—
190C2×#537D5311yescharacteristic
191#3007—532no—
192#2503—531no—
193#2040—534no—
194#2007—534no—
195#1821—531no—
196#1789—532no—
197C2×C53—22no—
198C2×C53—22no—
199C2×C53—41no—
200C2×C53—84no—
201C2×C53#261311yescharacteristic
202C2×C53C2×#45511yescharacteristic
203C2×C53#302011yescharacteristic
204C22×#41C2×D5312yes—
205C2×#137—534no—
206#537—24no—
207#537—22no—
208#537C2×D5314yes—
209C2×C22⋊Q8—22no—
210C2×C22⋊Q8—531no—
211C2×#121—22no—
212C2×#121—22no—
213C22×C4⋊C4C2×D5311yescharacteristic
214#452—1064no—
215#451—1062no—
216#450—1062no—
217C2×#104—538no—
218#417—534no—
219#407—532no—
220#400—1062no—
221#391—532no—
222#382—532no—
223#381—1062no—
224C4×C22⋊C4—531no—
225#343—534no—
226C53#2961811yescharacteristic, complemented, Sylow
227C2×#41—22no—
228C2×#41—22no—
229C2×#41—48no—
230C2×#41C22⋊D5314yes—
231C2×#41C22×D5314yes—
232C22×Q8—531no—
233C22×Q8C22⋊D5312yes—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4C22⋊D5312yes—
237C23×C4—22no—
238C23×C4C22⋊D5312yes—
239C23×C4C22×D5311yescharacteristic
240#137—2128no—
241C2×Q16—1064no—
242C2×SD16—1064no—
243C2×M16—534no—
244C22⋊Q8—28no—
245C22⋊Q8—28no—
246C22⋊Q8—1062no—
247C22⋊Q8—1062no—
248#121—28no—
249#121—28no—
250#121—28no—
251#121—28no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—531no—
256C2×C4⋊C4—1061no—
257C2×C4⋊C4C22⋊D5312yes—
258C2×C4⋊C4C22×D5312yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4—1061no—
262C2×C42—532no—
263#104—2128no—
264#103—1064no—
265#102—1064no—
266C22⋊C8—1062no—
267C22⋊C8—1062no—
268#95—1061no—
269#95—1064no—
270#95—1064no—
271C24—22no—
272C24C2×C22⋊D5311yescharacteristic
273#41—28no—
274#41—28no—
275#41—44no—
276#41—44no—
277#41—48no—
278#41—48no—
279C2×Q8—22no—
280C2×Q8—44no—
281C2×Q8—1062no—
282C2×Q8—1064no—
283C2×Q8C2×C22⋊D5314yes—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—28no—
287C2×D4—22no—
288C2×D4—44no—
289C2×D4—48no—
290C2×D4—84no—
291C2×D4C2×C22⋊D5314yes—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—1062no—
301C22×C4—1061no—
302C22×C4—1061no—
303C22×C4C2×C22⋊D5311yescharacteristic
304C22×C4C2×C22⋊D5312yes—
305C22×C4C2×C22⋊D5312yes—
306C22×C4C2×C22⋊D5314yes—
307C22×C4C23×D5311yescharacteristic
308Q16—2128no—
309SD16—2128no—
310M16—1068no—
311C2×C8—1064no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—1062no—
316C4⋊C4—2122no—
317C22⋊C4—216no—
318C22⋊C4—2122no—
319C22⋊C4—2121no—
320C22⋊C4—2121no—
321C42—1064no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—41no—
326C23—44no—
327C23—44no—
328C23—44no—
329C23#1313318611yescharacteristic
330C23#1313318611yescharacteristic
331C23C22×C22⋊D5311yescharacteristic
332Q8—24no—
333Q8—24no—
334Q8—44no—
335Q8—2124no—
336D4—24no—
337D4—24no—
338D4—44no—
339D4—48no—
340D4—48no—
341D4—88no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—82no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4#1313318612yes—
361C2×C4C22×C22⋊D5312yes—
362C2×C4#1313326214yes—
363C8—2124no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—44no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—88no—
376C22#1681080112yes—
377C22C2×#1313318611yescharacteristic
378C22#1681119012yes—
379C4—22no—
380C4—44no—
381C4—44no—
382C4—4242no—
383C4—4241no—
384C4—4241no—
385C4#1681118214yes—
386C2—22no—
387C2—22no—
388C2—41no—
389C2—84no—
390C2#2028042711yescharacteristic
391C2C2×#1681080111yescharacteristic
392C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#29618
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20283975C2semidirect
2. #20279196C2semidirect
3. #20275941C2semidirect
4. #20275762C2semidirect
5. #20274795C2semidirect
6. #20274659C2semidirect
8. C2×#13133247C22semidirect
13. C2×#13133144C22semidirect
14. #16810624C22semidirect
17. #16810528C22semidirect
18. #16810488C22semidirect
21. #16810392C22semidirect
226. C53#29618semidirect

Maximal subgroups

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

subgroupindex
#202839752
#202791962
#202759412
#202757622
#202747952
#202746592
C2×C53×#5372
#2961853