Finite Group Databasebeta

← #27489644

Subgroups — #27489644

Order 13568.

1358 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1358 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#20283967C212yesmaximal, complemented
2#20279203C211yesmaximal, characteristic, complemented
3#20275941C214yesmaximal, complemented
4#20275757C214yesmaximal, complemented
5#20274795C211yesmaximal, characteristic, complemented
6#20274659C212yesmaximal, complemented
7C2×C53×#537C211yesmaximal, characteristic, complemented
8C2×#13133242C2214yescomplemented
9C2×#13133191C2211yescharacteristic
10#16810768—24no—
11#16810762—22no—
12#16810757—22no—
13C2×#13133144C2218yescomplemented
14#16810624C2214yescomplemented
15#16810587C2212yes—
16#16810549—22no—
17#16810524C2212yescomplemented
18#16810502—22no—
19#16810488C2212yescomplemented
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×D212C2311yescharacteristic
31C22×C22⋊D53—21no—
32#13133242—48no—
33C2×#11719924—24no—
34C2×#11719922—24no—
35C2×#11719917C2314yes—
36#13133191—22no—
37#13133191—22no—
38C2×C4⋊Dic53C2311yescharacteristic
39C2×C22⋊Dic53—21no—
40C2×#11719921—21no—
41C2×#11719921—24no—
42C2×C4×Dic53C2312yes—
43#13133144—48no—
44#13133141—24no—
45#13133138—24no—
46C22⋊#88322—22no—
47C22⋊#88322—22no—
48#13133124—24no—
49#13133124—21no—
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
76C23×D53—21no—
77#41×C53—28no—
78#41×C53—28no—
79#41×C53—44no—
80#41×C53—44no—
81#41×C53—48no—
82#41×C53—48no—
83C2×Q8×C53—22no—
84C2×Q8×C53—44no—
85C2×Q8×C53C2×D414yes—
86C2×D4×C53—24no—
87C2×D4×C53—24no—
88C2×D4×C53—28no—
89C2×D4×C53—22no—
90C2×D4×C53—44no—
91C2×D4×C53—48no—
92C2×D4×C53—84no—
93C2×D4×C53C2×D414yes—
94C22×C4×C53—24no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—24no—
99C22×C4×C53—24no—
100C22×C4×C53—48no—
101C22×C4×C53—82no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D412yes—
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D414yes—
106C22×C4×C53C2411yescharacteristic
107C2×C22⋊D53—44no—
108C2×C22⋊D53—41no—
109C2×C22⋊D53—41no—
110C22×Dic53—21no—
111C22×Dic53—22no—
112C2×D212—22no—
113C2×D212—24no—
114C4⋊C4×C53—28no—
115C4⋊C4×C53—24no—
116C4⋊C4×C53—24no—
117C22⋊C4×C53—216no—
118C22⋊Dic53—42no—
119#11719924—48no—
120#11719922—48no—
121#11719921—42no—
122#11719921—84no—
123C4⋊Dic53—22no—
124C4×Dic53—24no—
125#11719917—28no—
126C2×#88322—24no—
127C23×C53—22no—
128C23×C53—22no—
129C23×C53—24no—
130C23×C53—41no—
131C23×C53—44no—
132C23×C53—44no—
133C23×C53—44no—
134C23×C53#12011yescharacteristic
135C23×C53#12011yescharacteristic
136C23×C53C22×D411yescharacteristic
137C22×D53—42no—
138C22×D53—42no—
139C22×D53—41no—
140C22×D53—42no—
141Q8×C53—24no—
142Q8×C53—24no—
143Q8×C53—44no—
144D4×C53—24no—
145D4×C53—24no—
146D4×C53—44no—
147D4×C53—48no—
148D4×C53—48no—
149D4×C53—88no—
150C2×C4×C53—22no—
151C2×C4×C53—24no—
152C2×C4×C53—24no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—48no—
156C2×C4×C53—44no—
157C2×C4×C53—44no—
158C2×C4×C53—48no—
159C2×C4×C53—44no—
160C2×C4×C53—82no—
161C2×C4×C53#12012yes—
162C2×C4×C53C22×D412yes—
163C2×C4×C53H(5,𝔽2)14yes—
164C22⋊D53—82no—
165C22⋊D53—82no—
166C2×Dic53—42no—
167C2×Dic53—42no—
168C2×Dic53—42no—
169C2×Dic53—41no—
170C2×Dic53—42no—
171D212—44no—
172#88322—44no—
173#29617—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—44no—
182C22×C53—44no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—88no—
186C22×C53#45412yes—
187C22×C53C2×#12011yescharacteristic
188C22×C53#53512yes—
189C2×D53—84no—
190C2×D53—81no—
191C2×D53—81no—
192C2×D53—81no—
193C4×C53—22no—
194C4×C53—44no—
195C4×C53—44no—
196C4×C53#53414yes—
197Dic53—82no—
198Dic53—81no—
199C2×#537D5311yescharacteristic, complemented
200#3006—532no—
201#2504—531no—
202#2040—534no—
203#2006—534no—
204#1821—531no—
205#1789—532no—
206C2×C53—22no—
207C2×C53—22no—
208C2×C53—41no—
209C2×C53—84no—
210C2×C53#261311yescharacteristic
211C2×C53C2×#45411yescharacteristic
212C2×C53#301911yescharacteristic
213D53—161no—
214C22×#41C2×D5312yes—
215C2×AGL(1,ℤ/8)—534no—
216#537—24no—
217#537—22no—
218#537C2×D5314yes—
219C2×C22⋊Q8—22no—
220C2×#121—22no—
221C2×#121—22no—
222C2×#121—531no—
223C22×C4⋊C4C2×D5311yescharacteristic
224#449—1064no—
225#448—1062no—
226#447—1062no—
227C2×#104—538no—
228#417—534no—
229#407—532no—
230#397—1062no—
231#390—532no—
232#386—1062no—
233#382—532no—
234C4×C22⋊C4—531no—
235#343—534no—
236C53#2961711yescharacteristic, complemented, Sylow
237C2×#41—22no—
238C2×#41—22no—
239C2×#41—48no—
240C2×#41C22⋊D5314yes—
241C2×#41C22×D5314yes—
242C22×Q8C22⋊D5312yes—
243C22×D4—22no—
244C22×D4—22no—
245C22×D4—531no—
246C22×D4—1061no—
247C22×D4C22⋊D5312yes—
248C23×C4—22no—
249C23×C4C22⋊D5312yes—
250C23×C4C22×D5311yescharacteristic
251AGL(1,ℤ/8)—2128no—
252C2×SD16—1064no—
253C2×D8—1064no—
254C2×M16—534no—
255C22⋊Q8—28no—
256C22⋊Q8—28no—
257#121—28no—
258#121—28no—
259#121—28no—
260#121—28no—
261#121—1062no—
262#121—1062no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—531no—
267C2×C4⋊C4C22⋊D5312yes—
268C2×C4⋊C4C22×D5312yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—1064no—
271C2×C22⋊C4—1061no—
272C2×C22⋊C4—1061no—
273C2×C42—532no—
274#104—2128no—
275#103—1064no—
276#102—1064no—
277C22⋊C8—1062no—
278C22⋊C8—1062no—
279#95—1061no—
280#95—1064no—
281C24—22no—
282C24—1061no—
283C24C2×C22⋊D5311yescharacteristic
284#41—28no—
285#41—28no—
286#41—44no—
287#41—44no—
288#41—48no—
289#41—48no—
290C2×Q8—22no—
291C2×Q8—44no—
292C2×Q8C2×C22⋊D5314yes—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—28no—
296C2×D4—22no—
297C2×D4—44no—
298C2×D4—48no—
299C2×D4—84no—
300C2×D4—1062no—
301C2×D4—1064no—
302C2×D4—2124no—
303C2×D4—2121no—
304C2×D4—2121no—
305C2×D4C2×C22⋊D5314yes—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—48no—
313C22×C4—82no—
314C22×C4—1061no—
315C22×C4—1062no—
316C22×C4C2×C22⋊D5311yescharacteristic
317C22×C4C2×C22⋊D5312yes—
318C22×C4C2×C22⋊D5312yes—
319C22×C4C2×C22⋊D5314yes—
320C22×C4C23×D5311yescharacteristic
321SD16—2128no—
322D8—2128no—
323M16—1068no—
324C2×C8—1064no—
325C4⋊C4—28no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—1062no—
329C22⋊C4—216no—
330C22⋊C4—4244no—
331C22⋊C4—2122no—
332C22⋊C4—2122no—
333C42—1064no—
334C23—22no—
335C23—22no—
336C23—24no—
337C23—41no—
338C23—44no—
339C23—44no—
340C23—44no—
341C23—2122no—
342C23—2122no—
343C23—2121no—
344C23—2122no—
345C23#1313318611yescharacteristic
346C23#1313318611yescharacteristic
347C23C22×C22⋊D5311yescharacteristic
348Q8—24no—
349Q8—24no—
350Q8—44no—
351D4—24no—
352D4—24no—
353D4—44no—
354D4—48no—
355D4—48no—
356D4—88no—
357D4—4242no—
358D4—4242no—
359D4—2124no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—82no—
371C2×C4—2122no—
372C2×C4—2122no—
373C2×C4—2122no—
374C2×C4—2121no—
375C2×C4—2122no—
376C2×C4#1313318612yes—
377C2×C4C22×C22⋊D5312yes—
378C2×C4#1313326214yes—
379C8—2124no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—44no—
391C22—88no—
392C22—4244no—
393C22—4241no—
394C22—4241no—
395C22—4241no—
396C22#1681079612yes—
397C22C2×#1313318611yescharacteristic
398C22#1681119012yes—
399C4—22no—
400C4—44no—
401C4—44no—
402C4—4242no—
403C4—4241no—
404C4#1681118214yes—
405C2—22no—
406C2—22no—
407C2—41no—
408C2—84no—
409C2—8481no—
410C2#2028042711yescharacteristic
411C2C2×#1681079611yescharacteristic
412C2#2028407211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#29617
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. #20283967C2semidirect
2. #20279203C2semidirect
3. #20275941C2semidirect
4. #20275757C2semidirect
5. #20274795C2semidirect
6. #20274659C2semidirect
7. C2×C53×#537C2semidirect
8. C2×#13133242C22semidirect
13. C2×#13133144C22semidirect
14. #16810624C22semidirect
17. #16810524C22semidirect
19. #16810488C22semidirect
21. #16810392C22semidirect
199. C2×#537D53semidirect
236. C53#29617semidirect

Maximal subgroups

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

subgroupindex
#202839672
#202792032
#202759412
#202757572
#202747952
#202746592
C2×C53×#5372
#2961753