Finite Group Databasebeta

← #27489637

Subgroups — #27489637

Order 13568.

1414 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1414 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#20279182C211yesmaximal, characteristic, complemented
3#20275936C214yesmaximal, complemented
4#20275757C214yesmaximal, complemented
5#20274795C211yesmaximal, characteristic, complemented
6#20274659C212yesmaximal, complemented
7C2×C53×#537C211yesmaximal, characteristic, complemented
8C2×#13133242C2214yescomplemented
9C2×#13133200—22no—
10C2×#13133191C2211yescharacteristic
11#16810768—24no—
12#16810762—22no—
13#16810757—22no—
14C2×#13133144C2218yescomplemented
15#16810624C2214yescomplemented
16#16810587C2212yes—
17#16810558—22no—
18#16810524C2212yescomplemented
19#16810492C2212yescomplemented
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#13133200—48no—
37#13133191—22no—
38#13133191—22no—
39C2×C4⋊Dic53C2311yescharacteristic
40C2×#11719919—24no—
41C2×C22⋊Dic53—21no—
42C2×#11719921—24no—
43C2×#11719921—21no—
44C2×C4×Dic53C2312yes—
45#13133144—48no—
46#13133141—24no—
47#13133138—24no—
48C22⋊#88322—22no—
49C22⋊#88322—22no—
50#13133124—21no—
51C2×#41×C53—22no—
52C2×#41×C53—22no—
53C2×#41×C53—48no—
54C2×#41×C53D414yes—
55C2×#41×C53C2314yes—
56C22×Q8×C53D412yes—
57C22×D4×C53—22no—
58C22×D4×C53—22no—
59C22×D4×C53D412yes—
60C23×C4×C53—22no—
61C23×C4×C53D412yes—
62C23×C4×C53C2311yescharacteristic
63C22⋊Q8×C53—28no—
64C22⋊Q8×C53—28no—
65#121×C53—28no—
66#121×C53—28no—
67#121×C53—28no—
68#121×C53—28no—
69C2×C4⋊C4×C53—22no—
70C2×C4⋊C4×C53—22no—
71C2×C4⋊C4×C53—22no—
72C2×C4⋊C4×C53D412yes—
73C2×C4⋊C4×C53C2312yes—
74C2×C22⋊C4×C53—24no—
75C24×C53—22no—
76C24×C53C2×D411yescharacteristic
77C23×D53—21no—
78#41×C53—28no—
79#41×C53—28no—
80#41×C53—44no—
81#41×C53—44no—
82#41×C53—48no—
83#41×C53—48no—
84C2×Q8×C53—22no—
85C2×Q8×C53—44no—
86C2×Q8×C53C2×D414yes—
87C2×D4×C53—24no—
88C2×D4×C53—24no—
89C2×D4×C53—28no—
90C2×D4×C53—22no—
91C2×D4×C53—44no—
92C2×D4×C53—48no—
93C2×D4×C53—84no—
94C2×D4×C53C2×D414yes—
95C22×C4×C53—24no—
96C22×C4×C53—22no—
97C22×C4×C53—22no—
98C22×C4×C53—22no—
99C22×C4×C53—24no—
100C22×C4×C53—24no—
101C22×C4×C53—48no—
102C22×C4×C53—44no—
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D412yes—
105C22×C4×C53C2×D412yes—
106C22×C4×C53C2×D414yes—
107C22×C4×C53C2411yescharacteristic
108C2×C22⋊D53—44no—
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C22×Dic53—22no—
112C22×Dic53—21no—
113C2×D212—22no—
114C2×D212—24no—
115C4⋊C4×C53—28no—
116C4⋊C4×C53—24no—
117C4⋊C4×C53—24no—
118C22⋊C4×C53—216no—
119C22⋊Dic53—24no—
120#11719924—48no—
121#11719922—48no—
122#11719921—48no—
123#11719921—42no—
124C4⋊Dic53—22no—
125#11719919—48no—
126C4×Dic53—24no—
127#11719917—28no—
128C2×#88322—24no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—24no—
132C23×C53—41no—
133C23×C53—44no—
134C23×C53—44no—
135C23×C53—44no—
136C23×C53#12011yescharacteristic
137C23×C53#12011yescharacteristic
138C23×C53C22×D411yescharacteristic
139C22×D53—42no—
140C22×D53—42no—
141C22×D53—41no—
142C22×D53—42no—
143Q8×C53—24no—
144Q8×C53—24no—
145Q8×C53—44no—
146D4×C53—24no—
147D4×C53—24no—
148D4×C53—44no—
149D4×C53—48no—
150D4×C53—48no—
151D4×C53—88no—
152C2×C4×C53—22no—
153C2×C4×C53—24no—
154C2×C4×C53—24no—
155C2×C4×C53—24no—
156C2×C4×C53—24no—
157C2×C4×C53—48no—
158C2×C4×C53—44no—
159C2×C4×C53—44no—
160C2×C4×C53—48no—
161C2×C4×C53—44no—
162C2×C4×C53—82no—
163C2×C4×C53#12012yes—
164C2×C4×C53C22×D412yes—
165C2×C4×C53H(5,𝔽2)14yes—
166C22⋊D53—82no—
167C22⋊D53—82no—
168C2×Dic53—24no—
169C2×Dic53—42no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—41no—
173D212—44no—
174#88322—44no—
175#29616—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—21no—
179C22×C53—21no—
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—44no—
186C22×C53—44no—
187C22×C53—88no—
188C22×C53#45312yes—
189C22×C53C2×#12011yescharacteristic
190C22×C53#53512yes—
191C2×D53—84no—
192C2×D53—81no—
193C2×D53—81no—
194C2×D53—81no—
195C4×C53—22no—
196C4×C53—44no—
197C4×C53—44no—
198C4×C53#53414yes—
199Dic53—44no—
200Dic53—81no—
201C2×#537D5311yescharacteristic, complemented
202#3006—532no—
203#2501—531no—
204#2039—534no—
205#2006—534no—
206#1821—531no—
207#1789—532no—
208C2×C53—22no—
209C2×C53—22no—
210C2×C53—41no—
211C2×C53—84no—
212C2×C53#261311yescharacteristic
213C2×C53C2×#45311yescharacteristic
214C2×C53#302011yescharacteristic
215D53—161no—
216C22×#41C2×D5312yes—
217C2×AGL(1,ℤ/8)—534no—
218#537—24no—
219#537—22no—
220#537C2×D5314yes—
221C2×#123—1062no—
222C2×C22⋊Q8—22no—
223C2×#121—22no—
224C2×#121—22no—
225C2×#121—531no—
226C22×C4⋊C4C2×D5311yescharacteristic
227#449—1064no—
228#448—1062no—
229#447—1062no—
230C2×#104—538no—
231#417—534no—
232#407—532no—
233#399—1062no—
234#390—532no—
235#384—532no—
236C4×C22⋊C4—531no—
237#343—534no—
238C53#2961611yescharacteristic, complemented, Sylow
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—48no—
242C2×#41C22⋊D5314yes—
243C2×#41C22×D5314yes—
244C22×Q8C22⋊D5312yes—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—531no—
248C22×D4—1061no—
249C22×D4C22⋊D5312yes—
250C23×C4—22no—
251C23×C4C22⋊D5312yes—
252C23×C4C22×D5311yescharacteristic
253AGL(1,ℤ/8)—2128no—
254C2×SD16—1064no—
255C2×D8—1064no—
256C2×M16—534no—
257#123—2128no—
258C22⋊Q8—28no—
259C22⋊Q8—28no—
260#121—28no—
261#121—28no—
262#121—28no—
263#121—28no—
264#121—1062no—
265#121—1062no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—531no—
270C2×C4⋊C4—1064no—
271C2×C4⋊C4C22⋊D5312yes—
272C2×C4⋊C4C22×D5312yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—1061no—
275C2×C22⋊C4—1061no—
276C2×C22⋊C4—1064no—
277C2×C42—532no—
278#104—2128no—
279#103—1064no—
280#102—1064no—
281C22⋊C8—1062no—
282C22⋊C8—1062no—
283#95—1061no—
284C24—22no—
285C24—1061no—
286C24C2×C22⋊D5311yescharacteristic
287#41—28no—
288#41—28no—
289#41—44no—
290#41—44no—
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—44no—
295C2×Q8C2×C22⋊D5314yes—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—28no—
299C2×D4—22no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—84no—
303C2×D4—1062no—
304C2×D4—1064no—
305C2×D4—2124no—
306C2×D4—2121no—
307C2×D4—2121no—
308C2×D4C2×C22⋊D5314yes—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—48no—
316C22×C4—44no—
317C22×C4—1062no—
318C22×C4—1061no—
319C22×C4C2×C22⋊D5311yescharacteristic
320C22×C4C2×C22⋊D5312yes—
321C22×C4C2×C22⋊D5312yes—
322C22×C4C2×C22⋊D5314yes—
323C22×C4C23×D5311yescharacteristic
324SD16—2128no—
325D8—2128no—
326M16—1068no—
327C2×C8—1064no—
328C4⋊C4—28no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—1062no—
332C4⋊C4—2128no—
333C22⋊C4—216no—
334C22⋊C4—1064no—
335C22⋊C4—2128no—
336C22⋊C4—2122no—
337C42—1064no—
338C23—22no—
339C23—22no—
340C23—24no—
341C23—41no—
342C23—44no—
343C23—44no—
344C23—44no—
345C23—2122no—
346C23—2122no—
347C23—2121no—
348C23—2122no—
349C23#1313318611yescharacteristic
350C23#1313318611yescharacteristic
351C23C22×C22⋊D5311yescharacteristic
352Q8—24no—
353Q8—24no—
354Q8—44no—
355D4—24no—
356D4—24no—
357D4—44no—
358D4—48no—
359D4—48no—
360D4—88no—
361D4—4242no—
362D4—4242no—
363D4—2124no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—48no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—44no—
374C2×C4—82no—
375C2×C4—1064no—
376C2×C4—2122no—
377C2×C4—2121no—
378C2×C4—2122no—
379C2×C4—2122no—
380C2×C4#1313318612yes—
381C2×C4C22×C22⋊D5312yes—
382C2×C4#1313326214yes—
383C8—2124no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—44no—
395C22—88no—
396C22—4244no—
397C22—4241no—
398C22—4241no—
399C22—4241no—
400C22#1681079112yes—
401C22C2×#1313318611yescharacteristic
402C22#1681119012yes—
403C4—22no—
404C4—44no—
405C4—44no—
406C4—4241no—
407C4—2124no—
408C4#1681118214yes—
409C2—22no—
410C2—22no—
411C2—41no—
412C2—84no—
413C2—8481no—
414C2#2028042711yescharacteristic
415C2C2×#1681079111yescharacteristic
416C2#2028407911yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#29616
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. #20279182C2semidirect
3. #20275936C2semidirect
4. #20275757C2semidirect
5. #20274795C2semidirect
6. #20274659C2semidirect
7. C2×C53×#537C2semidirect
8. C2×#13133242C22semidirect
14. C2×#13133144C22semidirect
15. #16810624C22semidirect
18. #16810524C22semidirect
19. #16810492C22semidirect
21. #16810392C22semidirect
201. C2×#537D53semidirect
238. C53#29616semidirect

Maximal subgroups

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

subgroupindex
#202839672
#202791822
#202759362
#202757572
#202747952
#202746592
C2×C53×#5372
#2961653