Finite Group Databasebeta

← #27358273

Subgroups — #27358273

Order 13568.

1230 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 183 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#20285521C212yesmaximal, complemented
2#20280095C211yesmaximal, characteristic, complemented
3#20273718C214yesmaximal, complemented
4#20273684C214yesmaximal, complemented
5#20273231C212yesmaximal, complemented
6#20273045C211yesmaximal, characteristic, complemented
7C2×C53×#537C211yesmaximal, characteristic
8C2×#13133196C2211yescharacteristic
9#16810958—22no—
10#16810934C2214yescomplemented
11#16810867—24no—
12#16810774—22no—
13#16810761C2214yescomplemented
14#16810587C2212yes—
15#16810553C2212yescomplemented
16#16810543C2212yescomplemented
17#16810518—22no—
18#16810496—21no—
19#16810496—21no—
20C4×C22⋊Dic53C2211yescharacteristic
21#16810357C2218yescomplemented
22#16810349C2214yescomplemented
23C22×#41×C53—21no—
24C53×#537—24no—
25C53×#537—22no—
26C53×#537C2214yes—
27C2×C22⋊Q8×C53—22no—
28C2×#121×C53—22no—
29C2×#121×C53—21no—
30C2×#121×C53C2212yes—
31C22×C4⋊C4×C53C2211yescharacteristic
32C22×Dic106C2311yescharacteristic
33C2×#11719923—24no—
34C2×#11719917—22no—
35#13133196—22no—
36#13133196—22no—
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—22no—
39C2×#11719919—21no—
40C2×#11719919—22no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133150—24no—
45#13133140—24no—
46#13133138—24no—
47C22⋊#88322—22no—
48C22⋊#88322C2314yes—
49#13133124—22no—
50#13133124—22no—
51#13133124C2312yes—
52C2×#41×C53—24no—
53C2×#41×C53—24no—
54C2×#41×C53—48no—
55C22×Q8×C53—21no—
56C22×D4×C53—21no—
57C22×D4×C53—22no—
58C22×D4×C53—21no—
59C22×D4×C53D412yes—
60C23×C4×C53—22no—
61C23×C4×C53D411yescharacteristic
62C23×C4×C53D411yescharacteristic
63C23×C4×C53C2311yescharacteristic
64C22⋊Q8×C53—28no—
65C22⋊Q8×C53—28no—
66#121×C53—28no—
67#121×C53—24no—
68#121×C53—28no—
69#121×C53—24no—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53D414yes—
73#121×C53C2314yes—
74C2×C4⋊C4×C53—21no—
75C2×C4⋊C4×C53—22no—
76C2×C4⋊C4×C53—22no—
77C2×C4⋊C4×C53D412yes—
78C2×C4⋊C4×C53D412yes—
79C2×C4⋊C4×C53C2312yes—
80C2×C22⋊C4×C53—22no—
81C2×C22⋊C4×C53—22no—
82C24×C53—22no—
83C24×C53C2×D411yescharacteristic
84#41×C53—416no—
85#41×C53—48no—
86#41×C53—48no—
87C2×Q8×C53—24no—
88C2×Q8×C53—44no—
89C2×D4×C53—24no—
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—22no—
94C2×D4×C53—44no—
95C2×D4×C53—48no—
96C2×D4×C53—82no—
97C2×D4×C53—82no—
98C2×D4×C53C2×D414yes—
99C22×C4×C53—21no—
100C22×C4×C53—21no—
101C22×C4×C53—24no—
102C22×C4×C53—28no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—48no—
107C22×C4×C53—81no—
108C22×C4×C53—81no—
109C22×C4×C53C2×D412yes—
110C22×C4×C53C2×D412yes—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D412yes—
113C22×C4×C53C2×D412yes—
114C22×C4×C53C2411yescharacteristic
115C22×Dic53—22no—
116C22×Dic53—21no—
117C22×Dic53—21no—
118C2×Dic106—22no—
119C2×Dic106—81no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—24no—
124C4⋊C4×C53C2×D414yes—
125C22⋊C4×C53—28no—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—24no—
128C22⋊Dic53—24no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131#11719923—84no—
132C4⋊Dic53—22no—
133#11719919—42no—
134#11719919—44no—
135#11719919—82no—
136C4×Dic53—81no—
137#11719917—82no—
138C2×#88322—24no—
139C23×C53—22no—
140C23×C53—22no—
141C23×C53—24no—
142C23×C53—41no—
143C23×C53—44no—
144C23×C53—44no—
145C23×C53—44no—
146C23×C53#12011yescharacteristic
147C23×C53#12011yescharacteristic
148C23×C53C22×D411yescharacteristic
149Q8×C53—48no—
150D4×C53—44no—
151D4×C53—48no—
152D4×C53—48no—
153D4×C53—48no—
154D4×C53—82no—
155D4×C53—84no—
156C2×C4×C53—24no—
157C2×C4×C53—24no—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—24no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—416no—
167C2×C4×C53—44no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—81no—
171C2×C4×C53#12012yes—
172C2×C4×C53AGL(1,ℤ/8)14yes—
173C2×C4×C53C22×D412yes—
174C2×Dic53—24no—
175C2×Dic53—42no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—41no—
179C2×Dic53—41no—
180C2×Dic53—42no—
181Dic106—44no—
182#88322—44no—
183#18325—531nomaximal, Sylow
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—21no—
187C22×C53—21no—
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—21no—
191C22×C53—44no—
192C22×C53—44no—
193C22×C53—44no—
194C22×C53—44no—
195C22×C53—88no—
196C22×C53#45612yes—
197C22×C53C2×#12011yescharacteristic
198C22×C53C2×AGL(1,ℤ/8)12yes—
199C4×C53—24no—
200C4×C53—22no—
201C4×C53—22no—
202C4×C53—44no—
203C4×C53—42no—
204Dic53—44no—
205Dic53—81no—
206Dic53—81no—
207C2×#537D5311yescharacteristic
208#3193—532no—
209#2583—531no—
210#1610—534no—
211#1604—534no—
212#1507—532no—
213#1463—531no—
214C2×C53—22no—
215C2×C53—22no—
216C2×C53—41no—
217C2×C53—84no—
218C2×C53#300311yescharacteristic
219C2×C53C2×#45611yescharacteristic
220C2×C53#302611yescharacteristic
221C22×#41—21no—
222#537—24no—
223#537—22no—
224#537C2×D5314yes—
225C2×C22⋊Q8—22no—
226C2×C22⋊Q8—531no—
227C2×#121—22no—
228C2×#121—21no—
229C2×#121C2×D5312yes—
230C22×C4⋊C4C2×D5311yescharacteristic
231#484—1062no—
232#480—534no—
233#468—1064no—
234#450—1062no—
235#448—534no—
236#407—532no—
237#398—532no—
238#395—532no—
239#389—1062no—
240#385—1061no—
241#385—1061no—
242C4×C22⋊C4—531no—
243#327—538no—
244#324—534no—
245C53#1832511yescharacteristic, complemented, Sylow
246C2×#41—24no—
247C2×#41—24no—
248C2×#41—48no—
249C22×Q8—21no—
250C22×Q8—531no—
251C22×D4—21no—
252C22×D4—22no—
253C22×D4—21no—
254C22×D4C22⋊D5312yes—
255C23×C4—22no—
256C23×C4C22⋊D5311yescharacteristic
257C23×C4C22⋊D5311yescharacteristic
258C23×C4C22×D5311yescharacteristic
259C2×SD16—1064no—
260C2×M16—1062no—
261C22⋊Q8—28no—
262C22⋊Q8—28no—
263C22⋊Q8—1062no—
264C22⋊Q8—1062no—
265#121—24no—
266#121—28no—
267#121—28no—
268#121—24no—
269#121—22no—
270#121—22no—
271#121C22⋊D5314yes—
272#121C22×D5314yes—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—531no—
277C2×C4⋊C4—1062no—
278C2×C4⋊C4—1061no—
279C2×C4⋊C4—1062no—
280C2×C4⋊C4C22⋊D5312yes—
281C2×C4⋊C4C22⋊D5312yes—
282C2×C4⋊C4C22×D5312yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—1061no—
286C2×C22⋊C4—1061no—
287C2×C42—1061no—
288#107—1064no—
289#103—1064no—
290#102—1064no—
291C22⋊C8—534no—
292C22⋊C8—1062no—
293#95—532no—
294#95—1062no—
295#95—1062no—
296C24—22no—
297C24C2×C22⋊D5311yescharacteristic
298#41—416no—
299#41—48no—
300#41—48no—
301C2×Q8—24no—
302C2×Q8—44no—
303C2×Q8—1062no—
304C2×Q8—4241no—
305C2×D4—24no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—24no—
309C2×D4—22no—
310C2×D4—44no—
311C2×D4—48no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4C2×C22⋊D5314yes—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—48no—
323C22×C4—81no—
324C22×C4—81no—
325C22×C4—1062no—
326C22×C4—1061no—
327C22×C4—1061no—
328C22×C4C2×C22⋊D5312yes—
329C22×C4C2×C22⋊D5312yes—
330C22×C4C2×C22⋊D5311yescharacteristic
331C22×C4C2×C22⋊D5312yes—
332C22×C4C2×C22⋊D5312yes—
333C22×C4C23×D5311yescharacteristic
334SD16—4244no—
335M16—4242no—
336C2×C8—1064no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—1062no—
342C4⋊C4—4242no—
343C4⋊C4—2122no—
344C4⋊C4—2124no—
345C4⋊C4C2×C22⋊D5314yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—1064no—
350C22⋊C4—2121no—
351C22⋊C4—2121no—
352C42—4241no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—41no—
357C23—44no—
358C23—44no—
359C23—44no—
360C23#1313318611yescharacteristic
361C23#1313318611yescharacteristic
362C23C22×C22⋊D5311yescharacteristic
363Q8—48no—
364Q8—2124no—
365D4—44no—
366D4—48no—
367D4—48no—
368D4—48no—
369D4—82no—
370D4—84no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—416no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—81no—
386C2×C4—1064no—
387C2×C4—2122no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2122no—
391C2×C4—2122no—
392C2×C4—2122no—
393C2×C4#1313318612yes—
394C2×C4#1313324314yes—
395C2×C4C22×C22⋊D5312yes—
396C8—2124no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—44no—
408C22—88no—
409C22#1681080612yes—
410C22C2×#1313318611yescharacteristic
411C22C2×#1313324312yes—
412C4—24no—
413C4—22no—
414C4—22no—
415C4—44no—
416C4—42no—
417C4—4241no—
418C4—4241no—
419C4—2124no—
420C2—22no—
421C2—22no—
422C2—41no—
423C2—84no—
424C2#2028394211yescharacteristic
425C2C2×#1681080611yescharacteristic
426C2#2028411211yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#537C2
socleC22×C53C2×#120
hypercenterC2×#537D53
hypocenterC53#18325
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #20285521C2semidirect
2. #20280095C2semidirect
3. #20273718C2semidirect
4. #20273684C2semidirect
5. #20273231C2semidirect
6. #20273045C2semidirect
10. #16810934C22semidirect
13. #16810761C22semidirect
15. #16810553C22semidirect
16. #16810543C22semidirect
21. #16810357C22semidirect
22. #16810349C22semidirect
245. C53#18325semidirect

Maximal subgroups

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

subgroupindex
#202855212
#202800952
#202737182
#202736842
#202732312
#202730452
C2×C53×#5372
#1832553