Finite Group Databasebeta

← #27299235

Subgroups — #27299235

Order 13568.

548 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 123 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#20277380C211yesmaximal, characteristic
2C2×#16810412C212yesmaximal
3C4×#13133134C211yesmaximal, characteristic
4#20272832C212yesmaximal
5C4×#99×C53C211yesmaximal, characteristic
6C2×#13133198C2211yescharacteristic
7C22×#11719919C411yescharacteristic, complemented
8C2×#13133134C2212yes—
9C4×#11719917—21no—
10#16810518—21no—
11#16810517—21no—
12#16810507—21no—
13#16810503C2211yescharacteristic
14#16810496—21no—
15#16810495C411yescharacteristic, complemented
16#16810484C411yescharacteristic, complemented
17C4×C22⋊Dic53C411yescharacteristic, complemented
18#16810412C418yescomplemented
19#16810392—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C2×#99×C53C2212yes—
22C4×C22⋊C4×C53—21no—
23C53×#342—21no—
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C2×#11719917—22no—
27#13133198—22no—
28#13133198C2×C414yes—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—41no—
34C2×#11719919—41no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53C2×C411yescharacteristic
39C2×C4×Dic53—21no—
40#13133134—22no—
41#13133134C2×C414yes—
42#13133128—41no—
43C4×#88322—41no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C22×D4×C53C2311yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53D411yescharacteristic
52C4×D4×C53—41no—
53C4×D4×C53—41no—
54C2×C4⋊C4×C53D411yescharacteristic
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53—41no—
59C2×C42×C53D411yescharacteristic
60#99×C53—22no—
61#99×C53C2×C414yes—
62#95×C53—41no—
63C24×C53C22×C411yescharacteristic
64C24×C53C2×D411yescharacteristic
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53C22×C412yes—
68C2×D4×C53#4112yes—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—42no—
74C22×C4×C53—81no—
75C22×C4×C53—81no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22⋊C411yescharacteristic
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—41no—
87C22×Dic53—41no—
88C22×Dic53—41no—
89C22×Dic53—81no—
90C22×Dic53C4212yescomplemented
91C4⋊C4×C53—41no—
92C22⋊C4×C53—24no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊C4×C53—42no—
96C42×C53—21no—
97C42×C53—21no—
98C42×C53—82no—
99C22⋊Dic53—21no—
100C22⋊Dic53—24no—
101C22⋊Dic53—41no—
102C22⋊Dic53—81no—
103C22⋊Dic53C4212yescomplemented
104#11719919—24no—
105#11719919—42no—
106#11719919—41no—
107#11719919—41no—
108#11719919—81no—
109C4×Dic53—81no—
110#11719917—24no—
111#11719917—42no—
112C2×#88322—42no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C4211yescharacteristic
122C23×C53C2×C22⋊C411yescharacteristic
123C23×C53#11711yescharacteristic
124C23×C53C4×D412yes—
125D4×C53—42no—
126D4×C53—42no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—44no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53—81no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53#9911yescharacteristic
144C2×C4×C53#9911yescharacteristic
145C2×C4×C53C4×D412yes—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—41no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—42no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—82no—
161#88322—44no—
162#12621—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—22no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—42no—
171C22×C53—42no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—41no—
176C22×C53—81no—
177C22×C53C4×C22⋊C411yescharacteristic
178C22×C53C2×#9911yescharacteristic
179C22×C53#41011yescharacteristic
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—22no—
183C4×C53—41no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—84no—
187Dic53—42no—
188Dic53—42no—
189Dic53—81no—
190Dic53—81no—
191#2322—531no—
192C2×#352—532no—
193C4×#100—531no—
194C4×#99D5311yescharacteristic
195#1390—532no—
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—42no—
199C2×C53—41no—
200C2×C53—41no—
201C2×C53C4×#9911yescharacteristic
202C2×C53#212111yescharacteristic
203C2×C53#212311yescharacteristic
204C2×#123—531no—
205C2×C4×D4C2×D5311yescharacteristic
206C22×C4⋊C4—531no—
207C2×#100—532no—
208C2×#99C2×D5312yes—
209C4×M16—1061no—
210#389—1061no—
211#388—1061no—
212#387—1061no—
213#386—531no—
214#385—531no—
215#385—1061no—
216#381—531no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—531no—
219#352—538no—
220#343—1061no—
221#342—21no—
222C53#1262111yescharacteristic, complemented, Sylow
223C22×D4C22×D5311yescharacteristic
224C23×C4—531no—
225C23×C4—531no—
226C23×C4C4×D5311yescharacteristic
227C23×C4C22⋊D5311yescharacteristic
228C2×M16—1062no—
229#123—534no—
230#123—1062no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—2121no—
238C2×C4⋊C4—2121no—
239C2×C4⋊C4D21211yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—531no—
242C2×C22⋊C4—531no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C22⋊C4C4×D5311yescharacteristic
246C2×C22⋊C4C22⋊D5311yescharacteristic
247C2×C42—41no—
248C2×C42—1061no—
249C2×C42D21211yescharacteristic
250#100—534no—
251#100—1062no—
252#99—22no—
253#99C4×D5314yes—
254#97—2121no—
255C4×C8—2121no—
256#95—41no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262C24C2×C4×D5311yescharacteristic
263C24C2×C22⋊D5311yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C2×C4×D5312yes—
267C2×D4#1171994512yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—42no—
273C22×C4—81no—
274C22×C4—81no—
275C22×C4—532no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—4241no—
282C22×C4—2121no—
283C22×C4—2121no—
284C22×C4—2121no—
285C22×C4#1171992111yescharacteristic
286C22×C4#1171992111yescharacteristic
287C22×C4#1171992111yescharacteristic
288C22×C4#1171992111yescharacteristic
289C22×C4C2×D21211yescharacteristic
290M16—1064no—
291M16—2122no—
292C2×C8—2122no—
293C4⋊C4—41no—
294C4⋊C4—1064no—
295C4⋊C4—4241no—
296C4⋊C4—2122no—
297C4⋊C4—2121no—
298C4⋊C4—2121no—
299C22⋊C4—24no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—532no—
304C22⋊C4—1061no—
305C22⋊C4—1064no—
306C22⋊C4—4241no—
307C22⋊C4—2121no—
308C42—21no—
309C42—21no—
310C42—82no—
311C42—4241no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C42×D5311yescharacteristic
321C23C2×#1171992111yescharacteristic
322C23#1313317411yescharacteristic
323C23C4×C22⋊D5312yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—44no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—4242no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4—2122no—
354C2×C4—2121no—
355C2×C4—2121no—
356C2×C4—2122no—
357C2×C4#1313313111yescharacteristic
358C2×C4#1313313111yescharacteristic
359C2×C4C4×D21212yes—
360C8—2124no—
361C22—21no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—21no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C4×#1171992111yescharacteristic
376C22C2×#1313313111yescharacteristic
377C22#1681059611yescharacteristic
378C4—22no—
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—84no—
385C4—4241no—
386C4—4241no—
387C4—2122no—
388C4—2122no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#1313313111yescharacteristic
395C2#2027635911yescharacteristic
396C2#2027636811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC23×C53C2×C42
FrattiniC22×D4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenterC4×#99D53
hypocenterC53#12621
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
7. C22×#11719919C4semidirect
15. #16810495C4semidirect
16. #16810484C4semidirect
17. C4×C22⋊Dic53C4semidirect
18. #16810412C4semidirect
90. C22×Dic53C42semidirect
103. C22⋊Dic53C42semidirect
222. C53#12621semidirect

Maximal subgroups

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

subgroupindex
#202773802
C2×#168104122
C4×#131331342
#202728322
C4×#99×C532
#1262153