Finite Group Databasebeta

← #27279642

Subgroups — #27279642

Order 13568.

560 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 119 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
2#20273033C211yesmaximal, characteristic
3C2×#16810348C212yesmaximal, complemented
4#20272561C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133198C411yescharacteristic
7C22×#11719919C2211yescharacteristic
8C2×C22⋊#88322C2212yes—
9#16810518—21no—
10#16810517—21no—
11#16810507—21no—
12#16810503C411yescharacteristic
13#16810496—21no—
14#16810495C411yescharacteristic
15#16810484C411yescharacteristic
16C4×C22⋊Dic53C2211yescharacteristic
17#16810349—21no—
18#16810348—24no—
19#16810347—21no—
20C2×C4×D4×C53C2211yescharacteristic
21C53×#435—24no—
22C53×#432—21no—
23C53×#407—21no—
24C2×C22⋊C8×C53C2212yes—
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27C22×#88322—22no—
28#13133198—22no—
29#13133198—22no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53C2×C411yescharacteristic
39C2×C22⋊Dic53C2×C411yescharacteristic
40C2×C4×Dic53—21no—
41C22⋊#88322—22no—
42C22⋊#88322—22no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C22×D4×C53C2×C411yescharacteristic
49C23×C4×C53D411yescharacteristic
50C23×C4×C53C2311yescharacteristic
51C2×M16×C53—22no—
52C22×C8×C53—22no—
53C4×D4×C53—22no—
54C4×D4×C53—22no—
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C4⋊C8×C53—22no—
60C22⋊C8×C53—22no—
61C22⋊C8×C53—22no—
62C22⋊C8×C53D414yes—
63#97×C53—22no—
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22×C411yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—22no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—81no—
73C22×C4×C53—81no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C2×D412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4112yes—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—22no—
84C22×Dic53—21no—
85C22×Dic53—22no—
86C22×Dic53—21no—
87C22×Dic53—21no—
88C22×Dic53—22no—
89C22×Dic53—81no—
90C22×Dic53M1612yescomplemented
91M16×C53—82no—
92C2×C8×C53—24no—
93C2×C8×C53—82no—
94C4⋊C4×C53—22no—
95C22⋊C4×C53—22no—
96C22⋊C4×C53—22no—
97C42×C53—21no—
98C42×C53—21no—
99C22⋊Dic53—24no—
100C22⋊Dic53—21no—
101C22⋊Dic53—22no—
102C22⋊Dic53—81no—
103C22⋊Dic53M1612yescomplemented
104#11719919—24no—
105#11719919—41no—
106#11719919—42no—
107#11719919—41no—
108#11719919—81no—
109C4×Dic53—81no—
110C2×#88322—42no—
111C2×#88322—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53C4×D412yes—
121C23×C53C2×M1611yescharacteristic
122C23×C53C2×M1611yescharacteristic
123C23×C53#13112yes—
124D4×C53—42no—
125D4×C53—42no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—81no—
139C2×C4×C53#12011yescharacteristic
140C2×C4×C53#12112yes—
141C2×C4×C53#12311yescharacteristic
142C2×Dic53—22no—
143C2×Dic53—22no—
144C2×Dic53—24no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—82no—
157C8×C53—44no—
158#88322—82no—
159#9110—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—22no—
162C22×C53—21no—
163C22×C53—22no—
164C22×C53—41no—
165C22×C53—41no—
166C22×C53—42no—
167C22×C53—42no—
168C22×C53—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—81no—
172C22×C53#38611yescharacteristic
173C22×C53#40711yescharacteristic
174C22×C53#40811yescharacteristic
175C22×C53#43512yes—
176C22×C53#43612yes—
177C4×C53—22no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—41no—
183Dic53—42no—
184Dic53—42no—
185Dic53—81no—
186Dic53—81no—
187#2973D5311yescharacteristic
188#2322—531no—
189#1459—531no—
190C2×#324—532no—
191#1282—532no—
192C2×C53—21no—
193C2×C53—21no—
194C2×C53—22no—
195C2×C53—22no—
196C2×C53—81no—
197C2×C53#187011yescharacteristic
198C2×C53#191311yescharacteristic
199C2×C53#191511yescharacteristic
200C2×#123—531no—
201C2×C4×D4C2×D5311yescharacteristic
202C22×C4⋊C4—531no—
203#435—24no—
204#432—21no—
205#407—21no—
206C2×C22⋊C8—532no—
207C2×C22⋊C8C2×D5312yes—
208#389—1061no—
209#388—1061no—
210#387—1061no—
211#386—531no—
212#385—531no—
213#385—1061no—
214#381—531no—
215C4×C22⋊C4—531no—
216#324—1061no—
217#324—1064no—
218#323—1061no—
219C53#911011yescharacteristic, complemented, Sylow
220C22×D4C4×D5311yescharacteristic
221C23×C4—531no—
222C23×C4—531no—
223C23×C4C22⋊D5311yescharacteristic
224C23×C4C22×D5311yescharacteristic
225C2×M16—22no—
226C22×C8—22no—
227C22×C8—1062no—
228#123—1062no—
229#123—1062no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4D21211yescharacteristic
239C2×C22⋊C4—531no—
240C2×C22⋊C4—531no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4—1061no—
243C2×C22⋊C4C4×D5311yescharacteristic
244C2×C22⋊C4D21211yescharacteristic
245C2×C42—1061no—
246C2×C42C22⋊D5311yescharacteristic
247C4⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8C22⋊D5314yes—
253#97—22no—
254#95—1061no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259C24#1171992111yescharacteristic
260C24C2×C4×D5311yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—532no—
270C22×C4—1061no—
271C22×C4—1061no—
272C22×C4—1062no—
273C22×C4—1061no—
274C22×C4—1062no—
275C22×C4—1062no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—4241no—
279C22×C4#1171992111yescharacteristic
280C22×C4#1171992111yescharacteristic
281C22×C4#1171992111yescharacteristic
282C22×C4C2×D21211yescharacteristic
283C22×C4#1171994512yes—
284C22×C4C2×C22⋊D5312yes—
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—82no—
287C2×C8—24no—
288C2×C8—82no—
289C2×C8—4242no—
290C2×C8—2122no—
291C4⋊C4—22no—
292C4⋊C4—1064no—
293C4⋊C4—4241no—
294C4⋊C4—2121no—
295C4⋊C4—2122no—
296C4⋊C4—2121no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—532no—
300C22⋊C4—1064no—
301C22⋊C4—1061no—
302C22⋊C4—1062no—
303C22⋊C4—4241no—
304C42—21no—
305C42—21no—
306C42—4241no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×#1171992111yescharacteristic
315C23C4×C22⋊D5312yes—
316C23M16×D5311yescharacteristic
317C23M16×D5311yescharacteristic
318C23#1313322512yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—1062no—
335C2×C4—1062no—
336C2×C4—1064no—
337C2×C4—4242no—
338C2×C4—2121no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4—2121no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2122no—
349C2×C4#1313318611yescharacteristic
350C2×C4#1313319112yes—
351C2×C4#1313320011yescharacteristic
352C8—44no—
353C8—4242no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1681050211yescharacteristic
367C22#1681058611yescharacteristic
368C22#1681059111yescharacteristic
369C22#1681069212yes—
370C22#1681069912yes—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—4241no—
378C4—4241no—
379C4—2122no—
380C4—2122no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2027501211yescharacteristic
387C2#2027525711yescharacteristic
388C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#9110
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
3. C2×#16810348C2semidirect
4. #20272561C2semidirect
90. C22×Dic53M16semidirect
103. C22⋊Dic53M16semidirect
219. C53#9110semidirect

Maximal subgroups

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

subgroupindex
#202773802
#202730332
C2×#168103482
#202725612
C53×#29732
#911053