Finite Group Databasebeta

← #27276706

Subgroups — #27276706

Order 13568.

910 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#13133210C211yesmaximal, characteristic
2#20274789C211yesmaximal, characteristic
3#20274787C212yesmaximal
4#20274787C212yesmaximal
5C53×#1819C211yesmaximal, characteristic
6C2×#13133210—24no—
7C2×#13133210—24no—
8C2×#13133210—22no—
9C2×#13133210—22no—
10C22×C4⋊Dic53C411yescharacteristic, complemented
11C22×#11719919C411yescharacteristic, complemented
12C22×#11719919C411yescharacteristic, complemented
13C22×C22⋊Dic53C2211yescharacteristic
14C22×C22⋊Dic53C2211yescharacteristic
15C22×C4×Dic53C411yescharacteristic, complemented
16#16810479C2212yes—
17#16810390—22no—
18#16810389—24no—
19#16810389—24no—
20C22×C22⋊C4×C53C2211yescharacteristic
21C53×#379C2212yes—
22C53×#342—22no—
23C23×Dic53C2×C411yescharacteristic
24C23×Dic53C2×C411yescharacteristic
25C23×Dic53C2×C411yescharacteristic
26C23×Dic53C2×C411yescharacteristic
27#13133210—216no—
28#13133210—216no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×#11719919—22no—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919—22no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
42C2×C22⋊Dic53—22no—
43C2×C22⋊Dic53—21no—
44C2×C22⋊Dic53—22no—
45C2×C22⋊Dic53—22no—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53D414yes—
48C2×C22⋊Dic53D414yes—
49C2×C4×Dic53—22no—
50C2×C4×Dic53—22no—
51C2×C4×Dic53—22no—
52C25×C53C2311yescharacteristic
53C23×C4×C53C2×C411yescharacteristic
54C23×C4×C53C2×C411yescharacteristic
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—22no—
60C2×C22⋊C4×C53D412yes—
61C2×C22⋊C4×C53D412yes—
62C24×C53—42no—
63C24×C53—42no—
64C24×C53—42no—
65C24×C53C22×C411yescharacteristic
66C24×C53C2×D411yescharacteristic
67C24×C53C2×D412yes—
68C24×C53C2×D412yes—
69C24×C53C2×D411yescharacteristic
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—42no—
76C22×C4×C53—42no—
77C22×C4×C53C22⋊C412yes—
78C22×C4×C53C22⋊C412yes—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—42no—
88C22×Dic53—42no—
89C22×Dic53—42no—
90C22×Dic53—42no—
91C22×Dic53—42no—
92C22×Dic53C22⋊C412yes—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C412yes—
95C22×Dic53C22⋊C412yes—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—44no—
99C22⋊C4×C53—44no—
100C22⋊C4×C53—44no—
101C22⋊Dic53—24no—
102C22⋊Dic53—24no—
103C22⋊Dic53—24no—
104C22⋊Dic53—24no—
105C22⋊Dic53—44no—
106C22⋊Dic53—44no—
107C22⋊Dic53—44no—
108C4⋊Dic53—28no—
109#11719919—28no—
110#11719919—28no—
111C4×Dic53—28no—
112C23×C53—22no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—24no—
117C23×C53—24no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—42no—
124C23×C53—42no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—88no—
128C23×C53C2×C22⋊C411yescharacteristic
129C23×C53C2×C22⋊C411yescharacteristic
130C23×C53C2×C22⋊C411yescharacteristic
131C23×C53#12012yes—
132C23×C53#12012yes—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—82no—
142C2×C4×C53—82no—
143C2×C4×C53—82no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—22no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—82no—
161C2×Dic53—82no—
162C2×Dic53—82no—
163#8545—531nomaximal, Sylow
164C22×C53—28no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—22no—
175C22×C53—22no—
176C22×C53—22no—
177C22×C53—82no—
178C22×C53—84no—
179C22×C53—84no—
180C22×C53—82no—
181C22×C53#37911yescharacteristic
182C22×C53#41012yes—
183C22×C53#41012yes—
184C22×C53#41012yes—
185C4×C53—42no—
186C4×C53—42no—
187C4×C53—162no—
188Dic53—42no—
189Dic53—42no—
190Dic53—42no—
191Dic53—42no—
192Dic53—162no—
193C22×#126—531no—
194#1819—532no—
195#1819—532no—
196#1819—531no—
197#1819D5311yescharacteristic
198C2×C53—22no—
199C2×C53—22no—
200C2×C53—22no—
201C2×C53—42no—
202C2×C53—42no—
203C2×C53#178211yescharacteristic
204C2×C53#178211yescharacteristic
205C2×C53#178211yescharacteristic
206C2×#126—1064no—
207C2×#126—1064no—
208C2×#126—1062no—
209C2×#126—1062no—
210C22×C4⋊C4—531no—
211C22×C4⋊C4—531no—
212C22×C4⋊C4—531no—
213C22×C22⋊C4—531no—
214C22×C22⋊C4—531no—
215C22×C22⋊C4C2×D5311yescharacteristic
216C22×C42—531no—
217#379—532no—
218#379C2×D5312yes—
219#342—22no—
220#342—1064no—
221#342—1064no—
222#342—1062no—
223C53#854511yescharacteristic, complemented, Sylow
224C25C22×D5311yescharacteristic
225C23×C4—531no—
226C23×C4—531no—
227C23×C4—531no—
228C23×C4—531no—
229C23×C4C4×D5311yescharacteristic
230C23×C4C4×D5311yescharacteristic
231#126—10616no—
232#126—10616no—
233C2×C4⋊C4—1062no—
234C2×C4⋊C4—1062no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4—1062no—
239C2×C4⋊C4—1062no—
240C2×C4⋊C4—1062no—
241C2×C4⋊C4—1062no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—534no—
248C2×C22⋊C4—534no—
249C2×C22⋊C4—1061no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1062no—
252C2×C22⋊C4—1062no—
253C2×C22⋊C4—1061no—
254C2×C22⋊C4—1061no—
255C2×C22⋊C4—1062no—
256C2×C22⋊C4—1062no—
257C2×C22⋊C4—1062no—
258C2×C22⋊C4D21212yes—
259C2×C22⋊C4C22⋊D5312yes—
260C2×C42—1062no—
261C2×C42—1062no—
262C2×C42—1062no—
263C24—42no—
264C24—42no—
265C24—42no—
266C24C2×C4×D5311yescharacteristic
267C24C2×D21211yescharacteristic
268C24D4×D5312yes—
269C24D4×D5312yes—
270C24C2×C22⋊D5311yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—532no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—1061no—
283C22×C4—1061no—
284C22×C4—1061no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4—2122no—
293C22×C4—2122no—
294C22×C4—2122no—
295C22×C4#1171992112yes—
296C22×C4#1171992112yes—
297C4⋊C4—1068no—
298C4⋊C4—1068no—
299C4⋊C4—1068no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—1064no—
306C22⋊C4—1064no—
307C22⋊C4—1064no—
308C22⋊C4—1064no—
309C22⋊C4—2124no—
310C22⋊C4—2124no—
311C22⋊C4—2124no—
312C42—1068no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—24no—
317C23—24no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23—41no—
328C23—88no—
329C23C22⋊C4×D5311yescharacteristic
330C23C22⋊C4×D5311yescharacteristic
331C23C2×#1171992111yescharacteristic
332C23#1313318512yes—
333C23#1313318712yes—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—4242no—
350C2×C4—4242no—
351C2×C4—4242no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C22—28no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—84no—
378C22—84no—
379C22—82no—
380C22—82no—
381C22#1681047811yescharacteristic
382C22#1681059512yes—
383C22#1681059512yes—
384C22#1681059612yes—
385C4—42no—
386C4—42no—
387C4—162no—
388C4—8482no—
389C4—2122no—
390C4—2122no—
391C4—2122no—
392C4—2122no—
393C2—22no—
394C2—22no—
395C2—22no—
396C2—42no—
397C2—42no—
398C2#2027463111yescharacteristic
399C2#2027463111yescharacteristic
400C2#2027463211yescharacteristic
subgroupG/H
centerC22#16810478
commutatorC24×C53C22×C4
FrattiniC25C22×D53
FittingC53×#1819C2
socleC22×C53#379
prime radical, cosocleC25×C53C23
hypercenter#1819D53
hypocenterC53#8545
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1819

Complements

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

NH ≅ G/Ntype
10. C22×C4⋊Dic53C4semidirect
11. C22×#11719919C4semidirect
12. C22×#11719919C4semidirect
15. C22×C4×Dic53C4semidirect
223. C53#8545semidirect

Maximal subgroups

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

subgroupindex
C22×#131332102
#202747892
#202747872
C53×#18192
#854553