Finite Group Databasebeta

← #27393033

Subgroups — #27393033

Order 13568.

1570 conjugacy classes of subgroups; only the 243 normal subgroups are shown.

normal subgroups only

This group has 1570 conjugacy classes of subgroups — too many to draw. Only its 243 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×#13133198C211yesmaximal, characteristic
2#20277093C212yesmaximal, complemented
3#20275170C218yesmaximal, complemented
4#20274787C212yesmaximal, complemented
5#20274630C211yesmaximal, characteristic
6C53×#1781C211yesmaximal, characteristic
7C24×Dic53C2211yescharacteristic
8C2×#13133198—24no—
9C2×#13133198—24no—
10C2×#13133198C414yescomplemented
11C2×#13133198C2214yes—
12C2×#13133173C2214yescomplemented
13C22×#11719919C412yescomplemented
14C22×C22⋊Dic53C412yescomplemented
15C22×C22⋊Dic53C2211yescharacteristic
16C2×#13133132C2214yes—
17#16810484C2214yescomplemented
18#16810481—24no—
19#16810479C2212yes—
20C22⋊C4×Dic53—24no—
21#16810389C2218yescomplemented
22C23×D4×C53C2211yescharacteristic
23C2×#99×C53C2214yes—
24C53×#379C2212yes—
25C23×Dic53—44no—
26C23×Dic53C2×C412yes—
27C23×Dic53C2×C412yes—
28C23×Dic53C2×C412yes—
29C23×Dic53D414yes—
30C23×Dic53D414yes—
31C23×Dic53C2312yes—
32#13133198—24no—
33#13133198—28no—
34#13133198—24no—
35#13133198—28no—
36#13133198—24no—
37#13133198—24no—
38#13133173—48no—
39C2×#11719919—24no—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719919—24no—
43C2×#11719919C2×C418yes—
44C2×C22⋊Dic53—24no—
45C2×C22⋊Dic53—24no—
46C2×C22⋊Dic53—22no—
47C2×C22⋊Dic53—22no—
48C2×C22⋊Dic53—24no—
49C2×C22⋊Dic53C2×C418yes—
50C2×C22⋊Dic53C2×C412yes—
51C2×C22⋊Dic53C2314yes—
52C2×C22⋊Dic53C2312yes—
53C2×C4×Dic53—24no—
54#13133132—44no—
55#13133124—28no—
56C25×C53C2×C411yescharacteristic
57C25×C53C2311yescharacteristic
58C22×D4×C53—24no—
59C22×D4×C53—44no—
60C22×D4×C53C2×C412yes—
61C22×D4×C53C2312yes—
62C23×C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—24no—
64C2×C22⋊C4×C53C2314yes—
65#99×C53—44no—
66C24×C53—24no—
67C24×C53—22no—
68C24×C53—44no—
69C24×C53—42no—
70C24×C53—44no—
71C24×C53C22×C411yescharacteristic
72C24×C53C22×C412yes—
73C24×C53C22×C411yescharacteristic
74C24×C53C2×D414yes—
75C24×C53C2411yescharacteristic
76C2×D4×C53—24no—
77C2×D4×C53—24no—
78C2×D4×C53—24no—
79C2×D4×C53—24no—
80C2×D4×C53—42no—
81C2×D4×C53—42no—
82C2×D4×C53—416no—
83C22×C4×C53—22no—
84C22×C4×C53—24no—
85C22×C4×C53—42no—
86C22×C4×C53C22×C412yes—
87C22×Dic53—28no—
88C22×Dic53—22no—
89C22×Dic53—22no—
90C22×Dic53—24no—
91C22×Dic53—44no—
92C22×Dic53—44no—
93C22×Dic53—48no—
94C22×Dic53—48no—
95C22×Dic53—44no—
96C22×Dic53C22⋊C412yes—
97C22×Dic53C22⋊C414yes—
98C22×Dic53C22⋊C412yes—
99C22×Dic53C22⋊C412yes—
100C22×Dic53C22⋊C414yes—
101C22×Dic53C22⋊C412yes—
102C22×Dic53C22×C414yes—
103C22×Dic53C22×C414yes—
104C22×Dic53C2×D414yes—
105C22×Dic53C2×D412yes—
106C22×Dic53C2×D412yes—
107C22⋊C4×C53—28no—
108C22⋊C4×C53—48no—
109C22⋊Dic53—24no—
110C22⋊Dic53—24no—
111C22⋊Dic53—28no—
112C22⋊Dic53—28no—
113C22⋊Dic53—28no—
114C22⋊Dic53—48no—
115#11719919—28no—
116#11719919—28no—
117#11719919—48no—
118C4×Dic53—48no—
119C23×C53—22no—
120C23×C53—24no—
121C23×C53—22no—
122C23×C53—28no—
123C23×C53—28no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—22no—
127C23×C53—42no—
128C23×C53—44no—
129C23×C53—42no—
130C23×C53—416no—
131C23×C53—44no—
132C23×C53—44no—
133C23×C53—44no—
134C23×C53—44no—
135C23×C53—88no—
136C23×C53C2×C22⋊C412yes—
137C23×C53C2×C22⋊C412yes—
138C23×C53C23×C411yescharacteristic
139C23×C53C22×D412yes—
140C23×C53H(5,𝔽2)14yes—
141D4×C53—48no—
142D4×C53—48no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—44no—
147C2×C4×C53—48no—
148C2×C4×C53—44no—
149C2×Dic53—28no—
150C2×Dic53—28no—
151C2×Dic53—24no—
152C2×Dic53—24no—
153C2×Dic53—24no—
154C2×Dic53—42no—
155C2×Dic53—44no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—44no—
159C2×Dic53—42no—
160C2×Dic53—44no—
161C2×Dic53—44no—
162C2×Dic53—44no—
163C2×Dic53—48no—
164C2×Dic53—44no—
165C2×Dic53C2×C22⋊C414yes—
166C2×Dic53C2×C22⋊C414yes—
167#21747—531nomaximal, Sylow
168C22×C53—28no—
169C22×C53—22no—
170C22×C53—24no—
171C22×C53—22no—
172C22×C53—22no—
173C22×C53—22no—
174C22×C53—24no—
175C22×C53—44no—
176C22×C53—416no—
177C22×C53—44no—
178C22×C53—42no—
179C22×C53—44no—
180C22×C53—82no—
181C22×C53—82no—
182C22×C53—88no—
183C22×C53#41014yes—
184C22×C53C22×C22⋊C411yescharacteristic
185C22×C53#51812yes—
186C22×C53#53414yes—
187C4×C53—42no—
188C4×C53—84no—
189Dic53—24no—
190Dic53—24no—
191Dic53—44no—
192Dic53—84no—
193C22×#123—531no—
194#2280—532no—
195#1899—538no—
196#1819—532no—
197#1781—531no—
198#1781D5311yescharacteristic
199C2×C53—24no—
200C2×C53—22no—
201C2×C53—24no—
202C2×C53—42no—
203C2×C53—82no—
204C2×C53#230811yescharacteristic
205C2×C53C2×#41012yes—
206C23×D4C2×D5311yescharacteristic
207C24×C4—531no—
208C2×#123—534no—
209C2×#123—534no—
210C2×#123—1064no—
211C2×#123—1064no—
212C2×#117—534no—
213C22×C4⋊C4—532no—
214C22×C22⋊C4—532no—
215C22×C22⋊C4—531no—
216C2×#99—534no—
217C2×#99C2×D5314yes—
218#381—534no—
219#380—1064no—
220#379—532no—
221#379C2×D5312yes—
222C4×C22⋊C4—1064no—
223#342—538no—
224C53#2174711yescharacteristic, complemented, Sylow
225C25C4×D5311yescharacteristic
226C25C22×D5311yescharacteristic
227C22×D4—24no—
228C22×D4—44no—
229C22×D4C4×D5312yes—
230C22×D4C22×D5312yes—
231C23×C4—532no—
232C23×C4—532no—
233C23×C4—532no—
234C23×C4—534no—
235C23×C4—534no—
236C23×C4—532no—
237C23×C4—2124no—
238C23×C4C4×D5311yescharacteristic
239#123—1064no—
240#123—1068no—
241#123—1064no—
242#123—1068no—
243#123—1064no—
244#123—1064no—
245#117—2128no—
246C2×C4⋊C4—538no—
247C2×C4⋊C4—1062no—
248C2×C4⋊C4—1064no—
249C2×C4⋊C4—1062no—
250C2×C4⋊C4—1064no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—538no—
253C2×C22⋊C4—532no—
254C2×C22⋊C4—532no—
255C2×C22⋊C4—534no—
256C2×C22⋊C4—1062no—
257C2×C22⋊C4—1064no—
258C2×C22⋊C4—1062no—
259C2×C22⋊C4—1064no—
260C2×C22⋊C4—1064no—
261C2×C22⋊C4C22×D5314yes—
262C2×C42—1064no—
263#99—44no—
264#99—2124no—
265#95—1068no—
266C24—24no—
267C24—22no—
268C24—44no—
269C24—42no—
270C24—44no—
271C24C2×C4×D5311yescharacteristic
272C24C2×C4×D5312yes—
273C24C2×C4×D5311yescharacteristic
274C24D4×D5314yes—
275C24C23×D5311yescharacteristic
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—416no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—42no—
286C22×C4—534no—
287C22×C4—534no—
288C22×C4—532no—
289C22×C4—532no—
290C22×C4—532no—
291C22×C4—534no—
292C22×C4—532no—
293C22×C4—532no—
294C22×C4—534no—
295C22×C4—532no—
296C22×C4—534no—
297C22×C4—1068no—
298C22×C4—1062no—
299C22×C4—1062no—
300C22×C4—1064no—
301C22×C4—2124no—
302C22×C4—2124no—
303C22×C4—2128no—
304C22×C4—2128no—
305C22×C4—2124no—
306C22×C4C2×C4×D5312yes—
307C4⋊C4—1068no—
308C4⋊C4—1068no—
309C4⋊C4—2128no—
310C22⋊C4—28no—
311C22⋊C4—48no—
312C22⋊C4—1068no—
313C22⋊C4—1068no—
314C22⋊C4—1064no—
315C22⋊C4—1064no—
316C22⋊C4—1068no—
317C22⋊C4—2128no—
318C42—2128no—
319C23—22no—
320C23—24no—
321C23—22no—
322C23—28no—
323C23—28no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—42no—
328C23—44no—
329C23—42no—
330C23—416no—
331C23—44no—
332C23—44no—
333C23—44no—
334C23—44no—
335C23—88no—
336C23C22⋊C4×D5312yes—
337C23C22⋊C4×D5312yes—
338C23C22×C4×D5311yescharacteristic
339C23C2×D4×D5312yes—
340C23#1313326214yes—
341D4—48no—
342D4—48no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—44no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—534no—
350C2×C4—534no—
351C2×C4—1068no—
352C2×C4—1064no—
353C2×C4—1064no—
354C2×C4—1068no—
355C2×C4—1064no—
356C2×C4—2122no—
357C2×C4—2124no—
358C2×C4—2122no—
359C2×C4—2122no—
360C2×C4—2124no—
361C2×C4—2122no—
362C2×C4—2128no—
363C2×C4—2124no—
364C2×C4—2124no—
365C2×C4—2124no—
366C2×C4—2124no—
367C22—28no—
368C22—24no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—24no—
373C22—22no—
374C22—44no—
375C22—416no—
376C22—44no—
377C22—44no—
378C22—42no—
379C22—82no—
380C22—82no—
381C22—88no—
382C22#1681059514yes—
383C22C2×C22⋊C4×D5311yescharacteristic
384C22#1681109612yes—
385C22#1681118314yes—
386C4—42no—
387C4—84no—
388C4—1064no—
389C4—1064no—
390C4—4244no—
391C4—2124no—
392C2—24no—
393C2—22no—
394C2—24no—
395C2—42no—
396C2—82no—
397C2#2027729111yescharacteristic
398C2C2×#1681059512yes—
subgroupG/H
centerC22C2×C22⋊C4×D53
commutatorC23×C53C23×C4
FrattiniC24C23×D53
FittingC53×#1781C2
socleC22×C53C22×C22⋊C4
prime radicalC23×D4×C53C22
cosocleC24×C53C24
hypercenter#1781D53
hypocenterC53#21747
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C25 ◁ #1781

Complements

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

NH ≅ G/Ntype
2. #20277093C2semidirect
3. #20275170C2semidirect
4. #20274787C2semidirect
10. C2×#13133198C4semidirect
12. C2×#13133173C22semidirect
13. C22×#11719919C4semidirect
14. C22×C22⋊Dic53C4semidirect
17. #16810484C22semidirect
21. #16810389C22semidirect
224. C53#21747semidirect

Maximal subgroups

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

subgroupindex
C22×#131331982
#202770932
#202751702
#202747872
#202746302
C53×#17812
#2174753