Finite Group Databasebeta

← #27276142

Subgroups — #27276142

Order 13568.

884 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 147 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#20275544C211yesmaximal, characteristic, complemented
2C4×#13133144C212yesmaximal, complemented
3#20273307C211yesmaximal, characteristic, complemented
4#20273154C212yesmaximal, complemented
5C53×#3481C211yesmaximal, characteristic
6#16810647C2211yescharacteristic, complemented
7C2×#13133144—22no—
8C2×#13133144—22no—
9C4×#11719917C2212yescomplemented
10C42×Dic53C2211yescharacteristic, complemented
11#16810382—22no—
12#16810363—22no—
13Q8⋊#88322—22no—
14D4⋊#88322—22no—
15C53×#559C411yescharacteristic, complemented
16C53×#554C412yescomplemented
17C53×#553—22no—
18C53×#551—22no—
19C53×#546—22no—
20D42×C53C411yescharacteristic, complemented
21C53×#532C2211yescharacteristic
22C53×#529—22no—
23C53×#525—22no—
24C4×#41×C53C2212yes—
25C2×C4×D4×C53—22no—
26C2×#11719917—22no—
27C2×#11719917—21no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53—22no—
30C4⋊#88322—22no—
31#13133144—28no—
32#13133144—44no—
33#13133128—22no—
34C4×#88322—22no—
35C2×#41×C53D412yes—
36C2×#41×C53D412yes—
37C22×D4×C53—22no—
38C23×C4×C53—22no—
39C4⋊Q8×C53C2×C411yescharacteristic
40#127×C53C2×C411yescharacteristic
41#126×C53—24no—
42#125×C53—22no—
43#124×C53—22no—
44#124×C53D412yes—
45#123×C53—22no—
46#123×C53—24no—
47#123×C53—22no—
48C22⋊Q8×C53—24no—
49C22⋊Q8×C53—22no—
50#121×C53—24no—
51#121×C53—22no—
52#121×C53—22no—
53#120×C53—22no—
54#120×C53—22no—
55C4×Q8×C53C2×C412yes—
56C4×Q8×C53D412yes—
57C4×D4×C53—24no—
58C4×D4×C53—24no—
59C4×D4×C53—22no—
60C4×D4×C53—22no—
61C4×D4×C53—22no—
62C4×D4×C53—22no—
63C4×D4×C53—22no—
64C4×D4×C53C2×C412yes—
65C4×D4×C53D412yes—
66#117×C53—22no—
67#117×C53—22no—
68#117×C53D412yes—
69C2×C4⋊C4×C53—22no—
70C2×C22⋊C4×C53—24no—
71C2×C42×C53—22no—
72C2×C42×C53C2311yescharacteristic
73C24×C53—22no—
74#41×C53—24no—
75#41×C53—42no—
76#41×C53—42no—
77#41×C53—42no—
78C2×Q8×C53C22⋊C412yes—
79C2×Q8×C53C22⋊C412yes—
80C2×D4×C53—24no—
81C2×D4×C53—22no—
82C2×D4×C53—22no—
83C2×D4×C53—22no—
84C2×D4×C53—44no—
85C2×D4×C53—44no—
86C2×D4×C53C22⋊C412yes—
87C2×D4×C53C22⋊C412yes—
88C22×C4×C53—24no—
89C22×C4×C53—22no—
90C22×C4×C53—22no—
91C22×C4×C53—22no—
92C22×C4×C53—22no—
93C22×C4×C53—22no—
94C22×C4×C53—82no—
95C22×C4×C53—82no—
96C22×C4×C53C2×D412yes—
97C22×C4×C53C2×D411yescharacteristic
98C22×Dic53—41no—
99C4⋊C4×C53—24no—
100C4⋊C4×C53—24no—
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53C22⋊C412yes—
105C4⋊C4×C53C22⋊C412yes—
106C22⋊C4×C53—24no—
107C22⋊C4×C53—24no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—22no—
111C22⋊C4×C53—22no—
112C22⋊C4×C53—22no—
113C22⋊C4×C53—24no—
114C42×C53—22no—
115C42×C53—22no—
116C42×C53—22no—
117C42×C53C22×C411yescharacteristic
118C42×C53C2×D412yes—
119C42×C53C2×D411yescharacteristic
120C4×Dic53—24no—
121C4×Dic53—22no—
122C4×Dic53—42no—
123C4×Dic53—41no—
124#11719917—24no—
125#11719917—42no—
126#11719917—42no—
127C2×#88322—42no—
128C2×#88322—41no—
129C23×C53—22no—
130C23×C53—22no—
131C23×C53—48no—
132C23×C53—82no—
133C23×C53#12011yescharacteristic
134Q8×C53—22no—
135Q8×C53—22no—
136Q8×C53—42no—
137D4×C53—22no—
138D4×C53—42no—
139D4×C53—44no—
140D4×C53—48no—
141D4×C53—42no—
142D4×C53—42no—
143D4×C53—82no—
144D4×C53#10414yescomplemented
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—22no—
150C2×C4×C53—22no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—44no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—42no—
159C2×C4×C53—42no—
160C2×C4×C53—82no—
161C2×C4×C53—84no—
162C2×C4×C53C2×C22⋊C411yescharacteristic
163C2×C4×C53C2×C22⋊C412yes—
164C2×C4×C53#12012yes—
165C2×C4×C53#12011yescharacteristic
166C2×Dic53—42no—
167C2×Dic53—44no—
168#88322—44no—
169#88322—81no—
170#8440—531nomaximal, Sylow
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—24no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—44no—
178C22×C53—88no—
179C22×C53#37911yescharacteristic
180C4×C53—21no—
181C4×C53—22no—
182C4×C53—42no—
183C4×C53—42no—
184C4×C53—42no—
185C4×C53—42no—
186C4×C53C2×#10412yes—
187C4×C53#42112yes—
188C4×C53#45311yescharacteristic
189C4×C53#45511yescharacteristic
190Dic53—44no—
191#3481D5311yescharacteristic
192#1962—531no—
193C4×#104—532no—
194#1525—531no—
195#1489—532no—
196C2×C53—41no—
197C2×C53—44no—
198C2×C53—82no—
199C2×C53#178912yes—
200C2×C53#179211yescharacteristic
201#559Dic5311yescharacteristic, complemented
202#554Dic5312yescomplemented
203#553—22no—
204#551—22no—
205#546—22no—
206D42Dic5311yescharacteristic, complemented
207#532C2×D5311yescharacteristic
208#529—22no—
209#525—22no—
210C4×#41C2×D5312yes—
211C2×C4×D4—22no—
212#423—531no—
213C2×#104—1062no—
214C2×#104—1062no—
215C4×M16—532no—
216C43—531no—
217#339—1062no—
218#329—1062no—
219Q8⋊C8—1062no—
220D4⋊C8—1062no—
221C53#844011yescharacteristic, complemented, Sylow
222C2×#41C22⋊D5312yes—
223C2×#41C22⋊D5312yes—
224C22×D4—22no—
225C23×C4—22no—
226C2×M16—1062no—
227C2×M16—1061no—
228C4⋊Q8C2×Dic5311yescharacteristic
229#127C2×Dic5311yescharacteristic
230#126—24no—
231#125—22no—
232#124—22no—
233#124C22⋊D5312yes—
234#123—22no—
235#123—24no—
236#123—22no—
237C22⋊Q8—24no—
238C22⋊Q8—22no—
239#121—24no—
240#121—22no—
241#121—22no—
242#120—22no—
243#120—22no—
244C4×Q8C2×Dic5312yes—
245C4×Q8C22⋊D5312yes—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4C2×Dic5312yes—
254C4×D4C22⋊D5312yes—
255#117—22no—
256#117—22no—
257#117C22⋊D5312yes—
258C2×C4⋊C4—22no—
259C2×C22⋊C4—24no—
260C2×C42—22no—
261C2×C42—1062no—
262C2×C42—1061no—
263C2×C42C22×D5311yescharacteristic
264C4⋊C8—1062no—
265#104—1068no—
266#104—2124no—
267#97—1062no—
268C4×C8—1062no—
269C24—22no—
270#41—24no—
271#41—42no—
272#41—42no—
273#41—42no—
274C2×Q8C22⋊Dic5312yes—
275C2×Q8C22⋊Dic5312yes—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4C22⋊Dic5312yes—
283C2×D4C22⋊Dic5312yes—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—2121no—
293C22×C4C2×C22⋊D5312yes—
294C22×C4C2×C22⋊D5311yescharacteristic
295M16—1064no—
296M16—2122no—
297M16—2122no—
298C2×C8—2121no—
299C2×C8—2122no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4C22⋊Dic5312yes—
306C4⋊C4C22⋊Dic5312yes—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C22⋊C4—22no—
312C22⋊C4—22no—
313C22⋊C4—22no—
314C22⋊C4—24no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—1064no—
319C42—1062no—
320C42—2121no—
321C42—2122no—
322C42C22×Dic5311yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C42C2×C22⋊D5312yes—
325C23—22no—
326C23—22no—
327C23—48no—
328C23—82no—
329C23#1313318611yescharacteristic
330Q8—22no—
331Q8—22no—
332Q8—42no—
333D4—22no—
334D4—42no—
335D4—44no—
336D4—48no—
337D4—42no—
338D4—42no—
339D4—82no—
340D4#1313314414yescomplemented
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—84no—
358C2×C4—2124no—
359C2×C4—2122no—
360C2×C4C2×C22⋊Dic5311yescharacteristic
361C2×C4C2×C22⋊Dic5312yes—
362C2×C4#1313318612yes—
363C2×C4#1313318611yescharacteristic
364C8—4241no—
365C8—2124no—
366C22—22no—
367C22—24no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22—88no—
374C22#1681047911yescharacteristic
375C4—22no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—2124no—
382C4C2×#1313314412yes—
383C4#1681064012yes—
384C4#1681079111yescharacteristic
385C4#1681080111yescharacteristic
386C2—41no—
387C2—44no—
388C2—82no—
389C2#2027465912yes—
390C2#2027466911yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3481C2
socleC22×C53#379
cosocleC2×C42×C53C23
hypercenter#3481D53
hypocenterC53#8440
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3481

Complements

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

NH ≅ G/Ntype
1. #20275544C2semidirect
2. C4×#13133144C2semidirect
3. #20273307C2semidirect
4. #20273154C2semidirect
6. #16810647C22semidirect
9. C4×#11719917C22semidirect
10. C42×Dic53C22semidirect
15. C53×#559C4semidirect
16. C53×#554C4semidirect
20. D42×C53C4semidirect
144. D4×C53#104semidirect
201. #559Dic53semidirect
202. #554Dic53semidirect
206. D42Dic53semidirect
221. C53#8440semidirect
340. D4#13133144semidirect

Maximal subgroups

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

subgroupindex
#202755442
C4×#131331442
#202733072
#202731542
C53×#34812
#844053