Finite Group Databasebeta

← #27367371

Subgroups — #27367371

Order 13568.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285866C211yesmaximal, characteristic, complemented
2#20285658C211yesmaximal, characteristic
3C2×#16810971C211yesmaximal, characteristic
4#20283645C211yesmaximal, characteristic, complemented
5#20274258C214yesmaximal, complemented
6#20274134C214yesmaximal, complemented
7#20273461C211yesmaximal, characteristic, complemented
8#20273384C211yesmaximal, characteristic
9C53×#1548C211yesmaximal, characteristic, complemented
10#16811180C2211yescharacteristic
11C2×#13133214C2211yescharacteristic
12#16811026—21no—
13#16810996—21no—
14#16810975—21no—
15#16810971—21no—
16#16810971—21no—
17#16810971C2212yes—
18#16810971C2212yes—
19#16810970—21no—
20#16810958—22no—
21#16810915C2214yescomplemented
22#16810824C2214yes—
23#16810776—22no—
24#16810750—21no—
25#16810750C2212yescomplemented
26#16810747—21no—
27#16810747C2212yescomplemented
28#16810676C2211yescharacteristic
29#16810651—21no—
30C2×#13133137—21no—
31C2×#13133137—21no—
32#16810581—21no—
33C2×C4×#88322C2211yescharacteristic
34#16810375C2212yescomplemented
35#16810373C2212yescomplemented
36#16810366C2214yes—
37C2×C4⋊Q8×C53C2211yescharacteristic
38C53×#431C2211yescharacteristic
39C53×#423C2211yescharacteristic
40C53×#333C2214yescomplemented
41C22×D212—21no—
42C22×D212—21no—
43C22×D212—21no—
44C2×#11719913—24no—
45C22×#88322D411yescharacteristic
46C22×#88322D411yescharacteristic
47#13133217D411yescharacteristic
48#13133217D411yescharacteristic
49#13133214—21no—
50#13133214—21no—
51#13133214—21no—
52#13133214C2312yes—
53#13133209—21no—
54#13133196—21no—
55#13133196D412yes—
56C4×Dic106—21no—
57C4×Dic106C2312yes—
58#13133176—21no—
59#13133150—24no—
60C4⋊#88322C2312yes—
61#13133146—22no—
62#13133146—22no—
63#13133140—24no—
64#13133139—24no—
65#13133137—22no—
66#13133137—22no—
67#13133137—22no—
68#13133137—22no—
69C22⋊#88322—21no—
70#13133127—22no—
71C4×#88322—21no—
72C4×#88322D411yescharacteristic
73C4×#88322D411yescharacteristic
74C4×#88322C2311yescharacteristic
75C4×#88322C2311yescharacteristic
76C8×Dic53—22no—
77C22×Q8×C53—21no—
78C2×M16×C53—21no—
79C4⋊Q8×C53—21no—
80C4⋊Q8×C53—21no—
81C4⋊Q8×C53—21no—
82C4⋊Q8×C53C2312yes—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C42×C53C2311yescharacteristic
86C4⋊C8×C53—21no—
87C4⋊C8×C53C2312yes—
88C4⋊C8×C53C2312yes—
89C22⋊C8×C53—21no—
90C4×C8×C53—21no—
91C23×D53—41no—
92C2×Q8×C53—22no—
93C2×Q8×C53—41no—
94C2×Q8×C53—82no—
95C22×C4×C53—41no—
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53#4111yescharacteristic
99C2×D212—22no—
100C2×D212—22no—
101C2×D212—22no—
102C2×D212—42no—
103C2×D212—42no—
104C2×D212—41no—
105C2×D212—41no—
106C2×D212—41no—
107C2×D212—82no—
108C2×D212—82no—
109C2×Dic106—21no—
110C2×Dic106C2×D412yes—
111M16×C53—42no—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53—41no—
117C4⋊C4×C53—41no—
118C42×C53C2×D411yescharacteristic
119C42×C53C2×D411yescharacteristic
120C42×C53#4111yescharacteristic
121C42×C53C2411yescharacteristic
122C22⋊Dic53—22no—
123C4⋊Dic53—21no—
124C4⋊Dic53—21no—
125C4⋊Dic53—21no—
126C4⋊Dic53C2×D412yes—
127#11719919—22no—
128#11719919—22no—
129C4×Dic53—22no—
130C2×#88322—22no—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322C2×D411yescharacteristic
134C2×#88322C2×D411yescharacteristic
135C2×#88322C2×D411yescharacteristic
136C2×#88322C2×D411yescharacteristic
137#11719913—84no—
138C23×C53#12111yescharacteristic
139C22×D53—42no—
140C22×D53—82no—
141C22×D53—82no—
142C22×D53—82no—
143Q8×C53—82no—
144Q8×C53—82no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—42no—
148C2×C4×C53—82no—
149C2×C4×C53#12111yescharacteristic
150C2×C4×C53#12111yescharacteristic
151C2×C4×C53#12111yescharacteristic
152C2×C4×C53#13512yes—
153C2×C4×C53#13712yes—
154C2×C4×C53C22×D411yescharacteristic
155C2×C4×C53C22×D411yescharacteristic
156C2×C4×C53C2×#4111yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—22no—
159D212—42no—
160D212—42no—
161D212—44no—
162D212—82no—
163D212—82no—
164D212—82no—
165D212—82no—
166Dic106—24no—
167Dic106—42no—
168C8×C53—42no—
169C8×C53—42no—
170#88322—21no—
171#88322—21no—
172#88322—21no—
173#88322—21no—
174#88322—42no—
175#19212—531nomaximal, Sylow
176C22×C53—21no—
177C22×C53—21no—
178C22×C53#46412yes—
179C22×C53C2×#12111yescharacteristic
180C2×D53—82no—
181C2×D53—82no—
182C2×D53—82no—
183C2×D53—164no—
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—82no—
187C4×C53#46412yes—
188C4×C53#47212yes—
189C4×C53C2×#13511yescharacteristic
190C4×C53C2×#13711yescharacteristic
191C4×C53#57712yes—
192Dic53—42no—
193Dic53—42no—
194#3230—531no—
195#3208—531no—
196C2×#486—531no—
197#2960—531no—
198#1698—534no—
199#1678—534no—
200#1564—531no—
201#1548D5311yescharacteristic, complemented
202#1545—531no—
203C2×C53—22no—
204C2×C53C2×#46411yescharacteristic
205C2×C53#305711yescharacteristic
206C2×C53C2×#47211yescharacteristic
207D53—162no—
208#533—531no—
209C2×C4⋊Q8C2×D5311yescharacteristic
210C2×#127—531no—
211#500—1061no—
212#492—1061no—
213#487—1061no—
214#486—532no—
215#486—532no—
216#486—1061no—
217#486—1061no—
218#486—1061no—
219#484—1062no—
220#477—534no—
221#460—534no—
222#450—1062no—
223#446—532no—
224#446—1061no—
225C8×Q8—532no—
226C8×Q8—1061no—
227#431—531no—
228#431C2×D5311yescharacteristic
229#424—1061no—
230#423C2×D5311yescharacteristic
231C2×#102—1061no—
232C2×#102—1061no—
233#405—1061no—
234C2×C4×C8—531no—
235#335—532no—
236#334—532no—
237#333C2×D5314yescomplemented
238#331—534no—
239C53#1921211yescharacteristic, complemented, Sylow
240C22×Q8—21no—
241C22×D4—1061no—
242C22×D4—1061no—
243C22×D4—1061no—
244C2×SD16—1064no—
245C2×M16—21no—
246C22×C8—531no—
247C22×C8—531no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—531no—
252C4⋊Q8—531no—
253C4⋊Q8C22×D5312yes—
254#127—532no—
255#127—1061no—
256#127—1061no—
257#127—1061no—
258#125—1061no—
259C22⋊Q8—532no—
260C22⋊Q8—1061no—
261C4×Q8—532no—
262C4×Q8—1061no—
263#117—1061no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C22×D5311yescharacteristic
267#107—1064no—
268C4⋊C8—21no—
269C4⋊C8—532no—
270C4⋊C8—1062no—
271C4⋊C8—1062no—
272C4⋊C8C22×D5312yes—
273C4⋊C8C22×D5312yes—
274#103—1064no—
275#102—1062no—
276#102—1062no—
277#102—1062no—
278#102—1062no—
279#102—1064no—
280C22⋊C8—21no—
281C22⋊C8—1061no—
282#97—1062no—
283C4×C8—21no—
284C4×C8—531no—
285C4×C8—531no—
286C4×C8—531no—
287C4×C8—531no—
288C4×C8—1062no—
289C4×C8—1061no—
290C24—2121no—
291C2×Q8—22no—
292C2×Q8—41no—
293C2×Q8—82no—
294C2×Q8—532no—
295C2×Q8—1061no—
296C2×D4—1062no—
297C2×D4—1062no—
298C2×D4—1062no—
299C2×D4—4242no—
300C2×D4—4242no—
301C2×D4—2121no—
302C2×D4—2121no—
303C2×D4—2121no—
304C2×D4—2122no—
305C2×D4—2122no—
306C22×C4—41no—
307C22×C4#1171994511yescharacteristic
308C22×C4D4×D5311yescharacteristic
309C22×C4D4×D5311yescharacteristic
310SD16—4244no—
311M16—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—531no—
315C2×C8—531no—
316C2×C8—531no—
317C2×C8—531no—
318C2×C8—1062no—
319C2×C8—1062no—
320C2×C8—1062no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—532no—
326C4⋊C4—1061no—
327C4⋊C4—1061no—
328C4⋊C4—1062no—
329C4⋊C4—1061no—
330C4⋊C4—1062no—
331C22⋊C4—1062no—
332C42—1062no—
333C42#1171994511yescharacteristic
334C42D4×D5311yescharacteristic
335C42D4×D5311yescharacteristic
336C42C23×D5311yescharacteristic
337C23—4242no—
338C23—4242no—
339C23—4242no—
340C23—2122no—
341C23#1313319011yescharacteristic
342Q8—82no—
343Q8—82no—
344Q8—1064no—
345Q8—2122no—
346D4—4242no—
347D4—4242no—
348D4—4242no—
349D4—4242no—
350D4—2122no—
351D4—2122no—
352D4—2124no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—1062no—
358C2×C4—1062no—
359C2×C4#1313319011yescharacteristic
360C2×C4#1313319011yescharacteristic
361C2×C4#1313319011yescharacteristic
362C2×C4#1313323812yes—
363C2×C4#1313324912yes—
364C2×C4C2×D4×D5311yescharacteristic
365C2×C4C2×D4×D5311yescharacteristic
366C2×C4C2×#1171994511yescharacteristic
367C8—42no—
368C8—42no—
369C8—1061no—
370C8—1061no—
371C8—1061no—
372C8—1061no—
373C8—2122no—
374C22—21no—
375C22—21no—
376C22—8484no—
377C22—4242no—
378C22—4242no—
379C22—4242no—
380C22#1681084812yes—
381C22C2×#1313319011yescharacteristic
382C4—21no—
383C4—21no—
384C4—82no—
385C4—2122no—
386C4—2122no—
387C4#1681084812yes—
388C4#1681089212yes—
389C4C2×#1313323811yescharacteristic
390C4C2×#1313324911yescharacteristic
391C4#1681143212yes—
392C2—22no—
393C2—8482no—
394C2C2×#1681084811yescharacteristic
395C2#2028435711yescharacteristic
396C2C2×#1681089211yescharacteristic
subgroupG/H
centerC22C2×#13133190
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#1548C2
socleC22×C53C2×#121
prime radicalC2×#13133214C22
hypercenter#1548D53
hypocenterC53#19212
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
1. #20285866C2semidirect
4. #20283645C2semidirect
5. #20274258C2semidirect
6. #20274134C2semidirect
7. #20273461C2semidirect
9. C53×#1548C2semidirect
21. #16810915C22semidirect
25. #16810750C22semidirect
27. #16810747C22semidirect
34. #16810375C22semidirect
35. #16810373C22semidirect
40. C53×#333C22semidirect
201. #1548D53semidirect
237. #333C2×D53semidirect
239. C53#19212semidirect

Maximal subgroups

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

subgroupindex
#202858662
#202856582
C2×#168109712
#202836452
#202742582
#202741342
#202734612
#202733842
C53×#15482
#1921253