Finite Group Databasebeta

← #27388133

Subgroups — #27388133

Order 13568.

1562 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#20288048C211yesmaximal, characteristic
2C2×#16810639C212yesmaximal, complemented
3C22×#13133143C212yesmaximal, complemented
4#20277199C211yesmaximal, characteristic, complemented
5#20275634C218yesmaximal, complemented
6C53×#2873C211yesmaximal, characteristic, complemented
7C22×#11719945C2212yes—
8#16811204—22no—
9#16811199C414yescomplemented
10#16811192C414yescomplemented
11#16811185—22no—
12#16811173C2214yes—
13#16811135—24no—
14C2×#13133176C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16#16810639—28no—
17C2×#13133143—28no—
18C2×#13133143C2218yescomplemented
19C2×#13133143C2218yescomplemented
20#16810522—22no—
21#16810498—22no—
22C4×C4⋊Dic53C2214yescomplemented
23C22×M16×C53C2212yes—
24C2×#117×C53C2211yescharacteristic
25C53×#405—24no—
26C4×M16×C53C2214yescomplemented
27C2×#11719945—22no—
28C2×#11719945—22no—
29C2×#11719945—24no—
30C2×#11719945—24no—
31C2×#11719945—24no—
32C2×#11719945—24no—
33C2×#11719945C2×C414yes—
34C2×#11719945C2314yes—
35C22×Dic106C2×C412yes—
36C22×D212C2×C412yes—
37C22×C22⋊D53—22no—
38C23×Dic53—21no—
39C22×C4×D53—22no—
40#13133217C2×C414yescomplemented
41#13133214C2×C414yescomplemented
42#13133205C2×C418yescomplemented
43#13133201—28no—
44#13133197—28no—
45#13133192—28no—
46#13133191—28no—
47#13133187—28no—
48C4×D212—28no—
49#13133176—24no—
50C2×C4⋊Dic53—24no—
51C2×C4⋊Dic53—21no—
52C2×C22⋊Dic53—21no—
53C2×C4×Dic53—21no—
54C2×C4×Dic53—24no—
55C2×C4×Dic53C2312yes—
56C2×C4×Dic53C2312yes—
57#13133143—216no—
58#13133143—216no—
59#13133124—24no—
60C23×C4×C53C2311yescharacteristic
61C2×M16×C53—22no—
62C2×M16×C53—24no—
63C2×M16×C53D418yescomplemented
64C2×M16×C53C2314yes—
65C22×C8×C53—22no—
66#117×C53C2×C414yes—
67#117×C53D414yes—
68C2×C4⋊C4×C53D412yes—
69C2×C22⋊C4×C53D412yes—
70C2×C42×C53C2312yes—
71#97×C53—24no—
72C4×C8×C53—24no—
73C24×C53C2×D411yescharacteristic
74C23×D53—22no—
75C22×C4×C53C22×C412yes—
76C22×C4×C53C2×D414yes—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D412yes—
79C22×C4×C53C2×D414yes—
80C22×C4×C53#4114yes—
81C22×C4×C53C2411yescharacteristic
82C2×C22⋊D53—24no—
83C2×C22⋊D53—24no—
84C2×C22⋊D53—44no—
85C2×C22⋊D53—44no—
86C22×Dic53—22no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—44no—
90C22×Dic53—42no—
91#11719945—28no—
92#11719945—28no—
93#11719945—24no—
94#11719945—24no—
95#11719945—24no—
96#11719945—24no—
97#11719945—44no—
98#11719945—44no—
99#11719945—44no—
100#11719945—44no—
101C2×D212—24no—
102C2×D212—24no—
103C2×D212—22no—
104C2×D212—28no—
105C2×D212C22×C414yes—
106C2×C4×D53—24no—
107C2×C4×D53—24no—
108C2×C4×D53—44no—
109C2×C4×D53—44no—
110C2×Dic106—24no—
111C2×Dic106—24no—
112C2×Dic106—22no—
113C2×Dic106C22×C414yes—
114M16×C53—28no—
115M16×C53—24no—
116M16×C53C2×D418yes—
117C2×C8×C53—28no—
118C2×C8×C53—24no—
119C4⋊C4×C53C22⋊C418yes—
120C22⋊C4×C53C22⋊C418yes—
121C42×C53C22×C414yes—
122C42×C53C2×D414yes—
123C22⋊Dic53—42no—
124#11719921—216no—
125C4⋊Dic53—28no—
126C4⋊Dic53—48no—
127C4⋊Dic53—42no—
128C4×Dic53—24no—
129C4×Dic53—24no—
130C4×Dic53—42no—
131C23×C53—42no—
132C23×C53C2×C22⋊C412yes—
133C23×C53#12412yes—
134C23×C53#12712yes—
135C23×C53C22×D411yescharacteristic
136C22×D53—24no—
137C22×D53—44no—
138C22×D53—84no—
139C2×C4×C53—44no—
140C2×C4×C53C2×C22⋊C418yes—
141C2×C4×C53C2×C22⋊C412yes—
142C2×C4×C53C4×D414yes—
143C2×C4×C53C4×D414yes—
144C2×C4×C53#12114yes—
145C2×C4×C53#12114yes—
146C2×C4×C53#12412yes—
147C2×C4×C53#12712yes—
148C2×C4×C53C23×C411yescharacteristic
149C2×C4×C53C22×D411yescharacteristic
150C2×C4×C53C22×D412yes—
151C2×C4×C53C2×#4112yes—
152C22⋊D53—44no—
153C22⋊D53—44no—
154C22⋊D53—48no—
155C2×Dic53—24no—
156C2×Dic53—24no—
157C2×Dic53—42no—
158C2×Dic53—48no—
159C2×Dic53—42no—
160C2×Dic53—44no—
161C2×Dic53—44no—
162C2×Dic53—42no—
163D212—24no—
164D212—24no—
165D212—44no—
166D212—88no—
167C4×D53—44no—
168C4×D53—44no—
169C4×D53—48no—
170Dic106—24no—
171Dic106—24no—
172Dic106—44no—
173C8×C53—28no—
174#21253—531nomaximal, Sylow
175C22×C53—22no—
176C22×C53—24no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53#39014yes—
180C22×C53C22×C22⋊C411yescharacteristic
181C22×C53C2×#12411yescharacteristic
182C22×C53C2×#12711yescharacteristic
183C2×D53—48no—
184C2×D53—44no—
185C2×D53—88no—
186C4×C53—28no—
187C4×C53#39014yes—
188C4×C53C2×C4×D412yes—
189C4×C53C2×#12112yes—
190Dic53—44no—
191Dic53—44no—
192Dic53—82no—
193#3449—531no—
194C2×#421—532no—
195C22×#104—532no—
196#2873D5311yescharacteristic, complemented
197#2296—531no—
198#1980—538no—
199C2×C53—21no—
200C2×C53—24no—
201C2×C53—21no—
202C2×C53C2×#39011yescharacteristic
203C2×C53#290112yes—
204D53—84no—
205C22×#41—532no—
206C22×M16C2×D5312yes—
207#537—1062no—
208#536—534no—
209#535—534no—
210#534—1062no—
211#532—534no—
212#525—1064no—
213C2×#117—531no—
214C2×#117C2×D5311yescharacteristic
215C22×C42—531no—
216#421—1068no—
217C2×#104—538no—
218C2×#104—538no—
219C2×#104—1068no—
220#405—24no—
221C4×M16C2×D5314yescomplemented
222#389—1062no—
223#385—1062no—
224C4×C4⋊C4—534no—
225C53#2125311yescharacteristic, complemented, Sylow
226C2×#41—534no—
227C2×#41—534no—
228C2×#41—1062no—
229C2×#41—1062no—
230C2×#41—1064no—
231C2×#41—1064no—
232C2×#41—1064no—
233C2×#41—1064no—
234C22×Q8—532no—
235C22×D4—532no—
236C22×D4—1062no—
237C23×C4—1062no—
238C23×C4—1061no—
239C23×C4C22×D5311yescharacteristic
240C2×M16—24no—
241C2×M16—22no—
242C2×M16D21218yescomplemented
243C2×M16C22×D5314yes—
244C22×C8—22no—
245C4⋊Q8—534no—
246#127—534no—
247#124—538no—
248#123—1068no—
249C22⋊Q8—1068no—
250#121—1068no—
251#121—1068no—
252#120—1068no—
253C4×D4—1068no—
254#117—1064no—
255#117C4×D5314yes—
256#117C22⋊D5314yes—
257C2×C4⋊C4—1061no—
258C2×C4⋊C4—1064no—
259C2×C4⋊C4D21212yes—
260C2×C22⋊C4—1061no—
261C2×C22⋊C4D21212yes—
262C2×C42—532no—
263C2×C42—532no—
264C2×C42—1064no—
265C2×C42—1061no—
266C2×C42C22×D5312yes—
267#104—10616no—
268#104—10616no—
269#97—24no—
270C4×C8—24no—
271#95—1064no—
272C24—1062no—
273C24C2×D21211yescharacteristic
274#41—1068no—
275#41—1068no—
276#41—1064no—
277#41—1064no—
278#41—1064no—
279#41—1064no—
280#41—2124no—
281#41—2124no—
282#41—2124no—
283#41—2124no—
284C2×Q8—534no—
285C2×Q8—1062no—
286C2×Q8—1064no—
287C2×Q8—1064no—
288C2×D4—534no—
289C2×D4—1064no—
290C2×D4—1064no—
291C2×D4—1068no—
292C2×D4—1062no—
293C2×D4—1064no—
294C2×D4—1064no—
295C2×D4—2124no—
296C2×D4—2124no—
297C22×C4—1062no—
298C22×C4—1062no—
299C22×C4—1064no—
300C22×C4—1064no—
301C22×C4—1062no—
302C22×C4—2124no—
303C22×C4—2122no—
304C22×C4—2124no—
305C22×C4—2124no—
306C22×C4C2×C4×D5312yes—
307C22×C4C2×D21214yes—
308C22×C4C2×D21211yescharacteristic
309C22×C4C2×D21214yes—
310C22×C4#1171994514yes—
311C22×C4C2×C22⋊D5312yes—
312C22×C4C23×D5311yescharacteristic
313M16—28no—
314M16—24no—
315M16C2×D21218yes—
316C2×C8—28no—
317C2×C8—24no—
318C4⋊C4—1068no—
319C4⋊C4—2128no—
320C4⋊C4—2122no—
321C4⋊C4#1171992118yes—
322C22⋊C4—10616no—
323C22⋊C4—2122no—
324C22⋊C4#1171992118yes—
325C42—1064no—
326C42—1064no—
327C42—2122no—
328C42C2×C4×D5314yes—
329C42C2×C22⋊D5314yes—
330C23—42no—
331C23—1064no—
332C23—4244no—
333C23—2124no—
334C23C2×#1171992112yes—
335C23#1313320512yes—
336C23#1313321412yes—
337C23C22×D21211yescharacteristic
338Q8—1064no—
339Q8—1064no—
340Q8—2124no—
341D4—1064no—
342D4—1064no—
343D4—4248no—
344D4—2128no—
345D4—2124no—
346D4—2124no—
347D4—2124no—
348C2×C4—44no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4—2124no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2128no—
355C2×C4—2124no—
356C2×C4—2122no—
357C2×C4—2128no—
358C2×C4—2124no—
359C2×C4—2124no—
360C2×C4C2×#1171992118yes—
361C2×C4C2×#1171992112yes—
362C2×C4C4×D21214yes—
363C2×C4C4×D21214yes—
364C2×C4#1313319114yes—
365C2×C4#1313319114yes—
366C2×C4#1313320512yes—
367C2×C4#1313321412yes—
368C2×C4C22×C4×D5311yescharacteristic
369C2×C4C22×C22⋊D5311yescharacteristic
370C2×C4C22×D21212yes—
371C2×C4C2×#1171994512yes—
372C8—28no—
373C22—22no—
374C22—24no—
375C22—42no—
376C22—42no—
377C22—4248no—
378C22—2128no—
379C22—2124no—
380C22#1681052514yes—
381C22C22×#1171992111yescharacteristic
382C22C2×#1313320511yescharacteristic
383C22C2×#1313321411yescharacteristic
384C4—28no—
385C4—4242no—
386C4—2124no—
387C4—2124no—
388C4#1681052514yes—
389C4C2×C4×D21212yes—
390C4C2×#1313319112yes—
391C2—21no—
392C2—24no—
393C2—21no—
394C2—4244no—
395C2C2×#1681052511yescharacteristic
396C2#2028326012yes—
subgroupG/H
centerC2×C4C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#2873C2
socleC22×C53C22×C22⋊C4
prime radical#20288048C2
cosocleC22×C4×C53C24
hypercenter#2873D53
hypocenterC53#21253
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. C2×#16810639C2semidirect
3. C22×#13133143C2semidirect
4. #20277199C2semidirect
5. #20275634C2semidirect
6. C53×#2873C2semidirect
9. #16811199C4semidirect
10. #16811192C4semidirect
18. C2×#13133143C22semidirect
19. C2×#13133143C22semidirect
22. C4×C4⋊Dic53C22semidirect
26. C4×M16×C53C22semidirect
40. #13133217C2×C4semidirect
41. #13133214C2×C4semidirect
42. #13133205C2×C4semidirect
63. C2×M16×C53D4semidirect
196. #2873D53semidirect
221. C4×M16C2×D53semidirect
225. C53#21253semidirect
242. C2×M16D212semidirect

Maximal subgroups

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

subgroupindex
#202880482
C2×#168106392
C22×#131331432
#202771992
#202756342
C53×#28732
#2125353