Finite Group Databasebeta

← #27387766

Subgroups — #27387766

Order 13568.

1638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811171C211yesmaximal, characteristic
2C2×#16810621C214yesmaximal, complemented
3C2×#16810575C211yesmaximal, characteristic, complemented
4#382×D53C211yesmaximal, characteristic, complemented
5#20275540C211yesmaximal, characteristic, complemented
6#20275535C211yesmaximal, characteristic, complemented
7#20275534C211yesmaximal, characteristic, complemented
8#20275525C214yesmaximal, complemented
9C53×#1961C211yesmaximal, characteristic, complemented
10C22×#11719947C2212yes—
11C22×#11719912C2212yes—
12#16811171—24no—
13#16811171C418yescomplemented
14C2×#13133217C2211yescharacteristic
15C2×#13133204C2212yes—
16C2×#13133188—22no—
17C2×D4×Dic53—22no—
18C2×C4⋊C4×D53C2212yes—
19C2×C42×D53C2211yescharacteristic
20#16810621—216no—
21C2×#13133142C2214yes—
22C2×#13133140C2214yes—
23C2×#13133138C2214yes—
24#16810575—24no—
25C2×#13133128C2211yescharacteristic
26C2×C8×Dic53C2212yes—
27#16810489C2211yescharacteristic
28#16810488C2212yes—
29#16810487—22no—
30#95×D53—22no—
31C2×#127×C53C2211yescharacteristic
32C2×#102×C53C2214yes—
33C2×#97×C53C2211yescharacteristic
34C53×#382C2211yescharacteristic
35C2×#11719947—28no—
36C2×#11719947—24no—
37C2×#11719947—28no—
38C2×#11719947C2×C418yes—
39C22×Dic106C2312yes—
40C22×C22⋊D53—22no—
41C23×Dic53—22no—
42C22×C4×D53—22no—
43C22×C4×D53D412yes—
44C22×C4×D53C2311yescharacteristic
45C2×#11719912—24no—
46C2×#11719912—24no—
47C2×#11719912D418yescomplemented
48C22×#88322C2312yes—
49#13133217—22no—
50#13133217C2×C414yescomplemented
51#13133204—24no—
52#13133204C2×C418yescomplemented
53#13133188—216no—
54D4×Dic53—216no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C4⋊C4×D53—28no—
59C2×C22⋊Dic53—24no—
60C2×C4×Dic53D412yes—
61C2×C4×Dic53C2311yescharacteristic
62C42×D53C2×C414yes—
63C42×D53D414yes—
64#13133142—28no—
65#13133140—28no—
66#13133138—28no—
67#13133128—22no—
68C8×Dic53—24no—
69#13133124—22no—
70#13133124—24no—
71C22×D4×C53—22no—
72C22×D4×C53C2312yes—
73C22×C8×C53C2312yes—
74#127×C53—22no—
75#127×C53C2×C414yescomplemented
76C2×C4⋊C4×C53C2312yes—
77C2×C42×C53C2311yescharacteristic
78#102×C53—28no—
79#97×C53—22no—
80#95×C53—22no—
81C24×C53—22no—
82C23×D53C2×D411yescharacteristic
83C2×D4×C53—28no—
84C2×D4×C53—24no—
85C2×D4×C53—22no—
86C2×D4×C53—24no—
87C2×D4×C53—48no—
88C2×D4×C53C22×C414yes—
89C22×C4×C53—22no—
90C22×C4×C53C2×D412yes—
91C22×C4×C53C2411yescharacteristic
92C2×C22⋊D53—28no—
93C2×C22⋊D53—44no—
94C2×C22⋊D53—44no—
95C22×Dic53—22no—
96C22×Dic53—28no—
97C22×Dic53—22no—
98C22×Dic53—44no—
99C22×Dic53—44no—
100C22×Dic53C2×D412yes—
101C22×Dic53C2×D411yescharacteristic
102#11719947—232no—
103#11719947—416no—
104C2×C4×D53—44no—
105C2×C4×D53—48no—
106C2×C4×D53—44no—
107C2×C4×D53C22⋊C418yes—
108C2×C4×D53C22×C412yes—
109C2×C4×D53C2×D414yes—
110C2×C4×D53C2×D412yes—
111C2×C4×D53#4114yes—
112C2×Dic106—24no—
113C2×Dic106—24no—
114C2×Dic106—22no—
115C2×Dic106C22×C414yes—
116C2×C8×C53—22no—
117C2×C8×C53—22no—
118C2×C8×C53C2×D414yes—
119C4⋊C4×C53—24no—
120C42×C53C22×C412yes—
121C42×C53C2×D412yes—
122C22⋊Dic53—216no—
123C4⋊Dic53—28no—
124C4⋊Dic53—24no—
125#11719919—44no—
126C4×Dic53C22⋊C418yes—
127C4×Dic53C22×C412yes—
128C4×Dic53C2×D412yes—
129C2×#88322—22no—
130C2×#88322—22no—
131C2×#88322C2×D414yes—
132#11719912—216no—
133C23×C53—24no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—48no—
137C23×C53C22×D411yescharacteristic
138C22×D53—24no—
139C22×D53C2×C22⋊C412yes—
140C22×D53#12412yes—
141C22×D53#12712yes—
142D4×C53—28no—
143D4×C53—416no—
144D4×C53—44no—
145C2×C4×C53—24no—
146C2×C4×C53—42no—
147C2×C4×C53—44no—
148C2×C4×C53C2×C22⋊C414yes—
149C2×C4×C53C23×C411yescharacteristic
150C2×C4×C53C22×D412yes—
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C53C2×#4112yes—
153C22⋊D53—416no—
154C2×Dic53—24no—
155C2×Dic53—24no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—416no—
159C2×Dic53—42no—
160C2×Dic53—44no—
161C2×Dic53C2×C22⋊C414yes—
162C2×Dic53C2×C22⋊C412yes—
163C2×Dic53#12412yes—
164C2×Dic53#12712yes—
165C4×D53—48no—
166C4×D53—88no—
167C4×D53C4×D418yes—
168C4×D53#12118yes—
169Dic106—28no—
170Dic106—44no—
171C8×C53—24no—
172#88322—24no—
173#21201—531nomaximal, Sylow
174C22×C53—48no—
175C22×C53—44no—
176C22×C53—84no—
177C22×C53—84no—
178C22×C53C22×C22⋊C411yescharacteristic
179C22×C53C2×#12411yescharacteristic
180C22×C53C2×#12711yescharacteristic
181C22×C53#57514yes—
182C2×D53—24no—
183C2×D53—24no—
184C2×D53—24no—
185C2×D53#39014yes—
186C4×C53—24no—
187C4×C53—82no—
188C4×C53C2×C4×D412yes—
189C4×C53C2×#12112yes—
190Dic53—24no—
191Dic53—44no—
192Dic53—82no—
193Dic53#39014yes—
194C2×#532—531no—
195C2×#417—534no—
196C2×C4×M16—531no—
197C2×#382—531no—
198#1961—531no—
199#1961D5311yescharacteristic, complemented
200#1960—531no—
201#1960—531no—
202#1958—534no—
203C2×C53—84no—
204C2×C53C2×#39011yescharacteristic
205C2×C53#289612yes—
206C2×C53#314212yes—
207C2×C53#315312yes—
208D53—24no—
209C22×#41—532no—
210C22×M16—532no—
211#532—538no—
212#532—1064no—
213C2×C4⋊Q8—531no—
214C2×#127C2×D5311yescharacteristic
215C2×#124—532no—
216C2×#121—1062no—
217C2×C4×D4—1062no—
218C22×C4⋊C4—532no—
219C22×C42—531no—
220#417—10616no—
221C2×#103—534no—
222C2×#103—534no—
223C2×#102—534no—
224C2×#102C2×D5314yes—
225C4×M16—1064no—
226C2×#97—531no—
227C2×#97C2×D5311yescharacteristic
228C2×C4×C8—532no—
229#382—532no—
230#382—531no—
231#382—1062no—
232#382C2×D5311yescharacteristic
233C2×#95—1062no—
234C53#2120111yescharacteristic, complemented, Sylow
235C2×#41—538no—
236C2×#41—1064no—
237C2×#41—1068no—
238C2×#41—1068no—
239C22×Q8—532no—
240C22×D4—22no—
241C22×D4—1062no—
242C22×D4C22×D5312yes—
243C23×C4—531no—
244C23×C4—532no—
245C23×C4—1062no—
246C23×C4—1062no—
247C2×M16—538no—
248C2×M16—1064no—
249C2×M16—1064no—
250C22×C8—532no—
251C22×C8C22×D5312yes—
252C4⋊Q8—534no—
253C4⋊Q8—1062no—
254#127—22no—
255#127C4×D5314yescomplemented
256#124—538no—
257#124—1064no—
258#121—10616no—
259C4×D4—10616no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—1068no—
262C2×C4⋊C4—1062no—
263C2×C4⋊C4—1062no—
264C2×C4⋊C4C22×D5312yes—
265C2×C22⋊C4—1064no—
266C2×C42—532no—
267C2×C42—531no—
268C2×C42—534no—
269C2×C42—534no—
270C2×C42C22×D5311yescharacteristic
271#103—1068no—
272#103—1068no—
273#102—28no—
274#102—1068no—
275#97—22no—
276#97—1062no—
277C4×C8—1064no—
278#95—22no—
279#95—1064no—
280#95—1062no—
281C24—22no—
282C24—531no—
283#41—10632no—
284#41—21216no—
285C2×Q8—534no—
286C2×Q8—1062no—
287C2×Q8—1064no—
288C2×Q8—1064no—
289C2×D4—28no—
290C2×D4—24no—
291C2×D4—22no—
292C2×D4—24no—
293C2×D4—48no—
294C2×D4—1068no—
295C2×D4—2124no—
296C2×D4—2124no—
297C2×D4C2×C4×D5314yes—
298C22×C4—22no—
299C22×C4—531no—
300C22×C4—532no—
301C22×C4—532no—
302C22×C4—534no—
303C22×C4—534no—
304C22×C4—532no—
305C22×C4—538no—
306C22×C4—1068no—
307C22×C4—1062no—
308C22×C4—1062no—
309C22×C4—2124no—
310C22×C4—2124no—
311C22×C4—2128no—
312C22×C4—2124no—
313C22×C4—2124no—
314C22×C4D4×D5312yes—
315C22×C4C23×D5311yescharacteristic
316M16—10616no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—534no—
320C2×C8—1062no—
321C2×C8—1062no—
322C2×C8D4×D5314yes—
323C4⋊C4—24no—
324C4⋊C4—1068no—
325C4⋊C4—1064no—
326C4⋊C4—2124no—
327C22⋊C4—10616no—
328C42—538no—
329C42—532no—
330C42—532no—
331C42C2×C4×D5312yes—
332C42D4×D5312yes—
333C23—24no—
334C23—42no—
335C23—42no—
336C23—48no—
337C23—532no—
338C23—532no—
339C23—532no—
340C23—1064no—
341C23C2×D4×D5311yescharacteristic
342Q8—1068no—
343Q8—2124no—
344D4—28no—
345D4—416no—
346D4—44no—
347D4—21216no—
348C2×C4—24no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—532no—
352C2×C4—532no—
353C2×C4—532no—
354C2×C4—538no—
355C2×C4—538no—
356C2×C4—534no—
357C2×C4—1064no—
358C2×C4—1064no—
359C2×C4—4248no—
360C2×C4—2128no—
361C2×C4—21216no—
362C2×C4—2122no—
363C2×C4—2124no—
364C2×C4—2122no—
365C2×C4—2122no—
366C2×C4C22⋊C4×D5314yes—
367C2×C4C22×C4×D5311yescharacteristic
368C2×C4C2×D4×D5312yes—
369C2×C4C2×D4×D5311yescharacteristic
370C2×C4#41×D5312yes—
371C8—24no—
372C8—1064no—
373C22—48no—
374C22—44no—
375C22—84no—
376C22—84no—
377C22—534no—
378C22—1064no—
379C22—1064no—
380C22—1064no—
381C22C2×C22⋊C4×D5311yescharacteristic
382C22#124×D5311yescharacteristic
383C22#127×D5311yescharacteristic
384C22#1681141914yes—
385C4—24no—
386C4—82no—
387C4—534no—
388C4—1064no—
389C4—4242no—
390C4—2124no—
391C4C4×D4×D5312yes—
392C4#121×D5312yes—
393C2—84no—
394C2—1064no—
395C2#390×D5311yescharacteristic
396C2#2028322912yes—
397C2#2028511412yes—
398C2#2028518012yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1961C2
socleC23×C53C22×D4
prime radicalC2×#16811171C2
cosocleC22×C4×C53C24
hypercenter#1961D53
hypocenterC53#21201
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1961

Complements

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

NH ≅ G/Ntype
2. C2×#16810621C2semidirect
3. C2×#16810575C2semidirect
4. #382×D53C2semidirect
5. #20275540C2semidirect
6. #20275535C2semidirect
7. #20275534C2semidirect
8. #20275525C2semidirect
9. C53×#1961C2semidirect
13. #16811171C4semidirect
47. C2×#11719912D4semidirect
50. #13133217C2×C4semidirect
52. #13133204C2×C4semidirect
75. #127×C53C2×C4semidirect
199. #1961D53semidirect
234. C53#21201semidirect
255. #127C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168111712
C2×#168106212
C2×#168105752
#382×D532
#202755402
#202755352
#202755342
#202755252
C53×#19612
#2120153