Finite Group Databasebeta

← #27393631

Subgroups — #27393631

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811178C211yesmaximal, characteristic
2C2×#16810678C211yesmaximal, characteristic
3#20277108C211yesmaximal, characteristic
4#20275689C211yesmaximal, characteristic, complemented
5#20275688C211yesmaximal, characteristic, complemented
6#20275687C211yesmaximal, characteristic, complemented
7#20275671C214yesmaximal, complemented
8#20274809C214yesmaximal
9C53×#1993C211yesmaximal, characteristic, complemented
10#16811178—24no—
11#16811178C418yescomplemented
12C2×#13133217C2211yescharacteristic
13C2×#13133208C2212yes—
14C2×#13133197—21no—
15C2×#13133193—21no—
16C2×Q8×Dic53—21no—
17C2×#13133182—21no—
18C2×#13133175C2212yes—
19C2×C42×D53C2211yescharacteristic
20#16810678—24no—
21C2×#13133146C2212yes—
22C2×#13133129C2212yes—
23C2×#13133128C2211yescharacteristic
24#16810490—21no—
25#16810489C2211yescharacteristic
26#16810485—21no—
27#16810480C2212yes—
28C4⋊C4×Dic53C2212yes—
29#16810470—21no—
30#16810468—21no—
31#16810386C2214yes—
32#16810385C2214yes—
33#16810385C2214yes—
34C2×C4⋊Q8×C53C2211yescharacteristic
35C2×#97×C53C2211yescharacteristic
36C53×#382C2211yescharacteristic
37C53×#340C2214yes—
38C22×Dic106—21no—
39C22×C4×D53D411yescharacteristic
40C22×C4×D53D411yescharacteristic
41C22×C4×D53C2311yescharacteristic
42C22×#88322C2312yes—
43#13133217—22no—
44#13133217C2×C414yes—
45#13133208—24no—
46#13133208C2×C418yes—
47#13133197—28no—
48#13133193—28no—
49Q8×Dic53—28no—
50#13133182—28no—
51#13133175—24no—
52#13133175C2×C418yes—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—22no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2312yes—
57C2×#11719919—22no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—22no—
62C2×C4×Dic53D411yescharacteristic
63C2×C4×Dic53D411yescharacteristic
64C2×C4×Dic53C2311yescharacteristic
65C42×D53C2×C414yes—
66C42×D53D414yes—
67#13133146—24no—
68#13133129—24no—
69#13133128—22no—
70#13133124—21no—
71#13133124—21no—
72#13133124—22no—
73#13133124—22no—
74C22×Q8×C53—21no—
75C22×C8×C53C2312yes—
76C4⋊Q8×C53—22no—
77C4⋊Q8×C53C2×C414yes—
78C2×C4⋊C4×C53—21no—
79C2×C4⋊C4×C53C2312yes—
80C2×C4⋊C4×C53C2312yes—
81C2×C42×C53C2311yescharacteristic
82#97×C53—22no—
83#95×C53—21no—
84#95×C53—21no—
85C23×D53C2×D411yescharacteristic
86C2×Q8×C53—24no—
87C2×Q8×C53—44no—
88C22×C4×C53—22no—
89C22×C4×C53—22no—
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2411yescharacteristic
93C22×Dic53—22no—
94C22×Dic53—22no—
95C22×Dic53C2×D411yescharacteristic
96C22×Dic53C2×D411yescharacteristic
97C22×Dic53C2×D411yescharacteristic
98C2×C4×D53—22no—
99C2×C4×D53—22no—
100C2×C4×D53—42no—
101C2×C4×D53—42no—
102C2×C4×D53C22⋊C414yes—
103C2×C4×D53C22⋊C414yes—
104C2×C4×D53C22×C412yes—
105C2×C4×D53C2×D412yes—
106C2×Dic106—24no—
107C2×Dic106—44no—
108C2×C8×C53—22no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C4⋊C4×C53—24no—
112C4⋊C4×C53—22no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53C22×C414yes—
115C42×C53C22×C412yes—
116C42×C53C2×D412yes—
117#11719921—28no—
118#11719921—44no—
119C4⋊Dic53—24no—
120C4⋊Dic53—22no—
121C4⋊Dic53—28no—
122C4⋊Dic53—24no—
123C4⋊Dic53C22×C414yes—
124#11719919—28no—
125C4×Dic53—28no—
126C4×Dic53—44no—
127C4×Dic53C22⋊C414yes—
128C4×Dic53C22⋊C414yes—
129C4×Dic53C22×C412yes—
130C4×Dic53C2×D412yes—
131C2×#88322—22no—
132C2×#88322—22no—
133C2×#88322—22no—
134C23×C53C22×D411yescharacteristic
135C22×D53—21no—
136C22×D53—21no—
137C22×D53—22no—
138C22×D53—22no—
139C22×D53C2×C22⋊C412yes—
140Q8×C53—48no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—24no—
144C2×C4×C53—44no—
145C2×C4×C53—42no—
146C2×C4×C53—42no—
147C2×C4×C53—42no—
148C2×C4×C53C2×C22⋊C412yes—
149C2×C4×C53C2×C22⋊C412yes—
150C2×C4×C53C23×C411yescharacteristic
151C2×C4×C53C22×D411yescharacteristic
152C2×C4×C532−1+412yes—
153C2×C4×C532−1+412yes—
154C2×Dic53—21no—
155C2×Dic53—21no—
156C2×Dic53—22no—
157C2×Dic53—22no—
158C2×Dic53—24no—
159C2×Dic53—44no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162C2×Dic53—42no—
163C2×Dic53C2×C22⋊C412yes—
164C2×Dic53C2×C22⋊C412yes—
165C2×Dic53C2×C22⋊C412yes—
166C4×D53—24no—
167C4×D53—24no—
168C4×D53—44no—
169C4×D53—44no—
170Dic106—48no—
171C8×C53—42no—
172#88322—42no—
173#21842—531nomaximal, Sylow
174C22×C53C22×C22⋊C411yescharacteristic
175C22×C53#53611yescharacteristic
176C22×C53#53611yescharacteristic
177C22×C53#57512yes—
178C22×C53#57512yes—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53—42no—
183C2×D53—42no—
184C4×C53—22no—
185C4×C53—22no—
186C4×C53—44no—
187C4×C53—82no—
188C4×C53#51912yes—
189C4×C53#53612yes—
190Dic53—22no—
191Dic53—22no—
192Dic53—22no—
193Dic53—44no—
194Dic53—82no—
195C2×#533—531no—
196C2×#432—531no—
197#2283—531no—
198#1993—531no—
199#1993—531no—
200#1993—531no—
201#1993D5311yescharacteristic, complemented
202#1989—534no—
203#1825—534no—
204C2×C53#231011yescharacteristic
205C2×C53#289611yescharacteristic
206C2×C53#289611yescharacteristic
207C2×C53#324012yes—
208C2×C53#324012yes—
209D53—42no—
210#533—538no—
211#533—1064no—
212C2×C4⋊Q8—531no—
213C2×C4⋊Q8C2×D5311yescharacteristic
214C2×#125—532no—
215C2×C22⋊Q8—1061no—
216C2×C22⋊Q8—1061no—
217C2×C4×Q8—1061no—
218C2×C4×Q8—1061no—
219C2×#117—532no—
220C22×C42—531no—
221#432—1064no—
222C2×C4⋊C8—532no—
223C2×C22⋊C8—532no—
224C2×#97—531no—
225C2×#97C2×D5311yescharacteristic
226#383—1061no—
227#382—531no—
228#382C2×D5311yescharacteristic
229#381—1061no—
230#380—532no—
231C4×C4⋊C4—532no—
232C4×C22⋊C4—1061no—
233#376—1061no—
234#340—534no—
235#340—534no—
236#340—534no—
237#340C2×D5314yes—
238C53#2184211yescharacteristic, complemented, Sylow
239C22×Q8—21no—
240C22×Q8—1061no—
241C23×C4—531no—
242C23×C4—531no—
243C23×C4—531no—
244C22×C8—532no—
245C22×C8C22×D5312yes—
246C4⋊Q8—22no—
247C4⋊Q8—534no—
248C4⋊Q8—1062no—
249C4⋊Q8C4×D5314yes—
250#125—538no—
251#125—1064no—
252C22⋊Q8—1068no—
253C22⋊Q8—1068no—
254C4×Q8—1068no—
255C4×Q8—1068no—
256#117—538no—
257#117—1064no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—532no—
260C2×C4⋊C4—532no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1062no—
263C2×C4⋊C4—1061no—
264C2×C4⋊C4C22×D5312yes—
265C2×C4⋊C4C22×D5312yes—
266C2×C22⋊C4—1062no—
267C2×C22⋊C4—1062no—
268C2×C42—531no—
269C2×C42—531no—
270C2×C42—531no—
271C2×C42—534no—
272C2×C42—534no—
273C2×C42—1062no—
274C2×C42—1062no—
275C2×C42C22×D5311yescharacteristic
276C4⋊C8—1064no—
277C22⋊C8—1064no—
278#97—22no—
279#97—1062no—
280#95—21no—
281#95—21no—
282#95—1062no—
283#95—1062no—
284#95—1061no—
285#95—1061no—
286C24—531no—
287C2×Q8—24no—
288C2×Q8—44no—
289C2×Q8—1064no—
290C2×Q8—2124no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—531no—
294C22×C4—531no—
295C22×C4—531no—
296C22×C4—532no—
297C22×C4—532no—
298C22×C4—534no—
299C22×C4—534no—
300C22×C4—1062no—
301C22×C4—1062no—
302C22×C4—1062no—
303C22×C4—1062no—
304C22×C4—2122no—
305C22×C4—2122no—
306C22×C4D4×D5311yescharacteristic
307C22×C4D4×D5311yescharacteristic
308C22×C4C23×D5311yescharacteristic
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—1062no—
313C2×C8—1062no—
314C2×C8—1062no—
315C4⋊C4—24no—
316C4⋊C4—22no—
317C4⋊C4—24no—
318C4⋊C4—534no—
319C4⋊C4—1068no—
320C4⋊C4—1068no—
321C4⋊C4—1064no—
322C4⋊C4—1062no—
323C4⋊C4—1064no—
324C4⋊C4C2×C4×D5314yes—
325C22⋊C4—1068no—
326C22⋊C4—2124no—
327C42—534no—
328C42—534no—
329C42—532no—
330C42—532no—
331C42—1068no—
332C42—2124no—
333C42C2×C4×D5312yes—
334C42D4×D5312yes—
335C23—532no—
336C23—1062no—
337C23—1062no—
338C23—1061no—
339C23—1061no—
340C23C2×D4×D5311yescharacteristic
341Q8—48no—
342Q8—2128no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—532no—
351C2×C4—532no—
352C2×C4—532no—
353C2×C4—1062no—
354C2×C4—1062no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1064no—
358C2×C4—1064no—
359C2×C4—1064no—
360C2×C4—2124no—
361C2×C4—2124no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2124no—
366C2×C4C22⋊C4×D5312yes—
367C2×C4C22⋊C4×D5312yes—
368C2×C4C22×C4×D5311yescharacteristic
369C2×C4C2×D4×D5311yescharacteristic
370C2×C4#1313326412yes—
371C2×C4#1313326512yes—
372C8—42no—
373C8—2122no—
374C22—1062no—
375C22—1062no—
376C22—1062no—
377C22—2122no—
378C22—2122no—
379C22C2×C22⋊C4×D5311yescharacteristic
380C22#1681119311yescharacteristic
381C22#1681119811yescharacteristic
382C22#1681141912yes—
383C22#1681142212yes—
384C4—22no—
385C4—22no—
386C4—44no—
387C4—82no—
388C4—1062no—
389C4—1062no—
390C4—1062no—
391C4—4242no—
392C4—2124no—
393C4#1681109912yes—
394C4#1681119612yes—
395C2—2122no—
396C2#2027729811yescharacteristic
397C2#2028322911yescharacteristic
398C2#2028323311yescharacteristic
399C2#2028595212yes—
400C2#2028595812yes—
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1993C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1993D53
hypocenterC53#21842
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
4. #20275689C2semidirect
5. #20275688C2semidirect
6. #20275687C2semidirect
7. #20275671C2semidirect
9. C53×#1993C2semidirect
11. #16811178C4semidirect
201. #1993D53semidirect
238. C53#21842semidirect

Maximal subgroups

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

subgroupindex
C2×#168111782
C2×#168106782
#202771082
#202756892
#202756882
#202756872
#202756712
#202748092
C53×#19932
#2184253