Finite Group Databasebeta

← #27397920

Subgroups — #27397920

Order 13568.

922 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811007C211yesmaximal, characteristic
2#20277426C212yesmaximal, complemented
3#20275531C212yesmaximal, complemented
4#20275069C212yesmaximal, complemented
5#20275062C212yesmaximal
6#20275055C212yesmaximal, complemented
7#20275048C212yesmaximal, complemented
8#20274843C211yesmaximal, characteristic, complemented
9C53×#1746C211yesmaximal, characteristic, complemented
10C22×Dic212C2211yescharacteristic
11C22×#11719913C2212yes—
12C22×D424C2211yescharacteristic
13C2×#13133205C2212yes—
14C2×C4×Dic106C2212yes—
15C2×C4×D212C2212yes—
16#16811007C418yescomplemented
17C2×#13133142C2212yes—
18C2×#13133142C2212yes—
19C2×#13133139C2212yes—
20C2×#13133139C2212yes—
21#16810529C2212yes—
22#16810525C2212yes—
23#16810515—22no—
24#16810515—22no—
25#16810497—22no—
26#16810497—22no—
27#16810493C2212yes—
28#16810386C2211yescharacteristic
29#16810386C2212yes—
30#16810386C2211yescharacteristic
31C2×#97×C53C2211yescharacteristic
32C2×C4×C8×C53C2211yescharacteristic
33C53×#376C2211yescharacteristic
34C53×#336C2212yes—
35C53×#336C2212yes—
36C22×Dic106C2312yes—
37C22×D212C2312yes—
38C22×C4×D53—22no—
39C2×Dic212—42no—
40C2×Dic212C2×C414yescomplemented
41C2×#11719913—42no—
42C2×#11719913—42no—
43C2×#11719913C2×C418yescomplemented
44C2×D424—42no—
45C2×D424C2×C414yes—
46#13133205—24no—
47#13133205C2×C418yescomplemented
48C4×Dic106—42no—
49C4×Dic106—42no—
50C4×D212—42no—
51C4×D212—42no—
52C2×C4⋊Dic53—22no—
53C2×C4⋊Dic53—22no—
54C2×C4⋊Dic53C2312yes—
55C2×C4⋊Dic53C2312yes—
56C2×#11719919—22no—
57C2×#11719921—22no—
58C2×#11719921—22no—
59C2×#11719921—22no—
60C2×#11719921—22no—
61C2×#11719921—22no—
62C2×C4×Dic53—22no—
63#13133142—42no—
64#13133142—42no—
65#13133139—42no—
66#13133139—42no—
67#13133124—22no—
68#13133124—22no—
69C22×C8×C53C2312yes—
70C22×C8×C53C2311yescharacteristic
71C22×C8×C53C2311yescharacteristic
72C2×C42×C53C2312yes—
73C2×C42×C53C2311yescharacteristic
74#97×C53—22no—
75C4×C8×C53C2×C414yescomplemented
76#95×C53D411yescharacteristic
77#95×C53D412yes—
78#95×C53D411yescharacteristic
79C23×D53—22no—
80C22×C4×C53C2×D412yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2411yescharacteristic
85C22×Dic53—22no—
86C22×Dic53—22no—
87C22×Dic53—22no—
88C2×D212—22no—
89C2×D212—22no—
90C2×D212—22no—
91C2×D212—22no—
92C2×D212—22no—
93C2×D212C22×C414yes—
94C2×C4×D53—44no—
95C2×C4×D53—82no—
96C2×C4×D53—82no—
97C2×Dic106—22no—
98C2×Dic106—22no—
99C2×Dic106—22no—
100C2×Dic106—22no—
101C2×Dic106—22no—
102C2×Dic106C22×C414yes—
103C2×C8×C53—22no—
104C2×C8×C53—22no—
105C2×C8×C53—22no—
106C2×C8×C53—22no—
107C2×C8×C53—22no—
108C2×C8×C53C22×C412yes—
109C2×C8×C53C22×C412yes—
110C42×C53—22no—
111C42×C53—22no—
112C42×C53C22×C412yes—
113C42×C53#4112yes—
114#11719921—28no—
115#11719921—28no—
116#11719921—44no—
117#11719921—82no—
118#11719921—82no—
119C4⋊Dic53—22no—
120C4⋊Dic53—22no—
121C4⋊Dic53—22no—
122C4⋊Dic53—22no—
123C4⋊Dic53—82no—
124C4⋊Dic53—82no—
125#11719919—44no—
126C4×Dic53—44no—
127Dic212—44no—
128D424—44no—
129#11719913—48no—
130C23×C53C22×D411yescharacteristic
131C22×D53—24no—
132C22×D53—42no—
133C22×D53—42no—
134C22×D53—42no—
135C22×D53—42no—
136C22×D53—42no—
137C22×D53—42no—
138C2×C4×C53—21no—
139C2×C4×C53—22no—
140C2×C4×C53—21no—
141C2×C4×C53—22no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53C4×D412yes—
148C2×C4×C53C4×D412yes—
149C2×C4×C53C23×C411yescharacteristic
150C2×C4×C53C2×#4111yescharacteristic
151C2×C4×C53H(5,𝔽2)12yes—
152C2×C4×C53H(5,𝔽2)12yes—
153C2×Dic53—24no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—42no—
162D212—24no—
163D212—24no—
164D212—42no—
165D212—42no—
166C4×D53—88no—
167Dic106—24no—
168Dic106—24no—
169Dic106—42no—
170Dic106—42no—
171C8×C53—22no—
172C8×C53—22no—
173C8×C53—42no—
174#22274—531nomaximal, Sylow
175C22×C53C2×C4×D411yescharacteristic
176C22×C53#53512yes—
177C22×C53#57611yescharacteristic
178C22×C53#57712yes—
179C22×C53#57811yescharacteristic
180C2×D53—44no—
181C2×D53—44no—
182C2×D53—44no—
183C2×D53—82no—
184C2×D53—82no—
185C2×D53—82no—
186C2×D53—82no—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—42no—
190C4×C53—42no—
191C4×C53#51811yescharacteristic
192C4×C53#51811yescharacteristic
193C4×C53#54012yes—
194Dic53—44no—
195Dic53—82no—
196Dic53—82no—
197C2×#495—531no—
198#2326—532no—
199#1959—532no—
200#1880—532no—
201#1879—532no—
202#1878—532no—
203#1877—532no—
204#1834—531no—
205#1746D5311yescharacteristic, complemented
206C2×C53#232811yescharacteristic
207C2×C53#295111yescharacteristic
208C2×C53#295211yescharacteristic
209C2×C53#322412yes—
210C2×C53#322512yes—
211D53—84no—
212C22×Q16—531no—
213C22×SD16—532no—
214C22×D8—531no—
215C2×#124—532no—
216C2×C4×Q8—532no—
217C2×C4×D4—532no—
218#495—538no—
219C2×#103—532no—
220C2×#103—532no—
221C2×#102—532no—
222C2×#102—532no—
223C2×#97C2×D5311yescharacteristic
224C2×C4×C8C2×D5311yescharacteristic
225#391—532no—
226#390—532no—
227#388—1062no—
228#388—1062no—
229#385—1062no—
230#385—1062no—
231#384—532no—
232#376C2×D5311yescharacteristic
233#340—531no—
234#340—532no—
235#340—531no—
236#336C2×D5312yes—
237#336C2×D5312yes—
238C53#2227411yescharacteristic, complemented, Sylow
239C22×Q8—532no—
240C22×D4—532no—
241C23×C4—1062no—
242C2×Q16—534no—
243C2×Q16—2122no—
244C2×SD16—538no—
245C2×SD16—2122no—
246C2×SD16—2122no—
247C2×D8—534no—
248C2×D8—2122no—
249C22×C8C22×D5312yes—
250C22×C8C22×D5311yescharacteristic
251C22×C8C22×D5311yescharacteristic
252#124—538no—
253#124—1064no—
254C4×Q8—2122no—
255C4×Q8—2122no—
256C4×D4—2122no—
257C4×D4—2122no—
258C2×C4⋊C4—532no—
259C2×C4⋊C4—532no—
260C2×C4⋊C4—1062no—
261C2×C4⋊C4—1062no—
262C2×C4⋊C4—1062no—
263C2×C22⋊C4—1062no—
264C2×C22⋊C4—1062no—
265C2×C22⋊C4—1062no—
266C2×C22⋊C4—1062no—
267C2×C22⋊C4—1062no—
268C2×C42—1062no—
269C2×C42C22×D5312yes—
270C2×C42C22×D5311yescharacteristic
271#103—2122no—
272#103—2122no—
273#102—2122no—
274#102—2122no—
275#97—22no—
276C4×C8C4×D5314yescomplemented
277#95—1062no—
278#95—1062no—
279#95D21211yescharacteristic
280#95D21212yes—
281#95D21211yescharacteristic
282C24—1062no—
283C2×Q8—534no—
284C2×Q8—1062no—
285C2×Q8—1062no—
286C2×Q8—1062no—
287C2×Q8—1062no—
288C2×Q8—1062no—
289C2×D4—534no—
290C2×D4—1062no—
291C2×D4—1062no—
292C2×D4—1062no—
293C2×D4—1062no—
294C2×D4—1062no—
295C22×C4—1062no—
296C22×C4—1062no—
297C22×C4—1062no—
298C22×C4—4242no—
299C22×C4—4242no—
300C22×C4—2124no—
301C22×C4C2×D21212yes—
302C22×C4C2×D21212yes—
303C22×C4C2×D21211yescharacteristic
304C22×C4C2×D21211yescharacteristic
305C22×C4C23×D5311yescharacteristic
306Q16—2124no—
307SD16—2128no—
308D8—2124no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8C2×C4×D5312yes—
315C2×C8C2×C4×D5312yes—
316C4⋊C4—1062no—
317C4⋊C4—1062no—
318C4⋊C4—1062no—
319C4⋊C4—1062no—
320C4⋊C4—4242no—
321C4⋊C4—4242no—
322C4⋊C4—2124no—
323C22⋊C4—1068no—
324C22⋊C4—1068no—
325C22⋊C4—4242no—
326C22⋊C4—4242no—
327C22⋊C4—2124no—
328C42—22no—
329C42—22no—
330C42—2124no—
331C42C2×C4×D5312yes—
332C42#1171994512yes—
333C23—1064no—
334C23—2122no—
335C23—2122no—
336C23—2122no—
337C23—2122no—
338C23—2122no—
339C23—2122no—
340C23C22×D21211yescharacteristic
341Q8—1064no—
342Q8—1064no—
343Q8—2122no—
344Q8—2122no—
345D4—1064no—
346D4—1064no—
347D4—2122no—
348D4—2122no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—1064no—
359C2×C4—4248no—
360C2×C4—2122no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2122no—
366C2×C4—2122no—
367C2×C4—2122no—
368C2×C4C4×D21212yes—
369C2×C4C4×D21212yes—
370C2×C4C22×C4×D5311yescharacteristic
371C2×C4C2×#1171994511yescharacteristic
372C2×C4#1313326312yes—
373C2×C4#1313326312yes—
374C8—22no—
375C8—22no—
376C8—42no—
377C22—4242no—
378C22—4242no—
379C22—4242no—
380C22—4242no—
381C22—2124no—
382C22—2124no—
383C22—2124no—
384C22C2×C4×D21211yescharacteristic
385C22#1681119212yes—
386C22#1681142811yescharacteristic
387C22#1681143512yes—
388C22#1681144011yescharacteristic
389C4—22no—
390C4—22no—
391C4—42no—
392C4—42no—
393C4—4242no—
394C4—4242no—
395C4—2124no—
396C4#1681109711yescharacteristic
397C4#1681109711yescharacteristic
398C4#1681122812yes—
399C2—4244no—
400C2#2027744011yescharacteristic
401C2#2028359211yescharacteristic
402C2#2028360011yescharacteristic
403C2#2028581112yes—
404C2#2028582612yes—
subgroupG/H
centerC23C22×D212
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1746C2
socleC23×C53C22×D4
prime radicalC22×D424C22
cosocleC22×C4×C53C24
hypercenter#1746D53
hypocenterC53#22274
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ #376 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #20277426C2semidirect
3. #20275531C2semidirect
4. #20275069C2semidirect
6. #20275055C2semidirect
7. #20275048C2semidirect
8. #20274843C2semidirect
9. C53×#1746C2semidirect
16. #16811007C4semidirect
40. C2×Dic212C2×C4semidirect
43. C2×#11719913C2×C4semidirect
47. #13133205C2×C4semidirect
75. C4×C8×C53C2×C4semidirect
205. #1746D53semidirect
238. C53#22274semidirect
276. C4×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168110072
#202774262
#202755312
#202750692
#202750622
#202750552
#202750482
#202748432
C53×#17462
#2227453