Finite Group Databasebeta

← #28304231

Subgroups — #28304231

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D53⋊C8C211yesmaximal, characteristic
2#20290570C211yesmaximal, characteristic
3C2×#16811473C212yesmaximal, complemented
4#20289551C212yesmaximal, complemented
5#20277245C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×C4⋊#88324C2212yes—
8C2×#13133274—21no—
9C4×D53⋊C8—24no—
10C2×C4×#88324—21no—
11C2×C4×#88324C2212yes—
12#16811489—21no—
13#16811473—24no—
14C2×#13133197C411yescharacteristic
15C2×#13133182C2211yescharacteristic
16C2×C42×D53C2211yescharacteristic
17#16810529C411yescharacteristic
18#16810506—21no—
19#16810494—21no—
20#16810480C411yescharacteristic
21C4⋊C4×Dic53—21no—
22C4⋊C4×Dic53C2211yescharacteristic
23#16810470—21no—
24C53×#389C411yescharacteristic
25C2×D53⋊C8—24no—
26C2×D53⋊C8—24no—
27C2×D53⋊C8—24no—
28C22×#88324—22no—
29C22×#88324—22no—
30C4⋊#88324—22no—
31C4⋊#88324—22no—
32#13133274—42no—
33C4×#88324—24no—
34C4×#88324—22no—
35C4×#88324D414yes—
36C22×Dic106C2×C411yescharacteristic
37C22×C4×D53C2×C411yescharacteristic
38C22×C4×D53D411yescharacteristic
39C22×C4×D53D411yescharacteristic
40#13133197—22no—
41#13133197C814yescomplemented
42#13133182—22no—
43#13133182—22no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53C2×C411yescharacteristic
46C2×#11719919—21no—
47C2×#11719921—21no—
48C2×#11719921—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53—21no—
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C2×C4×Dic53C2311yescharacteristic
54C42×D53—22no—
55C42×D53—22no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59#13133124—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C42×C53C2×C411yescharacteristic
64#95×C53—21no—
65C23×D53C22⋊C411yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C2×C4×D53—22no—
77C2×C4×D53—22no—
78C2×C4×D53—22no—
79C2×C4×D53—22no—
80C2×C4×D53—21no—
81C2×C4×D53—22no—
82C2×C4×D53—21no—
83C2×C4×D53—22no—
84C2×C4×D53C2×C812yes—
85C2×C4×D53M1612yes—
86C2×C4×D53SD1612yes—
87C2×C4×D53Q1612yes—
88C2×Dic106—22no—
89C2×Dic106—22no—
90C2×Dic106—21no—
91C2×Dic106C2×C812yescomplemented
92C2×#88324—24no—
93C2×#88324—24no—
94C2×#88324—24no—
95C2×#88324—42no—
96C2×#88324—42no—
97C2×#88324—42no—
98D53⋊C8—216no—
99C4⋊C4×C53—21no—
100C4⋊C4×C53—22no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53C2×C812yescomplemented
103C42×C53—22no—
104#11719921—24no—
105#11719921—42no—
106C4⋊Dic53—24no—
107C4⋊Dic53—22no—
108#11719919—41no—
109#11719919—41no—
110C4×Dic53—21no—
111C4×Dic53—22no—
112C4×Dic53—21no—
113C4×Dic53—42no—
114C4×Dic53—42no—
115C4×Dic53C2×D412yes—
116C4×Dic53#4112yes—
117C23×C53C2×C22⋊C411yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—21no—
122C22×D53C22⋊C812yes—
123C22×D53#10312yes—
124C22×D53#10412yes—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—22no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53#10312yes—
136C2×C4×C53#10412yes—
137C2×C4×C53C4×D412yes—
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53#13112yes—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—22no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—41no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53C22⋊C812yes—
152C2×Dic53#12011yescharacteristic
153C2×Dic53#12112yes—
154C2×Dic53#12311yescharacteristic
155C2×Dic53C2×SD1611yescharacteristic
156C2×Dic53C2×Q1611yescharacteristic
157C2×Dic53#13512yes—
158C4×D53—24no—
159C4×D53—24no—
160C4×D53—24no—
161C4×D53—24no—
162C4×D53—24no—
163C4×D53—24no—
164Dic106—24no—
165Dic106—42no—
166#88324—28no—
167#88324—82no—
168#9071—531nomaximal, Sylow
169C22×C53#38611yescharacteristic
170C22×C53C2×C22⋊C811yescharacteristic
171C22×C53#40811yescharacteristic
172C22×C53C2×#10311yescharacteristic
173C22×C53#41611yescharacteristic
174C22×C53C2×#10411yescharacteristic
175C22×C53C2×#10411yescharacteristic
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—22no—
181C2×D53—22no—
182C2×D53Q8⋊C814yescomplemented
183C4×C53—22no—
184C4×C53—22no—
185C4×C53—42no—
186C4×C53—81no—
187C4×C53C8×D412yes—
188C4×C53#43512yes—
189Dic53—22no—
190Dic53—21no—
191Dic53—21no—
192Dic53—42no—
193Dic53—81no—
194Dic53#46512yes—
195Dic53#46712yes—
196C22×C4×C8—531no—
197C4×C22⋊Q8—531no—
198#1797—531no—
199C2×Q8⋊C8—532no—
200#1279—532no—
201C2×C53C2×Q8⋊C811yescharacteristic
202C2×C53#147611yescharacteristic
203C2×C53#149311yescharacteristic
204C2×C53#149411yescharacteristic
205C2×C53#186911yescharacteristic
206C2×C53#189211yescharacteristic
207C2×C53#189611yescharacteristic
208D53—24no—
209C23×C8—1061no—
210C2×C22⋊Q8—531no—
211C2×C4×Q8—531no—
212C22×C42—531no—
213C2×C4⋊C8—532no—
214C2×C22⋊C8—1061no—
215C2×C4×C8—532no—
216C2×C4×C8—1064no—
217C2×C4×C8—1061no—
218#391—531no—
219#389#615711yescharacteristic
220#387—1061no—
221#385—1061no—
222#380—531no—
223C4×C4⋊C4—531no—
224C4×C4⋊C4—1061no—
225C4×C22⋊C4—1061no—
226#336—1061no—
227Q8⋊C8—1064no—
228C53#907111yescharacteristic, complemented, Sylow
229C22×Q8—531no—
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C22×C8—1062no—
234C22×C8—1064no—
235C22×C8—1064no—
236C22×C8—1064no—
237C22×C8—1062no—
238C22⋊Q8—534no—
239C22⋊Q8—1062no—
240C4×Q8—1062no—
241C4×Q8—1062no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4C2×#615711yescharacteristic
247C2×C4⋊C4C2×#615711yescharacteristic
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—1062no—
254C2×C42—1062no—
255C2×C42—1061no—
256C2×C42—1061no—
257C2×C42C2×#615711yescharacteristic
258C4⋊C8—1062no—
259C4⋊C8—1062no—
260C22⋊C8—2122no—
261C4×C8—534no—
262C4×C8—1062no—
263C4×C8—1064no—
264#95—21no—
265#95—1061no—
266#95—1061no—
267#95—1061no—
268#95—1061no—
269C24—531no—
270C2×Q8—532no—
271C2×Q8—1061no—
272C2×Q8—1062no—
273C2×Q8—1062no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—531no—
279C22×C4—532no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—1061no—
284C22×C4—1062no—
285C22×C4—1061no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—1062no—
290C22×C4—1062no—
291C22×C4—1061no—
292C22×C4—1061no—
293C22×C4C22⋊#615711yescharacteristic
294C22×C4C22⋊#615711yescharacteristic
295C22×C4C22×#615711yescharacteristic
296C2×C8—1064no—
297C2×C8—1064no—
298C2×C8—1064no—
299C2×C8—10616no—
300C2×C8—2122no—
301C2×C8—2122no—
302C2×C8—2122no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—42no—
306C4⋊C4—1064no—
307C4⋊C4—1062no—
308C4⋊C4—2121no—
309C4⋊C4—2121no—
310C4⋊C4D53⋊C812yescomplemented
311C22⋊C4—1064no—
312C22⋊C4—2122no—
313C42—22no—
314C42—532no—
315C42—532no—
316C42—1061no—
317C42—1062no—
318C42—1061no—
319C42—2122no—
320C42—2122no—
321C23—532no—
322C23—532no—
323C23—532no—
324C23—1061no—
325C23—1061no—
326C23—1061no—
327C23—1061no—
328C23C2×C22⋊#615711yescharacteristic
329Q8—1064no—
330Q8—2122no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—532no—
342C2×C4—531no—
343C2×C4—531no—
344C2×C4—532no—
345C2×C4—531no—
346C2×C4—531no—
347C2×C4—532no—
348C2×C4—1064no—
349C2×C4—1064no—
350C2×C4—1064no—
351C2×C4—1064no—
352C2×C4—1061no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1064no—
357C2×C4—1064no—
358C2×C4—1062no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4#1313328012yes—
365C2×C4#1313328112yes—
366C2×C4D4×#615712yes—
367C2×C4C2×D53⋊C811yescharacteristic
368C2×C4D53⋊M1611yescharacteristic
369C2×C4#1313330512yes—
370C8—1068no—
371C8—4242no—
372C22—534no—
373C22—1062no—
374C22—1062no—
375C22—1062no—
376C22—1062no—
377C22—1062no—
378C22—1062no—
379C22#1681157111yescharacteristic
380C22D53⋊(C22⋊C8)11yescharacteristic
381C22#1681159811yescharacteristic
382C22C2×#1313328011yescharacteristic
383C22#1681161311yescharacteristic
384C22C2×#1313328111yescharacteristic
385C22D53⋊#10411yescharacteristic
386C4—22no—
387C4—22no—
388C4—42no—
389C4—81no—
390C4—532no—
391C4—532no—
392C4—1062no—
393C4—1061no—
394C4—1061no—
395C4—4241no—
396C4—2122no—
397C4#1681165112yes—
398C4#1681165312yes—
399C2—1064no—
400C2D53⋊(Q8⋊C8)11yescharacteristic
401C2#2029000711yescharacteristic
402C2#2029006511yescharacteristic
403C2#2029006911yescharacteristic
404C2#2029069111yescharacteristic
405C2#2029072811yescharacteristic
406C2#2029073311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#9071
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. C2×#16811473C2semidirect
4. #20289551C2semidirect
41. #13133197C8semidirect
91. C2×Dic106C2×C8semidirect
102. C4⋊C4×C53C2×C8semidirect
182. C2×D53Q8⋊C8semidirect
228. C53#9071semidirect
310. C4⋊C4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D53⋊C82
#202905702
C2×#168114732
#202895512
#202772452
#907153