Finite Group Databasebeta

← #28301848

Subgroups — #28301848

Order 13568.

640 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133274C211yesmaximal, characteristic
2C2×#16811477C212yesmaximal
3#20289891C211yesmaximal, characteristic
4#20289588C212yesmaximal
5#20277253C211yesmaximal, characteristic
6C2×#13133274C2212yes—
7C2×C4×#88324—21no—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4—21no—
10C2×(C2×#6157)⋊C4C2212yes—
11#16811489—21no—
12#16811477C418yescomplemented
13C2×#13133193C2211yescharacteristic
14C2×Q8×Dic53C411yescharacteristic, complemented
15C2×C42×D53C2211yescharacteristic
16#16810521C411yescharacteristic, complemented
17#16810510—21no—
18#16810497—21no—
19#16810480C2211yescharacteristic
20C4⋊C4×Dic53C411yescharacteristic, complemented
21#16810474—21no—
22#16810470—21no—
23C53×#391C411yescharacteristic, complemented
24C23×#6157—22no—
25C22×#88324—22no—
26#13133274—22no—
27#13133274C2×C414yes—
28C4×#88324—42no—
29(C2×#6157)⋊C4—22no—
30(C2×#6157)⋊C4—42no—
31(C2×#6157)⋊C4—42no—
32(C2×#6157)⋊C4C2×C414yes—
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×C4×D53C2311yescharacteristic
36#13133193—22no—
37#13133193C2×C414yes—
38Q8×Dic53—42no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53C2×C411yescharacteristic
41C2×C4⋊Dic53C2×C411yescharacteristic
42C2×#11719919—21no—
43C2×#11719919—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×C4×Dic53—21no—
47C2×C4×Dic53—21no—
48C2×C4×Dic53C2×C411yescharacteristic
49C2×C4×Dic53D411yescharacteristic
50C2×C4×Dic53D411yescharacteristic
51C42×D53—42no—
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55C22×Q8×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C42×C53C2×C411yescharacteristic
58#95×C53—21no—
59#95×C53—21no—
60C23×D53C2×D411yescharacteristic
61C22×#6157—24no—
62C22×#6157—42no—
63C22×#6157—42no—
64C22×#6157—42no—
65C22×#6157—42no—
66C22×#6157—42no—
67C22×#6157—42no—
68C2×Q8×C53—21no—
69C2×Q8×C53—41no—
70C2×Q8×C53—41no—
71C2×Q8×C53C4212yes—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C22⋊C411yescharacteristic
81C22×Dic53C2×D411yescharacteristic
82C2×C4×D53—22no—
83C2×C4×D53—22no—
84C2×C4×D53—41no—
85C2×C4×D53—41no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53SD1612yes—
90C2×C4×D53Q1612yes—
91C2×C4×D53C22×C412yes—
92C2×C4×D53#4112yes—
93C2×#88324—24no—
94C2×#88324—42no—
95C2×#88324—42no—
96C4⋊C4×C53—41no—
97C42×C53—41no—
98#11719921—24no—
99#11719921—42no—
100C4⋊Dic53—21no—
101C4⋊Dic53—42no—
102C4⋊Dic53—41no—
103C4⋊Dic53C4212yes—
104#11719919—24no—
105#11719919—42no—
106C4×Dic53—22no—
107C4×Dic53—22no—
108C4×Dic53—41no—
109C4×Dic53—42no—
110C4×Dic53—42no—
111C23×C53C2×C22⋊C411yescharacteristic
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53C4×D412yes—
117C22×D53C2×SD1611yescharacteristic
118C22×D53C2×Q1611yescharacteristic
119C22×D53#13512yes—
120C2×#6157—44no—
121C2×#6157—44no—
122C2×#6157—44no—
123C2×#6157—82no—
124C2×#6157—82no—
125C2×#6157—82no—
126C2×#6157—82no—
127Q8×C53—42no—
128Q8×C53—42no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53#10312yes—
143C2×C4×C53C2×C4211yescharacteristic
144C2×C4×C53#11711yescharacteristic
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—22no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53#9912yes—
158C2×Dic53#10312yes—
159C2×Dic53C4×D412yes—
160C4×D53—42no—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—42no—
164C4×D53—44no—
165#88324—44no—
166#8221—531nomaximal, Sylow
167C22×C53C4×C22⋊C411yescharacteristic
168C22×C53C2×#9911yescharacteristic
169C22×C53#41011yescharacteristic
170C22×C53C2×#10311yescharacteristic
171C22×C53#41611yescharacteristic
172C22×C53#42111yescharacteristic
173C22×C53#42111yescharacteristic
174C2×D53—22no—
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—21no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—22no—
182C2×D53C4×SD1612yes—
183C2×D53C4×Q1612yes—
184#6157—84no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—42no—
189C4×C53—41no—
190C4×C53—81no—
191Dic53—22no—
192Dic53—22no—
193Dic53—22no—
194Dic53—42no—
195Dic53—81no—
196C4×C22⋊Q8—531no—
197C4×C22⋊C8—531no—
198C2×#328—532no—
199#1434—531no—
200#1294—532no—
201C2×C53#151611yescharacteristic
202C2×C53#151811yescharacteristic
203C2×C53#152011yescharacteristic
204C2×C53#152111yescharacteristic
205C2×C53C4×#9911yescharacteristic
206C2×C53#175911yescharacteristic
207C2×C53C4×#10311yescharacteristic
208D53—22no—
209D53—22no—
210C2×C22⋊Q8—531no—
211C2×C4×Q8—531no—
212C22×C42—531no—
213C2×C22⋊C8—532no—
214C2×C4×C8—1061no—
215#391#615711yescharacteristic, complemented
216#389—531no—
217#387—1061no—
218#385—1061no—
219#380—531no—
220C4×C4⋊C4—531no—
221C4×C4⋊C4—1061no—
222C4×C22⋊C4—1061no—
223C2×#95—532no—
224C2×#95—1061no—
225C2×#95—1061no—
226#336—1061no—
227#328—538no—
228C53#822111yescharacteristic, complemented, Sylow
229C22×Q8C2×#615711yescharacteristic
230C23×C4—531no—
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—1062no—
234C22×C8—1062no—
235C22⋊Q8—534no—
236C22⋊Q8—1062no—
237C4×Q8—2122no—
238C2×C4⋊C4—531no—
239C2×C4⋊C4—531no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4C2×#615711yescharacteristic
244C2×C22⋊C4—1061no—
245C2×C22⋊C4—1061no—
246C2×C42—531no—
247C2×C42—531no—
248C2×C42—531no—
249C2×C42—1061no—
250C2×C42—1061no—
251C2×C42—2122no—
252C2×C42C2×#615711yescharacteristic
253C22⋊C8—534no—
254C22⋊C8—1062no—
255C4×C8—2122no—
256#95—21no—
257#95—21no—
258#95—534no—
259#95—1061no—
260#95—1061no—
261#95—1062no—
262#95—1061no—
263#95—2122no—
264#95—2122no—
265C24—531no—
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C4×#615712yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—532no—
276C22×C4—532no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—1062no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—1064no—
283C22×C4—1061no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2122no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4—2122no—
293C22×C4—2122no—
294C22×C4—2122no—
295C22×C4C22⋊#615711yescharacteristic
296C22×C4C22⋊#615711yescharacteristic
297C22×C4C22×#615711yescharacteristic
298C2×C8—1064no—
299C2×C8—2122no—
300C2×C8—2122no—
301C4⋊C4—41no—
302C4⋊C4—532no—
303C4⋊C4—1064no—
304C4⋊C4—1061no—
305C4⋊C4—2121no—
306C4⋊C4—2122no—
307C4⋊C4—2122no—
308C22⋊C4—1064no—
309C22⋊C4—2122no—
310C42—41no—
311C42—1062no—
312C42—1062no—
313C42—2121no—
314C42—2122no—
315C42—2122no—
316C23—532no—
317C23—531no—
318C23—531no—
319C23—532no—
320C23—1061no—
321C23—1061no—
322C23—1061no—
323C23—1061no—
324C23C2×C22⋊#615711yescharacteristic
325Q8—42no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—532no—
341C2×C4—532no—
342C2×C4—532no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1061no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1062no—
350C2×C4—4242no—
351C2×C4—4242no—
352C2×C4—4242no—
353C2×C4—4242no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2124no—
359C2×C4—2124no—
360C2×C4—2124no—
361C2×C4—2124no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4#1313328012yes—
368C2×C4C2×C4×#615711yescharacteristic
369C2×C4#1313329611yescharacteristic
370C8—2124no—
371C22—532no—
372C22—532no—
373C22—1062no—
374C22—1061no—
375C22—1061no—
376C22—1061no—
377C22—1061no—
378C22—1062no—
379C22—1062no—
380C22—1062no—
381C22C4×C22⋊#615711yescharacteristic
382C22D53⋊#9911yescharacteristic
383C22#1681160311yescharacteristic
384C22C2×#1313328011yescharacteristic
385C22#1681161311yescharacteristic
386C22#1681162011yescharacteristic
387C22#1681162211yescharacteristic
388C4—22no—
389C4—21no—
390C4—21no—
391C4—41no—
392C4—42no—
393C4—81no—
394C4—1062no—
395C4—1062no—
396C4—1062no—
397C4—4244no—
398C4—4241no—
399C4—2122no—
400C2—1062no—
401C2—1062no—
402C2#2029014211yescharacteristic
403C2#2029014911yescharacteristic
404C2#2029015311yescharacteristic
405C2#2029015611yescharacteristic
406C2#2029046211yescharacteristic
407C2#2029047911yescharacteristic
408C2C4×#1313328011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenter#391#6157
hypocenterC53#8221
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
12. #16811477C4semidirect
14. C2×Q8×Dic53C4semidirect
16. #16810521C4semidirect
20. C4⋊C4×Dic53C4semidirect
23. C53×#391C4semidirect
215. #391#6157semidirect
228. C53#8221semidirect

Maximal subgroups

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

subgroupindex
C4×#131332742
C2×#168114772
#202898912
#202895882
#202772532
#822153