Finite Group Databasebeta

← #28301838

Subgroups — #28301838

Order 13568.

720 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 130 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
1#20290442C211yesmaximal, characteristic
2C2×#16811477C212yesmaximal
3C4×(C2×#6157)⋊C4C211yesmaximal, characteristic
4#20289588C212yesmaximal
5#20277253C211yesmaximal, characteristic
6C22×C4×#6157—21no—
7C2×#13133274C2212yes—
8C2×#13133272—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—
26C2×C4×#6157—24no—
27C2×C4×#6157—42no—
28C2×C4×#6157—42no—
29#13133274—22no—
30#13133274C2×C414yes—
31#13133272—42no—
32(C2×#6157)⋊C4—24no—
33(C2×#6157)⋊C4—22no—
34(C2×#6157)⋊C4C2×C414yes—
35C22×C4×D53D411yescharacteristic
36C22×C4×D53D411yescharacteristic
37C22×C4×D53C2311yescharacteristic
38#13133193—22no—
39#13133193C2×C414yes—
40Q8×Dic53—42no—
41C2×C4⋊Dic53—21no—
42C2×C4⋊Dic53C2×C411yescharacteristic
43C2×C4⋊Dic53C2×C411yescharacteristic
44C2×#11719919—21no—
45C2×#11719919—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C42×D53—42no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C22×Q8×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C42×C53C2×C411yescharacteristic
60#95×C53—21no—
61#95×C53—21no—
62C23×D53C2×D411yescharacteristic
63C22×#6157—24no—
64C22×#6157—24no—
65C22×#6157—24no—
66C22×#6157—42no—
67C22×#6157—42no—
68C22×#6157—42no—
69C22×#6157—42no—
70C2×Q8×C53—21no—
71C2×Q8×C53—41no—
72C2×Q8×C53—41no—
73C2×Q8×C53C4212yescomplemented
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53C22⋊C411yescharacteristic
82C22×Dic53C22⋊C411yescharacteristic
83C22×Dic53C2×D411yescharacteristic
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53—22no—
87C2×C4×D53—22no—
88C2×C4×D53—22no—
89C2×C4×D53—22no—
90C2×C4×D53—41no—
91C2×C4×D53—41no—
92C2×C4×D53—41no—
93C2×C4×D53C22×C412yes—
94C2×C4×D53#4112yes—
95C2×#88324—24no—
96C2×#88324—42no—
97C2×#88324—42no—
98C4×#6157—48no—
99C4⋊C4×C53—41no—
100C42×C53—41no—
101#11719921—24no—
102#11719921—41no—
103#11719921—41no—
104C4⋊Dic53—21no—
105C4⋊Dic53—42no—
106C4⋊Dic53—41no—
107C4⋊Dic53C4212yescomplemented
108#11719919—24no—
109#11719919—42no—
110C4×Dic53—22no—
111C4×Dic53—22no—
112C4×Dic53—41no—
113C4×Dic53—42no—
114C4×Dic53—42no—
115C23×C53C2×C22⋊C411yescharacteristic
116C22×D53—21no—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—21no—
120C22×D53C4×D412yes—
121C22×D53AGL(1,ℤ/8)11yescharacteristic
122C22×D53#13712yes—
123C22×D53#13711yescharacteristic
124C2×#6157—28no—
125C2×#6157—44no—
126C2×#6157—44no—
127C2×#6157—44no—
128C2×#6157—44no—
129C2×#6157—44no—
130C2×#6157—44no—
131Q8×C53—42no—
132Q8×C53—42no—
133C2×C4×C53—21no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—22no—
138C2×C4×C53—21no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53#10412yes—
147C2×C4×C53C2×C4211yescharacteristic
148C2×C4×C53#11711yescharacteristic
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—22no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53#9912yes—
162C2×Dic53#10412yes—
163C2×Dic53C4×D412yes—
164C4×D53—24no—
165C4×D53—24no—
166C4×D53—42no—
167C4×D53—42no—
168C4×D53—44no—
169#88324—44no—
170#8219—531nomaximal, Sylow
171C22×C53C4×C22⋊C411yescharacteristic
172C22×C53C2×#9911yescharacteristic
173C22×C53#41011yescharacteristic
174C22×C53#41711yescharacteristic
175C22×C53#41911yescharacteristic
176C22×C53C2×#10411yescharacteristic
177C22×C53C2×#10411yescharacteristic
178C2×D53—22no—
179C2×D53—22no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—22no—
183C2×D53—22no—
184C2×D53—21no—
185C2×D53—21no—
186C2×D53#44012yes—
187C2×D53#44112yes—
188#6157—48no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—22no—
192C4×C53—42no—
193C4×C53—41no—
194C4×C53—81no—
195Dic53—22no—
196Dic53—22no—
197Dic53—22no—
198Dic53—42no—
199Dic53—81no—
200C4×C22⋊Q8—531no—
201#1749—531no—
202C2×#328—532no—
203C4×#95—531no—
204#1294—532no—
205C2×C53#150411yescharacteristic
206C2×C53#150611yescharacteristic
207C2×C53#150811yescharacteristic
208C2×C53#150911yescharacteristic
209C2×C53C4×#9911yescharacteristic
210C2×C53C4×#10411yescharacteristic
211C2×C53#176311yescharacteristic
212D53—22no—
213D53—22no—
214C2×C22⋊Q8—531no—
215C2×C4×Q8—531no—
216C22×C42—531no—
217C22×C42—1061no—
218C2×C22⋊C8—532no—
219C2×#97—1061no—
220#391#615711yescharacteristic, complemented
221#389—531no—
222#387—1061no—
223#385—1061no—
224#380—531no—
225C4×C4⋊C4—531no—
226C4×C4⋊C4—1061no—
227C4×C22⋊C4—1061no—
228C2×#95—532no—
229C2×#95—1061no—
230#336—1061no—
231#328—538no—
232C53#821911yescharacteristic, complemented, Sylow
233C22×Q8C2×#615711yescharacteristic
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—531no—
237C23×C4—1062no—
238C22×C8—1062no—
239C22⋊Q8—534no—
240C22⋊Q8—1062no—
241C4×Q8—2122no—
242C2×C4⋊C4—531no—
243C2×C4⋊C4—531no—
244C2×C4⋊C4—1061no—
245C2×C4⋊C4—1061no—
246C2×C4⋊C4—1061no—
247C2×C4⋊C4C2×#615711yescharacteristic
248C2×C22⋊C4—1061no—
249C2×C22⋊C4—1061no—
250C2×C42—531no—
251C2×C42—531no—
252C2×C42—531no—
253C2×C42—1061no—
254C2×C42—1061no—
255C2×C42—1064no—
256C2×C42—2122no—
257C2×C42—2122no—
258C2×C42—2122no—
259C2×C42C2×#615711yescharacteristic
260C22⋊C8—534no—
261C22⋊C8—1062no—
262#97—2122no—
263#95—21no—
264#95—21no—
265#95—534no—
266#95—1061no—
267#95—1061no—
268#95—1062no—
269#95—1064no—
270#95—1061no—
271C24—531no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8C4×#615712yescomplemented
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—531no—
279C22×C4—531no—
280C22×C4—531no—
281C22×C4—532no—
282C22×C4—532no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1061no—
286C22×C4—1061no—
287C22×C4—1062no—
288C22×C4—1062no—
289C22×C4—1064no—
290C22×C4—1064no—
291C22×C4—1064no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—2121no—
295C22×C4—2121no—
296C22×C4—2121no—
297C22×C4—2122no—
298C22×C4—2122no—
299C22×C4—2122no—
300C22×C4—2122no—
301C22×C4C22⋊#615711yescharacteristic
302C22×C4C22⋊#615711yescharacteristic
303C22×C4C22×#615711yescharacteristic
304C2×C8—1064no—
305C2×C8—2122no—
306C2×C8—2122no—
307C4⋊C4—41no—
308C4⋊C4—532no—
309C4⋊C4—1064no—
310C4⋊C4—1061no—
311C4⋊C4—2121no—
312C4⋊C4—2122no—
313C4⋊C4—2122no—
314C22⋊C4—1064no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—41no—
318C42—1062no—
319C42—1062no—
320C42—2121no—
321C42—2122no—
322C42—2122no—
323C42—2128no—
324C23—532no—
325C23—531no—
326C23—531no—
327C23—532no—
328C23—1061no—
329C23—1061no—
330C23—1061no—
331C23—1061no—
332C23C2×C22⋊#615711yescharacteristic
333Q8—42no—
334Q8—42no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—532no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—1064no—
352C2×C4—1064no—
353C2×C4—1061no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1068no—
360C2×C4—1062no—
361C2×C4—2122no—
362C2×C4—2122no—
363C2×C4—2124no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2124no—
368C2×C4—2124no—
369C2×C4—2124no—
370C2×C4—2124no—
371C2×C4—2124no—
372C2×C4—2124no—
373C2×C4—2121no—
374C2×C4—2121no—
375C2×C4#1313328212yes—
376C2×C4C2×C4×#615711yescharacteristic
377C2×C4#1313329611yescharacteristic
378C8—2124no—
379C22—532no—
380C22—532no—
381C22—1061no—
382C22—1061no—
383C22—1062no—
384C22—1062no—
385C22—1061no—
386C22—1061no—
387C22—1062no—
388C22—1062no—
389C22C4×C22⋊#615711yescharacteristic
390C22D53⋊#9911yescharacteristic
391C22#1681160311yescharacteristic
392C22#1681161411yescharacteristic
393C22#1681161611yescharacteristic
394C22D53⋊#10411yescharacteristic
395C22C2×#1313328211yescharacteristic
396C4—22no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—81no—
402C4—1062no—
403C4—1062no—
404C4—1062no—
405C4—4241no—
406C4—2128no—
407C4—2122no—
408C2—1062no—
409C2—1062no—
410C2#2029010211yescharacteristic
411C2#2029010911yescharacteristic
412C2#2029011311yescharacteristic
413C2#2029011611yescharacteristic
414C2#2029046211yescharacteristic
415C2C4×#1313328211yescharacteristic
416C2#2029048711yescharacteristic
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#8219
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
73. C2×Q8×C53C42semidirect
107. C4⋊Dic53C42semidirect
220. #391#6157semidirect
232. C53#8219semidirect
275. C2×Q8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
#202904422
C2×#168114772
C4×(C2×#6157)⋊C42
#202895882
#202772532
#821953