Finite Group Databasebeta

← #28301825

Subgroups — #28301825

Order 13568.

628 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 126 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
3#20289891C211yesmaximal, characteristic
4#20289588C212yesmaximal
5C4×#13133193C211yesmaximal, characteristic
6C2×#13133274C2212yes—
7C2×#13133272—21no—
8C2×(C2×#6157)⋊C4—21no—
9C2×(C2×#6157)⋊C4—21no—
10C2×(C2×#6157)⋊C4C2212yes—
11#16811489—21no—
12#16811477C418yescomplemented
13C2×#13133193C2211yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810530C411yescharacteristic, complemented
16#16810519C411yescharacteristic, complemented
17#16810504—21no—
18#16810494—21no—
19#16810480C2211yescharacteristic
20C4×#11719919—21no—
21C4×C4⋊Dic53C411yescharacteristic, complemented
22C4×#11719921—21no—
23C2×C4×Q8×C53C411yescharacteristic, complemented
24C23×#6157—22no—
25C22×#88324—22no—
26#13133274—22no—
27#13133274C2×C414yes—
28#13133272—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—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719921—21no—
43C2×#11719921—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C42×D53—42no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C22×Q8×C53C2×C411yescharacteristic
55C4×Q8×C53—42no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53—21no—
59C2×C42×C53C2×C411yescharacteristic
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—21no—
83C2×C4×D53—22no—
84C2×C4×D53—21no—
85C2×C4×D53—22no—
86C2×C4×D53—41no—
87C2×C4×D53—41no—
88C2×C4×D53—41no—
89C2×C4×D53—41no—
90C2×C4×D53—41no—
91C2×C4×D53C22×C412yes—
92C2×C4×D53#4112yes—
93C2×#88324—24no—
94C2×#88324—42no—
95C2×#88324—42no—
96C4⋊C4×C53—42no—
97C4⋊C4×C53—41no—
98C42×C53—22no—
99C42×C53—42no—
100#11719921—24no—
101#11719921—41no—
102#11719921—41no—
103C4⋊Dic53—21no—
104C4⋊Dic53—41no—
105C4⋊Dic53C4212yes—
106#11719919—24no—
107#11719919—42no—
108C4×Dic53—22no—
109C4×Dic53—41no—
110C4×Dic53—41no—
111C4×Dic53—42no—
112C23×C53C2×C22⋊C411yescharacteristic
113C22×D53—21no—
114C22×D53—21no—
115C22×D53—21no—
116C22×D53—21no—
117C22×D53C4×D412yes—
118C22×D53AGL(1,ℤ/8)11yescharacteristic
119C22×D53#13712yes—
120C22×D53#13711yescharacteristic
121C2×#6157—44no—
122C2×#6157—44no—
123C2×#6157—44no—
124C2×#6157—82no—
125C2×#6157—82no—
126C2×#6157—82no—
127C2×#6157—82no—
128Q8×C53—42no—
129Q8×C53—42no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53#9912yes—
144C2×C4×C53C2×C4211yescharacteristic
145C2×C4×C53#11711yescharacteristic
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—22no—
149C2×Dic53—21no—
150C2×Dic53—21no—
151C2×Dic53—21no—
152C2×Dic53—21no—
153C2×Dic53—21no—
154C2×Dic53—21no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53C4×D412yes—
161C4×D53—42no—
162C4×D53—42no—
163C4×D53—44no—
164C4×D53—42no—
165C4×D53—42no—
166#88324—44no—
167#8217—531nomaximal, Sylow
168C22×C53C4×C22⋊C411yescharacteristic
169C22×C53C2×#9911yescharacteristic
170C22×C53#41011yescharacteristic
171C22×C53#41711yescharacteristic
172C22×C53#41911yescharacteristic
173C22×C53#42111yescharacteristic
174C22×C53#42111yescharacteristic
175C2×D53—22no—
176C2×D53—22no—
177C2×D53—22no—
178C2×D53—21no—
179C2×D53—21no—
180C2×D53—21no—
181C2×D53—21no—
182C2×D53—22no—
183C2×D53#44012yes—
184C2×D53#44112yes—
185#6157—84no—
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—22no—
189C4×C53—42no—
190C4×C53—41no—
191C4×C53—81no—
192Dic53—22no—
193Dic53—42no—
194Dic53—41no—
195Dic53—41no—
196Dic53—81no—
197C4×C22⋊Q8—531no—
198#1749—531no—
199C2×#328—532no—
200#1434—531no—
201#1294—532no—
202C2×C53#151111yescharacteristic
203C2×C53#151211yescharacteristic
204C2×C53#151311yescharacteristic
205C2×C53#151511yescharacteristic
206C2×C53C4×#9911yescharacteristic
207C2×C53#175911yescharacteristic
208C2×C53#176311yescharacteristic
209D53—22no—
210D53—22no—
211C2×C22⋊Q8—531no—
212C2×C4×Q8#615711yescharacteristic, complemented
213C22×C42—531no—
214C2×C22⋊C8—532no—
215C2×#97—1061no—
216#391—531no—
217#389—531no—
218#387—1061no—
219#385—1061no—
220#380—531no—
221C4×C4⋊C4—531no—
222C4×C4⋊C4—1061no—
223C4×C22⋊C4—1061no—
224C2×#95—532no—
225C2×#95—1061no—
226C2×#95—1061no—
227#336—1061no—
228#328—538no—
229C53#821711yescharacteristic, complemented, Sylow
230C22×Q8C2×#615711yescharacteristic
231C23×C4—531no—
232C23×C4—531no—
233C23×C4—531no—
234C23×C4—1062no—
235C22×C8—1062no—
236C22⋊Q8—534no—
237C22⋊Q8—1062no—
238C4×Q8—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—531no—
241C2×C4⋊C4—531no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4C2×#615711yescharacteristic
245C2×C22⋊C4—1061no—
246C2×C22⋊C4—1061no—
247C2×C42—21no—
248C2×C42—531no—
249C2×C42—531no—
250C2×C42—531no—
251C2×C42—1061no—
252C2×C42—2122no—
253C2×C42C2×#615711yescharacteristic
254C22⋊C8—534no—
255C22⋊C8—1062no—
256#97—2122no—
257#95—534no—
258#95—1061no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262#95—1062no—
263#95—1061no—
264#95—2122no—
265#95—2122no—
266C24—531no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C4×#615712yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—531no—
274C22×C4—532no—
275C22×C4—532no—
276C22×C4—531no—
277C22×C4—531no—
278C22×C4—1061no—
279C22×C4—1062no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1062no—
283C22×C4—1064no—
284C22×C4—1061no—
285C22×C4—2121no—
286C22×C4—2121no—
287C22×C4—2121no—
288C22×C4—2121no—
289C22×C4—2121no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4—2122no—
293C22×C4—2122no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4C22⋊#615711yescharacteristic
297C22×C4C22⋊#615711yescharacteristic
298C22×C4C22×#615711yescharacteristic
299C2×C8—1064no—
300C2×C8—2122no—
301C2×C8—2122no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—532no—
305C4⋊C4—1064no—
306C4⋊C4—1061no—
307C4⋊C4—2121no—
308C4⋊C4—2122no—
309C22⋊C4—1064no—
310C22⋊C4—2121no—
311C22⋊C4—2121no—
312C42—22no—
313C42—42no—
314C42—1062no—
315C42—2121no—
316C42—2121no—
317C42—2122no—
318C23—532no—
319C23—531no—
320C23—531no—
321C23—532no—
322C23—1061no—
323C23—1061no—
324C23—1061no—
325C23—1061no—
326C23C2×C22⋊#615711yescharacteristic
327Q8—42no—
328Q8—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—532no—
343C2×C4—1061no—
344C2×C4—1061no—
345C2×C4—1062no—
346C2×C4—1061no—
347C2×C4—1061no—
348C2×C4—1061no—
349C2×C4—1061no—
350C2×C4—1061no—
351C2×C4—1061no—
352C2×C4—4242no—
353C2×C4—4242no—
354C2×C4—4242no—
355C2×C4—4242no—
356C2×C4—2122no—
357C2×C4—2122no—
358C2×C4—2124no—
359C2×C4—2122no—
360C2×C4—2122no—
361C2×C4—2124no—
362C2×C4—2124no—
363C2×C4—2124no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4#1313327512yes—
370C2×C4C2×C4×#615711yescharacteristic
371C2×C4#1313329611yescharacteristic
372C8—2124no—
373C22—532no—
374C22—532no—
375C22—1062no—
376C22—1061no—
377C22—1061no—
378C22—1061no—
379C22—1061no—
380C22—1062no—
381C22—1062no—
382C22—1062no—
383C22C4×C22⋊#615711yescharacteristic
384C22C2×#1313327511yescharacteristic
385C22#1681160211yescharacteristic
386C22#1681161411yescharacteristic
387C22#1681161611yescharacteristic
388C22#1681162011yescharacteristic
389C22#1681162211yescharacteristic
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—42no—
395C4—81no—
396C4—1062no—
397C4—4244no—
398C4—4241no—
399C4—2121no—
400C4—2121no—
401C4—2122no—
402C2—1062no—
403C2—1062no—
404C2#2029012511yescharacteristic
405C2#2029012811yescharacteristic
406C2#2029013011yescharacteristic
407C2#2029013711yescharacteristic
408C2C4×#1313327511yescharacteristic
409C2#2029047911yescharacteristic
410C2#2029048711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#8217
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
12. #16811477C4semidirect
15. #16810530C4semidirect
16. #16810519C4semidirect
21. C4×C4⋊Dic53C4semidirect
23. C2×C4×Q8×C53C4semidirect
212. C2×C4×Q8#6157semidirect
229. C53#8217semidirect

Maximal subgroups

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

subgroupindex
#202904422
C2×#168114772
#202898912
#202895882
C4×#131331932
#821753