Finite Group Databasebeta

← #27299066

Subgroups — #27299066

Order 13568.

826 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 826 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×#16810595C212yesmaximal
2#20277305C211yesmaximal, characteristic
3#20272840C211yesmaximal, characteristic, complemented
4#20272839C212yesmaximal, complemented
5C53×#1393C211yesmaximal, characteristic, complemented
6C22×#11719947C2211yescharacteristic
7C2×#13133174C411yescharacteristic, complemented
8C2×#13133173—22no—
9C2×C4⋊C4×D53C411yescharacteristic, complemented
10#16810595—24no—
11#16810595C418yescomplemented
12C2×#13133132C2212yes—
13C2×#13133131—22no—
14#16810529C411yescharacteristic, complemented
15#16810528C411yescharacteristic, complemented
16#16810526C2211yescharacteristic
17#16810499—22no—
18#16810481—22no—
19C2×#99×C53C2212yes—
20C53×#390C2211yescharacteristic
21C2×#11719947—42no—
22C2×#11719947—42no—
23C2×#11719947C2×C414yes—
24C2×#11719947D412yes—
25C2×#11719947Q812yes—
26C22×Dic106C2×C411yescharacteristic
27C22×C22⋊D53D412yes—
28C23×Dic53D412yes—
29C22×C4×D53—21no—
30C22×C4×D53C2×C411yescharacteristic
31#13133174—42no—
32#13133173—28no—
33#13133173—44no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×#11719919—42no—
38C4⋊C4×D53—42no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—22no—
41C2×#11719921—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44#13133132—22no—
45#13133132C2×C414yescomplemented
46#13133131—28no—
47#13133124—22no—
48#13133124—22no—
49C22×D4×C53C2311yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53—22no—
53C2×C42×C53C2×C411yescharacteristic
54#99×C53—22no—
55#99×C53C2×C414yescomplemented
56C24×C53C2×D412yes—
57C23×D53C22⋊C411yescharacteristic
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53C22×C412yes—
61C2×D4×C53C2×D411yescharacteristic
62C2×D4×C53C2×Q811yescharacteristic
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—42no—
66C22×C4×C53C22×C411yescharacteristic
67C2×C22⋊D53—42no—
68C2×C22⋊D53—42no—
69C2×C22⋊D53C22⋊C414yes—
70C2×C22⋊D53C4⋊C414yes—
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—24no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—42no—
77C22×Dic53C22⋊C412yes—
78C22×Dic53C22⋊C414yes—
79C22×Dic53C22⋊C411yescharacteristic
80C22×Dic53C4⋊C414yes—
81#11719947—44no—
82#11719947—44no—
83#11719947—44no—
84#11719947—44no—
85C2×C4×D53—22no—
86C2×C4×D53—22no—
87C2×C4×D53—42no—
88C2×C4×D53—82no—
89C2×C4×D53C4212yes—
90C2×C4×D53C4⋊C412yes—
91C2×Dic106—41no—
92C2×Dic106—41no—
93C2×Dic106C4212yes—
94C2×Dic106C4⋊C412yes—
95C4⋊C4×C53—41no—
96C22⋊C4×C53—24no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—82no—
99C42×C53—41no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102C22⋊Dic53—82no—
103#11719921—42no—
104C4⋊Dic53—41no—
105#11719919—48no—
106#11719919—42no—
107#11719919—42no—
108C4×Dic53—48no—
109C4×Dic53—41no—
110C23×C53—22no—
111C23×C53—22no—
112C23×C53—42no—
113C23×C53—42no—
114C23×C53C2×C22⋊C412yes—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53C2×C4⋊C412yes—
117C22×D53—21no—
118C22×D53—21no—
119C22×D53—41no—
120C22×D53—41no—
121C22×D53#9512yes—
122D4×C53—42no—
123D4×C53—41no—
124D4×C53—41no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—44no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—82no—
134C2×C4×C53—82no—
135C2×C4×C53C2×C4211yescharacteristic
136C2×C4×C53C2×C4⋊C411yescharacteristic
137C22⋊D53—44no—
138C22⋊D53—44no—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—24no—
142C2×Dic53—24no—
143C2×Dic53—24no—
144C2×Dic53—24no—
145C2×Dic53—42no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—44no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53#9514yes—
154C2×Dic53#9512yes—
155C2×Dic53#9912yes—
156C2×Dic53#9912yes—
157C4×D53—24no—
158C4×D53—24no—
159C4×D53—84no—
160Dic106—42no—
161Dic106—42no—
162#12583—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—22no—
166C22×C53—22no—
167C22×C53—42no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—82no—
172C22×C53C2×#9511yescharacteristic
173C22×C53C2×#9911yescharacteristic
174C22×C53C2×#9911yescharacteristic
175C2×D53—22no—
176C2×D53—42no—
177C2×D53—41no—
178C2×D53—41no—
179C2×D53—42no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—81no—
183C4×C53—84no—
184C4×C53—81no—
185Dic53—24no—
186Dic53—81no—
187Dic53—84no—
188Dic53—81no—
189Dic53#34214yes—
190C2×#410—532no—
191#2311—531no—
192#1393—532no—
193#1393—531no—
194#1393D5311yescharacteristic, complemented
195C2×C53—21no—
196C2×C53—21no—
197C2×C53—42no—
198C2×C53—42no—
199C2×C53C2×#34211yescharacteristic
200C2×C53#212811yescharacteristic
201C2×C53#212911yescharacteristic
202D53—42no—
203C22×#41—531no—
204C2×#117—531no—
205C2×#117—1062no—
206C22×C4⋊C4—531no—
207#410—538no—
208#410—1064no—
209C2×#99—532no—
210C2×#99—1062no—
211C2×#99C2×D5312yes—
212#391—531no—
213#391—531no—
214#390—531no—
215#390C2×D5311yescharacteristic
216#386—1062no—
217#380—1062no—
218C53#1258311yescharacteristic, complemented, Sylow
219C2×#41—534no—
220C2×#41—532no—
221C2×#41—532no—
222C2×#41—2122no—
223C2×#41—2122no—
224C22×Q8—531no—
225C22×D4—532no—
226C22×D4C22×D5311yescharacteristic
227C23×C4—532no—
228C23×C4—531no—
229C23×C4—1061no—
230#117—1068no—
231#117—2122no—
232#117—2124no—
233C2×C4⋊C4—531no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—2122no—
237C2×C4⋊C4—2122no—
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4—1061no—
244C2×C42—531no—
245C2×C42—2122no—
246C2×C42C4×D5311yescharacteristic
247#99—22no—
248#99—534no—
249#99—1068no—
250#99—1062no—
251#99C4×D5314yescomplemented
252#95—1062no—
253#95—1062no—
254C24—531no—
255C24D4×D5312yes—
256#41—2124no—
257#41—2124no—
258#41—2124no—
259#41—2124no—
260C2×Q8—532no—
261C2×Q8—532no—
262C2×Q8—2121no—
263C2×Q8—2121no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—534no—
267C2×D4—534no—
268C2×D4—2122no—
269C2×D4—2122no—
270C2×D4C2×C4×D5312yes—
271C2×D4D4×D5311yescharacteristic
272C2×D4Q8×D5311yescharacteristic
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—531no—
277C22×C4—534no—
278C22×C4—532no—
279C22×C4—534no—
280C22×C4—532no—
281C22×C4—532no—
282C22×C4—1062no—
283C22×C4—1062no—
284C22×C4—1062no—
285C22×C4—1062no—
286C22×C4—1064no—
287C22×C4—1061no—
288C22×C4—1061no—
289C22×C4—4242no—
290C22×C4—2122no—
291C22×C4—2122no—
292C22×C4C2×C4×D5311yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—2121no—
295C4⋊C4—2122no—
296C4⋊C4—2122no—
297C4⋊C4—2128no—
298C22⋊C4—24no—
299C22⋊C4—42no—
300C22⋊C4—82no—
301C22⋊C4—1064no—
302C22⋊C4—4242no—
303C22⋊C4—2122no—
304C22⋊C4—2122no—
305C42—41no—
306C42—2121no—
307C42—2128no—
308C23—22no—
309C23—22no—
310C23—42no—
311C23—42no—
312C23—532no—
313C23—1061no—
314C23—1061no—
315C23—2121no—
316C23—2121no—
317C23C22⋊C4×D5312yes—
318C23C22⋊C4×D5311yescharacteristic
319C23C4⋊C4×D5312yes—
320Q8—2122no—
321Q8—2122no—
322D4—42no—
323D4—41no—
324D4—41no—
325D4—2124no—
326D4—2124no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—532no—
338C2×C4—532no—
339C2×C4—532no—
340C2×C4—534no—
341C2×C4—1064no—
342C2×C4—1064no—
343C2×C4—1064no—
344C2×C4—1064no—
345C2×C4—1064no—
346C2×C4—1064no—
347C2×C4—1062no—
348C2×C4—1062no—
349C2×C4—4244no—
350C2×C4—4242no—
351C2×C4—4242no—
352C2×C4—2124no—
353C2×C4—2121no—
354C2×C4—2122no—
355C2×C4—2121no—
356C2×C4—2121no—
357C2×C4—2121no—
358C2×C4C42×D5311yescharacteristic
359C2×C4C4⋊C4×D5311yescharacteristic
360C22—21no—
361C22—21no—
362C22—22no—
363C22—22no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—82no—
369C22—1062no—
370C22—2122no—
371C22—2122no—
372C22—2121no—
373C22—2121no—
374C22#95×D5311yescharacteristic
375C22#99×D5311yescharacteristic
376C22#99×D5311yescharacteristic
377C4—21no—
378C4—21no—
379C4—81no—
380C4—84no—
381C4—81no—
382C4—534no—
383C4—1064no—
384C4—4241no—
385C4—4244no—
386C4—4241no—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—2122no—
392C2#342×D5311yescharacteristic
393C2#2027638511yescharacteristic
394C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C2×C42
FrattiniC22×D4C22×D53
FittingC53×#1393C2
socleC22×C53C2×#99
prime radicalC22×#11719947C22
cosocleC22×D4×C53C23
hypercenter#1393D53
hypocenterC53#12583
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1393

Complements

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

NH ≅ G/Ntype
3. #20272840C2semidirect
4. #20272839C2semidirect
5. C53×#1393C2semidirect
7. C2×#13133174C4semidirect
9. C2×C4⋊C4×D53C4semidirect
11. #16810595C4semidirect
14. #16810529C4semidirect
15. #16810528C4semidirect
45. #13133132C2×C4semidirect
55. #99×C53C2×C4semidirect
194. #1393D53semidirect
218. C53#12583semidirect
251. #99C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168105952
#202773052
#202728402
#202728392
C53×#13932
#1258353