Finite Group Databasebeta

← #27300082

Subgroups — #27300082

Order 13568.

588 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277439C211yesmaximal, characteristic
2C2×#16810415C212yesmaximal
3#20274701C212yesmaximal, complemented
4#20272840C211yesmaximal, characteristic, complemented
5C53×#1393C211yesmaximal, characteristic, complemented
6C2×#13133213C2211yescharacteristic
7C2×D4×Dic53C2211yescharacteristic
8C2×#13133179C411yescharacteristic, complemented
9C2×#13133132C2212yes—
10#16810525C411yescharacteristic, complemented
11#16810500—22no—
12#16810499—22no—
13#16810488C411yescharacteristic, complemented
14#16810481—22no—
15#16810469C411yescharacteristic, complemented
16#16810415—24no—
17#16810389—22no—
18C2×#99×C53C2212yes—
19C53×#390C2211yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C22⋊D53—22no—
22C23×Dic53D412yes—
23C22×C4×D53—21no—
24#13133213—22no—
25#13133213D412yes—
26#13133213Q812yes—
27D4×Dic53—42no—
28#13133179—42no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—42no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53D412yes—
33C2×#11719921—21no—
34C2×#11719921—22no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37#13133132—22no—
38#13133132—22no—
39#13133124—21no—
40#13133124—22no—
41#13133124—21no—
42#13133124—42no—
43C22×D4×C53C2311yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C42×C53C2×C411yescharacteristic
48#99×C53—22no—
49#99×C53—22no—
50C24×C53C2×D412yes—
51C23×D53—21no—
52C2×D4×C53—41no—
53C2×D4×C53—41no—
54C2×D4×C53C2×D411yescharacteristic
55C2×D4×C53C2×Q811yescharacteristic
56C2×D4×C53#4112yes—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—42no—
60C22×C4×C53C22×C411yescharacteristic
61C2×C22⋊D53—24no—
62C2×C22⋊D53—24no—
63C2×C22⋊D53—82no—
64C2×C22⋊D53—82no—
65C22×Dic53—21no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53—82no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C412yes—
72C22×Dic53C22⋊C411yescharacteristic
73C2×D212—21no—
74C2×D212—41no—
75C2×D212—41no—
76C2×D212C4⋊C412yescomplemented
77C2×C4×D53—42no—
78C2×C4×D53—82no—
79C4⋊C4×C53—41no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—82no—
83C42×C53—41no—
84C22⋊Dic53—42no—
85C22⋊Dic53—42no—
86C22⋊Dic53—42no—
87#11719921—42no—
88#11719921—82no—
89C4⋊Dic53—41no—
90#11719919—84no—
91C4×Dic53—21no—
92C4×Dic53—22no—
93C4×Dic53C4⋊C412yescomplemented
94C23×C53—22no—
95C23×C53—22no—
96C23×C53—42no—
97C23×C53—42no—
98C23×C53C2×C22⋊C411yescharacteristic
99C23×C53#12112yes—
100C23×C53C22⋊Q812yes—
101C22×D53—22no—
102C22×D53—42no—
103C22×D53—42no—
104C22×D53—41no—
105C22×D53—41no—
106D4×C53—41no—
107D4×C53—41no—
108D4×C53—42no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—82no—
117C2×C4×C53—82no—
118C2×C4×C53—82no—
119C2×C4×C53C2×C4⋊C411yescharacteristic
120C2×C4×C53#11711yescharacteristic
121C22⋊D53—84no—
122C22⋊D53—84no—
123C2×Dic53—22no—
124C2×Dic53—21no—
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—44no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—42no—
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136C2×Dic53#9912yes—
137D212—42no—
138D212—42no—
139C4×D53—84no—
140#12799—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—42no—
149C22×C53—82no—
150C22×C53#38011yescharacteristic
151C22×C53C2×#9911yescharacteristic
152C22×C53#41011yescharacteristic
153C2×D53—44no—
154C2×D53—42no—
155C2×D53—82no—
156C2×D53—81no—
157C2×D53—81no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—81no—
161C4×C53—81no—
162C4×C53—162no—
163Dic53—22no—
164Dic53—42no—
165Dic53—42no—
166Dic53—41no—
167Dic53—162no—
168#2328—531no—
169C2×#353—532no—
170#1799—532no—
171#1393—531no—
172#1393D5311yescharacteristic, complemented
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—42no—
176C2×C53—42no—
177C2×C53#179911yescharacteristic
178C2×C53#212711yescharacteristic
179C2×C53#212811yescharacteristic
180D53—82no—
181C2×#127—531no—
182C2×C4×D4—531no—
183C2×C4×D4—531no—
184C2×#99—532no—
185C2×#99C2×D5312yes—
186#390—531no—
187#390C2×D5311yescharacteristic
188#386—1062no—
189#386—1062no—
190#382—531no—
191#380—1062no—
192#376—531no—
193#353—1064no—
194#342—1062no—
195C53#1279911yescharacteristic, complemented, Sylow
196C22×D4—531no—
197C22×D4—1062no—
198C22×D4C22×D5311yescharacteristic
199C23×C4—532no—
200C23×C4—1061no—
201#127—532no—
202#127—532no—
203#127—1062no—
204C4×D4—2122no—
205C4×D4—2122no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—2122no—
208C2×C4⋊C4C4×D5311yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—532no—
212C2×C22⋊C4—1062no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1062no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42C4×D5311yescharacteristic
218#99—22no—
219#99—22no—
220#99—1062no—
221#99—1062no—
222#95—1061no—
223#95—1061no—
224#95—1062no—
225#95—2122no—
226C24—1061no—
227C24D4×D5312yes—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—532no—
231C2×D4—1064no—
232C2×D4—1064no—
233C2×D4—1061no—
234C2×D4—4242no—
235C2×D4—4242no—
236C2×D4—2121no—
237C2×D4—2121no—
238C2×D4#1171994512yes—
239C2×D4D4×D5311yescharacteristic
240C2×D4Q8×D5311yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—532no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—1061no—
250C22×C4—4242no—
251C22×C4—4242no—
252C22×C4—2122no—
253C22×C4—2122no—
254C22×C4C2×C4×D5311yescharacteristic
255C4⋊C4—41no—
256C4⋊C4—4244no—
257C4⋊C4—2121no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C22⋊C4—4242no—
262C22⋊C4—2122no—
263C22⋊C4—2122no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C42—41no—
267C42—532no—
268C42—1062no—
269C42—1061no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—42no—
274C23—1062no—
275C23—2122no—
276C23—2122no—
277C23—2121no—
278C23—2121no—
279C23C22⋊C4×D5311yescharacteristic
280C23#1313319012yes—
281C23#1313319512yes—
282D4—42no—
283D4—41no—
284D4—41no—
285D4—4244no—
286D4—4244no—
287D4—2122no—
288D4—2122no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—532no—
300C2×C4—1061no—
301C2×C4—1062no—
302C2×C4—1062no—
303C2×C4—4244no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—2121no—
309C2×C4—2124no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2122no—
313C2×C4—2121no—
314C2×C4C4⋊C4×D5311yescharacteristic
315C2×C4#1313317411yescharacteristic
316C22—21no—
317C22—21no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—4242no—
326C22—4241no—
327C22—4241no—
328C22—2124no—
329C22—2122no—
330C22#1681048211yescharacteristic
331C22#99×D5311yescharacteristic
332C22#1681059511yescharacteristic
333C4—21no—
334C4—21no—
335C4—81no—
336C4—81no—
337C4—162no—
338C4—8482no—
339C4—1062no—
340C4—2122no—
341C4—2122no—
342C4—2121no—
343C2—21no—
344C2—21no—
345C2—42no—
346C2—42no—
347C2—4242no—
348C2#2027470211yescharacteristic
349C2#2027638311yescharacteristic
350C2#2027638511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1393C2
socleC22×C53C2×#99
prime radicalC2×#13133213C22
cosocleC22×D4×C53C23
hypercenter#1393D53
hypocenterC53#12799
LCSG ▷ C22×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. #20274701C2semidirect
4. #20272840C2semidirect
5. C53×#1393C2semidirect
8. C2×#13133179C4semidirect
10. #16810525C4semidirect
13. #16810488C4semidirect
15. #16810469C4semidirect
76. C2×D212C4⋊C4semidirect
93. C4×Dic53C4⋊C4semidirect
172. #1393D53semidirect
195. C53#12799semidirect

Maximal subgroups

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

subgroupindex
#202774392
C2×#168104152
#202747012
#202728402
C53×#13932
#1279953