Finite Group Databasebeta

← #27300086

Subgroups — #27300086

Order 13568.

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

normal subgroups only

This group has 524 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#20277425C211yesmaximal, characteristic
2C2×#16810417C212yesmaximal
3#20274701C212yesmaximal
4#20272840C211yesmaximal, characteristic
5C53×#1393C211yesmaximal, characteristic
6C2×#13133204C2211yescharacteristic
7C2×#13133182C411yescharacteristic, complemented
8C2×D4×Dic53C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810529C411yescharacteristic, complemented
11#16810517—22no—
12#16810495—22no—
13#16810492C411yescharacteristic, complemented
14#16810481—22no—
15#16810469C411yescharacteristic, complemented
16#16810417—24no—
17#16810389—22no—
18C2×#99×C53C2212yes—
19C53×#390C2211yescharacteristic
20C22×Dic106C2×C411yescharacteristic
21C23×Dic53D412yes—
22#13133204—22no—
23#13133204D412yes—
24#13133204Q812yes—
25#13133182—42no—
26D4×Dic53—42no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—21no—
29C2×#11719919—22no—
30C2×#11719919—42no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53D412yes—
34C2×C4×Dic53—21no—
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—
51C2×D4×C53—41no—
52C2×D4×C53—41no—
53C2×D4×C53C2×D411yescharacteristic
54C2×D4×C53C2×Q811yescharacteristic
55C2×D4×C53#4112yes—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—42no—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—42no—
65C22×Dic53—82no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C412yes—
68C22×Dic53C22⋊C411yescharacteristic
69C2×Dic106—21no—
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106C4⋊C412yescomplemented
73C4⋊C4×C53—41no—
74C22⋊C4×C53—42no—
75C22⋊C4×C53—42no—
76C22⋊C4×C53—82no—
77C42×C53—41no—
78C22⋊Dic53—24no—
79C22⋊Dic53—24no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C22⋊Dic53—42no—
83C4⋊Dic53—41no—
84#11719919—42no—
85#11719919—82no—
86#11719919—84no—
87C4×Dic53—21no—
88C4×Dic53—22no—
89C4×Dic53—42no—
90C4×Dic53C4⋊C412yescomplemented
91C23×C53—22no—
92C23×C53—22no—
93C23×C53—42no—
94C23×C53—42no—
95C23×C53C2×C22⋊C411yescharacteristic
96C23×C53#12112yes—
97C23×C53C22⋊Q812yes—
98D4×C53—41no—
99D4×C53—41no—
100D4×C53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—82no—
109C2×C4×C53—82no—
110C2×C4×C53—82no—
111C2×C4×C53C2×C4⋊C411yescharacteristic
112C2×C4×C53#11711yescharacteristic
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—21no—
117C2×Dic53—41no—
118C2×Dic53—44no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—82no—
126C2×Dic53—82no—
127C2×Dic53—82no—
128C2×Dic53—82no—
129C2×Dic53#9912yes—
130Dic106—42no—
131Dic106—42no—
132#12800—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—42no—
140C22×C53—42no—
141C22×C53—82no—
142C22×C53#38011yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53#41011yescharacteristic
145C4×C53—21no—
146C4×C53—21no—
147C4×C53—81no—
148C4×C53—81no—
149C4×C53—162no—
150Dic53—22no—
151Dic53—42no—
152Dic53—42no—
153Dic53—42no—
154Dic53—41no—
155Dic53—162no—
156#2326—531no—
157C2×#354—532no—
158#1799—532no—
159#1393—531no—
160#1393D5311yescharacteristic
161C2×C53—21no—
162C2×C53—21no—
163C2×C53—42no—
164C2×C53—42no—
165C2×C53#179911yescharacteristic
166C2×C53#212711yescharacteristic
167C2×C53#212911yescharacteristic
168C2×#124—531no—
169C2×C4×Q8—531no—
170C2×C4×D4—531no—
171C2×#99—532no—
172C2×#99C2×D5312yes—
173#391—531no—
174#390C2×D5311yescharacteristic
175#388—1062no—
176#385—1062no—
177#384—531no—
178#380—1062no—
179#376—531no—
180#354—1064no—
181#342—1062no—
182C53#1280011yescharacteristic, complemented, Sylow
183C22×Q8—531no—
184C22×D4C22×D5311yescharacteristic
185C23×C4—532no—
186#124—532no—
187#124—532no—
188#124—1062no—
189C4×Q8—2122no—
190C4×D4—2122no—
191C2×C4⋊C4—531no—
192C2×C4⋊C4—1062no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—2122no—
195C2×C4⋊C4C4×D5311yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—532no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4—1062no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—1061no—
204C2×C42C4×D5311yescharacteristic
205#99—22no—
206#99—22no—
207#99—1062no—
208#99—1062no—
209#95—1061no—
210#95—1061no—
211#95—1062no—
212#95—2122no—
213C24D4×D5312yes—
214C2×Q8—532no—
215C2×Q8—1061no—
216C2×Q8—2121no—
217C2×Q8—2121no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4#1171994512yes—
221C2×D4D4×D5311yescharacteristic
222C2×D4Q8×D5311yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4—4242no—
234C22×C4—2122no—
235C22×C4C2×C4×D5311yescharacteristic
236C4⋊C4—41no—
237C4⋊C4—4242no—
238C4⋊C4—4244no—
239C4⋊C4—2121no—
240C4⋊C4—2122no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C22⋊C4—1064no—
245C22⋊C4—1064no—
246C22⋊C4—2122no—
247C22⋊C4—2122no—
248C22⋊C4—2122no—
249C42—41no—
250C42—532no—
251C42—1062no—
252C42—1061no—
253C42—2122no—
254C23—22no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23C22⋊C4×D5311yescharacteristic
259C23#1313319012yes—
260C23#1313319512yes—
261Q8—2122no—
262Q8—2122no—
263D4—42no—
264D4—41no—
265D4—41no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—532no—
277C2×C4—1061no—
278C2×C4—1062no—
279C2×C4—1062no—
280C2×C4—1062no—
281C2×C4—4242no—
282C2×C4—4242no—
283C2×C4—4242no—
284C2×C4—4242no—
285C2×C4—2121no—
286C2×C4—2124no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4—2121no—
291C2×C4—2122no—
292C2×C4—2121no—
293C2×C4C4⋊C4×D5311yescharacteristic
294C2×C4#1313317411yescharacteristic
295C22—21no—
296C22—21no—
297C22—22no—
298C22—22no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22#1681048211yescharacteristic
305C22#99×D5311yescharacteristic
306C22#1681059511yescharacteristic
307C4—21no—
308C4—21no—
309C4—81no—
310C4—81no—
311C4—162no—
312C4—8482no—
313C4—1062no—
314C4—2122no—
315C4—2122no—
316C4—2122no—
317C4—2121no—
318C2—21no—
319C2—21no—
320C2—42no—
321C2—42no—
322C2#2027470211yescharacteristic
323C2#2027638111yescharacteristic
324C2#2027638811yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#1393C2
socleC22×C53C2×#99
prime radical, cosocleC22×D4×C53C23
hypercenter#1393D53
hypocenterC53#12800
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
7. C2×#13133182C4semidirect
10. #16810529C4semidirect
13. #16810492C4semidirect
15. #16810469C4semidirect
72. C2×Dic106C4⋊C4semidirect
90. C4×Dic53C4⋊C4semidirect
182. C53#12800semidirect

Maximal subgroups

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

subgroupindex
#202774252
C2×#168104172
#202747012
#202728402
C53×#13932
#1280053