Finite Group Databasebeta

← #27300078

Subgroups — #27300078

Order 13568.

544 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 95 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#20277262C211yesmaximal, 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#16810511—22no—
12#16810488C411yescharacteristic, complemented
13#16810481—22no—
14C22⋊C4×Dic53—22no—
15C42×Dic53C411yescharacteristic, 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—41no—
26#13133182—41no—
27D4×Dic53—41no—
28D4×Dic53—41no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—42no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53D412yes—
35C2×C4×Dic53—21no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40#13133132—22no—
41#13133132—22no—
42#13133124—22no—
43#13133124—22no—
44#13133124—42no—
45C22×D4×C53C2311yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—22no—
49C2×C42×C53C2×C411yescharacteristic
50#99×C53—22no—
51#99×C53—22no—
52C24×C53C2×D412yes—
53C2×D4×C53—41no—
54C2×D4×C53—41no—
55C2×D4×C53C2×D411yescharacteristic
56C2×D4×C53C2×Q811yescharacteristic
57C2×D4×C53#4112yes—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—42no—
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—22no—
65C22×Dic53—22no—
66C22×Dic53—44no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C412yes—
70C22×Dic53C22⋊C411yescharacteristic
71C2×Dic106—21no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74C2×Dic106C4⋊C412yescomplemented
75C4⋊C4×C53—41no—
76C22⋊C4×C53—44no—
77C22⋊C4×C53—42no—
78C22⋊C4×C53—42no—
79C42×C53—41no—
80C22⋊Dic53—24no—
81C22⋊Dic53—24no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C22⋊Dic53—42no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—84no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—42no—
92C4×Dic53—42no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C4×Dic53C4⋊C412yescomplemented
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—42no—
99C23×C53—42no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53#12112yes—
102C23×C53C22⋊Q812yes—
103D4×C53—41no—
104D4×C53—41no—
105D4×C53—42no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—22no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—82no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53C2×C4⋊C411yescharacteristic
117C2×C4×C53#11711yescharacteristic
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—21no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—44no—
128C2×Dic53—42no—
129C2×Dic53—41no—
130C2×Dic53—82no—
131C2×Dic53—82no—
132C2×Dic53—82no—
133C2×Dic53—82no—
134C2×Dic53#9911yescharacteristic
135C2×Dic53#9911yescharacteristic
136Dic106—42no—
137Dic106—42no—
138#12798—531nomaximal, Sylow
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53—42no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—82no—
148C22×C53#38011yescharacteristic
149C22×C53C2×#9911yescharacteristic
150C22×C53#41011yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—42no—
154C4×C53—81no—
155C4×C53—162no—
156Dic53—21no—
157Dic53—42no—
158Dic53—42no—
159Dic53—42no—
160Dic53—41no—
161Dic53—162no—
162Dic53#35412yescomplemented
163C4×#124—531no—
164C2×#354—532no—
165#1799—532no—
166#1393—531no—
167#1393D5311yescharacteristic
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—42no—
171C2×C53—42no—
172C2×C53#179911yescharacteristic
173C2×C53C2×#35411yescharacteristic
174C2×C53#212611yescharacteristic
175C2×#124—531no—
176C2×C4×Q8—531no—
177C2×C4×D4—531no—
178C2×#99—532no—
179C2×#99C2×D5312yes—
180#391—531no—
181#390C2×D5311yescharacteristic
182#388—1062no—
183#382—531no—
184#380—1062no—
185C4×C22⋊C4—1062no—
186C43—531no—
187#354—1064no—
188#342—1062no—
189C53#1279811yescharacteristic, complemented, Sylow
190C22×Q8—531no—
191C22×D4C22×D5311yescharacteristic
192C23×C4—532no—
193#124—532no—
194#124—532no—
195#124—1062no—
196C4×Q8—2121no—
197C4×Q8—2121no—
198C4×D4—2121no—
199C4×D4—2121no—
200C2×C4⋊C4—531no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—2122no—
203C2×C4⋊C4C4×D5311yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—532no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—1061no—
212C2×C42—1061no—
213C2×C42—1061no—
214C2×C42C4×D5311yescharacteristic
215#99—22no—
216#99—22no—
217#99—1062no—
218#99—1062no—
219#95—1062no—
220#95—1062no—
221#95—2122no—
222C24D4×D5312yes—
223C2×Q8—532no—
224C2×Q8—1061no—
225C2×Q8—2121no—
226C2×Q8—2121no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4#1171994512yes—
230C2×D4D4×D5311yescharacteristic
231C2×D4Q8×D5311yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—531no—
236C22×C4—532no—
237C22×C4—531no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—2124no—
243C22×C4—2122no—
244C22×C4C2×C4×D5311yescharacteristic
245C4⋊C4—41no—
246C4⋊C4—4244no—
247C4⋊C4—2121no—
248C4⋊C4—2122no—
249C22⋊C4—44no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—1064no—
253C22⋊C4—1064no—
254C22⋊C4—2122no—
255C22⋊C4—2122no—
256C22⋊C4—2122no—
257C42—41no—
258C42—532no—
259C42—1061no—
260C42—1061no—
261C42—1061no—
262C42—2122no—
263C42—2121no—
264C42—2121no—
265C42—2122no—
266C23—22no—
267C23—22no—
268C23—42no—
269C23—42no—
270C23C22⋊C4×D5311yescharacteristic
271C23#1313319012yes—
272C23#1313319512yes—
273Q8—2122no—
274Q8—2122no—
275D4—42no—
276D4—41no—
277D4—41no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—531no—
289C2×C4—531no—
290C2×C4—1062no—
291C2×C4—1061no—
292C2×C4—1062no—
293C2×C4—1062no—
294C2×C4—1062no—
295C2×C4—4242no—
296C2×C4—4242no—
297C2×C4—4242no—
298C2×C4—4242no—
299C2×C4—2124no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2122no—
306C2×C4C4⋊C4×D5311yescharacteristic
307C2×C4#1313317411yescharacteristic
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22—42no—
316C22—82no—
317C22#1681048211yescharacteristic
318C22#99×D5311yescharacteristic
319C22#1681059511yescharacteristic
320C4—21no—
321C4—21no—
322C4—42no—
323C4—81no—
324C4—162no—
325C4—532no—
326C4—8482no—
327C4—1061no—
328C4—2122no—
329C4—2122no—
330C4—2122no—
331C4—2121no—
332C2—21no—
333C2—21no—
334C2—42no—
335C2—42no—
336C2#2027470211yescharacteristic
337C2#354×D5311yescharacteristic
338C2#2027637711yescharacteristic
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#12798
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
12. #16810488C4semidirect
15. C42×Dic53C4semidirect
74. C2×Dic106C4⋊C4semidirect
95. C4×Dic53C4⋊C4semidirect
162. Dic53#354semidirect
189. C53#12798semidirect

Maximal subgroups

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

subgroupindex
#202772622
C2×#168104172
#202747012
#202728402
C53×#13932
#1279853