Finite Group Databasebeta

← #27293377

Subgroups — #27293377

Order 13568.

524 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 87 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#20277438C211yesmaximal, characteristic
2#20273032C212yesmaximal, complemented
3#20272659C211yesmaximal, characteristic, complemented
4#20272646C212yesmaximal, complemented
5C53×#1324C211yesmaximal, characteristic, complemented
6C2×#13133214C411yescharacteristic
7C2×#13133179C411yescharacteristic
8C2×#13133179C2211yescharacteristic
9C22⋊C8×D53—22no—
10#16810524C411yescharacteristic
11#16810524C2211yescharacteristic
12#16810501—22no—
13#16810501—22no—
14#16810469C411yescharacteristic
15#16810349C2212yes—
16#16810346—22no—
17C53×#382C2211yescharacteristic
18C53×#324C2212yes—
19C22×D212—22no—
20C22×D212C2×C411yescharacteristic
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23C22×C4×D53D412yes—
24C2×C8×D53—42no—
25#13133214—22no—
26#13133214C814yescomplemented
27#13133179—42no—
28#13133179—42no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—21no—
32C2×#11719921D412yes—
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53C2×C411yescharacteristic
35C22⋊#88322—22no—
36#13133129—42no—
37#13133124—21no—
38#13133124—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42C22⋊C8×C53—22no—
43#95×C53—22no—
44C23×D53—21no—
45C23×D53C22⋊C412yes—
46C22×C4×C53—21no—
47C22×C4×C53—21no—
48C22×C4×C53C22×C411yescharacteristic
49C22×C4×C53C2×D412yes—
50C22×Dic53—21no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C2×D212—21no—
54C2×D212—28no—
55C2×D212—41no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212—41no—
59C2×D212—82no—
60C2×D212—82no—
61C2×D212C2×C812yes—
62C2×D212C2×C812yescomplemented
63C2×D212M1612yes—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—42no—
67C2×C4×D53—82no—
68C2×C4×D53—82no—
69C2×C8×C53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C42×C53—21no—
73C42×C53C2×C812yescomplemented
74#11719921—42no—
75#11719921—42no—
76#11719921—82no—
77#11719921—82no—
78C4×Dic53—22no—
79C4×Dic53—41no—
80C2×#88322—42no—
81C8×D53—84no—
82C23×C53C2×C22⋊C411yescharacteristic
83C22×D53—22no—
84C22×D53—22no—
85C22×D53—22no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53C22⋊C814yes—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53C22×C811yescharacteristic
103C2×C4×C53C2×M1611yescharacteristic
104C2×Dic53—21no—
105C2×Dic53—41no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53#9912yes—
111D212—42no—
112D212—42no—
113D212—42no—
114D212—42no—
115D212—88no—
116C4×D53—44no—
117C4×D53—82no—
118C4×D53—84no—
119C8×C53—82no—
120#88322—82no—
121#11479—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53C2×C22⋊C811yescharacteristic
125C22×C53C2×#9911yescharacteristic
126C22×C53#41311yescharacteristic
127C2×D53—24no—
128C2×D53—42no—
129C2×D53—42no—
130C2×D53—44no—
131C2×D53—44no—
132C2×D53—42no—
133C2×D53—82no—
134C2×D53—82no—
135C2×D53—81no—
136C2×D53—81no—
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53—42no—
141C4×C53—81no—
142Dic53—22no—
143Dic53—41no—
144Dic53—81no—
145#2328—531no—
146#1459—532no—
147#1324—531no—
148#1324D5311yescharacteristic, complemented
149#1318—532no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53#145911yescharacteristic
153C2×C53#212711yescharacteristic
154C2×C53#213411yescharacteristic
155D53—44no—
156D53—82no—
157C2×#127—531no—
158C2×C4×D4—531no—
159C2×C4×D4—531no—
160C2×C22⋊C8—1062no—
161#390—531no—
162#390—531no—
163#386—1062no—
164#386—1062no—
165#382C2×D5311yescharacteristic
166#376—531no—
167#324—532no—
168#324C2×D5312yes—
169#323—1062no—
170C53#1147911yescharacteristic, complemented, Sylow
171C22×D4—531no—
172C22×D4—531no—
173C22×D4—1062no—
174C23×C4—532no—
175C23×C4—1061no—
176C22×C8—2122no—
177#127—534no—
178#127—1062no—
179C4×D4—2122no—
180C4×D4—2122no—
181C2×C4⋊C4C4×D5311yescharacteristic
182C2×C4⋊C4C22×D5311yescharacteristic
183C2×C22⋊C4—532no—
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42C4×D5311yescharacteristic
190C22⋊C8—22no—
191C22⋊C8—1062no—
192C22⋊C8—2122no—
193#95—22no—
194#95—1061no—
195#95—1061no—
196C24—532no—
197C24—1061no—
198C2×D4—532no—
199C2×D4—532no—
200C2×D4—532no—
201C2×D4—1068no—
202C2×D4—1061no—
203C2×D4—4242no—
204C2×D4—4242no—
205C2×D4—2121no—
206C2×D4—2121no—
207C2×D4—2121no—
208C2×D4—2121no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—531no—
212C22×C4—531no—
213C22×C4—1062no—
214C22×C4—1062no—
215C22×C4—1061no—
216C22×C4—4242no—
217C22×C4—4242no—
218C22×C4—2122no—
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4D4×D5312yes—
221C2×C8—42no—
222C2×C8—4244no—
223C2×C8—2122no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C22⋊C4—4242no—
227C22⋊C4—4242no—
228C22⋊C4—2122no—
229C22⋊C4—2122no—
230C42—21no—
231C42—1062no—
232C42—2121no—
233C42C8×D5312yescomplemented
234C23—534no—
235C23—1062no—
236C23—1062no—
237C23—1062no—
238C23—2122no—
239C23—2122no—
240C23—2122no—
241C23—2122no—
242C23—2121no—
243C23—2121no—
244C23C22⋊C4×D5311yescharacteristic
245D4—4248no—
246D4—2122no—
247D4—2122no—
248D4—2122no—
249D4—2122no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—532no—
260C2×C4—1061no—
261C2×C4—4242no—
262C2×C4—4244no—
263C2×C4—2124no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4C2×C8×D5311yescharacteristic
270C2×C4M16×D5311yescharacteristic
271C8—82no—
272C8—4242no—
273C22—21no—
274C22—21no—
275C22—1064no—
276C22—4242no—
277C22—4242no—
278C22—4241no—
279C22—4241no—
280C22—2124no—
281C22—2124no—
282C22—2122no—
283C22—2122no—
284C22—2122no—
285C22C22⋊C8×D5311yescharacteristic
286C22#99×D5311yescharacteristic
287C22#1681060611yescharacteristic
288C4—21no—
289C4—21no—
290C4—42no—
291C4—42no—
292C4—81no—
293C4—1062no—
294C4—4241no—
295C4—2121no—
296C2—21no—
297C2—21no—
298C2—4242no—
299C2—2124no—
300C2#2027303211yescharacteristic
301C2#2027638311yescharacteristic
302C2#2027640511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1324C2
socleC22×C53C2×#99
prime radicalC2×#13133214C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1324D53
hypocenterC53#11479
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
2. #20273032C2semidirect
3. #20272659C2semidirect
4. #20272646C2semidirect
5. C53×#1324C2semidirect
26. #13133214C8semidirect
62. C2×D212C2×C8semidirect
73. C42×C53C2×C8semidirect
148. #1324D53semidirect
170. C53#11479semidirect
233. C42C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202774382
#202730322
#202726592
#202726462
C53×#13242
#1147953