Finite Group Databasebeta

← #28299882

Subgroups — #28299882

Order 13568.

390 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 390 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290432C211yesmaximal, characteristic
2#20290430C211yesmaximal, characteristic
3#20289536C212yesmaximal
4#20289534C212yesmaximal
5#20283098C211yesmaximal, characteristic
6C2×#13133272—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810529C411yescharacteristic
13#16810528C411yescharacteristic
14#16810528C2211yescharacteristic
15#16810522C411yescharacteristic
16#16810519C2211yescharacteristic
17#16810509—21no—
18#16810506—21no—
19#16810504—21no—
20#16810490—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133272—24no—
26C4×#88324—24no—
27C22×Dic106C2×C411yescharacteristic
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C2×Dic106—22no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×#88324—24no—
65C2×#88324—24no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C42×C53—41no—
74C4⋊Dic53—22no—
75C4⋊Dic53—42no—
76#11719919—42no—
77C4×Dic53—22no—
78C4×Dic53—22no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—41no—
83C23×C53C2×C22⋊C411yescharacteristic
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—41no—
92C2×C4×C53—41no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53#10412yes—
98C2×C4×C53#13112yes—
99C2×Dic53—21no—
100C2×Dic53—21no—
101C2×Dic53—21no—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53#10412yes—
115C2×Dic53#13512yes—
116C2×Dic53#13712yes—
117Dic106—82no—
118#88324—44no—
119#88324—44no—
120#7693—531nomaximal, Sylow
121C22×C53#40811yescharacteristic
122C22×C53#41011yescharacteristic
123C22×C53#41311yescharacteristic
124C22×C53#41611yescharacteristic
125C22×C53#41911yescharacteristic
126C22×C53C2×#10411yescharacteristic
127C22×C53C2×#10411yescharacteristic
128C4×C53—22no—
129C4×C53—22no—
130C4×C53—41no—
131C4×C53—81no—
132C4×C53—81no—
133Dic53—22no—
134Dic53—22no—
135Dic53—22no—
136Dic53—41no—
137Dic53—81no—
138Dic53—81no—
139#2863—531no—
140#1745—531no—
141#1744—531no—
142#1272—532no—
143#1271—532no—
144C2×C53#147311yescharacteristic
145C2×C53#148511yescharacteristic
146C2×C53#149411yescharacteristic
147C2×C53#150011yescharacteristic
148C2×C53#150611yescharacteristic
149C2×C53#152411yescharacteristic
150C2×C53#155411yescharacteristic
151C2×#97—1061no—
152C2×C4×C8—1061no—
153#391—531no—
154#391—531no—
155#391—531no—
156#389—531no—
157#389—531no—
158#389#615711yescharacteristic
159#387—1061no—
160#387—1061no—
161#387—1061no—
162#383—1061no—
163C43—531no—
164#336—532no—
165#336—532no—
166#336—1061no—
167#336—1061no—
168C53#769311yescharacteristic, complemented, Sylow
169C22×Q8—531no—
170C22×C8—1062no—
171C22×C8—1062no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—531no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4—1061no—
176C2×C4⋊C4C2×#615711yescharacteristic
177C2×C4⋊C4C2×#615711yescharacteristic
178C2×C42—531no—
179C2×C42—531no—
180C2×C42—531no—
181C2×C42—531no—
182C2×C42—531no—
183C2×C42—531no—
184C2×C42C2×#615711yescharacteristic
185#97—1064no—
186C4×C8—1064no—
187#95—21no—
188#95—1061no—
189#95—1061no—
190#95—1061no—
191#95—1061no—
192#95—1061no—
193#95—1061no—
194#95—1061no—
195#95—1061no—
196C2×Q8—1062no—
197C2×Q8—2121no—
198C2×Q8—2121no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—531no—
205C22×C4—1061no—
206C22×C4—1061no—
207C22×C4C22⋊#615711yescharacteristic
208C22×C4C22⋊#615711yescharacteristic
209C22×C4C22×#615711yescharacteristic
210C2×C8—1064no—
211C2×C8—1064no—
212C2×C8—2122no—
213C2×C8—2122no—
214C2×C8—2122no—
215C2×C8—2122no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—1062no—
220C4⋊C4—2122no—
221C4⋊C4—2122no—
222C42—41no—
223C42—1062no—
224C42—1062no—
225C42—1062no—
226C42—1062no—
227C42—1062no—
228C42—2121no—
229C23C2×C22⋊#615711yescharacteristic
230Q8—4242no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—532no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—1061no—
248C2×C4—1061no—
249C2×C4—1061no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4#1313328112yes—
263C2×C4#1313330512yes—
264C8—2124no—
265C8—2124no—
266C22#1681159811yescharacteristic
267C22#1681160211yescharacteristic
268C22#1681161011yescharacteristic
269C22#1681161311yescharacteristic
270C22#1681161611yescharacteristic
271C22C2×#1313328111yescharacteristic
272C22D53⋊#10411yescharacteristic
273C4—22no—
274C4—22no—
275C4—41no—
276C4—81no—
277C4—81no—
278C4—1062no—
279C4—1062no—
280C4—1062no—
281C4—4241no—
282C4—4241no—
283C4—2121no—
284C2#2028999811yescharacteristic
285C2#2029003911yescharacteristic
286C2#2029006911yescharacteristic
287C2#2029008611yescharacteristic
288C2#2029010811yescharacteristic
289C2#2029016611yescharacteristic
290C2#2029024111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7693
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
168. C53#7693semidirect

Maximal subgroups

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

subgroupindex
#202904322
#202904302
#202895362
#202895342
#202830982
#769353