Finite Group Databasebeta

← #28299899

Subgroups — #28299899

Order 13568.

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

normal subgroups only

This group has 378 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#20290631C211yesmaximal, characteristic
2#20290501C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289534C212yesmaximal
5#20283100C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810522C411yescharacteristic
13#16810522C411yescharacteristic
14#16810519C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810510—21no—
17#16810510—21no—
18#16810506—21no—
19#16810506—21no—
20#16810489C411yescharacteristic
21C42×Dic53C2211yescharacteristic
22C53×#382C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—24no—
27#13133272—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×C4×Dic53C2×C411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53D411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53C2311yescharacteristic
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C42×C53C2×C411yescharacteristic
49#95×C53—21no—
50#95×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22⋊C411yescharacteristic
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C22×Dic53C2×D411yescharacteristic
61C22×Dic53C2×D411yescharacteristic
62C2×#88324—24no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C42×C53—22no—
71C4⋊Dic53—22no—
72C4⋊Dic53—22no—
73C4⋊Dic53—42no—
74C4⋊Dic53—42no—
75#11719919—41no—
76#11719919—41no—
77#11719919—41no—
78#11719919—41no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C4×Dic53—21no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C23×C53C2×C22⋊C411yescharacteristic
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53#10112yes—
102C2×C4×C53#13112yes—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53#10412yes—
119C2×Dic53AGL(1,ℤ/8)11yescharacteristic
120C2×Dic53AGL(1,ℤ/8)11yescharacteristic
121C2×Dic53#13711yescharacteristic
122C2×Dic53#13711yescharacteristic
123#88324—44no—
124#88324—82no—
125#7697—531nomaximal, Sylow
126C22×C53#40811yescharacteristic
127C22×C53#41011yescharacteristic
128C22×C53C2×#10111yescharacteristic
129C22×C53#41711yescharacteristic
130C22×C53#41711yescharacteristic
131C22×C53C2×#10411yescharacteristic
132C22×C53#42111yescharacteristic
133C4×C53—22no—
134C4×C53—22no—
135C4×C53—41no—
136C4×C53—81no—
137C4×C53—81no—
138Dic53—22no—
139Dic53—21no—
140Dic53—21no—
141Dic53—21no—
142Dic53—21no—
143Dic53—41no—
144Dic53—81no—
145Dic53—81no—
146#2864—531no—
147#1831—531no—
148#1767—531no—
149#1271—532no—
150#1271—532no—
151C2×C53#147011yescharacteristic
152C2×C53#148311yescharacteristic
153C2×C53#149511yescharacteristic
154C2×C53#150911yescharacteristic
155C2×C53#151211yescharacteristic
156C2×C53#153111yescharacteristic
157C2×C53#156111yescharacteristic
158C2×C4⋊C8—1061no—
159C2×C4⋊C8—1061no—
160C2×#97—1061no—
161#389—531no—
162#389—531no—
163#389—531no—
164#389—531no—
165#387—1061no—
166#387—1061no—
167#387—1061no—
168#387—1061no—
169#382—531no—
170#382#615711yescharacteristic
171C43—531no—
172#336—532no—
173#336—532no—
174#336—1061no—
175C53#769711yescharacteristic, complemented, Sylow
176C22×C8—1062no—
177C22×C8—1062no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C2×#615711yescharacteristic
185C2×C4⋊C4C2×#615711yescharacteristic
186C2×C42—531no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42C2×#615711yescharacteristic
193C4⋊C8—1064no—
194C4⋊C8—2122no—
195#97—2122no—
196#95—21no—
197#95—21no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4C22⋊#615711yescharacteristic
213C22×C4C22⋊#615711yescharacteristic
214C22×C4C22×#615711yescharacteristic
215C2×C8—1064no—
216C2×C8—2122no—
217C2×C8—2122no—
218C2×C8—2122no—
219C2×C8—2122no—
220C2×C8—2122no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—1062no—
224C4⋊C4—1062no—
225C4⋊C4—2121no—
226C4⋊C4—2121no—
227C4⋊C4—2121no—
228C4⋊C4—2121no—
229C4⋊C4—2122no—
230C4⋊C4—2122no—
231C42—22no—
232C42—1061no—
233C42—1062no—
234C42—1061no—
235C42—1061no—
236C42—1061no—
237C42—1062no—
238C42—2121no—
239C42—2121no—
240C23C2×C22⋊#615711yescharacteristic
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—531no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—531no—
258C2×C4—532no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—1061no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4#1313327812yes—
275C2×C4#1313330512yes—
276C8—4242no—
277C8—2124no—
278C22#1681159811yescharacteristic
279C22#1681160311yescharacteristic
280C22C2×#1313327811yescharacteristic
281C22#1681161411yescharacteristic
282C22#1681161411yescharacteristic
283C22D53⋊#10411yescharacteristic
284C22#1681162011yescharacteristic
285C4—22no—
286C4—22no—
287C4—41no—
288C4—81no—
289C4—81no—
290C4—1061no—
291C4—1061no—
292C4—1061no—
293C4—1061no—
294C4—1062no—
295C4—4241no—
296C4—4241no—
297C4—2121no—
298C2#2028998911yescharacteristic
299C2#2029003311yescharacteristic
300C2#2029007311yescharacteristic
301C2#2029011611yescharacteristic
302C2#2029012711yescharacteristic
303C2#2029018711yescharacteristic
304C2#2029025411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7697
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
175. C53#7697semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202905012
#202895342
#202831002
#769753