Finite Group Databasebeta

← #28299907

Subgroups — #28299907

Order 13568.

394 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 88 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#20290642C211yesmaximal, characteristic
2#20290432C211yesmaximal, characteristic
3#20289534C212yesmaximal
4#20289533C212yesmaximal
5#20283074C211yesmaximal, characteristic
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133207—21no—
13#16810522C2211yescharacteristic
14#16810521C411yescharacteristic
15#16810520—21no—
16#16810519C2211yescharacteristic
17#16810518—21no—
18#16810492C411yescharacteristic
19#16810492C411yescharacteristic
20#16810487—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133272—24no—
26#13133207—44no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
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×#88324—24no—
62C2×#88324—42no—
63C2×#88324—42no—
64C2×#88324—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—41no—
70C42×C53—22no—
71C4⋊Dic53—22no—
72C4⋊Dic53—42no—
73C4⋊Dic53—41no—
74#11719919—42no—
75#11719919—42no—
76#11719919—42no—
77#11719919—42no—
78#11719919—42no—
79#11719919—42no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—41no—
85C4×Dic53—41no—
86C23×C53C2×C22⋊C411yescharacteristic
87C2×C4×C53—21no—
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#13112yes—
102C2×Dic53—21no—
103C2×Dic53—21no—
104C2×Dic53—21no—
105C2×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53#10112yes—
117C2×Dic53#10412yes—
118C2×Dic53AGL(1,ℤ/8)12yes—
119C2×Dic53#13712yes—
120#88324—44no—
121#88324—82no—
122#7700—531nomaximal, Sylow
123C22×C53#40811yescharacteristic
124C22×C53#41011yescharacteristic
125C22×C53C2×#10111yescharacteristic
126C22×C53#41811yescharacteristic
127C22×C53#41911yescharacteristic
128C22×C53C2×#10411yescharacteristic
129C22×C53#42111yescharacteristic
130C4×C53—22no—
131C4×C53—41no—
132C4×C53—41no—
133C4×C53—81no—
134C4×C53—81no—
135Dic53—22no—
136Dic53—22no—
137Dic53—22no—
138Dic53—22no—
139Dic53—81no—
140Dic53—81no—
141#2859—531no—
142#1839—531no—
143#1745—531no—
144#1271—532no—
145#1271—532no—
146C2×C53#147011yescharacteristic
147C2×C53#148511yescharacteristic
148C2×C53#149611yescharacteristic
149C2×C53#150511yescharacteristic
150C2×C53#151511yescharacteristic
151C2×C53#152811yescharacteristic
152C2×C53#155611yescharacteristic
153C2×#125—1061no—
154C2×#97—1061no—
155#389—531no—
156#389—531no—
157#389—531no—
158#389—1061no—
159#389—1061no—
160#389#615711yescharacteristic
161#384—531no—
162#384—531no—
163#382—1061no—
164C43—531no—
165#336—532no—
166#336—532no—
167#336—1061no—
168#336—1061no—
169#336—1061no—
170C53#770011yescharacteristic, complemented, Sylow
171C22×C8—1062no—
172C22×C8—1062no—
173#125—2124no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—531no—
176C2×C4⋊C4—531no—
177C2×C4⋊C4—1061no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4—1061no—
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
193#97—1064no—
194#95—21no—
195#95—1061no—
196#95—1061no—
197#95—1061no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—531no—
201C22×C4—531no—
202C22×C4—531no—
203C22×C4—531no—
204C22×C4—1061no—
205C22×C4—1061no—
206C22×C4C22⋊#615711yescharacteristic
207C22×C4C22⋊#615711yescharacteristic
208C22×C4C22×#615711yescharacteristic
209C2×C8—1064no—
210C2×C8—2122no—
211C2×C8—2122no—
212C2×C8—2122no—
213C2×C8—2122no—
214C2×C8—2122no—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4—42no—
218C4⋊C4—1062no—
219C4⋊C4—2121no—
220C4⋊C4—2122no—
221C4⋊C4—2122no—
222C4⋊C4—2122no—
223C4⋊C4—2122no—
224C4⋊C4—2122no—
225C4⋊C4—2122no—
226C4⋊C4—2122no—
227C42—22no—
228C42—1062no—
229C42—1062no—
230C42—1062no—
231C42—1062no—
232C42—2121no—
233C42—2121no—
234C23C2×C22⋊#615711yescharacteristic
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—532no—
250C2×C4—532no—
251C2×C4—532no—
252C2×C4—532no—
253C2×C4—1061no—
254C2×C4—1061no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—1061no—
258C2×C4—1061no—
259C2×C4—1061no—
260C2×C4—1061no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4#1313330412yes—
268C8—4242no—
269C8—2124no—
270C22#1681159611yescharacteristic
271C22#1681160211yescharacteristic
272C22D53⋊#10111yescharacteristic
273C22#1681161511yescharacteristic
274C22#1681161611yescharacteristic
275C22D53⋊#10411yescharacteristic
276C22#1681162011yescharacteristic
277C4—22no—
278C4—41no—
279C4—41no—
280C4—81no—
281C4—81no—
282C4—1062no—
283C4—1062no—
284C4—1062no—
285C4—1062no—
286C4—4241no—
287C4—4241no—
288C2#2028998711yescharacteristic
289C2#2029003711yescharacteristic
290C2#2029007411yescharacteristic
291C2#2029010511yescharacteristic
292C2#2029013711yescharacteristic
293C2#2029017811yescharacteristic
294C2#2029024511yescharacteristic
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#7700
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
170. C53#7700semidirect

Maximal subgroups

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

subgroupindex
#202906422
#202904322
#202895342
#202895332
#202830742
#770053