Finite Group Databasebeta

← #28300025

Subgroups — #28300025

Order 13568.

366 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 94 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#20290630C211yesmaximal, characteristic
3#20289536C212yesmaximal
4#20289533C212yesmaximal
5#20283098C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C4⋊#88324—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810529C411yescharacteristic
13#16810528C411yescharacteristic
14#16810528C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810519C411yescharacteristic
17#16810509—21no—
18#16810506—21no—
19#16810504—21no—
20#16810490—21no—
21C42×Dic53C2211yescharacteristic
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C4⋊#88324—42no—
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—42no—
65C2×#88324—42no—
66C2×#88324—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C4⋊C4×C53—22no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C42×C53—22no—
74C4⋊Dic53—41no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—42no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—41no—
81C4×Dic53—41no—
82C4×Dic53—41no—
83C4×Dic53—41no—
84C4×Dic53SD1612yes—
85C4×Dic53Q1612yes—
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—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53#10312yes—
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—21no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53#10312yes—
118C2×Dic53C2×SD1611yescharacteristic
119C2×Dic53C2×Q1611yescharacteristic
120C2×Dic53AGL(1,ℤ/8)11yescharacteristic
121C2×Dic53#13711yescharacteristic
122Dic106—82no—
123#88324—82no—
124#88324—82no—
125#7733—531nomaximal, Sylow
126C22×C53#40811yescharacteristic
127C22×C53#41011yescharacteristic
128C22×C53#41311yescharacteristic
129C22×C53C2×#10311yescharacteristic
130C22×C53#41711yescharacteristic
131C22×C53#42111yescharacteristic
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#2863—531no—
147#1831—531no—
148#1831—531no—
149#1272—532no—
150#1271—532no—
151C2×C53#147311yescharacteristic
152C2×C53#148111yescharacteristic
153C2×C53#149511yescharacteristic
154C2×C53#151211yescharacteristic
155C2×C53#151811yescharacteristic
156C2×C53#153511yescharacteristic
157C2×C53#155011yescharacteristic
158C2×C4⋊C8—1061no—
159C2×C4⋊C8—1061no—
160#391—531no—
161#391—531no—
162#391—531no—
163#389—531no—
164#389—531no—
165#389#615711yescharacteristic
166#387—1061no—
167#387—1061no—
168#387—1061no—
169#383—1061no—
170C43—531no—
171#336—532no—
172#336—532no—
173#336—1061no—
174#336—1061no—
175C53#773311yescharacteristic, complemented, Sylow
176C22×Q8—531no—
177C22×C8—1062no—
178C22×C8—1062no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—1061no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4C2×#615711yescharacteristic
184C2×C4⋊C4C2×#615711yescharacteristic
185C2×C42—531no—
186C2×C42—531no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42C2×#615711yescharacteristic
192C4⋊C8—2122no—
193C4⋊C8—2122no—
194#95—21no—
195#95—1061no—
196#95—1061no—
197#95—1061no—
198#95—1061no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203C2×Q8—1062no—
204C2×Q8—2121no—
205C2×Q8—2121no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—531no—
209C22×C4—531no—
210C22×C4—531no—
211C22×C4—531no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4C22⋊#615711yescharacteristic
215C22×C4C22⋊#615711yescharacteristic
216C22×C4C22×#615711yescharacteristic
217C2×C8—2122no—
218C2×C8—2122no—
219C2×C8—2122no—
220C2×C8—2122no—
221C2×C8—2122no—
222C2×C8—2122no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—2121no—
227C4⋊C4—2121no—
228C4⋊C4—2121no—
229C4⋊C4—2122no—
230C42—22no—
231C42—532no—
232C42—532no—
233C42—1061no—
234C42—1061no—
235C42—2121no—
236C42—2121no—
237C42—2121no—
238C42—2121no—
239C23C2×C22⋊#615711yescharacteristic
240Q8—4242no—
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—532no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—531no—
258C2×C4—531no—
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#1313328012yes—
275C2×C4#1313330412yes—
276C8—4242no—
277C8—4242no—
278C22#1681159611yescharacteristic
279C22#1681160411yescharacteristic
280C22#1681160911yescharacteristic
281C22C2×#1313328011yescharacteristic
282C22#1681161411yescharacteristic
283C22#1681162011yescharacteristic
284C22#1681162111yescharacteristic
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#2028999911yescharacteristic
299C2#2029002311yescharacteristic
300C2#2029007011yescharacteristic
301C2#2029012911yescharacteristic
302C2#2029014711yescharacteristic
303C2#2029019911yescharacteristic
304C2#2029023211yescharacteristic
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#7733
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
175. C53#7733semidirect

Maximal subgroups

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

subgroupindex
#202906312
#202906302
#202895362
#202895332
#202830982
#773353