Finite Group Databasebeta

← #28299800

Subgroups — #28299800

Order 13568.

468 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 92 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#20290607C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3#20289533C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283026C211yesmaximal, characteristic
6C2×#13133274—21no—
7C2×#13133274—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133209C411yescharacteristic
13C2×C42×D53C2211yescharacteristic
14#16810522C2211yescharacteristic
15#16810522C2211yescharacteristic
16#16810508—21no—
17#16810508—21no—
18#16810497—21no—
19#16810497—21no—
20#16810480C411yescharacteristic
21#16810480C411yescharacteristic
22C53×#382C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—24no—
26#13133274—42no—
27C4×#88324—42no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31#13133209—42no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43C42×D53—42no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C42×C53C2×C411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53C23×D53C22⋊C411yescharacteristic
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—21no—
60C22×Dic53—21no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72C2×C4×D53—41no—
73C2×#88324—24no—
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—22no—
81C42×C53—22no—
82#11719921—42no—
83#11719921—42no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719919—42no—
89#11719919—42no—
90C4×Dic53—22no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—21no—
95C22×D53—21no—
96C22×D53—21no—
97C22×D53—21no—
98C22×D53—21no—
99C22×D53—21no—
100C22×D53#10412yes—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53#9912yes—
115C2×C4×C53#13112yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53#10112yes—
129C2×Dic53#13512yes—
130C2×Dic53#13512yes—
131C4×D53—44no—
132C4×D53—44no—
133C4×D53—44no—
134#88324—44no—
135#88324—82no—
136#7671—531nomaximal, Sylow
137C22×C53#40811yescharacteristic
138C22×C53C2×#9911yescharacteristic
139C22×C53C2×#10111yescharacteristic
140C22×C53#41611yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53C2×#10411yescharacteristic
143C22×C53#42111yescharacteristic
144C2×D53—22no—
145C2×D53—22no—
146C2×D53—22no—
147C2×D53—41no—
148C2×D53—41no—
149C2×D53—41no—
150C2×D53—41no—
151C4×C53—22no—
152C4×C53—22no—
153C4×C53—41no—
154C4×C53—81no—
155C4×C53—81no—
156Dic53—22no—
157Dic53—22no—
158Dic53—22no—
159Dic53—81no—
160Dic53—81no—
161#2851—531no—
162#1817—531no—
163C4×C22⋊C8—531no—
164#1271—532no—
165#1271—532no—
166C2×C53#147211yescharacteristic
167C2×C53#148211yescharacteristic
168C2×C53#149411yescharacteristic
169C2×C53#150111yescharacteristic
170C2×C53#151611yescharacteristic
171C2×C53#152811yescharacteristic
172C2×C53#154411yescharacteristic
173D53—42no—
174C2×#125—531no—
175C22×C42—531no—
176C2×C22⋊C8—1061no—
177C2×C22⋊C8—1061no—
178C2×C4×C8—1061no—
179#389—531no—
180#389—531no—
181#387—1061no—
182#387—1061no—
183#385—1061no—
184#385—1061no—
185#382#615711yescharacteristic
186#380—531no—
187#380—531no—
188#336—532no—
189#336—532no—
190#336—1061no—
191C53#767111yescharacteristic, complemented, Sylow
192C23×C4—531no—
193C23×C4—531no—
194C23×C4—531no—
195C22×C8—1062no—
196C22×C8—1062no—
197#125—2122no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C2×#615711yescharacteristic
205C2×C4⋊C4C2×#615711yescharacteristic
206C2×C22⋊C4—1061no—
207C2×C22⋊C4—1061no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—2122no—
212C2×C42C2×#615711yescharacteristic
213C22⋊C8—1064no—
214C22⋊C8—2122no—
215C4×C8—2122no—
216#95—21no—
217#95—21no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222C24—531no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—1062no—
229C22×C4—1062no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1062no—
234C22×C4—1061no—
235C22×C4—1061no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4—2121no—
239C22×C4C22⋊#615711yescharacteristic
240C22×C4C22⋊#615711yescharacteristic
241C22×C4C22×#615711yescharacteristic
242C2×C8—1064no—
243C2×C8—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—2121no—
251C4⋊C4—2121no—
252C4⋊C4—2122no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C22⋊C4—2122no—
257C22⋊C4—2122no—
258C42—22no—
259C42—1062no—
260C42—2121no—
261C42—2121no—
262C23—532no—
263C23—1061no—
264C23—1061no—
265C23—1061no—
266C23—1061no—
267C23—1061no—
268C23—1061no—
269C23C2×C22⋊#615711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—532no—
284C2×C4—532no—
285C2×C4—532no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—2124no—
293C2×C4—2124no—
294C2×C4—2124no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4#1313327612yes—
302C2×C4#1313330412yes—
303C8—4242no—
304C8—2124no—
305C22—1062no—
306C22—1062no—
307C22—1062no—
308C22—2121no—
309C22—2121no—
310C22—2121no—
311C22—2121no—
312C22#1681159711yescharacteristic
313C22C2×#1313327611yescharacteristic
314C22D53⋊#10111yescharacteristic
315C22#1681161311yescharacteristic
316C22#1681161311yescharacteristic
317C22D53⋊#10411yescharacteristic
318C22#1681162011yescharacteristic
319C4—22no—
320C4—22no—
321C4—41no—
322C4—81no—
323C4—81no—
324C4—1062no—
325C4—1062no—
326C4—1062no—
327C4—4241no—
328C4—4241no—
329C2—2122no—
330C2#2028999311yescharacteristic
331C2#2029002711yescharacteristic
332C2#2029006711yescharacteristic
333C2#2029009211yescharacteristic
334C2#2029014311yescharacteristic
335C2#2029017811yescharacteristic
336C2#2029022111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#7671
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
3. #20289533C2semidirect
4. #20289533C2semidirect
191. C53#7671semidirect

Maximal subgroups

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

subgroupindex
#202906072
C4×#131332742
#202895332
#202830262
#767153