Finite Group Databasebeta

← #28299947

Subgroups — #28299947

Order 13568.

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

normal subgroups only

This group has 500 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#20290569C211yesmaximal, characteristic
2#20290378C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283023C211yesmaximal, characteristic
6C22×#11719940—21no—
7C2×C4⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133208C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810510—21no—
17#16810506—21no—
18#16810498—21no—
19#16810495—21no—
20#16810488C411yescharacteristic
21#16810483C411yescharacteristic
22C53×#380C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—42no—
25C2×#11719940—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28C4⋊#88324—42no—
29C22⋊#88324—42no—
30C23×Dic53D411yescharacteristic
31C23×Dic53D411yescharacteristic
32#13133208—42no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×C22⋊Dic53—21no—
40C2×C4×Dic53—42no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53C2311yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C23×C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53—21no—
54#95×C53—21no—
55C24×C53C22⋊C411yescharacteristic
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—22no—
64C22×Dic53—22no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C22⋊C411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C22×Dic53C2×D411yescharacteristic
77#11719940—48no—
78C2×#88324—24no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C2×#88324—42no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊Dic53—42no—
88C4⋊Dic53—22no—
89C4⋊Dic53—42no—
90C4⋊Dic53—41no—
91C4⋊Dic53—41no—
92C4⋊Dic53—41no—
93#11719919—42no—
94#11719919—41no—
95#11719919—41no—
96C4×Dic53—22no—
97C4×Dic53—22no—
98C4×Dic53—41no—
99C4×Dic53—41no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53#10412yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—44no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53#10112yes—
120C2×C4×C53#13112yes—
121C2×Dic53—24no—
122C2×Dic53—24no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—21no—
126C2×Dic53—21no—
127C2×Dic53—21no—
128C2×Dic53—21no—
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53#9912yes—
141C2×Dic53AGL(1,ℤ/8)11yescharacteristic
142C2×Dic53AGL(1,ℤ/8)11yescharacteristic
143C2×Dic53#13711yescharacteristic
144C2×Dic53#13711yescharacteristic
145#88324—44no—
146#88324—82no—
147#7711—531nomaximal, Sylow
148C22×C53—22no—
149C22×C53—22no—
150C22×C53—22no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53#40811yescharacteristic
156C22×C53C2×#9911yescharacteristic
157C22×C53C2×#10111yescharacteristic
158C22×C53#41711yescharacteristic
159C22×C53#41711yescharacteristic
160C22×C53C2×#10411yescharacteristic
161C22×C53#42111yescharacteristic
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—81no—
165C4×C53—81no—
166Dic53—21no—
167Dic53—22no—
168Dic53—21no—
169Dic53—21no—
170Dic53—21no—
171Dic53—41no—
172Dic53—81no—
173Dic53—81no—
174#2851—531no—
175#1797—531no—
176#1728—531no—
177#1271—532no—
178#1271—532no—
179C2×C53—42no—
180C2×C53#147211yescharacteristic
181C2×C53#148311yescharacteristic
182C2×C53#149511yescharacteristic
183C2×C53#150411yescharacteristic
184C2×C53#151311yescharacteristic
185C2×C53#152811yescharacteristic
186C2×C53#156111yescharacteristic
187C22×M16—1061no—
188C2×#125—531no—
189C22×C42—531no—
190C2×C4⋊C8—1061no—
191C2×C22⋊C8—1061no—
192#389—531no—
193#389—531no—
194#387—1061no—
195#387—1061no—
196#385—1061no—
197#385—1061no—
198#382—531no—
199#380—531no—
200#380#615711yescharacteristic
201#336—532no—
202#336—532no—
203#336—1061no—
204C53#771111yescharacteristic, complemented, Sylow
205C23×C4—531no—
206C23×C4—531no—
207C23×C4C2×#615711yescharacteristic
208C2×M16—1064no—
209C2×M16—2122no—
210C2×M16—2122no—
211C22×C8—1062no—
212C22×C8—1062no—
213#125—2122no—
214C2×C4⋊C4—531no—
215C2×C4⋊C4—531no—
216C2×C4⋊C4—1061no—
217C2×C4⋊C4—1061no—
218C2×C4⋊C4—1061no—
219C2×C4⋊C4—1061no—
220C2×C4⋊C4C2×#615711yescharacteristic
221C2×C4⋊C4C2×#615711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—1061no—
224C2×C42—531no—
225C2×C42—531no—
226C2×C42—531no—
227C2×C42—531no—
228C2×C42—2122no—
229C4⋊C8—2122no—
230C22⋊C8—2122no—
231#95—21no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237C24C22⋊#615711yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—1062no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—1061no—
251C22×C4—1061no—
252C22×C4—1062no—
253C22×C4—1061no—
254C22×C4—1061no—
255C22×C4—2121no—
256C22×C4—2121no—
257C22×C4C22⋊#615711yescharacteristic
258C22×C4C22×#615711yescharacteristic
259M16—2128no—
260C2×C8—1064no—
261C2×C8—2122no—
262C2×C8—2122no—
263C2×C8—2122no—
264C2×C8—2122no—
265C2×C8—2122no—
266C4⋊C4—22no—
267C4⋊C4—41no—
268C4⋊C4—1062no—
269C4⋊C4—2121no—
270C4⋊C4—2122no—
271C4⋊C4—2121no—
272C4⋊C4—2121no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C4⋊C4—2122no—
276C22⋊C4—42no—
277C22⋊C4—2122no—
278C42—1062no—
279C42—1062no—
280C42—2121no—
281C42—2121no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23#1313328112yes—
289C23C2×C22⋊#615711yescharacteristic
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—532no—
302C2×C4—531no—
303C2×C4—531no—
304C2×C4—531no—
305C2×C4—531no—
306C2×C4—1064no—
307C2×C4—1064no—
308C2×C4—1061no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—1061no—
315C2×C4—1061no—
316C2×C4—1061no—
317C2×C4—2122no—
318C2×C4—2122no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4—2121no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4—2121no—
325C2×C4#1313327812yes—
326C2×C4#1313330412yes—
327C8—4242no—
328C8—2124no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22#1681159611yescharacteristic
337C22D53⋊#9911yescharacteristic
338C22C2×#1313327811yescharacteristic
339C22#1681161411yescharacteristic
340C22#1681161411yescharacteristic
341C22C2×#1313328111yescharacteristic
342C22#1681162211yescharacteristic
343C4—22no—
344C4—22no—
345C4—81no—
346C4—81no—
347C4—1061no—
348C4—1062no—
349C4—1061no—
350C4—1061no—
351C4—1061no—
352C4—4241no—
353C4—4241no—
354C4—2121no—
355C2—42no—
356C2#2028999411yescharacteristic
357C2#2029003011yescharacteristic
358C2#2029007211yescharacteristic
359C2#2029010011yescharacteristic
360C2#2029013311yescharacteristic
361C2#2029017711yescharacteristic
362C2#2029025411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7711
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
204. C53#7711semidirect

Maximal subgroups

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

subgroupindex
#202905692
#202903782
#202895342
#202895332
#202830232
#771153