Finite Group Databasebeta

← #28308405

Subgroups — #28308405

Order 13568.

394 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289978C211yesmaximal, characteristic
2#20289958C211yesmaximal, characteristic
3#20289646C212yesmaximal
4#20289645C212yesmaximal
5#20279701C211yesmaximal, characteristic
6#16811643—21no—
7#16811639—21no—
8#16811468—21no—
9#16811468—21no—
10#16811468C2212yes—
11#16811468C2212yes—
12C2×#13133215C411yescharacteristic
13C2×#13133215C411yescharacteristic
14C2×Q8×Dic53C2211yescharacteristic
15#16810556—21no—
16#16810556—21no—
17#16810530C2211yescharacteristic
18#16810529C2211yescharacteristic
19#16810518—21no—
20#16810518—21no—
21C4×#11719919C411yescharacteristic
22C2×C4⋊Q8×C53C411yescharacteristic
23#13133284—41no—
24C4⋊#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—41no—
28C4×#88324—41no—
29C22×Dic106C2×C411yescharacteristic
30#13133215—42no—
31#13133215—41no—
32#13133215—41no—
33Q8×Dic53—41no—
34Q8×Dic53—41no—
35C2×C4⋊Dic53D411yescharacteristic
36C2×C4⋊Dic53D411yescharacteristic
37C2×C4⋊Dic53C2311yescharacteristic
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53D411yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C22×Q8×C53C2×C411yescharacteristic
53C22×Q8×C53C2×C411yescharacteristic
54C4⋊Q8×C53—42no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C42×C53C2×C411yescharacteristic
58C2×Q8×C53—22no—
59C2×Q8×C53—22no—
60C2×Q8×C53—22no—
61C2×Q8×C53—41no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×Dic106—22no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C42×C53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—24no—
87#11719919—42no—
88#11719919—42no—
89#11719919—42no—
90#11719919—41no—
91#11719919—41no—
92C4×Dic53—21no—
93C4×Dic53—21no—
94C4×Dic53—21no—
95C4×Dic53—21no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—41no—
99C23×C53C2×C22⋊C411yescharacteristic
100Q8×C53—44no—
101Q8×C53—82no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#10012yes—
116C2×C4×C53#13112yes—
117C2×Dic53—22no—
118C2×Dic53—22no—
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—41no—
129C2×Dic53—41no—
130C2×Dic53#9911yescharacteristic
131C2×Dic53#9911yescharacteristic
132Dic106—82no—
133#88324—82no—
134#88324—82no—
135#11568—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53#40811yescharacteristic
139C22×C53C2×#9911yescharacteristic
140C22×C53C2×#10011yescharacteristic
141C4×C53—22no—
142C4×C53—22no—
143C4×C53—42no—
144C4×C53—41no—
145C4×C53—81no—
146Dic53—21no—
147Dic53—42no—
148Dic53—41no—
149Dic53—41no—
150Dic53—41no—
151Dic53—81no—
152Dic53#35612yes—
153#2546—531no—
154#1467—531no—
155#1461—531no—
156#1319—532no—
157#1319—532no—
158C2×C53—21no—
159C2×C53—21no—
160C2×C53#147111yescharacteristic
161C2×C53#212911yescharacteristic
162C2×C53C2×#35611yescharacteristic
163C2×C4⋊Q8—531no—
164C2×C4⋊Q8—531no—
165C2×C4⋊Q8#615711yescharacteristic
166C2×C4×Q8—531no—
167#432—1061no—
168#431—1061no—
169#398—1061no—
170#398—1061no—
171#391—531no—
172#391—531no—
173#389—1061no—
174#389—1061no—
175C4×C4⋊C4—531no—
176#324—532no—
177#324—532no—
178#324—1061no—
179#324—1061no—
180C53#1156811yescharacteristic, complemented, Sylow
181C22×Q8—531no—
182C22×Q8C2×#615711yescharacteristic
183C22×Q8C2×#615711yescharacteristic
184C4⋊Q8—42no—
185C4⋊Q8—2121no—
186C4⋊Q8—2121no—
187C4⋊Q8—2122no—
188C4×Q8—2121no—
189C4×Q8—2121no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—531no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4—1061no—
199C2×C4⋊C4—1061no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—531no—
204C2×C42C2×#615711yescharacteristic
205C4⋊C8—2121no—
206C4⋊C8—2121no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209#97—2121no—
210C4×C8—2121no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1062no—
223C2×Q8—2121no—
224C2×Q8—2121no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—531no—
228C22×C4—531no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—1061no—
232C22×C4—1061no—
233C22×C4C22⋊#615711yescharacteristic
234C22×C4C22⋊#615711yescharacteristic
235C22×C4C22×#615711yescharacteristic
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—1064no—
241C4⋊C4—2121no—
242C4⋊C4—2121no—
243C4⋊C4—2121no—
244C4⋊C4—2122no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C4⋊C4—2122no—
248C4⋊C4—2122no—
249C42—41no—
250C42—1061no—
251C42—1061no—
252C42—1061no—
253C42—1061no—
254C42—1061no—
255C42—1061no—
256C42—2121no—
257C23C2×C22⋊#615711yescharacteristic
258Q8—44no—
259Q8—82no—
260Q8—4242no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—531no—
275C2×C4—531no—
276C2×C4—1062no—
277C2×C4—1062no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4#1313327712yes—
290C2×C4#1313330412yes—
291C8—4242no—
292C8—4242no—
293C22—21no—
294C22—21no—
295C22#1681159611yescharacteristic
296C22D53⋊#9911yescharacteristic
297C22C2×#1313327711yescharacteristic
298C4—22no—
299C4—22no—
300C4—42no—
301C4—41no—
302C4—81no—
303C4—532no—
304C4—1061no—
305C4—4241no—
306C4—2122no—
307C4—2121no—
308C4—2121no—
309C4—2121no—
310C2—21no—
311C2—21no—
312C2#2028999111yescharacteristic
313C2#2029099811yescharacteristic
314C2D53⋊#35611yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11568
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
180. C53#11568semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202899582
#202896462
#202896452
#202797012
#1156853