Finite Group Databasebeta

← #28308157

Subgroups — #28308157

Order 13568.

662 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 80 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#20289966C212yesmaximal, complemented
2#20289641C212yesmaximal, complemented
3#20289639C212yesmaximal
4#20277098C211yesmaximal, characteristic
5#16811643—22no—
6#16811468—22no—
7#16811466C2214yes—
8C2×#13133176C411yescharacteristic
9C2×#13133176C2211yescharacteristic
10#16810486C411yescharacteristic
11#16810486C2211yescharacteristic
12#16810483—22no—
13#16810479C411yescharacteristic
14#16810479C2211yescharacteristic
15C4×C22⋊Dic53—22no—
16C24×C4×C53C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20#13133176—42no—
21#13133176—42no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53D412yes—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53C2×C411yescharacteristic
27C2×C22⋊Dic53C2×C411yescharacteristic
28C2×C22⋊Dic53C2×C411yescharacteristic
29C2×C22⋊Dic53C2311yescharacteristic
30C2×C4×Dic53—21no—
31C2×C4×Dic53D412yes—
32#13133124—22no—
33#13133124—22no—
34C25×C53C2×C411yescharacteristic
35C23×C4×C53—22no—
36C23×C4×C53—21no—
37C23×C4×C53—21no—
38C23×C4×C53—22no—
39C23×C4×C53—42no—
40C23×C4×C53—42no—
41C23×C4×C53C2×C411yescharacteristic
42C23×C4×C53C2×C411yescharacteristic
43C24×C53—21no—
44C24×C53—21no—
45C24×C53—41no—
46C24×C53—41no—
47C24×C53—44no—
48C24×C53M1612yes—
49C24×C53C22×C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—24no—
55C22×C4×C53—24no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—48no—
59C22×C4×C53—48no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53M1612yes—
69C22×Dic53—21no—
70C22×Dic53—21no—
71C22×Dic53C22⋊C412yes—
72C22×Dic53C2×D412yes—
73C2×#88324—44no—
74C22⋊Dic53—41no—
75C22⋊Dic53—41no—
76C22⋊Dic53—44no—
77C22⋊Dic53—44no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C4⋊Dic53—42no—
81C4⋊Dic53—42no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—42no—
85C23×C53—24no—
86C23×C53—22no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—42no—
90C23×C53—44no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53—42no—
94C23×C53—44no—
95C23×C53—44no—
96C23×C53—48no—
97C23×C53—44no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53C2×M1611yescharacteristic
102C23×C53C2×M1611yescharacteristic
103C2×C4×C53—24no—
104C2×C4×C53—24no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—22no—
110C2×C4×C53—48no—
111C2×C4×C53—48no—
112C2×C4×C53—42no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53#9912yes—
117C2×C4×C53#10012yes—
118C2×Dic53—22no—
119C2×Dic53—22no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126#88324—84no—
127#11501—531nomaximal, Sylow
128C22×C53—22no—
129C22×C53—24no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53—41no—
133C22×C53—41no—
134C22×C53—44no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—44no—
138C22×C53—44no—
139C22×C53—48no—
140C22×C53—44no—
141C22×C53—42no—
142C22×C53—42no—
143C22×C53#40711yescharacteristic
144C22×C53C2×#9911yescharacteristic
145C22×C53C2×#10011yescharacteristic
146C4×C53—22no—
147C4×C53—22no—
148C4×C53—42no—
149Dic53—42no—
150Dic53—42no—
151Dic53—81no—
152Dic53—81no—
153#2280—531no—
154#1464—532no—
155#1317—532no—
156#1316—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—44no—
160C2×C53—41no—
161C2×C53—41no—
162C2×C53#146211yescharacteristic
163C2×C53#212012yes—
164C24×C4#615711yescharacteristic
165C2×#117—531no—
166C2×#117—531no—
167#432—1062no—
168#381—531no—
169#381—531no—
170#380—1062no—
171#379—531no—
172#379—531no—
173C4×C22⋊C4—1062no—
174#324—1062no—
175#323—534no—
176C53#1150111yescharacteristic, complemented, Sylow
177C25C2×#615711yescharacteristic
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—21no—
181C23×C4—21no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4C2×#615711yescharacteristic
185C23×C4C2×#615711yescharacteristic
186#117—2122no—
187#117—2122no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—1061no—
190C2×C22⋊C4—531no—
191C2×C22⋊C4—531no—
192C2×C22⋊C4—531no—
193C2×C22⋊C4—531no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4—1062no—
196C2×C42—532no—
197C2×C42—1061no—
198C4⋊C8—2122no—
199C22⋊C8—1064no—
200#97—2122no—
201#95—1062no—
202#95—1062no—
203C24—21no—
204C24—21no—
205C24—41no—
206C24—41no—
207C24—44no—
208C24#1171994012yes—
209C24C22×#615711yescharacteristic
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—48no—
219C22×C4—48no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—532no—
227C22×C4—532no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4#1171994012yes—
231C22×C4C22⋊#615711yescharacteristic
232C22×C4C22⋊#615711yescharacteristic
233C2×C8—2124no—
234C4⋊C4—2122no—
235C4⋊C4—2122no—
236C22⋊C4—2121no—
237C22⋊C4—2121no—
238C22⋊C4—2124no—
239C22⋊C4—2124no—
240C22⋊C4—2121no—
241C22⋊C4—2121no—
242C42—1062no—
243C42—1062no—
244C42—2122no—
245C23—24no—
246C23—22no—
247C23—21no—
248C23—21no—
249C23—42no—
250C23—44no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—44no—
255C23—44no—
256C23—48no—
257C23—44no—
258C23—41no—
259C23—41no—
260C23C2×C22⋊#615711yescharacteristic
261C23C2×#1171994011yescharacteristic
262C23C2×#1171994011yescharacteristic
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—48no—
271C2×C4—48no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—1062no—
277C2×C4—1062no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2122no—
283C2×C4—2122no—
284C2×C4#1313327512yes—
285C2×C4#1313327712yes—
286C8—4244no—
287C22—22no—
288C22—24no—
289C22—21no—
290C22—21no—
291C22—41no—
292C22—41no—
293C22—44no—
294C22—42no—
295C22—42no—
296C22—44no—
297C22—44no—
298C22—48no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22#1681159311yescharacteristic
303C22C2×#1313327511yescharacteristic
304C22C2×#1313327711yescharacteristic
305C4—22no—
306C4—22no—
307C4—42no—
308C4—4241no—
309C4—4241no—
310C4—2122no—
311C4—2122no—
312C2—21no—
313C2—21no—
314C2—44no—
315C2—41no—
316C2—41no—
317C2#2028996011yescharacteristic
318C2#2029096612yes—
subgroupG/H
centerC22C2×#13133275
commutatorC24×C53C22×C4
FrattiniC24C22×#6157
FittingC24×C4×C53C4
socleC22×C53C2×#99
prime radicalC25×C53C2×C4
cosocleC2×C22⋊Dic53C23
hypercenterC24×C4#6157
hypocenterC53#11501
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #20289966C2semidirect
2. #20289641C2semidirect
176. C53#11501semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202896412
#202896392
#202770982
#1150153