Finite Group Databasebeta

← #28309575

Subgroups — #28309575

Order 13568.

372 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 76 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#20290582C211yesmaximal, characteristic
2#20289959C211yesmaximal, characteristic
3#20289714C212yesmaximal
4#20289713C212yesmaximal
5#20279926C211yesmaximal, characteristic
6#16811639—21no—
7#16811626—21no—
8C2×#13133278—21no—
9#16811503C2212yes—
10#16811469—21no—
11#16811469C2212yes—
12C2×#13133216C411yescharacteristic
13C2×#13133182C2211yescharacteristic
14#16810563C2211yescharacteristic
15#16810562C411yescharacteristic
16#16810561C2211yescharacteristic
17#16810527—21no—
18#16810527—21no—
19#16810527—21no—
20#16810487—21no—
21#16810474C411yescharacteristic
22C53×#400C411yescharacteristic
23C2×#11719940—22no—
24#13133284—41no—
25C4⋊#88324—41no—
26C4⋊#88324—41no—
27#13133278—42no—
28C22⋊#88324—22no—
29C4×#88324—41no—
30C22×Dic106—21no—
31C22×Dic106D411yescharacteristic
32#13133216—41no—
33#13133216—41no—
34#13133182—41no—
35#13133182—41no—
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919D411yescharacteristic
40C2×#11719919C2311yescharacteristic
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124C2×C411yescharacteristic
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53C22⋊C411yescharacteristic
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×Dic106—41no—
69C2×Dic106—41no—
70C2×Dic106—44no—
71C2×Dic106—41no—
72C2×Dic106—41no—
73C2×Dic106SD1612yes—
74C2×Dic106Q1612yes—
75#11719940—42no—
76#11719940—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊Dic53—41no—
82#11719919—41no—
83#11719919—41no—
84#11719919—42no—
85#11719919—42no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—41no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53#10312yes—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53#9911yescharacteristic
119C2×Dic53#9911yescharacteristic
120C2×Dic53#10312yes—
121C2×Dic53C2×SD1611yescharacteristic
122C2×Dic53C2×Q1611yescharacteristic
123Dic106—42no—
124Dic106—42no—
125Dic106—82no—
126Dic106—82no—
127#88324—82no—
128#88324—82no—
129#11907—531nomaximal, Sylow
130C22×C53—21no—
131C22×C53—21no—
132C22×C53C2×#9911yescharacteristic
133C22×C53C2×#10311yescharacteristic
134C22×C53#42111yescharacteristic
135C4×C53—22no—
136C4×C53—41no—
137C4×C53—41no—
138C4×C53—81no—
139C4×C53—81no—
140Dic53—21no—
141Dic53—21no—
142Dic53—22no—
143Dic53—21no—
144Dic53—41no—
145Dic53—81no—
146Dic53—81no—
147Dic53#35512yes—
148#2568—531no—
149#1801—531no—
150#1461—531no—
151#1353—532no—
152#1353—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53#152011yescharacteristic
156C2×C53#212311yescharacteristic
157C2×C53C2×#35511yescharacteristic
158C2×C4⋊Q8—531no—
159C2×C4×Q8—531no—
160#431—1061no—
161#423—1061no—
162C2×#101—1061no—
163#400—531no—
164#400—531no—
165#400—531no—
166#400#615711yescharacteristic
167#391—1061no—
168#391—1061no—
169#391—1061no—
170#382—1061no—
171C4×C4⋊C4—531no—
172#343—532no—
173#324—532no—
174#324—1061no—
175C53#1190711yescharacteristic, complemented, Sylow
176C22×Q8—531no—
177C22×Q8—1061no—
178C2×M16—1062no—
179C4⋊Q8—2121no—
180C4⋊Q8—2121no—
181C4×Q8—2121no—
182C4×Q8—2121no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C4⋊C4C2×#615711yescharacteristic
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—1061no—
194C4⋊C8—2121no—
195C4⋊C8—2121no—
196C4⋊C8—2121no—
197#101—2122no—
198C22⋊C8—1062no—
199C4×C8—2121no—
200#95—21no—
201#95—21no—
202#95—531no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208#95—1061no—
209#95C2×#615711yescharacteristic
210C2×Q8—532no—
211C2×Q8—532no—
212C2×Q8—2121no—
213C2×Q8—2121no—
214C2×Q8—2121no—
215C2×Q8—2121no—
216C2×Q8—2124no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—531no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—531no—
223C22×C4—1061no—
224C22×C4—1061no—
225C22×C4C22⋊#615711yescharacteristic
226C22×C4C22⋊#615711yescharacteristic
227C22×C4C22×#615711yescharacteristic
228M16—2122no—
229M16—2122no—
230C2×C8—2122no—
231C2×C8—2122no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C42—1061no—
240C42—1061no—
241C42—1061no—
242C42—1061no—
243C42—1061no—
244C42—1061no—
245C42—2122no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—4242no—
248Q8—4242no—
249Q8—2122no—
250Q8—2122no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—531no—
263C2×C4—531no—
264C2×C4—532no—
265C2×C4—531no—
266C2×C4—531no—
267C2×C4—1062no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2122no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4#1313328012yes—
280C8—4242no—
281C8—4242no—
282C22—21no—
283C22—21no—
284C22D53⋊#9911yescharacteristic
285C22C2×#1313328011yescharacteristic
286C22#1681162011yescharacteristic
287C4—22no—
288C4—41no—
289C4—41no—
290C4—81no—
291C4—81no—
292C4—532no—
293C4—1061no—
294C4—1062no—
295C4—1061no—
296C4—1061no—
297C4—4241no—
298C4—4241no—
299C4—2121no—
300C2—21no—
301C2—21no—
302C2#2029015311yescharacteristic
303C2#2029097811yescharacteristic
304C2D53⋊#35511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#400C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC2×#11719919C23
hypercenter#400#6157
hypocenterC53#11907
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #400

Complements

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

NH ≅ G/Ntype
175. C53#11907semidirect

Maximal subgroups

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

subgroupindex
#202905822
#202899592
#202897142
#202897132
#202799262
#1190753