Finite Group Databasebeta

← #28309177

Subgroups — #28309177

Order 13568.

378 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 68 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#20290468C211yesmaximal, characteristic
2#20289979C211yesmaximal, characteristic
3#20289714C212yesmaximal
4#20289713C212yesmaximal
5#20279926C211yesmaximal, characteristic
6#16811643—21no—
7C2×#13133278—21no—
8C4×#11719940—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—
25#13133278—24no—
26C22⋊#88324—22no—
27#13133272—41no—
28#13133272—41no—
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—22no—
69C2×Dic106—22no—
70C2×Dic106—41no—
71C2×Dic106—44no—
72C2×Dic106—41no—
73C2×Dic106—41no—
74#11719940—24no—
75#11719940—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊Dic53—41no—
81#11719919—41no—
82#11719919—41no—
83#11719919—42no—
84#11719919—42no—
85C4×Dic53—21no—
86C4×Dic53—21no—
87C4×Dic53—21no—
88C4×Dic53—21no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—42no—
92C23×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53—21no—
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×Dic53—21no—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—22no—
110C2×Dic53—21no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53#9911yescharacteristic
118C2×Dic53#9911yescharacteristic
119C2×Dic53#10412yes—
120C2×Dic53#13712yes—
121Dic106—44no—
122Dic106—82no—
123Dic106—82no—
124#88324—44no—
125#88324—82no—
126#11794—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53C2×#9911yescharacteristic
130C22×C53#41911yescharacteristic
131C22×C53C2×#10411yescharacteristic
132C4×C53—41no—
133C4×C53—41no—
134C4×C53—41no—
135C4×C53—81no—
136C4×C53—81no—
137Dic53—21no—
138Dic53—22no—
139Dic53—22no—
140Dic53—41no—
141Dic53—81no—
142Dic53—81no—
143Dic53#35512yes—
144#2568—531no—
145C4×#101—531no—
146#1467—531no—
147#1353—532no—
148#1353—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53#150811yescharacteristic
152C2×C53#212311yescharacteristic
153C2×C53C2×#35511yescharacteristic
154C2×C4⋊Q8—531no—
155C2×C4×Q8—531no—
156#432—1061no—
157C2×#101—1061no—
158C4×M16—1061no—
159#400—531no—
160#400—531no—
161#400—531no—
162#400#615711yescharacteristic
163#391—1061no—
164#391—1061no—
165#391—1061no—
166#382—1061no—
167C4×C4⋊C4—531no—
168#343—532no—
169#324—532no—
170#324—1061no—
171C53#1179411yescharacteristic, complemented, Sylow
172C22×Q8—531no—
173C22×Q8—1061no—
174C2×M16—1062no—
175C4⋊Q8—2121no—
176C4⋊Q8—2121no—
177C4×Q8—2121no—
178C4×Q8—2121no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C2×#615711yescharacteristic
185C2×C4⋊C4C2×#615711yescharacteristic
186C2×C42—531no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42—1061no—
190C4⋊C8—2121no—
191#101—1064no—
192C22⋊C8—1062no—
193#97—2121no—
194#97—2121no—
195C4×C8—2121no—
196#95—21no—
197#95—21no—
198#95—531no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205#95C2×#615711yescharacteristic
206C2×Q8—1062no—
207C2×Q8—1062no—
208C2×Q8—2121no—
209C2×Q8—2121no—
210C2×Q8—2121no—
211C2×Q8—2124no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—1061no—
219C22×C4—1061no—
220C22×C4C22⋊#615711yescharacteristic
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223M16—1064no—
224M16—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C4⋊C4—2121no—
232C4⋊C4—2122no—
233C4⋊C4—2122no—
234C42—1061no—
235C42—1061no—
236C42—1061no—
237C42—1061no—
238C42—1061no—
239C42—1061no—
240C42—2122no—
241C23C2×C22⋊#615711yescharacteristic
242Q8—4242no—
243Q8—4242no—
244Q8—2124no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—531no—
258C2×C4—531no—
259C2×C4—532no—
260C2×C4—532no—
261C2×C4—1062no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2122no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C8—4242no—
274C8—2124no—
275C22—21no—
276C22—21no—
277C22D53⋊#9911yescharacteristic
278C22#1681161611yescharacteristic
279C22D53⋊#10411yescharacteristic
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—532no—
286C4—1062no—
287C4—1062no—
288C4—1061no—
289C4—4241no—
290C4—4241no—
291C4—2121no—
292C2—21no—
293C2—21no—
294C2#2029011311yescharacteristic
295C2#2029097811yescharacteristic
296C2D53⋊#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#11794
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
171. C53#11794semidirect

Maximal subgroups

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

subgroupindex
#202904682
#202899792
#202897142
#202897132
#202799262
#1179453