Finite Group Databasebeta

← #27496216

Subgroups — #27496216

Order 13568.

582 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 163 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#20283100C211yesmaximal, characteristic
2#20283092C211yesmaximal, characteristic
3#20276288C214yesmaximal
4#20276285C214yesmaximal
5#20275689C211yesmaximal, characteristic
6#20275688C211yesmaximal, characteristic
7#20275687C211yesmaximal, characteristic
8#20274463C211yesmaximal, characteristic
9C53×#1993C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11C2×#13133128C2211yescharacteristic
12#16810530—22no—
13#16810528—22no—
14#16810522C2212yes—
15#16810519C2212yes—
16#16810510—22no—
17#16810506—22no—
18#16810489C2211yescharacteristic
19#16810488C2212yes—
20#16810488C2212yes—
21C42×Dic53C2211yescharacteristic
22#16810386C2214yes—
23#16810385C2214yes—
24#16810385C2214yes—
25#16810377C2212yes—
26#16810377C2212yes—
27C2×C4⋊Q8×C53C2211yescharacteristic
28C2×#97×C53C2211yescharacteristic
29C53×#382C2211yescharacteristic
30C53×#340C2214yes—
31C22×Dic106—21no—
32C22×#88322C2312yes—
33#13133217—22no—
34#13133217—22no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—22no—
37C2×C4⋊Dic53C2312yes—
38C2×C4⋊Dic53C2312yes—
39C2×#11719919—22no—
40C2×C4×Dic53D412yes—
41C2×C4×Dic53D412yes—
42C2×C4×Dic53C2311yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133128—22no—
45#13133124—21no—
46#13133124—21no—
47#13133124—22no—
48#13133124—22no—
49#13133124—24no—
50#13133124—24no—
51C22×Q8×C53—21no—
52C22×C8×C53C2312yes—
53C4⋊Q8×C53—22no—
54C4⋊Q8×C53—22no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2312yes—
57C2×C4⋊C4×C53C2312yes—
58C2×C42×C53C2311yescharacteristic
59#97×C53—22no—
60#95×C53—21no—
61#95×C53—21no—
62C2×Q8×C53—44no—
63C2×Q8×C53—42no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2411yescharacteristic
69C22×Dic53—22no—
70C22×Dic53—22no—
71C22×Dic53C2×D412yes—
72C22×Dic53C2×D412yes—
73C2×Dic106—44no—
74C2×Dic106—42no—
75C2×C8×C53—22no—
76C2×C8×C53—22no—
77C2×C8×C53—22no—
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C4⋊C4×C53—24no—
81C4⋊C4×C53—42no—
82C42×C53#4112yes—
83C42×C53#4112yes—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C4⋊Dic53—24no—
87C4⋊Dic53—44no—
88C4⋊Dic53—42no—
89#11719919—44no—
90C4×Dic53—22no—
91C4×Dic53—22no—
92C4×Dic53—42no—
93C4×Dic53—42no—
94C2×#88322—22no—
95C2×#88322—22no—
96C2×#88322—22no—
97C23×C53C22×D411yescharacteristic
98Q8×C53—84no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—44no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53#12412yes—
107C2×C4×C53#12412yes—
108C2×C4×C53C2×#4111yescharacteristic
109C2×C4×C53C2×#4111yescharacteristic
110C2×C4×C532−1+412yes—
111C2×C4×C532−1+412yes—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—22no—
117C2×Dic53—22no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—44no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123Dic106—84no—
124C8×C53—42no—
125#88322—42no—
126#30234—531nomaximal, Sylow
127C22×C53C2×#12411yescharacteristic
128C22×C53#53611yescharacteristic
129C22×C53#53611yescharacteristic
130C22×C53#57512yes—
131C22×C53#57512yes—
132C4×C53—22no—
133C4×C53—22no—
134C4×C53—82no—
135C4×C53—82no—
136C4×C53#54112yes—
137C4×C53#54112yes—
138Dic53—42no—
139Dic53—42no—
140Dic53—82no—
141Dic53—82no—
142#2864—531no—
143#2862—531no—
144#2101—534no—
145#2101—534no—
146#1993—531no—
147#1993—531no—
148#1993—531no—
149#1993D5311yescharacteristic
150#1745—531no—
151C2×C53#264211yescharacteristic
152C2×C53#314011yescharacteristic
153C2×C53#314011yescharacteristic
154C2×C53#324012yes—
155C2×C53#324012yes—
156C2×C4⋊Q8—531no—
157C2×C4⋊Q8C2×D5311yescharacteristic
158C2×#97—531no—
159C2×#97C2×D5311yescharacteristic
160#391—1062no—
161#391—1062no—
162#389—532no—
163#389—532no—
164#387—1062no—
165#387—1062no—
166#382—532no—
167#382—532no—
168#382—531no—
169#382C2×D5311yescharacteristic
170C43—531no—
171#340—534no—
172#340—534no—
173#340—534no—
174#340C2×D5314yes—
175#336—532no—
176#336—532no—
177C53#3023411yescharacteristic, complemented, Sylow
178C22×Q8—21no—
179C22×Q8—1061no—
180C22×C8—532no—
181C22×C8C22×D5312yes—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—1062no—
185C4⋊Q8—1062no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—532no—
188C2×C4⋊C4—532no—
189C2×C4⋊C4—1062no—
190C2×C4⋊C4—1062no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4C22×D5312yes—
193C2×C4⋊C4C22×D5312yes—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—532no—
197C2×C42—532no—
198C2×C42C22×D5311yescharacteristic
199#97—22no—
200#97—1062no—
201#95—21no—
202#95—21no—
203#95—1064no—
204#95—1064no—
205#95—1062no—
206#95—1062no—
207#95—1061no—
208#95—1061no—
209C2×Q8—44no—
210C2×Q8—42no—
211C2×Q8—2124no—
212C2×Q8—2122no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—532no—
216C22×C4—532no—
217C22×C4—1062no—
218C22×C4—1062no—
219C22×C4D4×D5311yescharacteristic
220C22×C4D4×D5311yescharacteristic
221C22×C4C23×D5311yescharacteristic
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—1062no—
226C2×C8—1062no—
227C2×C8—1062no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—42no—
232C4⋊C4—1062no—
233C4⋊C4—1062no—
234C4⋊C4—1064no—
235C4⋊C4—2124no—
236C4⋊C4—2124no—
237C4⋊C4—2122no—
238C42—1062no—
239C42—1062no—
240C42—2122no—
241C42—2122no—
242C42#1171994512yes—
243C42#1171994712yes—
244C23C2×D4×D5311yescharacteristic
245Q8—84no—
246Q8—4244no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1062no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1062no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2124no—
264C2×C4—2122no—
265C2×C4#1313320312yes—
266C2×C4#1313320312yes—
267C2×C4C2×#1171994711yescharacteristic
268C2×C4C2×#1171994511yescharacteristic
269C2×C4#1313326412yes—
270C2×C4#1313326512yes—
271C8—42no—
272C8—2122no—
273C22C2×#1313320311yescharacteristic
274C22#1681119311yescharacteristic
275C22#1681119811yescharacteristic
276C22#1681141912yes—
277C22#1681142212yes—
278C4—22no—
279C4—22no—
280C4—82no—
281C4—82no—
282C4—4242no—
283C4—4242no—
284C4—2122no—
285C4—2122no—
286C4#1681123112yes—
287C4#1681123312yes—
288C2#2028066411yescharacteristic
289C2#2028509911yescharacteristic
290C2#2028510311yescharacteristic
291C2#2028595012yes—
292C2#2028595812yes—
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1993C2
socle, prime radicalC23×C53C22×D4
hypercenter#1993D53
hypocenterC53#30234
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
177. C53#30234semidirect

Maximal subgroups

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

subgroupindex
#202831002
#202830922
#202762882
#202762852
#202756892
#202756882
#202756872
#202744632
C53×#19932
#3023453