Finite Group Databasebeta

← #27279818

Subgroups — #27279818

Order 13568.

694 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#16810469C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3C2×#16810360C212yesmaximal
4#20272593C212yesmaximal
5C53×#2855C211yesmaximal, characteristic
6C22×C4×Dic53—21no—
7C2×C22⋊#88322C2212yes—
8#16810469—24no—
9C2×#13133124—21no—
10C2×#13133124C2212yes—
11#16810378—22no—
12#16810360—24no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C42×C53C2211yescharacteristic
15C53×#391C412yescomplemented
16C53×#389C412yescomplemented
17C53×#387—22no—
18C53×#385—21no—
19C53×#381—21no—
20C53×#380C2211yescharacteristic
21C23×Dic53—22no—
22C22×#88322—22no—
23C2×C4×Dic53—28no—
24C2×C4×Dic53—24no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—24no—
28#13133124—22no—
29#13133124D414yes—
30C22×Q8×C53C2×C411yescharacteristic
31C23×C4×C53D411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53C2311yescharacteristic
34C22⋊Q8×C53—22no—
35C22⋊Q8×C53—22no—
36C2×C4⋊C4×C53—22no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C412yes—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C42×C53—22no—
43C2×C42×C53—22no—
44C2×C42×C53C2×C412yes—
45C2×C42×C53D412yes—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53—21no—
50C24×C53C2×D411yescharacteristic
51C2×Q8×C53—22no—
52C2×Q8×C53—42no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—24no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53#4112yes—
70C22×Dic53—28no—
71C22×Dic53—24no—
72C22×Dic53—44no—
73C22×Dic53—42no—
74C22×Dic53—42no—
75C4⋊C4×C53—22no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—44no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—41no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—42no—
82C42×C53—24no—
83C42×C53—24no—
84C4×Dic53—216no—
85C2×#88322—44no—
86C2×#88322—42no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C23×C53#13512yes—
93C23×C53#13712yes—
94Q8×C53—82no—
95C2×C4×C53—24no—
96C2×C4×C53—24no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53#10414yes—
116C2×C4×C53C4×D412yes—
117C2×C4×C53#12112yes—
118C2×Dic53—28no—
119C2×Dic53—44no—
120C2×Dic53—48no—
121C2×Dic53—48no—
122C2×Dic53—44no—
123#88322—82no—
124#9145—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53#38611yescharacteristic
132C22×C53#41012yes—
133C22×C53#41611yescharacteristic
134C22×C53#41911yescharacteristic
135C22×C53C2×#10412yes—
136C22×C53#45212yes—
137C22×C53#48512yes—
138C4×C53—24no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—42no—
142C4×C53—81no—
143C4×C53—81no—
144C4×C53—82no—
145Dic53—48no—
146#2855D5311yescharacteristic
147C2×#376—531no—
148#1817—531no—
149C2×#328—532no—
150#1294—532no—
151C2×C53—22no—
152C2×C53—22no—
153C2×C53#150012yes—
154C2×C53#150612yes—
155C2×C53#187311yescharacteristic
156C2×C53#189611yescharacteristic
157C2×C53#189911yescharacteristic
158C2×C22⋊Q8C2×D5311yescharacteristic
159C22×C42—1061no—
160C22×C42C2×D5311yescharacteristic
161C2×C22⋊C8—532no—
162#391Dic5312yescomplemented
163#389Dic5312yescomplemented
164#387—22no—
165#385—21no—
166#381—21no—
167#380C2×D5311yescharacteristic
168#376—1064no—
169C2×#95—532no—
170C2×#95—1061no—
171#336—1062no—
172#328—1064no—
173C53#914511yescharacteristic, complemented, Sylow
174C22×Q8C2×Dic5311yescharacteristic
175C23×C4—1062no—
176C23×C4C22⋊D5311yescharacteristic
177C23×C4C22⋊D5311yescharacteristic
178C23×C4C22×D5311yescharacteristic
179C22×C8—1062no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4C2×Dic5311yescharacteristic
185C2×C4⋊C4C2×Dic5312yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—1064no—
191C2×C42—1068no—
192C2×C42C2×Dic5312yes—
193C2×C42C22⋊D5312yes—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196#95—21no—
197#95—22no—
198#95—22no—
199#95—22no—
200#95—534no—
201#95—1064no—
202#95—1062no—
203C24C2×C22⋊D5311yescharacteristic
204C2×Q8—22no—
205C2×Q8—42no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—1068no—
218C22×C4—1064no—
219C22×C4—2124no—
220C22×C4—2122no—
221C22×C4—2122no—
222C22×C4C22⋊Dic5312yes—
223C22×C4C22⋊Dic5312yes—
224C22×C4D4×D5312yes—
225C22×C4#1171994712yes—
226C22×C4C22×Dic5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228C2×C8—2124no—
229C2×C8—2122no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C42—24no—
238C42—24no—
239C42—10616no—
240C23—22no—
241C23—22no—
242C23C2×C22⋊Dic5311yescharacteristic
243C23#1313318511yescharacteristic
244C23#1313319811yescharacteristic
245C23#1313323712yes—
246C23#1313324812yes—
247Q8—82no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—1068no—
269C2×C4—2128no—
270C2×C4—2128no—
271C2×C4—2124no—
272C2×C4—2124no—
273C2×C4#1313314414yes—
274C2×C4D4×Dic5312yes—
275C2×C4#1313318912yes—
276C8—4242no—
277C22—24no—
278C22—24no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22#1681050311yescharacteristic
284C22#1681059712yes—
285C22#1681061911yescharacteristic
286C22#1681063211yescharacteristic
287C22C2×#1313314412yes—
288C22#1681078312yes—
289C22#1681096112yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—82no—
295C4—81no—
296C4—81no—
297C4—2128no—
298C2—22no—
299C2—22no—
300C2#2027320112yes—
301C2#2027322712yes—
302C2#2027502811yescharacteristic
303C2#2027515911yescharacteristic
304C2#2027517411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic53
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2855C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2855D53
hypocenterC53#9145
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
15. C53×#391C4semidirect
16. C53×#389C4semidirect
162. #391Dic53semidirect
163. #389Dic53semidirect
173. C53#9145semidirect

Maximal subgroups

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

subgroupindex
C2×#168104692
#202747832
C2×#168103602
#202725932
C53×#28552
#914553