Finite Group Databasebeta

← #27280151

Subgroups — #27280151

Order 13568.

516 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 119 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#20277318C211yesmaximal, characteristic
2#20273287C212yesmaximal
3C2×#16810349C211yesmaximal, characteristic
4#20272582C212yesmaximal
5C53×#2627C211yesmaximal, characteristic
6C2×C22⋊#88322C2212yes—
7#16810509—24no—
8#16810486—21no—
9#16810483C2212yes—
10C4×C22⋊Dic53—21no—
11#16810360—24no—
12#16810349—22no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C4⋊C4×C53C2211yescharacteristic
15C53×#400C412yescomplemented
16C53×#398—22no—
17C53×#395C412yescomplemented
18C53×#385—21no—
19C53×#381—21no—
20C53×#380C2211yescharacteristic
21C22×#88322—22no—
22C2×C4⋊Dic53—22no—
23C2×C22⋊Dic53—22no—
24C2×C4×Dic53—22no—
25C22⋊#88322—22no—
26C22⋊#88322—22no—
27#13133124—22no—
28#13133124—24no—
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—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53C2×C412yes—
43C2×C4⋊C4×C53D412yes—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46#95×C53—22no—
47#95×C53—22no—
48#95×C53—22no—
49#95×C53C2×C412yes—
50C24×C53C2×D411yescharacteristic
51C2×Q8×C53—22no—
52C2×Q8×C53—81no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—22no—
57C22×C4×C53—22no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—81no—
63C22×C4×C53—81no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D412yes—
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70C22×Dic53—24no—
71C4⋊C4×C53—24no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—22no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—81no—
76C4⋊C4×C53—82no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—42no—
80C22⋊Dic53—42no—
81C22⋊Dic53—42no—
82C4⋊Dic53—82no—
83C4×Dic53—82no—
84C2×#88322—42no—
85C2×#88322—82no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53#12011yescharacteristic
90C23×C53#12311yescharacteristic
91C23×C53#13512yes—
92C23×C53#13712yes—
93Q8×C53—82no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—81no—
114C2×C4×C53C4×D412yes—
115C2×C4×C53#12112yes—
116C2×Dic53—44no—
117C2×Dic53—44no—
118C2×Dic53—44no—
119#88322—82no—
120#9202—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—42no—
128C22×C53—42no—
129C22×C53#38611yescharacteristic
130C22×C53#41611yescharacteristic
131C22×C53#41911yescharacteristic
132C22×C53#45212yes—
133C22×C53#48512yes—
134C4×C53—22no—
135C4×C53—22no—
136C4×C53—42no—
137C4×C53—42no—
138C4×C53—82no—
139C4×C53—81no—
140C4×C53—81no—
141Dic53—84no—
142#2627D5311yescharacteristic
143#2313—531no—
144#1520—532no—
145C2×#324—531no—
146#1289—532no—
147C2×C53—22no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—22no—
151C2×C53#187311yescharacteristic
152C2×C53#191211yescharacteristic
153C2×C53#191311yescharacteristic
154C2×C22⋊Q8C2×D5311yescharacteristic
155C22×C4⋊C4C2×D5311yescharacteristic
156C2×C22⋊C8—532no—
157#400Dic5312yescomplemented
158#398—22no—
159#395Dic5312yescomplemented
160#387—1064no—
161#385—21no—
162#381—21no—
163#381—1061no—
164#380—532no—
165#380C2×D5311yescharacteristic
166C4×C22⋊C4—1061no—
167#328—1064no—
168#324—1062no—
169C53#920211yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic5311yescharacteristic
171C23×C4C22⋊D5311yescharacteristic
172C23×C4C22⋊D5311yescharacteristic
173C23×C4C22×D5311yescharacteristic
174C22×C8—1062no—
175C22⋊Q8—22no—
176C22⋊Q8—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—1062no—
183C2×C4⋊C4C2×Dic5311yescharacteristic
184C2×C4⋊C4C2×Dic5312yes—
185C2×C4⋊C4C22⋊D5312yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—1062no—
189C2×C42—1062no—
190C22⋊C8—1062no—
191C22⋊C8—1062no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—534no—
196#95—1062no—
197#95—1064no—
198#95C2×Dic5312yes—
199C24C2×C22⋊D5311yescharacteristic
200C2×Q8—22no—
201C2×Q8—81no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4—1064no—
214C22×C4C22⋊Dic5312yes—
215C22×C4C22⋊Dic5312yes—
216C22×C4D4×D5312yes—
217C22×C4#1171994712yes—
218C22×C4C22×Dic5311yescharacteristic
219C22×C4C2×C22⋊D5311yescharacteristic
220C2×C8—4242no—
221C2×C8—2122no—
222C4⋊C4—24no—
223C4⋊C4—24no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4—82no—
228C4⋊C4—4242no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C22⋊C4—42no—
232C22⋊C4—2122no—
233C22⋊C4—2122no—
234C42—4242no—
235C23—41no—
236C23—41no—
237C23C2×C22⋊Dic5311yescharacteristic
238C23#1313318511yescharacteristic
239C23#1313319811yescharacteristic
240C23#1313323712yes—
241C23#1313324812yes—
242Q8—82no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—81no—
263C2×C4—2124no—
264C2×C4—2124no—
265C2×C4—2124no—
266C2×C4D4×Dic5312yes—
267C2×C4#1313318912yes—
268C8—4242no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—42no—
276C22—42no—
277C22#1681050311yescharacteristic
278C22#1681061911yescharacteristic
279C22#1681063211yescharacteristic
280C22#1681078312yes—
281C22#1681096112yes—
282C4—22no—
283C4—22no—
284C4—42no—
285C4—42no—
286C4—82no—
287C4—81no—
288C4—81no—
289C4—4244no—
290C2—21no—
291C2—22no—
292C2—21no—
293C2—22no—
294C2#2027502811yescharacteristic
295C2#2027525311yescharacteristic
296C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2627C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2627D53
hypocenterC53#9202
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
15. C53×#400C4semidirect
17. C53×#395C4semidirect
157. #400Dic53semidirect
159. #395Dic53semidirect
169. C53#9202semidirect

Maximal subgroups

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

subgroupindex
#202773182
#202732872
C2×#168103492
#202725822
C53×#26272
#920253