Finite Group Databasebeta

← #27279622

Subgroups — #27279622

Order 13568.

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

normal subgroups only

This group has 522 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#20277422C211yesmaximal, characteristic
2#20273107C212yesmaximal
3#20273039C211yesmaximal, characteristic, complemented
4#20272543C212yesmaximal, complemented
5C53×#2983C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×C4×D212C2211yescharacteristic
8#16810676C2212yes—
9#16810530C411yescharacteristic
10#16810524C411yescharacteristic
11#16810514—22no—
12#16810494—22no—
13#16810492C411yescharacteristic
14#16810469C2211yescharacteristic
15#16810352—24no—
16#16810349—22no—
17C2×C4×Q8×C53C2211yescharacteristic
18C53×#446—24no—
19C53×#432—22no—
20C53×#431C2212yes—
21C22×D212C2×C411yescharacteristic
22C22×C4×D53—21no—
23#13133206—22no—
24#13133206—22no—
25C4×D212—22no—
26C4×D212—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719919—22no—
29C2×#11719921—22no—
30C2×#11719921—22no—
31C2×#11719921—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C4⋊#88322—22no—
35C22⋊#88322—22no—
36C4×#88322—22no—
37#13133124—22no—
38#13133124—21no—
39#13133124—21no—
40C22×Q8×C53C2×C411yescharacteristic
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53D412yes—
45C2×C42×C53D412yes—
46C2×C42×C53C2311yescharacteristic
47C4⋊C8×C53—24no—
48C4⋊C8×C53—22no—
49C22⋊C8×C53—22no—
50#97×C53—24no—
51C4×C8×C53D414yes—
52C23×D53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C22×C4×C53C22⋊C412yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×C4×C53C2×D411yescharacteristic
61C22×C4×C53C2×D411yescharacteristic
62C22×Dic53—21no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C2×D212—22no—
66C2×D212—22no—
67C2×D212—21no—
68C2×D212M1612yes—
69C2×C4×D53—42no—
70C2×C4×D53—82no—
71C2×C8×C53—24no—
72C4⋊C4×C53—22no—
73C4⋊C4×C53—24no—
74C42×C53—22no—
75C42×C53—22no—
76C42×C53C2×D412yes—
77C42×C53#4112yes—
78#11719921—28no—
79#11719921—41no—
80#11719921—41no—
81#11719921—82no—
82C4⋊Dic53—22no—
83#11719919—82no—
84C4×Dic53—21no—
85C4×Dic53—22no—
86C4×Dic53M1612yes—
87C2×#88322—42no—
88C23×C53C2×C22⋊C411yescharacteristic
89C22×D53—22no—
90C22×D53—42no—
91C22×D53—42no—
92C22×D53—41no—
93C22×D53—41no—
94Q8×C53—42no—
95Q8×C53—42no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53C4×D412yes—
104C2×C4×C53#12011yescharacteristic
105C2×C4×C53#12112yes—
106C2×C4×C53#12311yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53C2×M1611yescharacteristic
109C2×C4×C53#13112yes—
110C2×Dic53—22no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—42no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118D212—24no—
119D212—42no—
120C4×D53—84no—
121C8×C53—44no—
122#88322—82no—
123#9106—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53#38611yescharacteristic
127C22×C53#40711yescharacteristic
128C22×C53#40811yescharacteristic
129C2×D53—44no—
130C2×D53—42no—
131C2×D53—82no—
132C2×D53—81no—
133C2×D53—81no—
134C4×C53—22no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53#43512yes—
140C4×C53#43612yes—
141Dic53—42no—
142Dic53—81no—
143Dic53—81no—
144#2983D5311yescharacteristic, complemented
145#2326—531no—
146#1478—532no—
147#1461—531no—
148#1274—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53#187011yescharacteristic
152C2×C53#191011yescharacteristic
153C2×C53#191211yescharacteristic
154D53—82no—
155C2×#124—531no—
156C2×C4×Q8C2×D5311yescharacteristic
157C2×C4×D4—531no—
158#446—24no—
159#432—22no—
160#431—532no—
161#431C2×D5312yes—
162#391—531no—
163#390—531no—
164#388—1062no—
165#385—1062no—
166#384—531no—
167#376—531no—
168D4⋊C8—1064no—
169#324—1062no—
170C53#910611yescharacteristic, complemented, Sylow
171C22×Q8C4×D5311yescharacteristic
172C22×D4—531no—
173C23×C4—1061no—
174#124—1062no—
175#124—1062no—
176C4×Q8—22no—
177C4×Q8—22no—
178C4×D4—1062no—
179C4×D4—1062no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—1062no—
182C2×C4⋊C4C4×D5311yescharacteristic
183C2×C4⋊C4D21212yes—
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—531no—
188C2×C42—531no—
189C2×C42C22⋊D5312yes—
190C2×C42C22×D5311yescharacteristic
191C4⋊C8—24no—
192C4⋊C8—22no—
193C4⋊C8—1062no—
194C22⋊C8—22no—
195C22⋊C8—1062no—
196#97—24no—
197C4×C8—1062no—
198C4×C8C22⋊D5314yes—
199#95—1061no—
200#95—1061no—
201#95—1062no—
202C24—1061no—
203C2×Q8—22no—
204C2×Q8—21no—
205C2×Q8—21no—
206C2×Q8—22no—
207C2×D4—532no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4—4242no—
215C22×C4—2122no—
216C22×C4#1171992112yes—
217C22×C4#1171992112yes—
218C22×C4C2×C4×D5311yescharacteristic
219C22×C4C2×D21211yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221C2×C8—24no—
222C2×C8—2122no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—1062no—
226C4⋊C4—4242no—
227C22⋊C4—1068no—
228C22⋊C4—4242no—
229C22⋊C4—2121no—
230C22⋊C4—2121no—
231C42—22no—
232C42—22no—
233C42—532no—
234C42—1061no—
235C42—1062no—
236C42#1171994512yes—
237C42C2×C22⋊D5312yes—
238C23—1062no—
239C23—2122no—
240C23—2122no—
241C23—2121no—
242C23—2121no—
243C23C2×#1171992111yescharacteristic
244Q8—42no—
245Q8—42no—
246D4—1064no—
247D4—2122no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—1062no—
256C2×C4—4244no—
257C2×C4—2121no—
258C2×C4—2122no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2121no—
264C2×C4C4×C22⋊D5312yes—
265C2×C4#1313318611yescharacteristic
266C2×C4#1313319112yes—
267C2×C4#1313320011yescharacteristic
268C2×C4M16×D5311yescharacteristic
269C2×C4M16×D5311yescharacteristic
270C2×C4#1313322512yes—
271C8—44no—
272C8—4242no—
273C22—21no—
274C22—21no—
275C22—4242no—
276C22—4241no—
277C22—4241no—
278C22—2124no—
279C22—2122no—
280C22#1681050211yescharacteristic
281C22#1681058611yescharacteristic
282C22#1681059111yescharacteristic
283C4—22no—
284C4—21no—
285C4—21no—
286C4—42no—
287C4—42no—
288C4—4241no—
289C4—4241no—
290C4—2122no—
291C4#1681069212yes—
292C4#1681069912yes—
293C2—21no—
294C2—21no—
295C2—4242no—
296C2#2027501211yescharacteristic
297C2#2027524211yescharacteristic
298C2#2027525211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9106
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
3. #20273039C2semidirect
4. #20272543C2semidirect
5. C53×#2983C2semidirect
144. #2983D53semidirect
170. C53#9106semidirect

Maximal subgroups

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

subgroupindex
#202774222
#202731072
#202730392
#202725432
C53×#29832
#910653