Finite Group Databasebeta

← #27272060

Subgroups — #27272060

Order 13568.

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

normal subgroups only

This group has 528 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#20277258C211yesmaximal, characteristic
2#20274459C212yesmaximal, complemented
3#20272541C211yesmaximal, characteristic, complemented
4#20272535C212yesmaximal
5C53×#1272C211yesmaximal, characteristic, complemented
6C2×#13133206C411yescharacteristic
7C2×Q8×Dic53C411yescharacteristic
8C2×#13133179C411yescharacteristic
9C2×C8×Dic53—22no—
10#16810525C2211yescharacteristic
11#16810515—22no—
12#16810488C411yescharacteristic
13#16810470—22no—
14C42×Dic53C2211yescharacteristic
15#16810378C2212yes—
16#16810377—22no—
17C53×#391C2211yescharacteristic
18C53×#336C2212yes—
19C22×D212C2×C411yescharacteristic
20C22×C4×D53—21no—
21C22×#88322—22no—
22#13133206—22no—
23#13133206C814yescomplemented
24Q8×Dic53—42no—
25#13133179—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×#11719921—22no—
29C2×#11719921—22no—
30C2×#11719921—21no—
31C2×C4×Dic53—21no—
32C2×C4×Dic53C2×C411yescharacteristic
33C2×C4×Dic53C2×C411yescharacteristic
34C2×C4×Dic53D412yes—
35C2×C4×Dic53D412yes—
36C8×Dic53—28no—
37#13133124—22no—
38C22×Q8×C53C2×C411yescharacteristic
39C22×C8×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2311yescharacteristic
42#95×C53—22no—
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—41no—
46C2×Q8×C53—41no—
47C2×Q8×C53C2×C812yes—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22×C411yescharacteristic
51C22×C4×C53C2×D412yes—
52C22×Dic53—21no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C412yes—
55C22×Dic53C22⋊C411yescharacteristic
56C2×D212—21no—
57C2×D212—41no—
58C2×D212—41no—
59C2×D212C2×C812yes—
60C2×C4×D53—42no—
61C2×C4×D53—44no—
62C2×C8×C53—24no—
63C2×C8×C53—42no—
64C2×C8×C53—42no—
65C4⋊C4×C53—41no—
66C42×C53—41no—
67#11719921—28no—
68#11719921—42no—
69#11719921—42no—
70#11719921—42no—
71C4⋊Dic53—42no—
72C4⋊Dic53—41no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—24no—
77C4×Dic53—42no—
78C4×Dic53C2×C812yes—
79C4×Dic53M1612yes—
80C2×#88322—24no—
81C2×#88322—42no—
82C2×#88322—42no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—22no—
85C22×D53—41no—
86C22×D53—41no—
87C22×D53—42no—
88C22×D53—42no—
89Q8×C53—42no—
90Q8×C53—42no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—21no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—41no—
99C2×C4×C53—42no—
100C2×C4×C53C22×C811yescharacteristic
101C2×C4×C53C2×M1611yescharacteristic
102C2×C4×C53#13512yes—
103C2×C4×C53#13512yes—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—21no—
107C2×Dic53—22no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53C22⋊C814yes—
111C2×Dic53#10012yes—
112C2×Dic53#10414yes—
113D212—42no—
114D212—42no—
115C4×D53—84no—
116C8×C53—44no—
117#88322—44no—
118#7567—531nomaximal, Sylow
119C22×C53C2×C22⋊C811yescharacteristic
120C22×C53#41011yescharacteristic
121C22×C53C2×#10011yescharacteristic
122C22×C53#41612yes—
123C22×C53C2×#10412yes—
124C2×D53—42no—
125C2×D53—44no—
126C2×D53—81no—
127C2×D53—82no—
128C2×D53—81no—
129C4×C53—21no—
130C4×C53—21no—
131C4×C53—22no—
132C4×C53—22no—
133C4×C53—42no—
134C4×C53—81no—
135Dic53—24no—
136Dic53—41no—
137Dic53—81no—
138Dic53#32914yes—
139C4×#124—531no—
140#1744—532no—
141#1272—531no—
142#1272D5311yescharacteristic, complemented
143#1270—532no—
144C2×C53#146011yescharacteristic
145C2×C53#147612yes—
146C2×C53#150012yes—
147C2×C53C2×#32911yescharacteristic
148C2×C53#154311yescharacteristic
149D53—82no—
150C2×#124—531no—
151C2×C4×Q8—531no—
152C2×C4×D4—531no—
153C2×C4×C8—1062no—
154#391C2×D5311yescharacteristic
155#390—531no—
156#388—1062no—
157#382—531no—
158C4×C22⋊C4—1062no—
159C43—531no—
160#336—532no—
161#336—1062no—
162#336C2×D5312yes—
163C53#756711yescharacteristic, complemented, Sylow
164C22×Q8C4×D5311yescharacteristic
165C22×D4—531no—
166C23×C4—1061no—
167C22×C8—22no—
168C22×C8—1062no—
169#124—534no—
170#124—1062no—
171C4×Q8—2122no—
172C4×D4—2122no—
173C2×C4⋊C4—531no—
174C2×C4⋊C4—1061no—
175C2×C4⋊C4C4×D5311yescharacteristic
176C2×C22⋊C4—1062no—
177C2×C22⋊C4—1061no—
178C2×C22⋊C4—1062no—
179C2×C42—531no—
180C2×C42—531no—
181C2×C42—532no—
182C2×C42—532no—
183C2×C42—1061no—
184C2×C42C22×D5311yescharacteristic
185C4×C8—1068no—
186#95—22no—
187#95—1062no—
188C24—1061no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8C8×D5312yes—
193C2×D4—532no—
194C2×D4—1061no—
195C2×D4—2121no—
196C2×D4—2121no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—531no—
200C22×C4—532no—
201C22×C4—531no—
202C22×C4—1061no—
203C22×C4—2122no—
204C22×C4—2124no—
205C22×C4C2×C4×D5311yescharacteristic
206C22×C4D4×D5312yes—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—1064no—
211C2×C8—2122no—
212C2×C8—2122no—
213C4⋊C4—41no—
214C4⋊C4—2121no—
215C4⋊C4—2122no—
216C22⋊C4—1068no—
217C22⋊C4—2122no—
218C22⋊C4—2122no—
219C22⋊C4—2122no—
220C42—41no—
221C42—532no—
222C42—532no—
223C42—1064no—
224C42—1062no—
225C42—1062no—
226C42—1062no—
227C42—2122no—
228C23—1062no—
229C23—2121no—
230C23—2121no—
231C23—2122no—
232C23—2122no—
233C23C22⋊C4×D5311yescharacteristic
234Q8—42no—
235Q8—42no—
236D4—2122no—
237D4—2122no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—532no—
248C2×C4—534no—
249C2×C4—534no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1061no—
253C2×C4—1062no—
254C2×C4—4244no—
255C2×C4—2121no—
256C2×C4—2122no—
257C2×C4C2×C8×D5311yescharacteristic
258C2×C4M16×D5311yescharacteristic
259C2×C4#1313323812yes—
260C2×C4#1313324012yes—
261C8—44no—
262C8—2124no—
263C22—4242no—
264C22—4241no—
265C22—4241no—
266C22—2122no—
267C22—2124no—
268C22C22⋊C8×D5311yescharacteristic
269C22#1681059511yescharacteristic
270C22#100×D5311yescharacteristic
271C22#1681061812yes—
272C22#104×D5312yes—
273C4—22no—
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—81no—
279C4—534no—
280C4—1064no—
281C4—4241no—
282C4—2121no—
283C2—4242no—
284C2#2027303411yescharacteristic
285C2#2027309612yes—
286C2#2027320012yes—
287C2#329×D5311yescharacteristic
288C2#2027337611yescharacteristic
subgroupG/H
centerC23C22⋊C4×D53
commutatorC2×C4×C53C22×C8
FrattiniC2×C42C22×D53
FittingC53×#1272C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1272D53
hypocenterC53#7567
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #20274459C2semidirect
3. #20272541C2semidirect
5. C53×#1272C2semidirect
23. #13133206C8semidirect
142. #1272D53semidirect
163. C53#7567semidirect

Maximal subgroups

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

subgroupindex
#202772582
#202744592
#202725412
#202725352
C53×#12722
#756753