Finite Group Databasebeta

← #27293579

Subgroups — #27293579

Order 13568.

424 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 75 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#20277423C211yesmaximal, characteristic
2#20273057C212yesmaximal, complemented
3#20272661C211yesmaximal, characteristic, complemented
4#20272646C212yesmaximal
5C53×#1325C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic
7C2×#13133182C2211yescharacteristic
8C2×#13133179C411yescharacteristic
9#16810677—22no—
10#16810528C411yescharacteristic
11#16810524C2211yescharacteristic
12#16810513—22no—
13#16810494—22no—
14#16810469C411yescharacteristic
15#16810349C2212yes—
16#16810348—22no—
17C53×#384C2211yescharacteristic
18C53×#324C2212yes—
19C22×Dic106C2×C411yescharacteristic
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133205—22no—
23#13133205—22no—
24#13133182—42no—
25#13133179—42no—
26C2×#11719919D412yes—
27C2×#11719921—22no—
28C2×#11719921—22no—
29C2×#11719921—21no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D412yes—
33#13133146—42no—
34C22⋊#88322—22no—
35#13133127—42no—
36#13133124—21no—
37#13133124—22no—
38#13133124—21no—
39C2×C4⋊C4×C53—22no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C42×C53C2×C411yescharacteristic
43C22⋊C8×C53—22no—
44C23×D53—21no—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22×C411yescharacteristic
48C22×C4×C53C2×D412yes—
49C22×Dic53—21no—
50C22×Dic53C22⋊C412yes—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C2×D212—21no—
54C2×D212—41no—
55C2×D212—41no—
56C2×D212M1612yes—
57C2×C4×D53—42no—
58C2×C4×D53—82no—
59C2×Dic106—21no—
60C2×Dic106—21no—
61C2×Dic106—41no—
62C2×Dic106—41no—
63C2×C8×C53—42no—
64C4⋊C4×C53—41no—
65C4⋊C4×C53—41no—
66C4⋊C4×C53—82no—
67C42×C53—21no—
68C42×C53M1612yes—
69#11719921—28no—
70#11719921—42no—
71#11719921—82no—
72#11719919—42no—
73C4×Dic53—22no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—41no—
77C2×#88322—42no—
78C23×C53C2×C22⋊C411yescharacteristic
79C22×D53—22no—
80C22×D53—42no—
81C22×D53—42no—
82C22×D53—41no—
83C22×D53—41no—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—41no—
89C2×C4×C53—41no—
90C2×C4×C53—42no—
91C2×C4×C53—42no—
92C2×C4×C53—41no—
93C2×C4×C53C2×M1611yescharacteristic
94C2×C4×C53C2×M1611yescharacteristic
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—21no—
98C2×Dic53—22no—
99C2×Dic53—41no—
100C2×Dic53—41no—
101C2×Dic53—41no—
102C2×Dic53—41no—
103C2×Dic53—41no—
104C2×Dic53#9912yes—
105D212—42no—
106D212—42no—
107C4×D53—84no—
108Dic106—42no—
109Dic106—42no—
110C8×C53—82no—
111#88322—82no—
112#11534—531nomaximal, Sylow
113C22×C53—21no—
114C22×C53—21no—
115C22×C53#40711yescharacteristic
116C22×C53C2×#9911yescharacteristic
117C22×C53#41311yescharacteristic
118C2×D53—44no—
119C2×D53—42no—
120C2×D53—82no—
121C2×D53—81no—
122C2×D53—81no—
123C4×C53—21no—
124C4×C53—21no—
125C4×C53—42no—
126C4×C53—42no—
127C4×C53—81no—
128Dic53—22no—
129Dic53—41no—
130Dic53—42no—
131Dic53—81no—
132#2326—531no—
133#1467—532no—
134#1325—531no—
135#1325D5311yescharacteristic, complemented
136#1318—532no—
137C2×C53—21no—
138C2×C53—21no—
139C2×C53#146511yescharacteristic
140C2×C53#212811yescharacteristic
141C2×C53#213311yescharacteristic
142D53—82no—
143C2×#124—531no—
144C2×C4×Q8—531no—
145C2×C4×D4—531no—
146#432—1062no—
147#391—531no—
148#390—531no—
149#388—1062no—
150#385—1062no—
151#384C2×D5311yescharacteristic
152#376—531no—
153#324—532no—
154#324—1062no—
155#324C2×D5312yes—
156C53#1153411yescharacteristic, complemented, Sylow
157C22×Q8—531no—
158C22×D4—531no—
159C23×C4—1061no—
160#124—1062no—
161#124—1062no—
162C4×Q8—2122no—
163C4×D4—2122no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—532no—
166C2×C4⋊C4C4×D5311yescharacteristic
167C2×C4⋊C4C22×D5311yescharacteristic
168C2×C22⋊C4—1062no—
169C2×C22⋊C4—1062no—
170C2×C22⋊C4—1061no—
171C2×C42—531no—
172C2×C42—532no—
173C2×C42—531no—
174C2×C42C4×D5311yescharacteristic
175C4⋊C8—2122no—
176C22⋊C8—22no—
177C22⋊C8—1062no—
178#97—2122no—
179#95—1061no—
180#95—1061no—
181#95—1062no—
182C24—1061no—
183C2×Q8—1061no—
184C2×Q8—1061no—
185C2×Q8—2121no—
186C2×Q8—2121no—
187C2×D4—532no—
188C2×D4—1061no—
189C2×D4—2121no—
190C2×D4—2121no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—531no—
194C22×C4—532no—
195C22×C4—531no—
196C22×C4—1061no—
197C22×C4—4242no—
198C22×C4—2122no—
199C22×C4C2×C4×D5311yescharacteristic
200C22×C4D4×D5312yes—
201C2×C8—42no—
202C2×C8—2122no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C4⋊C4—2122no—
207C22⋊C4—1068no—
208C22⋊C4—4242no—
209C22⋊C4—2122no—
210C42—21no—
211C42—1062no—
212C42—1062no—
213C42—1062no—
214C42—2121no—
215C42#1171991212yes—
216C23—1062no—
217C23—2122no—
218C23—2122no—
219C23—2121no—
220C23—2121no—
221C23C22⋊C4×D5311yescharacteristic
222Q8—2122no—
223Q8—2122no—
224D4—2122no—
225D4—2122no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—41no—
235C2×C4—532no—
236C2×C4—1062no—
237C2×C4—1061no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—4244no—
241C2×C4—2121no—
242C2×C4—2121no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4M16×D5311yescharacteristic
247C2×C4C2×#1171991211yescharacteristic
248C8—82no—
249C8—4242no—
250C22—21no—
251C22—21no—
252C22—4242no—
253C22—4241no—
254C22—4241no—
255C22—2124no—
256C22—2122no—
257C22#1681058511yescharacteristic
258C22#99×D5311yescharacteristic
259C22#1681060611yescharacteristic
260C4—21no—
261C4—21no—
262C4—42no—
263C4—42no—
264C4—81no—
265C4—1062no—
266C4—4241no—
267C4—2122no—
268C4—2121no—
269C2—21no—
270C2—21no—
271C2—4242no—
272C2#2027305211yescharacteristic
273C2#2027638511yescharacteristic
274C2#2027640311yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1325C2
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1325D53
hypocenterC53#11534
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1325

Complements

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

NH ≅ G/Ntype
2. #20273057C2semidirect
3. #20272661C2semidirect
5. C53×#1325C2semidirect
135. #1325D53semidirect
156. C53#11534semidirect

Maximal subgroups

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

subgroupindex
#202774232
#202730572
#202726612
#202726462
C53×#13252
#1153453