Finite Group Databasebeta

← #28308262

Subgroups — #28308262

Order 13568.

444 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289978C212yesmaximal, complemented
2#20289656C212yesmaximal, complemented
3#20289644C212yesmaximal
4#20277258C211yesmaximal, characteristic
5#16811643—22no—
6#16811468—22no—
7#16811468C2214yes—
8C2×#13133206C411yescharacteristic
9C2×Q8×Dic53C2211yescharacteristic
10C2×#13133179C411yescharacteristic
11#16810525C2211yescharacteristic
12#16810515—22no—
13#16810488C2211yescharacteristic
14#16810470—22no—
15C42×Dic53C411yescharacteristic
16C53×#391C411yescharacteristic
17#13133284—42no—
18C22⋊#88324—24no—
19#13133272—42no—
20C22×D212C2×C411yescharacteristic
21C22×C4×D53—21no—
22#13133206—22no—
23#13133206—22no—
24Q8×Dic53—41no—
25Q8×Dic53—41no—
26#13133179—41no—
27#13133179—41no—
28C2×C4⋊Dic53D412yes—
29C2×C4⋊Dic53C2311yescharacteristic
30C2×#11719921—22no—
31C2×#11719921—21no—
32C2×#11719921—22no—
33C2×C4×Dic53—21no—
34C2×C4×Dic53—21no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37C2×C4×Dic53D412yes—
38#13133124—22no—
39C22×Q8×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C42×C53C2×C411yescharacteristic
42#95×C53—22no—
43C23×D53—21no—
44C2×Q8×C53—21no—
45C2×Q8×C53—21no—
46C2×Q8×C53—41no—
47C2×Q8×C53—41no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C412yes—
51C22×C4×C53C22×C411yescharacteristic
52C22×Dic53—21no—
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D412yes—
56C2×D212—21no—
57C2×D212—41no—
58C2×D212—41no—
59C2×D212M1612yes—
60C2×C4×D53—44no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×#88324—44no—
64C4⋊C4×C53—41no—
65C42×C53—41no—
66#11719921—28no—
67#11719921—42no—
68#11719921—44no—
69C4⋊Dic53—42no—
70C4⋊Dic53—41no—
71C4×Dic53—21no—
72C4×Dic53—21no—
73C4×Dic53—21no—
74C4×Dic53—22no—
75C4×Dic53—22no—
76C4×Dic53—42no—
77C4×Dic53—42no—
78C4×Dic53M1612yes—
79C23×C53C2×C22⋊C411yescharacteristic
80C22×D53—22no—
81C22×D53—42no—
82C22×D53—42no—
83C22×D53—41no—
84C22×D53—41no—
85Q8×C53—42no—
86Q8×C53—42no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—22no—
90C2×C4×C53—22no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53—42no—
96C2×C4×C53—41no—
97C2×C4×C53C2×M1611yescharacteristic
98C2×C4×C53C2×M1611yescharacteristic
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—21no—
103C2×Dic53—41no—
104C2×Dic53—41no—
105C2×Dic53—41no—
106C2×Dic53—42no—
107C2×Dic53#9911yescharacteristic
108C2×Dic53#9911yescharacteristic
109D212—42no—
110D212—42no—
111C4×D53—82no—
112C4×D53—82no—
113#88324—84no—
114#11530—531nomaximal, Sylow
115C22×C53—21no—
116C22×C53—21no—
117C22×C53#40711yescharacteristic
118C22×C53C2×#9911yescharacteristic
119C22×C53#41311yescharacteristic
120C2×D53—44no—
121C2×D53—42no—
122C2×D53—82no—
123C2×D53—81no—
124C2×D53—81no—
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—81no—
130Dic53—21no—
131Dic53—42no—
132Dic53—42no—
133Dic53—41no—
134Dic53#35512yes—
135C4×#124—531no—
136#1467—532no—
137#1324—532no—
138#1318—532no—
139C2×C53—21no—
140C2×C53—21no—
141C2×C53#146511yescharacteristic
142C2×C53#212611yescharacteristic
143C2×C53C2×#35511yescharacteristic
144D53—82no—
145C2×#124—531no—
146C2×C4×Q8—531no—
147C2×C4×D4—531no—
148#432—1062no—
149#391#615711yescharacteristic
150#390—531no—
151#388—1062no—
152#382—531no—
153C4×C22⋊C4—1062no—
154C43—531no—
155#324—534no—
156#324—1062no—
157C53#1153011yescharacteristic, complemented, Sylow
158C22×Q8C2×#615711yescharacteristic
159C22×D4—531no—
160C23×C4—1061no—
161#124—1062no—
162#124—1062no—
163C4×Q8—2121no—
164C4×Q8—2121no—
165C4×D4—2121no—
166C4×D4—2121no—
167C2×C4⋊C4—532no—
168C2×C4⋊C4—531no—
169C2×C4⋊C4C2×#615711yescharacteristic
170C2×C22⋊C4—1062no—
171C2×C22⋊C4—1061no—
172C2×C22⋊C4—1062no—
173C2×C42—531no—
174C2×C42—532no—
175C2×C42—531no—
176C2×C42—1061no—
177C2×C42—1061no—
178C2×C42C2×#615711yescharacteristic
179C4⋊C8—2122no—
180C22⋊C8—1064no—
181#97—2122no—
182#95—22no—
183#95—1062no—
184C24—1061no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—532no—
190C2×D4—1061no—
191C2×D4—2121no—
192C2×D4—2121no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—531no—
196C22×C4—531no—
197C22×C4—532no—
198C22×C4—1061no—
199C22×C4—2121no—
200C22×C4—2121no—
201C22×C4—2124no—
202C22×C4C22⋊#615712yes—
203C22×C4C22×#615711yescharacteristic
204C2×C8—2124no—
205C4⋊C4—41no—
206C4⋊C4—2121no—
207C4⋊C4—2122no—
208C22⋊C4—1068no—
209C22⋊C4—2124no—
210C22⋊C4—2122no—
211C42—41no—
212C42—532no—
213C42—1062no—
214C42—1062no—
215C42—1061no—
216C42—1061no—
217C42—1061no—
218C42—2122no—
219C42—2122no—
220C23—1062no—
221C23—2122no—
222C23—2122no—
223C23—2121no—
224C23—2121no—
225C23C2×C22⋊#615711yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—2122no—
229D4—2122no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—531no—
241C2×C4—531no—
242C2×C4—1062no—
243C2×C4—1062no—
244C2×C4—1061no—
245C2×C4—1062no—
246C2×C4—4242no—
247C2×C4—4242no—
248C2×C4—2121no—
249C2×C4—2121no—
250C2×C4—2121no—
251C2×C4—2122no—
252C2×C4D53⋊M1611yescharacteristic
253C2×C4D53⋊M1611yescharacteristic
254C8—4244no—
255C22—21no—
256C22—21no—
257C22—4242no—
258C22—4241no—
259C22—4241no—
260C22—2124no—
261C22—2122no—
262C22#1681159511yescharacteristic
263C22D53⋊#9911yescharacteristic
264C22#1681161011yescharacteristic
265C4—21no—
266C4—21no—
267C4—42no—
268C4—42no—
269C4—81no—
270C4—532no—
271C4—1061no—
272C4—2122no—
273C4—2122no—
274C4—2121no—
275C2—21no—
276C2—21no—
277C2—4242no—
278C2#2028997311yescharacteristic
279C2#2029098911yescharacteristic
280C2D53⋊#35511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenter#391#6157
hypocenterC53#11530
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #20289978C2semidirect
2. #20289656C2semidirect
157. C53#11530semidirect

Maximal subgroups

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

subgroupindex
#202899782
#202896562
#202896442
#202772582
#1153053