Finite Group Databasebeta

← #28303992

Subgroups — #28303992

Order 13568.

542 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 102 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#20291611C211yesmaximal, characteristic, complemented
2#20290014C212yesmaximal
3#20289958C211yesmaximal, characteristic, complemented
4#20289543C212yesmaximal, complemented
5#124×Dic53C211yesmaximal, characteristic
6#16811674—24no—
7#16811643—22no—
8#16811639C2212yes—
9#16811639C2212yes—
10D4⋊#88324—24no—
11#16811468—22no—
12C2×Q8×Dic53C2211yescharacteristic
13C2×D4×Dic53C2211yescharacteristic
14#16810530C411yescharacteristic
15#16810526C411yescharacteristic
16#16810516—22no—
17#16810489C411yescharacteristic
18C22⋊C4×Dic53—22no—
19C42×Dic53C2211yescharacteristic
20C2×#124×C53C411yescharacteristic
21#13133284—24no—
22C4⋊#88324—22no—
23C4⋊#88324D414yes—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26#13133272—24no—
27C4×#88324—22no—
28C4×#88324—22no—
29C23×Dic53—21no—
30Q8×Dic53—22no—
31Q8×Dic53—22no—
32D4×Dic53—22no—
33D4×Dic53—22no—
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53D412yes—
37C2×C22⋊Dic53—22no—
38C2×C22⋊Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53—21no—
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53D412yes—
43C2×C4×Dic53C2311yescharacteristic
44#13133124—22no—
45#13133124—22no—
46C22×Q8×C53C2×C411yescharacteristic
47C22×D4×C53C2×C411yescharacteristic
48#124×C53—22no—
49#124×C53—22no—
50C2×C22⋊C4×C53—22no—
51C2×C42×C53C2×C411yescharacteristic
52C24×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53—21no—
55C2×Q8×C53—21no—
56C2×Q8×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—22no—
60C2×D4×C53M1612yes—
61C22×C4×C53—21no—
62C22×C4×C53C22⋊C412yes—
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—44no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53C22⋊C412yes—
70C22×Dic53C2×D411yescharacteristic
71C22×Dic53C2×D411yescharacteristic
72C2×#88324—24no—
73C2×#88324—42no—
74C22⋊C4×C53—28no—
75C42×C53—21no—
76C42×C53M1612yes—
77C22⋊Dic53—44no—
78C22⋊Dic53—42no—
79C4⋊Dic53—22no—
80C4⋊Dic53—24no—
81C4⋊Dic53—22no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—42no—
86C4×Dic53—42no—
87C4×Dic53C2×D412yes—
88C4×Dic53#4112yes—
89C23×C53—22no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95Q8×C53—42no—
96Q8×C53—42no—
97D4×C53—24no—
98D4×C53—42no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53C4×D412yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53#13112yes—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—41no—
118C2×Dic53—82no—
119C2×Dic53—82no—
120C2×Dic53#12011yescharacteristic
121C2×Dic53#12112yes—
122C2×Dic53#12311yescharacteristic
123#88324—44no—
124#88324—82no—
125#8964—531nomaximal, Sylow
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—44no—
129C22×C53—42no—
130C22×C53—82no—
131C22×C53—81no—
132C22×C53—81no—
133C22×C53#38611yescharacteristic
134C22×C53#40711yescharacteristic
135C22×C53#40811yescharacteristic
136C4×C53—42no—
137C4×C53—42no—
138C4×C53#43512yes—
139C4×C53#43612yes—
140Dic53—22no—
141Dic53—21no—
142Dic53—42no—
143Dic53—42no—
144Dic53—81no—
145Dic53#45412yes—
146#2983—531no—
147C4×#124—531no—
148#1478—532no—
149#1461—531no—
150#1276—532no—
151C2×C53—21no—
152C2×C53—21no—
153C2×C53—82no—
154C2×C53#187011yescharacteristic
155C2×C53#188411yescharacteristic
156C2×C53#188811yescharacteristic
157C2×#124#615711yescharacteristic
158C2×C4×Q8—531no—
159C2×C4×D4—531no—
160#446—1064no—
161#432—1062no—
162#431—532no—
163#431—532no—
164#391—531no—
165#390—531no—
166#388—1062no—
167#382—531no—
168C4×C22⋊C4—1062no—
169C43—531no—
170D4⋊C8—1064no—
171#324—1062no—
172C53#896411yescharacteristic, complemented, Sylow
173C22×Q8C2×#615711yescharacteristic
174C22×D4C2×#615711yescharacteristic
175C23×C4—1061no—
176#124—22no—
177#124—22no—
178C4×Q8—1062no—
179C4×Q8—1062no—
180C4×D4—1062no—
181C4×D4—1062no—
182C2×C4⋊C4—532no—
183C2×C4⋊C4—531no—
184C2×C4⋊C4—531no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—1061no—
187C2×C22⋊C4—1062no—
188C2×C42—531no—
189C2×C42—532no—
190C2×C42—531no—
191C2×C42—1061no—
192C2×C42—1061no—
193C2×C42C2×#615711yescharacteristic
194C4⋊C8—534no—
195C4⋊C8—1062no—
196C4⋊C8—1064no—
197C22⋊C8—1062no—
198C22⋊C8—1062no—
199#97—1064no—
200C4×C8—1062no—
201C4×C8—1062no—
202#95—1062no—
203#95—1062no—
204C24—21no—
205C2×Q8—22no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×D4—22no—
210C2×D4—21no—
211C2×D4—22no—
212C2×D4#1171994012yes—
213C22×C4—21no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—532no—
217C22×C4—1061no—
218C22×C4—1061no—
219C22×C4—2121no—
220C22×C4—2121no—
221C22×C4—2124no—
222C22×C4C22⋊#615712yes—
223C22×C4C22×#615711yescharacteristic
224C2×C8—1064no—
225C2×C8—2122no—
226C4⋊C4—1064no—
227C4⋊C4—1062no—
228C4⋊C4—1062no—
229C22⋊C4—28no—
230C22⋊C4—2122no—
231C22⋊C4—2124no—
232C42—21no—
233C42—532no—
234C42—532no—
235C42—1062no—
236C42—1062no—
237C42—1062no—
238C42—2122no—
239C42—2122no—
240C42#1171994012yes—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23C2×C22⋊#615711yescharacteristic
247Q8—42no—
248Q8—42no—
249D4—24no—
250D4—42no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—531no—
258C2×C4—532no—
259C2×C4—531no—
260C2×C4—1062no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—4242no—
264C2×C4—4242no—
265C2×C4—2121no—
266C2×C4—2122no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2122no—
271C2×C4D4×#615712yes—
272C2×C4C2×#1171994011yescharacteristic
273C2×C4C2×#1171994011yescharacteristic
274C2×C4#1313330412yes—
275C8—4242no—
276C8—2124no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22—42no—
281C22—82no—
282C22—81no—
283C22—81no—
284C22#1681157111yescharacteristic
285C22#1681159311yescharacteristic
286C22#1681159611yescharacteristic
287C4—42no—
288C4—42no—
289C4—532no—
290C4—1062no—
291C4—1061no—
292C4—4241no—
293C4—2122no—
294C4—2122no—
295C4#1681165412yes—
296C4#1681165812yes—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2029069211yescharacteristic
301C2#2029071911yescharacteristic
302C2#2029072311yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#124×C53C4
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×#124#6157
hypocenterC53#8964
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #20291611C2semidirect
3. #20289958C2semidirect
4. #20289543C2semidirect
172. C53#8964semidirect

Maximal subgroups

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

subgroupindex
#202916112
#202900142
#202899582
#202895432
#124×Dic532
#896453