Finite Group Databasebeta

← #28311195

Subgroups — #28311195

Order 13568.

500 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×#16811532C212yesmaximal
2#20290581C212yesmaximal, complemented
3#20289812C212yesmaximal, complemented
4C4×#13133205C211yesmaximal, characteristic
5#16811626—22no—
6C2×#13133278C2214yes—
7#16811532—24no—
8#16811503—22no—
9C2×#13133205C2211yescharacteristic
10C2×C4×Dic106C2211yescharacteristic
11C2×C4×D212C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810525C411yescharacteristic
14#16810514—22no—
15#16810489C411yescharacteristic
16C4×#11719921—22no—
17C43×C53C411yescharacteristic
18C2×#11719940—24no—
19#13133284—42no—
20C4⋊#88324—42no—
21#13133278—24no—
22#13133278—24no—
23C22×Dic106C2311yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26#13133205—22no—
27#13133205D412yes—
28#13133205Q812yes—
29C4×Dic106—41no—
30C4×Dic106—41no—
31C4×D212—41no—
32C4×D212—41no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919D412yes—
36C2×#11719921—21no—
37C2×#11719921—22no—
38C2×#11719921—22no—
39C2×C4×Dic53D412yes—
40#13133124—22no—
41#13133124—22no—
42C2×C42×C53—21no—
43C2×C42×C53—21no—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53C2×C411yescharacteristic
47C23×D53—21no—
48C22×C4×C53—21no—
49C22×C4×C53—21no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—21no—
54C22×Dic53C22⋊C412yes—
55C22×Dic53C2×D412yes—
56C2×D212—21no—
57C2×D212—41no—
58C2×D212—41no—
59C2×D212C4⋊C412yes—
60C2×C4×D53—44no—
61C2×C4×D53—41no—
62C2×C4×D53—41no—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C2×Dic106C2×D411yescharacteristic
66C2×Dic106C2×Q811yescharacteristic
67C2×Dic106#4112yes—
68#11719940—44no—
69#11719940—44no—
70C2×#88324—44no—
71C42×C53—21no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—42no—
75C42×C53—42no—
76C42×C53—41no—
77C42×C53C4⋊C412yes—
78#11719921—24no—
79#11719921—24no—
80#11719921—44no—
81#11719921—42no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84#11719919—42no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C23×C53C2×C22⋊C411yescharacteristic
88C22×D53—22no—
89C22×D53—42no—
90C22×D53—42no—
91C22×D53—41no—
92C22×D53—41no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#9911yescharacteristic
106C2×C4×C53C2×C4⋊C411yescharacteristic
107C2×C4×C53#11711yescharacteristic
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53#12112yes—
115C2×Dic53C22⋊Q812yes—
116D212—42no—
117D212—42no—
118C4×D53—82no—
119C4×D53—82no—
120Dic106—41no—
121Dic106—41no—
122Dic106—42no—
123#88324—84no—
124#12801—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53#38011yescharacteristic
128C22×C53C2×#9911yescharacteristic
129C22×C53#41011yescharacteristic
130C2×D53—44no—
131C2×D53—42no—
132C2×D53—82no—
133C2×D53—81no—
134C2×D53—81no—
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—42no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53#35512yescomplemented
142Dic53—22no—
143Dic53—22no—
144Dic53—42no—
145Dic53—81no—
146C4×#124—531no—
147C2×#355—532no—
148#1801—532no—
149#1394—532no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53#179911yescharacteristic
153C2×C53C2×#35511yescharacteristic
154C2×C53#213211yescharacteristic
155D53—82no—
156C2×#124—531no—
157C2×C4×Q8—531no—
158C2×C4×D4—531no—
159#423—1062no—
160C2×#101—534no—
161#391—531no—
162#390—531no—
163#388—1062no—
164#382—531no—
165C4×C22⋊C4—1062no—
166C43#615711yescharacteristic
167#355—1064no—
168#343—1062no—
169C53#1280111yescharacteristic, complemented, Sylow
170C22×Q8—531no—
171C22×D4—531no—
172C23×C4—1061no—
173C2×M16—1064no—
174#124—532no—
175#124—532no—
176#124—1062no—
177C4×Q8—2121no—
178C4×Q8—2121no—
179C4×D4—2121no—
180C4×D4—2121no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—531no—
184C2×C22⋊C4—1062no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—532no—
190C2×C42C2×#615711yescharacteristic
191C2×C42C2×#615711yescharacteristic
192C2×C42C2×#615711yescharacteristic
193C4⋊C8—2122no—
194C4⋊C8—2122no—
195#101—1064no—
196#101—1064no—
197#95—1062no—
198#95—1062no—
199C24—1061no—
200C2×Q8—531no—
201C2×Q8—531no—
202C2×Q8—532no—
203C2×Q8—2121no—
204C2×Q8—2121no—
205C2×D4—532no—
206C2×D4—1061no—
207C2×D4—2121no—
208C2×D4—2121no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—532no—
212C22×C4—532no—
213C22×C4—1061no—
214C22×C4—2124no—
215C22×C4—2121no—
216C22×C4—2121no—
217C22×C4C22⋊#615711yescharacteristic
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22×#615711yescharacteristic
220M16—2124no—
221M16—2124no—
222C2×C8—2124no—
223C4⋊C4—2122no—
224C4⋊C4—2121no—
225C4⋊C4—2121no—
226C22⋊C4—1064no—
227C22⋊C4—1064no—
228C22⋊C4—2122no—
229C22⋊C4—2124no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—41no—
234C42—42no—
235C42—42no—
236C42—1062no—
237C42—1062no—
238C42C4⋊#615712yes—
239C23—1062no—
240C23—2122no—
241C23—2122no—
242C23—2121no—
243C23—2121no—
244C23C2×C22⋊#615711yescharacteristic
245Q8—2121no—
246Q8—2121no—
247Q8—2122no—
248D4—2122no—
249D4—2122no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—532no—
262C2×C4—532no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1062no—
266C2×C4—4242no—
267C2×C4—4242no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2122no—
271C2×C4#1313327611yescharacteristic
272C2×C4#1313327611yescharacteristic
273C2×C4C2×C4⋊#615711yescharacteristic
274C2×C4#1313329611yescharacteristic
275C8—4244no—
276C22—21no—
277C22—21no—
278C22—4242no—
279C22—4241no—
280C22—4241no—
281C22—2124no—
282C22—2122no—
283C22#1681156511yescharacteristic
284C22C2×#1313327611yescharacteristic
285C22#1681160211yescharacteristic
286C4—21no—
287C4—21no—
288C4—21no—
289C4—42no—
290C4—42no—
291C4—41no—
292C4—1062no—
293C4—1062no—
294C4—4241no—
295C4—2122no—
296C4#1681153212yescomplemented
297C2—21no—
298C2—21no—
299C2—4242no—
300C2#2029057611yescharacteristic
301C2C2×#1681153211yescharacteristic
302C2#2029100811yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC43×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC22×Dic106C23
hypercenterC43#6157
hypocenterC53#12801
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C43

Complements

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

NH ≅ G/Ntype
2. #20290581C2semidirect
3. #20289812C2semidirect
141. C4×C53#355semidirect
169. C53#12801semidirect
296. C4#16811532semidirect

Maximal subgroups

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

subgroupindex
C2×#168115322
#202905812
#202898122
C4×#131332052
#1280153