Finite Group Databasebeta

← #28310629

Subgroups — #28310629

Order 13568.

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

normal subgroups only

This group has 532 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
1C2×#16811532C212yesmaximal
2#20290468C212yesmaximal, complemented
3#20289813C212yesmaximal, complemented
4#20277254C211yesmaximal, characteristic
5C2×#13133278C2214yes—
6C4×#11719940—22no—
7#16811532C418yescomplemented
8#16811503—22no—
9C2×#13133205C2211yescharacteristic
10C2×#13133182C2211yescharacteristic
11C2×#13133179C411yescharacteristic
12#16810528C2211yescharacteristic
13#16810524C411yescharacteristic
14#16810512—22no—
15#16810470—22no—
16C42×Dic53C411yescharacteristic
17C53×#382C411yescharacteristic
18C2×#11719940—24no—
19#13133278—24no—
20#13133278C2×C418yescomplemented
21#13133272—42no—
22C4×#88324—42no—
23C22×Dic106C2311yescharacteristic
24C22×D212C2×C411yescharacteristic
25C22×C4×D53—21no—
26#13133205—22no—
27#13133205C2×C414yes—
28#13133182—41no—
29#13133182—41no—
30#13133179—41no—
31#13133179—41no—
32C2×#11719919D412yes—
33C2×#11719921—22no—
34C2×#11719921—21no—
35C2×#11719921—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D412yes—
41#13133124—22no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45#95×C53—22no—
46C23×D53—21no—
47C22×C4×C53—21no—
48C22×C4×C53—21no—
49C22×C4×C53C22⋊C412yes—
50C22×C4×C53C22×C411yescharacteristic
51C22×Dic53—21no—
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C22⋊C411yescharacteristic
54C22×Dic53C2×D412yes—
55C2×D212—21no—
56C2×D212—41no—
57C2×D212—41no—
58C2×D212C4212yes—
59C2×C4×D53—44no—
60C2×C4×D53—41no—
61C2×C4×D53—41no—
62C2×Dic106—41no—
63C2×Dic106—41no—
64C2×Dic106C22×C412yes—
65C2×Dic106#4112yes—
66#11719940—28no—
67#11719940—44no—
68C2×#88324—44no—
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C42×C53—21no—
72C42×C53C4212yes—
73#11719921—28no—
74#11719921—44no—
75#11719921—42no—
76#11719919—42no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—22no—
80C4×Dic53—22no—
81C4×Dic53—42no—
82C4×Dic53—42no—
83C4×Dic53—41no—
84C23×C53C2×C22⋊C411yescharacteristic
85C22×D53—22no—
86C22×D53—42no—
87C22×D53—42no—
88C22×D53—41no—
89C22×D53—41no—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—41no—
99C2×C4×C53C2×C4211yescharacteristic
100C2×C4×C53#11711yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—21no—
106C2×Dic53—41no—
107C2×Dic53—41no—
108C2×Dic53—41no—
109C2×Dic53#9911yescharacteristic
110C2×Dic53#9911yescharacteristic
111C2×Dic53C4×D414yes—
112D212—42no—
113D212—42no—
114C4×D53—82no—
115C4×D53—82no—
116Dic106—42no—
117Dic106—42no—
118#88324—48no—
119C4×#355—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53C4×C22⋊C411yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41011yescharacteristic
125C2×D53—44no—
126C2×D53—42no—
127C2×D53—82no—
128C2×D53—81no—
129C2×D53—81no—
130C4×C53—21no—
131C4×C53—21no—
132C4×C53—42no—
133C4×C53—42no—
134C4×C53—81no—
135Dic53—21no—
136Dic53—24no—
137Dic53—42no—
138Dic53—41no—
139Dic53#35512yescomplemented
140C4×#124—531no—
141C2×#355—532no—
142C4×#101—532no—
143#1394—532no—
144C2×C53—21no—
145C2×C53—21no—
146C2×C53C4×#9911yescharacteristic
147C2×C53C2×#35511yescharacteristic
148C2×C53#213211yescharacteristic
149D53—82no—
150C2×#124—531no—
151C2×C4×Q8—531no—
152C2×C4×D4—531no—
153C2×#101—534no—
154C4×M16—1062no—
155#391—531no—
156#390—531no—
157#388—1062no—
158#382#615711yescharacteristic
159C4×C22⋊C4—1062no—
160C43—531no—
161#355—538no—
162#343—1062no—
163C53C4×#35511yescharacteristic, complemented, Sylow
164C22×Q8—531no—
165C22×D4—531no—
166C23×C4—1061no—
167C2×M16—1064no—
168#124—534no—
169#124—1062no—
170C4×Q8—2121no—
171C4×Q8—2121no—
172C4×D4—2121no—
173C4×D4—2121no—
174C2×C4⋊C4—532no—
175C2×C4⋊C4C2×#615711yescharacteristic
176C2×C4⋊C4C2×#615711yescharacteristic
177C2×C22⋊C4—1062no—
178C2×C22⋊C4—1061no—
179C2×C22⋊C4—1062no—
180C2×C42—532no—
181C2×C42—531no—
182C2×C42—531no—
183C2×C42—1061no—
184C2×C42—1061no—
185C2×C42C2×#615711yescharacteristic
186#101—538no—
187#101—1064no—
188#97—2122no—
189C4×C8—2122no—
190#95—22no—
191#95—1062no—
192C24—1061no—
193C2×Q8—532no—
194C2×Q8—532no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C2×D4—532no—
198C2×D4—1061no—
199C2×D4—2121no—
200C2×D4—2121no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—531no—
204C22×C4—532no—
205C22×C4—531no—
206C22×C4—1061no—
207C22×C4—2121no—
208C22×C4—2121no—
209C22×C4—2124no—
210C22×C4C22⋊#615712yes—
211C22×C4C22×#615711yescharacteristic
212M16—1068no—
213M16—2124no—
214C2×C8—2124no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—2122no—
218C22⋊C4—1068no—
219C22⋊C4—2122no—
220C22⋊C4—2124no—
221C42—21no—
222C42—1062no—
223C42—1062no—
224C42—1061no—
225C42—1061no—
226C42—2121no—
227C42—2122no—
228C42—2122no—
229C42C4×#615712yes—
230C23—1062no—
231C23—2122no—
232C23—2122no—
233C23—2121no—
234C23—2121no—
235C23C2×C22⋊#615711yescharacteristic
236Q8—2122no—
237Q8—2122no—
238D4—2122no—
239D4—2122no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—531no—
250C2×C4—534no—
251C2×C4—531no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1062no—
255C2×C4—1062no—
256C2×C4—1061no—
257C2×C4—4242no—
258C2×C4—4242no—
259C2×C4—2121no—
260C2×C4—2121no—
261C2×C4—2121no—
262C2×C4C2×C4×#615711yescharacteristic
263C2×C4#1313329611yescharacteristic
264C8—2128no—
265C22—21no—
266C22—21no—
267C22—4242no—
268C22—4241no—
269C22—4241no—
270C22—2124no—
271C22—2122no—
272C22C4×C22⋊#615711yescharacteristic
273C22D53⋊#9911yescharacteristic
274C22#1681160411yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—81no—
280C4—532no—
281C4—1064no—
282C4—1061no—
283C4—2122no—
284C4—2121no—
285C2—21no—
286C2—21no—
287C2—4242no—
288C2#2029046011yescharacteristic
289C2D53⋊#35511yescharacteristic
290C2#2029100911yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC22×Dic106C23
hypercenter#382#6157
hypocenterC53C4×#355
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #382

Complements

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

NH ≅ G/Ntype
2. #20290468C2semidirect
3. #20289813C2semidirect
7. #16811532C4semidirect
20. #13133278C2×C4semidirect
139. Dic53#355semidirect
163. C53C4×#355semidirect

Maximal subgroups

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

subgroupindex
C2×#168115322
#202904682
#202898132
#202772542
C4×#35553