Finite Group Databasebeta

← #28305696

Subgroups — #28305696

Order 13568.

582 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 140 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#20290647C211yesmaximal, characteristic, complemented
2#20290429C211yesmaximal, characteristic, complemented
3C2×#16811473C212yesmaximal, complemented
4C2×#16811471C212yesmaximal
5#20277254C211yesmaximal, characteristic
6C2×C4⋊#88324—22no—
7C2×C4⋊#88324C2212yes—
8C2×C4×#88324C2212yes—
9#16811489—22no—
10#16811473—24no—
11#16811471—24no—
12C2×#13133205C411yescharacteristic
13C2×#13133182C2211yescharacteristic
14C2×#13133179C2211yescharacteristic
15#16810528C411yescharacteristic
16#16810524C411yescharacteristic
17#16810512—22no—
18#16810470—22no—
19C42×Dic53C2211yescharacteristic
20C53×#382C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23C4⋊#88324—22no—
24C4⋊#88324—22no—
25C4⋊#88324—44no—
26C4×#88324—22no—
27C4×#88324D414yes—
28C22×Dic106C2×C411yescharacteristic
29C22×D212C2×C411yescharacteristic
30C22×C4×D53—21no—
31#13133205—22no—
32#13133205C814yescomplemented
33#13133182—22no—
34#13133182—22no—
35#13133179—22no—
36#13133179—22no—
37C2×#11719919—21no—
38C2×#11719921—22no—
39C2×#11719921—22no—
40C2×#11719921—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53D412yes—
44C2×C4×Dic53D412yes—
45C2×C4×Dic53C2311yescharacteristic
46#13133124—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53C2×C411yescharacteristic
50#95×C53—22no—
51C23×D53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53C22⋊C412yes—
58C22×Dic53C2×D411yescharacteristic
59C22×Dic53C2×D411yescharacteristic
60C2×D212—22no—
61C2×D212—22no—
62C2×D212—21no—
63C2×D212C2×C812yes—
64C2×C4×D53—41no—
65C2×C4×D53—41no—
66C2×C4×D53—42no—
67C2×C4×D53—42no—
68C2×Dic106—22no—
69C2×Dic106—22no—
70C2×Dic106—21no—
71C2×Dic106C2×C812yes—
72C2×#88324—24no—
73C2×#88324—44no—
74C2×#88324—42no—
75C2×#88324—44no—
76C4⋊C4×C53—22no—
77C4⋊C4×C53—22no—
78C42×C53C2×C812yes—
79C42×C53M1612yes—
80#11719921—28no—
81#11719921—44no—
82#11719921—42no—
83#11719919—42no—
84C4×Dic53—22no—
85C4×Dic53—22no—
86C4×Dic53—22no—
87C4×Dic53—42no—
88C4×Dic53—41no—
89C4×Dic53—41no—
90C4×Dic53C2×D412yes—
91C4×Dic53#4112yes—
92C23×C53C2×C22⋊C411yescharacteristic
93C22×D53—22no—
94C22×D53—41no—
95C22×D53—41no—
96C22×D53—42no—
97C22×D53—42no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53C22⋊C814yes—
105C2×C4×C53C4×D412yes—
106C2×C4×C53C22×C811yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53#13112yes—
109C2×Dic53—22no—
110C2×Dic53—22no—
111C2×Dic53—22no—
112C2×Dic53—22no—
113C2×Dic53—22no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53#12112yes—
117C2×Dic53#12411yescharacteristic
118C2×Dic53#12711yescharacteristic
119C2×Dic53AGL(1,ℤ/8)12yes—
120C2×Dic53#13712yes—
121D212—24no—
122D212—42no—
123C4×D53—82no—
124C4×D53—81no—
125C4×D53—81no—
126Dic106—24no—
127Dic106—42no—
128#88324—44no—
129#88324—82no—
130#9640—531nomaximal, Sylow
131C22×C53#39011yescharacteristic
132C22×C53C2×C22⋊C811yescharacteristic
133C22×C53#40811yescharacteristic
134C22×C53#41712yes—
135C22×C53#42112yes—
136C2×D53—42no—
137C2×D53—44no—
138C2×D53—81no—
139C2×D53—81no—
140C2×D53—82no—
141C4×C53—24no—
142C4×C53—81no—
143C4×C53—81no—
144C4×C53C8×D412yes—
145C4×C53#43612yes—
146Dic53—22no—
147Dic53—42no—
148Dic53—42no—
149Dic53#48912yes—
150Dic53#49611yescharacteristic
151Dic53#49711yescharacteristic
152C4×#124—531no—
153#1842—531no—
154#1744—531no—
155C2×Q8⋊C8—532no—
156C2×D4⋊C8—532no—
157C2×C53#147712yes—
158C2×C53#149512yes—
159C2×C53#195411yescharacteristic
160C2×C53#195911yescharacteristic
161C2×C53#196311yescharacteristic
162D53—82no—
163C2×#124—531no—
164C2×C4×Q8—531no—
165C2×C4×D4—531no—
166C2×C4⋊C8—532no—
167C2×C4⋊C8—1062no—
168C2×C4×C8—532no—
169#391—531no—
170#390—531no—
171#388—1062no—
172#382#615711yescharacteristic
173C4×C22⋊C4—1062no—
174C43—531no—
175#336—1062no—
176Q8⋊C8—1064no—
177D4⋊C8—1064no—
178C53#964011yescharacteristic, complemented, Sylow
179C22×Q8—531no—
180C22×D4—531no—
181C23×C4—1061no—
182C22×C8—1062no—
183C22×C8—1062no—
184#124—534no—
185#124—1062no—
186C4×Q8—1062no—
187C4×Q8—1062no—
188C4×D4—1062no—
189C4×D4—1062no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4C2×#615711yescharacteristic
192C2×C4⋊C4C2×#615711yescharacteristic
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1061no—
195C2×C22⋊C4—1062no—
196C2×C42—531no—
197C2×C42—532no—
198C2×C42—531no—
199C2×C42—532no—
200C2×C42—1061no—
201C2×C42C2×#615711yescharacteristic
202C4⋊C8—1062no—
203C4⋊C8—1062no—
204C4⋊C8—2124no—
205C4×C8—534no—
206C4×C8—1062no—
207#95—22no—
208#95—1062no—
209C24—1061no—
210C2×Q8—532no—
211C2×Q8—1061no—
212C2×Q8—1062no—
213C2×Q8—1062no—
214C2×D4—532no—
215C2×D4—1062no—
216C2×D4—1061no—
217C2×D4—1062no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—531no—
221C22×C4—531no—
222C22×C4—532no—
223C22×C4—1061no—
224C22×C4—2121no—
225C22×C4—2121no—
226C22×C4—2122no—
227C22×C4—2122no—
228C22×C4C22⋊#615712yes—
229C22×C4C22×#615711yescharacteristic
230C2×C8—1064no—
231C2×C8—2122no—
232C2×C8—2124no—
233C2×C8—2124no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—2122no—
237C22⋊C4—1068no—
238C22⋊C4—2122no—
239C22⋊C4—2124no—
240C42—532no—
241C42—532no—
242C42—1062no—
243C42—1062no—
244C42—1062no—
245C42—2122no—
246C42—2121no—
247C42—2121no—
248C42D53⋊C812yes—
249C42#1171993612yes—
250C23—1062no—
251C23—2121no—
252C23—2121no—
253C23—2122no—
254C23—2122no—
255C23C2×C22⋊#615711yescharacteristic
256Q8—1064no—
257Q8—2122no—
258D4—1064no—
259D4—2122no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—531no—
267C2×C4—531no—
268C2×C4—532no—
269C2×C4—532no—
270C2×C4—532no—
271C2×C4—1062no—
272C2×C4—1062no—
273C2×C4—1062no—
274C2×C4—1062no—
275C2×C4—1062no—
276C2×C4—4241no—
277C2×C4—4242no—
278C2×C4—4241no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4#1313327414yes—
282C2×C4D4×#615712yes—
283C2×C4C2×D53⋊C811yescharacteristic
284C2×C4C2×#1171993611yescharacteristic
285C2×C4#1313330512yes—
286C8—4242no—
287C8—2124no—
288C22—4242no—
289C22—4241no—
290C22—4241no—
291C22—2122no—
292C22—2124no—
293C22#1681157511yescharacteristic
294C22C2×#1313327411yescharacteristic
295C22#1681159811yescharacteristic
296C22#1681161412yes—
297C22#1681162012yes—
298C4—24no—
299C4—81no—
300C4—81no—
301C4—531no—
302C4—532no—
303C4—531no—
304C4—1062no—
305C4—2122no—
306C4—2122no—
307C4#1681165112yes—
308C4#1681165712yes—
309C2—4242no—
310C2#2029001212yes—
311C2#2029007312yes—
312C2#2029083711yescharacteristic
313C2#2029084511yescharacteristic
314C2#2029085211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#382C4
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C4×Dic53C23
hypercenter#382#6157
hypocenterC53#9640
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #382

Complements

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

NH ≅ G/Ntype
1. #20290647C2semidirect
2. #20290429C2semidirect
3. C2×#16811473C2semidirect
32. #13133205C8semidirect
178. C53#9640semidirect

Maximal subgroups

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

subgroupindex
#202906472
#202904292
C2×#168114732
C2×#168114712
#202772542
#964053