Finite Group Databasebeta

← #28299851

Subgroups — #28299851

Order 13568.

562 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 116 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
1C4×#13133274C211yesmaximal, characteristic
2#20290379C211yesmaximal, characteristic
3#20289533C214yesmaximal, complemented
4#20283068C211yesmaximal, characteristic
5C22×#11719936—21no—
6C2×#13133274—21no—
7C2×C4×#88324—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×#13133217C411yescharacteristic
12C2×#13133197—21no—
13C2×C42×D53C2211yescharacteristic
14#16810528—21no—
15#16810522—21no—
16#16810522C2212yes—
17#16810480—21no—
18#16810480C412yes—
19C53×#384C411yescharacteristic
20C2×#11719936—24no—
21C2×#11719936—42no—
22C2×#11719936—42no—
23C22×#88324—22no—
24C22×#88324—22no—
25#13133274—24no—
26C4×#88324—42no—
27C22×Dic106—21no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53D411yescharacteristic
30C22×C4×D53D411yescharacteristic
31#13133217—42no—
32#13133197—42no—
33#13133197—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53—22no—
36C2×C4⋊Dic53C2×C412yes—
37C2×#11719921—22no—
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41C42×D53—42no—
42#13133124—22no—
43#13133124—22no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C42×C53C2×C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53C22⋊C411yescharacteristic
51C22×C4×C53C22⋊C411yescharacteristic
52C22×C4×C53C22×C411yescharacteristic
53C22×Dic53—22no—
54C22×Dic53C22⋊C411yescharacteristic
55C22×Dic53C2×D411yescharacteristic
56C22×Dic53C2×D411yescharacteristic
57C2×C4×D53—22no—
58C2×C4×D53—22no—
59C2×C4×D53—22no—
60C2×C4×D53—22no—
61C2×C4×D53—22no—
62C2×C4×D53—22no—
63C2×C4×D53—41no—
64C2×C4×D53—41no—
65C2×C4×D53D812yes—
66C2×C4×D53SD1612yes—
67C2×Dic106—42no—
68C2×Dic106—42no—
69C2×Dic106—42no—
70C2×#88324—24no—
71C2×#88324—24no—
72C2×#88324—42no—
73C2×#88324—42no—
74C2×#88324—42no—
75C2×#88324—42no—
76#11719936—48no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—42no—
80C42×C53—22no—
81#11719921—44no—
82C4⋊Dic53—42no—
83C4⋊Dic53—42no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4×Dic53—22no—
87C4×Dic53—41no—
88C4×Dic53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—21no—
91C22×D53—21no—
92C22×D53—21no—
93C22×D53—21no—
94C22×D53—21no—
95C22×D53—21no—
96C22×D53#10212yes—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—42no—
104C2×C4×C53#9912yes—
105C2×C4×C53#10212yes—
106C2×C4×C53#10412yes—
107C2×C4×C53#10412yes—
108C2×C4×C53#13112yes—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—42no—
118C2×Dic53#10112yes—
119C2×Dic53C2×D811yescharacteristic
120C2×Dic53C2×SD1611yescharacteristic
121C2×Dic53#13712yes—
122C4×D53—24no—
123C4×D53—24no—
124C4×D53—44no—
125C4×D53—44no—
126Dic106—84no—
127#88324—44no—
128#88324—44no—
129#7685—531nomaximal, Sylow
130C22×C53#40811yescharacteristic
131C22×C53C2×#9911yescharacteristic
132C22×C53C2×#10111yescharacteristic
133C22×C53C2×#10211yescharacteristic
134C22×C53#41911yescharacteristic
135C22×C53C2×#10411yescharacteristic
136C22×C53C2×#10411yescharacteristic
137C2×D53—22no—
138C2×D53—22no—
139C2×D53—22no—
140C2×D53—41no—
141C2×D53—41no—
142C2×D53—41no—
143C2×D53—41no—
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—82no—
147C4×C53#32714yes—
148Dic53—22no—
149Dic53—21no—
150Dic53—21no—
151Dic53—22no—
152Dic53—82no—
153#2857—531no—
154C4×C22⋊C8—531no—
155#1728—531no—
156#1271—534no—
157C2×C53#147211yescharacteristic
158C2×C53#148511yescharacteristic
159C2×C53#149211yescharacteristic
160C2×C53C2×#32711yescharacteristic
161C2×C53#150811yescharacteristic
162C2×C53#152611yescharacteristic
163C2×C53#154611yescharacteristic
164D53—42no—
165C22×M16—1061no—
166C2×C4⋊Q8—531no—
167C2×C22⋊Q8—1061no—
168C22×C42—531no—
169C2×C22⋊C8—1061no—
170C2×C4×C8—1061no—
171#391—1061no—
172#389—532no—
173#389—1061no—
174#384#615711yescharacteristic
175#380—532no—
176#380—1061no—
177#336—532no—
178#336—532no—
179#336—1061no—
180C53#768511yescharacteristic, complemented, Sylow
181C22×Q8—1061no—
182C23×C4—531no—
183C23×C4—531no—
184C23×C4—531no—
185C2×M16—1064no—
186C2×M16—2122no—
187C2×M16—2122no—
188C22×C8—1062no—
189C22×C8—1062no—
190C4⋊Q8—2122no—
191C22⋊Q8—2122no—
192C22⋊Q8—2122no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—532no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1061no—
198C2×C4⋊C4C2×#615712yes—
199C2×C22⋊C4—1062no—
200C2×C42—531no—
201C2×C42—531no—
202C2×C42—531no—
203C2×C42—2122no—
204C2×C42C2×#615711yescharacteristic
205C22⋊C8—1064no—
206C4×C8—2122no—
207#95—1062no—
208#95—1062no—
209C24—531no—
210C2×Q8—2122no—
211C2×Q8—2122no—
212C2×Q8—2122no—
213C22×C4—22no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—532no—
218C22×C4—532no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—1062no—
222C22×C4—1062no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—1062no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4C22⋊#615711yescharacteristic
229C22×C4C22⋊#615711yescharacteristic
230C22×C4C22×#615711yescharacteristic
231M16—2128no—
232C2×C8—1064no—
233C2×C8—1064no—
234C2×C8—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—24no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—2122no—
242C4⋊C4—2122no—
243C4⋊C4—2122no—
244C4⋊C4—2122no—
245C22⋊C4—2124no—
246C42—22no—
247C42—1062no—
248C42—2121no—
249C42—2121no—
250C23—532no—
251C23—1061no—
252C23—1061no—
253C23—1061no—
254C23—1061no—
255C23—1061no—
256C23—1061no—
257C23C2×C22⋊#615711yescharacteristic
258Q8—4244no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—532no—
267C2×C4—531no—
268C2×C4—531no—
269C2×C4—532no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—1061no—
273C2×C4—1061no—
274C2×C4—1061no—
275C2×C4—1061no—
276C2×C4—1064no—
277C2×C4—1064no—
278C2×C4—2124no—
279C2×C4—2124no—
280C2×C4—2122no—
281C2×C4—2122no—
282C2×C4—2122no—
283C2×C4#1313327612yes—
284C2×C4#1313327912yes—
285C2×C4#1313328112yes—
286C2×C4#1313328212yes—
287C2×C4#1313330412yes—
288C8—2124no—
289C8—2124no—
290C22—1062no—
291C22—1062no—
292C22—1062no—
293C22—2121no—
294C22—2121no—
295C22—2121no—
296C22—2121no—
297C22#1681159711yescharacteristic
298C22C2×#1313327611yescharacteristic
299C22D53⋊#10111yescharacteristic
300C22C2×#1313327911yescharacteristic
301C22#1681161611yescharacteristic
302C22C2×#1313328111yescharacteristic
303C22C2×#1313328211yescharacteristic
304C4—22no—
305C4—22no—
306C4—82no—
307C4—1061no—
308C4—1061no—
309C4—1062no—
310C4—1062no—
311C4—4242no—
312C4#1681147514yes—
313C2—2122no—
314C2#2028999311yescharacteristic
315C2#2029003811yescharacteristic
316C2#2029006111yescharacteristic
317C2C2×#1681147511yescharacteristic
318C2#2029011411yescharacteristic
319C2#2029017311yescharacteristic
320C2#2029022511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#384#6157
hypocenterC53#7685
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
3. #20289533C2semidirect
180. C53#7685semidirect

Maximal subgroups

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

subgroupindex
C4×#131332742
#202903792
#202895332
#202830682
#768553