Finite Group Databasebeta

← #28299588

Subgroups — #28299588

Order 13568.

408 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 96 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#20290643C211yesmaximal, characteristic
2#20290432C211yesmaximal, characteristic
3#20289535C212yesmaximal
4#20289533C212yesmaximal
5#20277193C211yesmaximal, characteristic
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12#16810530C411yescharacteristic
13#16810530C2211yescharacteristic
14#16810522C411yescharacteristic
15#16810522C2211yescharacteristic
16#16810509—21no—
17#16810509—21no—
18C4×C4⋊Dic53—21no—
19C4×C4⋊Dic53C411yescharacteristic
20#16810469—21no—
21C42×Dic53C2211yescharacteristic
22C2×C4×Q8×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133272—24no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×C4⋊Dic53C2×C411yescharacteristic
30C2×C4×Dic53—21no—
31C2×C4×Dic53C2×C411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53D411yescharacteristic
36C2×C4×Dic53C2311yescharacteristic
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44C22×Q8×C53C2×C411yescharacteristic
45C4×Q8×C53—22no—
46C4×Q8×C53—22no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53C2×C411yescharacteristic
51C2×Q8×C53—21no—
52C2×Q8×C53—41no—
53C2×Q8×C53—41no—
54C2×Q8×C53M1612yes—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—21no—
61C22×Dic53—21no—
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C22⋊C411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C22×Dic53C2×D411yescharacteristic
66C2×#88324—24no—
67C2×#88324—42no—
68C2×#88324—42no—
69C2×#88324—42no—
70C2×#88324—42no—
71C2×#88324—42no—
72C4⋊C4×C53—24no—
73C4⋊C4×C53—21no—
74C4⋊C4×C53M1612yes—
75C42×C53—24no—
76C42×C53—21no—
77C42×C53—21no—
78C4⋊Dic53—42no—
79C4⋊Dic53—42no—
80C4⋊Dic53—41no—
81C4⋊Dic53—41no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—22no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53—41no—
88C4×Dic53—42no—
89C23×C53C2×C22⋊C411yescharacteristic
90Q8×C53—42no—
91Q8×C53—42no—
92C2×C4×C53—21no—
93C2×C4×C53—21no—
94C2×C4×C53—21no—
95C2×C4×C53—21no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53#10412yes—
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53#13512yes—
125C2×Dic53#13712yes—
126#88324—44no—
127#88324—82no—
128#7611—531nomaximal, Sylow
129C22×C53#40711yescharacteristic
130C22×C53#41011yescharacteristic
131C22×C53#41311yescharacteristic
132C22×C53#41611yescharacteristic
133C22×C53#41911yescharacteristic
134C22×C53C2×#10411yescharacteristic
135C22×C53#42111yescharacteristic
136C4×C53—21no—
137C4×C53—21no—
138C4×C53—22no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—42no—
142Dic53—22no—
143Dic53—22no—
144Dic53—41no—
145Dic53—41no—
146Dic53—81no—
147Dic53—81no—
148#2295—531no—
149#1839—531no—
150#1745—531no—
151#1272—532no—
152#1271—532no—
153C2×C53#146411yescharacteristic
154C2×C53#148811yescharacteristic
155C2×C53#149111yescharacteristic
156C2×C53#150011yescharacteristic
157C2×C53#151511yescharacteristic
158C2×C53#152911yescharacteristic
159C2×C53#155411yescharacteristic
160C2×C4×Q8#615711yescharacteristic
161C2×#97—1061no—
162#391—531no—
163#391—531no—
164#389—531no—
165#389—531no—
166#387—1061no—
167#387—1061no—
168C4×C4⋊C4—531no—
169C4×C4⋊C4—1061no—
170#376—1061no—
171C43—531no—
172#336—532no—
173#336—532no—
174#336—1061no—
175#336—1061no—
176#336—1061no—
177C53#761111yescharacteristic, complemented, Sylow
178C22×Q8C2×#615711yescharacteristic
179C22×C8—1062no—
180C22×C8—1062no—
181C4×Q8—22no—
182C4×Q8—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4C2×#615711yescharacteristic
189C2×C42—21no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—531no—
193C2×C42—531no—
194C2×C42—531no—
195C2×C42—531no—
196C2×C42—1061no—
197C2×C42C2×#615711yescharacteristic
198#97—1064no—
199#95—1061no—
200#95—1061no—
201#95—1061no—
202#95—1061no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8#1171994012yes—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—531no—
213C22×C4—531no—
214C22×C4—531no—
215C22×C4—531no—
216C22×C4—1061no—
217C22×C4—1061no—
218C22×C4C22⋊#615711yescharacteristic
219C22×C4C22⋊#615711yescharacteristic
220C22×C4C22×#615711yescharacteristic
221C2×C8—1064no—
222C2×C8—2122no—
223C2×C8—2122no—
224C2×C8—2122no—
225C2×C8—2122no—
226C2×C8—2122no—
227C4⋊C4—24no—
228C4⋊C4—21no—
229C4⋊C4—2121no—
230C4⋊C4—2121no—
231C4⋊C4—2122no—
232C4⋊C4—2122no—
233C4⋊C4#1171994012yes—
234C42—24no—
235C42—21no—
236C42—21no—
237C42—1062no—
238C42—1062no—
239C42—1062no—
240C42—2121no—
241C42—2121no—
242C42—2121no—
243C42—2122no—
244C23C2×C22⋊#615711yescharacteristic
245Q8—42no—
246Q8—42no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—532no—
261C2×C4—532no—
262C2×C4—1061no—
263C2×C4—1061no—
264C2×C4—1061no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—1061no—
268C2×C4—1061no—
269C2×C4—1061no—
270C2×C4—1061no—
271C2×C4—1061no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4#1313328212yes—
279C2×C4C2×#1171994011yescharacteristic
280C2×C4C2×#1171994011yescharacteristic
281C8—4242no—
282C8—2124no—
283C22#1681159311yescharacteristic
284C22#1681160211yescharacteristic
285C22#1681160911yescharacteristic
286C22#1681161311yescharacteristic
287C22#1681161611yescharacteristic
288C22C2×#1313328211yescharacteristic
289C22#1681162011yescharacteristic
290C4—22no—
291C4—21no—
292C4—21no—
293C4—42no—
294C4—41no—
295C4—42no—
296C4—1062no—
297C4—1062no—
298C4—4241no—
299C4—4241no—
300C4—2121no—
301C4—2121no—
302C2#2028996611yescharacteristic
303C2#2029004611yescharacteristic
304C2#2029005711yescharacteristic
305C2#2029008811yescharacteristic
306C2#2029013711yescharacteristic
307C2#2029018211yescharacteristic
308C2#2029024011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7611
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
177. C53#7611semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202904322
#202895352
#202895332
#202771932
#761153