Finite Group Databasebeta

← #28303418

Subgroups — #28303418

Order 13568.

384 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 72 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#20289966C211yesmaximal, characteristic
2#20289966C211yesmaximal, characteristic
3#20289566C212yesmaximal, complemented
4#20289566C212yesmaximal, complemented
5#20277211C211yesmaximal, characteristic
6#16811643C2212yes—
7#16811643C2212yes—
8#16811468—21no—
9#16811468—21no—
10#16811466—21no—
11#16811466—21no—
12C2×#13133176C2211yescharacteristic
13#16810517—21no—
14#16810511—21no—
15#16810508—21no—
16#16810506—21no—
17C4×#11719919C2211yescharacteristic
18C4×#11719919C2211yescharacteristic
19C4×C22⋊Dic53C411yescharacteristic
20#16810469C411yescharacteristic
21C42×Dic53C411yescharacteristic
22C4×C22⋊C4×C53C411yescharacteristic
23#13133284—22no—
24#13133284—22no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—22no—
28#13133272—22no—
29#13133176—22no—
30#13133176—22no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53D411yescharacteristic
37C2×C22⋊Dic53D411yescharacteristic
38C2×C4×Dic53—21no—
39C2×C4×Dic53—21no—
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C2×C4×Dic53C2311yescharacteristic
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53#95×C53—21no—
54C24×C53C22⋊C411yescharacteristic
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×#88324—42no—
70C2×#88324—42no—
71C22⋊C4×C53—42no—
72C42×C53—41no—
73C42×C53—41no—
74C22⋊Dic53—22no—
75C22⋊Dic53—22no—
76C22⋊Dic53—42no—
77C4⋊Dic53—22no—
78C4⋊Dic53—22no—
79#11719919—42no—
80#11719919—42no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—42no—
84C4×Dic53—42no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C4×Dic53#4112yes—
88C4×Dic53#4112yes—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53C2×C22⋊C411yescharacteristic
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53#11711yescharacteristic
109C2×C4×C53#11711yescharacteristic
110C2×Dic53—22no—
111C2×Dic53—21no—
112C2×Dic53—21no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53#12312yes—
123C2×Dic53#12312yes—
124#88324—82no—
125#88324—82no—
126#8744—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—81no—
133C22×C53—81no—
134C22×C53#38111yescharacteristic
135C22×C53#41311yescharacteristic
136C22×C53#41311yescharacteristic
137C4×C53—21no—
138C4×C53—21no—
139C4×C53—42no—
140C4×C53—41no—
141C4×C53—81no—
142Dic53—22no—
143Dic53—22no—
144Dic53—42no—
145Dic53—41no—
146Dic53—41no—
147Dic53—81no—
148#2298—531no—
149#1464—531no—
150#1464—531no—
151#1285—532no—
152#1285—532no—
153C2×C53—21no—
154C2×C53—21no—
155C2×C53—81no—
156C2×C53#182211yescharacteristic
157C2×C53#182711yescharacteristic
158C2×C53#182711yescharacteristic
159C2×#117—531no—
160#432—532no—
161#432—532no—
162#388—1061no—
163#388—1061no—
164#387—1061no—
165#387—1061no—
166C4×C4⋊C4—531no—
167C4×C4⋊C4—531no—
168C4×C22⋊C4—531no—
169C4×C22⋊C4#615711yescharacteristic
170#376—531no—
171C43—531no—
172#324—1061no—
173#324—1061no—
174#323—1061no—
175#323—1061no—
176C53#874411yescharacteristic, complemented, Sylow
177C23×C4C2×#615711yescharacteristic
178#117—1062no—
179#117—1062no—
180C2×C4⋊C4—531no—
181C2×C4⋊C4—531no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—1061no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—531no—
187C2×C22⋊C4—1061no—
188C2×C42—531no—
189C2×C42—531no—
190C2×C42—531no—
191C2×C42—531no—
192C2×C42—1061no—
193C2×C42—1061no—
194C2×C42C2×#615711yescharacteristic
195C2×C42C2×#615711yescharacteristic
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198C22⋊C8—1062no—
199C22⋊C8—1062no—
200#97—1062no—
201#97—1062no—
202#95—21no—
203#95—1061no—
204#95—1061no—
205#95—1061no—
206#95—1061no—
207#95—1061no—
208C24C22⋊#615711yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—531no—
216C22×C4—531no—
217C22×C4—531no—
218C22×C4—531no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4C22⋊#615711yescharacteristic
222C22×C4C22×#615711yescharacteristic
223C2×C8—2122no—
224C2×C8—2122no—
225C4⋊C4—1062no—
226C4⋊C4—1062no—
227C4⋊C4—2122no—
228C4⋊C4—2122no—
229C22⋊C4—42no—
230C22⋊C4—1062no—
231C22⋊C4—1062no—
232C22⋊C4—2122no—
233C42—41no—
234C42—41no—
235C42—532no—
236C42—532no—
237C42—1061no—
238C42—1061no—
239C42—2121no—
240C42—2121no—
241C42—2122no—
242C42—2122no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—41no—
248C23C2×C22⋊#615711yescharacteristic
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—532no—
263C2×C4—532no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—1061no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4#1313329611yescharacteristic
277C2×C4#1313329611yescharacteristic
278C8—4242no—
279C8—4242no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22#1681156611yescharacteristic
288C22#1681161011yescharacteristic
289C22#1681161011yescharacteristic
290C4—21no—
291C4—21no—
292C4—42no—
293C4—41no—
294C4—81no—
295C4—1062no—
296C4—1062no—
297C4—4241no—
298C4—2122no—
299C4—2121no—
300C4—2121no—
301C2—21no—
302C2—21no—
303C2—81no—
304C2#2029061711yescharacteristic
305C2#2029062411yescharacteristic
306C2#2029062511yescharacteristic
subgroupG/H
centerC22#16811566
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC4×C22⋊C4×C53C4
socleC22×C53#381
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C22⋊C4#6157
hypocenterC53#8744
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289566C2semidirect
4. #20289566C2semidirect
176. C53#8744semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202895662
#202772112
#874453