Finite Group Databasebeta

← #28308315

Subgroups — #28308315

Order 13568.

380 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 68 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#20289946C211yesmaximal, characteristic
3#20289660C212yesmaximal
4#20289641C212yesmaximal
5#20279002C211yesmaximal, characteristic
6#16811643—21no—
7#16811639—21no—
8#16811468—21no—
9#16811468—21no—
10#16811466C2212yes—
11#16811466C2212yes—
12C2×#13133176C2211yescharacteristic
13#16810562—21no—
14#16810553—21no—
15#16810504—21no—
16#16810504—21no—
17#16810496C411yescharacteristic
18#16810496C2211yescharacteristic
19#16810486C411yescharacteristic
20#16810486C411yescharacteristic
21#16810486C2211yescharacteristic
22C22×C4⋊C4×C53C411yescharacteristic
23#13133284—41no—
24C4⋊#88324—41no—
25C22⋊#88324—22no—
26C22⋊#88324—22no—
27#13133272—41no—
28C4×#88324—41no—
29#13133176—41no—
30#13133176—41no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×C22⋊Dic53C2×C411yescharacteristic
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53C2×C411yescharacteristic
38C2×C22⋊Dic53C2311yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49C23×C4×C53C2×C411yescharacteristic
50C23×C4×C53C2×C411yescharacteristic
51C23×C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—41no—
55C2×C4⋊C4×C53—41no—
56C24×C53C22×C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—41no—
64C22×C4×C53—42no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53—41no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×Dic53—21no—
71C22×Dic53—21no—
72C22×Dic53C22⋊C411yescharacteristic
73C22×Dic53C22⋊C411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C2×#88324—42no—
77C2×#88324—42no—
78C4⋊C4×C53—42no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊Dic53—22no—
82C22⋊Dic53—41no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87#11719919—42no—
88#11719919—42no—
89C4×Dic53—21no—
90C4×Dic53—21no—
91C4×Dic53—21no—
92C4×Dic53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#13112yes—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—82no—
115C2×C4×C53#9911yescharacteristic
116C2×C4×C53#9911yescharacteristic
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131#88324—82no—
132#88324—82no—
133#11544—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—22no—
137C22×C53—42no—
138C22×C53—42no—
139C22×C53—41no—
140C22×C53—41no—
141C22×C53#40811yescharacteristic
142C22×C53C2×#9911yescharacteristic
143C22×C53#41311yescharacteristic
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—42no—
147C4×C53—41no—
148C4×C53—81no—
149Dic53—41no—
150Dic53—41no—
151Dic53—41no—
152Dic53—41no—
153Dic53—81no—
154Dic53—81no—
155#2484—531no—
156#1464—531no—
157#1458—531no—
158#1326—532no—
159#1317—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53#146811yescharacteristic
164C2×C53#212011yescharacteristic
165C2×C53#212311yescharacteristic
166C2×#117—531no—
167C22×C4⋊C4#615711yescharacteristic
168#432—1061no—
169#431—1061no—
170#400—1061no—
171#398—1061no—
172#387—1061no—
173#387—1061no—
174#385—531no—
175#385—531no—
176#381—531no—
177#381—531no—
178#381—531no—
179#324—1061no—
180#324—1061no—
181#323—532no—
182#323—532no—
183C53#1154411yescharacteristic, complemented, Sylow
184C23×C4C2×#615711yescharacteristic
185C23×C4C2×#615711yescharacteristic
186C23×C4C2×#615711yescharacteristic
187#117—2121no—
188#117—2121no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—531no—
194C2×C4⋊C4—531no—
195C2×C4⋊C4—1061no—
196C2×C4⋊C4—1061no—
197C2×C22⋊C4—531no—
198C2×C22⋊C4—531no—
199C2×C22⋊C4—531no—
200C2×C22⋊C4—531no—
201C2×C42—531no—
202C2×C42—531no—
203C4⋊C8—2121no—
204C4⋊C8—2121no—
205C22⋊C8—1062no—
206C22⋊C8—1062no—
207#97—2121no—
208C4×C8—2121no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214#95—1061no—
215#95—1061no—
216#95—1061no—
217C24C22×#615711yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—41no—
225C22×C4—42no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—531no—
230C22×C4—531no—
231C22×C4—531no—
232C22×C4—531no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4C22⋊#615711yescharacteristic
236C22×C4C22⋊#615711yescharacteristic
237C2×C8—2122no—
238C2×C8—2122no—
239C4⋊C4—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2121no—
243C4⋊C4—2121no—
244C4⋊C4—2122no—
245C4⋊C4—2122no—
246C22⋊C4—1062no—
247C22⋊C4—2121no—
248C22⋊C4—2121no—
249C22⋊C4—2121no—
250C42—1061no—
251C42—1061no—
252C42—1061no—
253C42—1061no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊#615711yescharacteristic
259C23#1313330412yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—82no—
276C2×C4—1061no—
277C2×C4—1061no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4—2121no—
288C2×C4—2121no—
289C2×C4—2121no—
290C2×C4#1313327511yescharacteristic
291C2×C4#1313327511yescharacteristic
292C8—4242no—
293C8—4242no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22#1681159611yescharacteristic
302C22C2×#1313327511yescharacteristic
303C22#1681161011yescharacteristic
304C4—21no—
305C4—21no—
306C4—42no—
307C4—41no—
308C4—81no—
309C4—4241no—
310C4—4241no—
311C4—2121no—
312C4—2121no—
313C4—2121no—
314C4—2121no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2028998011yescharacteristic
319C2#2029096611yescharacteristic
320C2#2029097711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
cosocleC2×C22⋊Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11544
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
183. C53#11544semidirect

Maximal subgroups

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

subgroupindex
#202899662
#202899462
#202896602
#202896412
#202790022
#1154453