Finite Group Databasebeta

← #28307990

Subgroups — #28307990

Order 13568.

434 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289946C211yesmaximal, characteristic
2#20289946C211yesmaximal, characteristic
3#20289660C214yesmaximal
4#20277339C211yesmaximal, characteristic
5#16811639—21no—
6#16811639—21no—
7#16811468—21no—
8#16811468—21no—
9#16811466C2212yes—
10#16811466C2212yes—
11C2×#13133176C411yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810519—21no—
14#16810519—21no—
15#16810496C412yes—
16#16810496C2212yes—
17#16810492—21no—
18#16810488—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C4⋊#88324—41no—
21C4⋊#88324—41no—
22C22⋊#88324—22no—
23C22⋊#88324—22no—
24C4×#88324—41no—
25C4×#88324—41no—
26#13133176—22no—
27#13133176—41no—
28#13133176—41no—
29#13133176C814yescomplemented
30C2×C4⋊Dic53—21no—
31C2×C4⋊Dic53D411yescharacteristic
32C2×C4⋊Dic53D411yescharacteristic
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×C22⋊Dic53C2×C412yes—
36C2×C22⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53C2311yescharacteristic
38C2×C4×Dic53—21no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41#13133124—22no—
42#13133124—22no—
43C23×C4×C53C2×C411yescharacteristic
44C23×C4×C53C2×C412yes—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—41no—
48C2×C4⋊C4×C53—41no—
49C24×C53C22×C411yescharacteristic
50C22×C4×C53—21no—
51C22×C4×C53—21no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—42no—
55C22×C4×C53—82no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C22⋊C411yescharacteristic
58C22×C4×C53C2×C812yes—
59C22×C4×C53M1612yes—
60C22×Dic53—22no—
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C22⋊C411yescharacteristic
63C22×Dic53C2×D411yescharacteristic
64C22×Dic53C2×D411yescharacteristic
65C2×#88324—42no—
66C2×#88324—42no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—41no—
69C4⋊C4×C53—41no—
70C22⋊Dic53—22no—
71C22⋊Dic53—41no—
72C22⋊Dic53—41no—
73C22⋊Dic53C2×C814yes—
74C4⋊Dic53—24no—
75C4⋊Dic53—41no—
76C4⋊Dic53—41no—
77#11719919—82no—
78#11719919—82no—
79C4×Dic53—24no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82C4×Dic53—21no—
83C4×Dic53—21no—
84C23×C53—21no—
85C23×C53—21no—
86C23×C53—41no—
87C23×C53—41no—
88C23×C53C2×C22⋊C411yescharacteristic
89C23×C53C22×C811yescharacteristic
90C23×C53C2×M1611yescharacteristic
91C2×C4×C53—21no—
92C2×C4×C53—21no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—84no—
101C2×C4×C53C22⋊C812yes—
102C2×C4×C53C22⋊C812yes—
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#10112yes—
106C2×Dic53—24no—
107C2×Dic53—21no—
108C2×Dic53—21no—
109C2×Dic53—21no—
110C2×Dic53—21no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—41no—
116C2×Dic53—41no—
117#88324—82no—
118#88324—82no—
119#11454—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—41no—
125C22×C53—41no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53C2×C22⋊C811yescharacteristic
130C22×C53C2×#9911yescharacteristic
131C22×C53C2×#10111yescharacteristic
132C4×C53—22no—
133C4×C53—82no—
134C4×C53#32412yes—
135C4×C53#32412yes—
136Dic53—44no—
137Dic53—41no—
138Dic53—41no—
139Dic53—41no—
140Dic53—41no—
141#2317—531no—
142#1458—531no—
143#1458—531no—
144#1326—534no—
145C2×C53—21no—
146C2×C53—21no—
147C2×C53—41no—
148C2×C53—41no—
149C2×C53C2×#32411yescharacteristic
150C2×C53#212211yescharacteristic
151C2×C53#212311yescharacteristic
152C2×#117—531no—
153C2×#117—531no—
154C22×C4⋊C4#615711yescharacteristic
155#431—1061no—
156#431—1061no—
157#389—1061no—
158#389—1061no—
159#385—532no—
160#385—532no—
161#384—1061no—
162#382—1061no—
163#324—1061no—
164#324—1061no—
165#323—532no—
166#323—532no—
167C53#1145411yescharacteristic, complemented, Sylow
168C23×C4C2×#615712yes—
169C23×C4C2×#615711yescharacteristic
170#117—534no—
171#117—1062no—
172#117—2121no—
173#117—2121no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—531no—
179C2×C4⋊C4—531no—
180C2×C4⋊C4—1061no—
181C2×C4⋊C4—1062no—
182C2×C4⋊C4—1062no—
183C2×C22⋊C4—532no—
184C2×C22⋊C4—531no—
185C2×C22⋊C4—531no—
186C2×C42—531no—
187C2×C42—531no—
188C2×C42—1061no—
189C4⋊C8—2121no—
190C4⋊C8—2121no—
191C22⋊C8—1062no—
192C22⋊C8—1062no—
193C4×C8—2121no—
194C4×C8—2121no—
195#95—1062no—
196#95—1062no—
197C24C22×#615711yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—531no—
205C22×C4—531no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—1062no—
209C22×C4D53⋊C812yes—
210C22×C4#1171993612yes—
211C22×C4C22⋊#615711yescharacteristic
212C22×C4C22⋊#615711yescharacteristic
213C2×C8—2122no—
214C2×C8—2122no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—1064no—
219C4⋊C4—4242no—
220C4⋊C4—4242no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C22⋊C4—534no—
224C22⋊C4—1062no—
225C22⋊C4—2121no—
226C22⋊C4—2121no—
227C42—1064no—
228C42—1061no—
229C42—1061no—
230C42—1061no—
231C42—1061no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×C22⋊#615711yescharacteristic
237C23C2×D53⋊C811yescharacteristic
238C23C2×#1171993611yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—84no—
249C2×C4—1064no—
250C2×C4—1061no—
251C2×C4—1061no—
252C2×C4—1061no—
253C2×C4—1061no—
254C2×C4—2122no—
255C2×C4—2122no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4—2121no—
259C2×C4—2121no—
260C2×C4#1313327412yes—
261C2×C4#1313327412yes—
262C2×C4#1313327511yescharacteristic
263C2×C4#1313327511yescharacteristic
264C2×C4#1313327812yes—
265C8—4242no—
266C8—4242no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×#1313327411yescharacteristic
277C22C2×#1313327511yescharacteristic
278C22C2×#1313327811yescharacteristic
279C4—22no—
280C4—82no—
281C4—2124no—
282C4—2121no—
283C4—2121no—
284C4—2121no—
285C4—2121no—
286C4#1681146912yes—
287C4#1681146912yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1681146911yescharacteristic
293C2#2029097411yescharacteristic
294C2#2029097711yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22×C4
cosocleC2×C22⋊Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#11454
LCSG ▷ C23×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
29. #13133176C8semidirect
167. C53#11454semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896602
#202773392
#1145453