Finite Group Databasebeta

← #28308409

Subgroups — #28308409

Order 13568.

424 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 74 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#20289972C211yesmaximal, characteristic
2#20289952C211yesmaximal, characteristic
3#20289656C214yesmaximal, complemented
4#20279889C211yesmaximal, characteristic
5#16811594—21no—
6D53⋊(C22⋊C8)—21no—
7#16811468C2212yes—
8#16811468C2212yes—
9#16811467—21no—
10#16811467—21no—
11C2×C4×D212C2211yescharacteristic
12#16810534—21no—
13#16810534—21no—
14#16810524C412yes—
15#16810488C2212yes—
16#16810480—21no—
17#16810480—21no—
18C4×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C2×#11719936—41no—
21C2×D53⋊C8—41no—
22#13133274—41no—
23#13133274—41no—
24C22⋊#88324—22no—
25C22⋊#88324—22no—
26C22×D212C2×C411yescharacteristic
27C22×C4×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C4×D212—41no—
30C4×D212—41no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2311yescharacteristic
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921D411yescharacteristic
36C2×#11719921D411yescharacteristic
37C2×C4×Dic53C2×C412yes—
38#13133124—21no—
39#13133124—22no—
40#13133124—22no—
41C22×Q8×C53—21no—
42C4⋊Q8×C53—41no—
43C4⋊Q8×C53—41no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C42×C53C2×C411yescharacteristic
47C23×D53C22⋊C411yescharacteristic
48C23×D53C22⋊C411yescharacteristic
49C2×Q8×C53—42no—
50C2×Q8×C53—82no—
51C22×C4×C53—22no—
52C22×C4×C53C22⋊C411yescharacteristic
53C22×C4×C53C22⋊C411yescharacteristic
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—22no—
56C22×Dic53C2×D411yescharacteristic
57C22×Dic53C2×D411yescharacteristic
58C2×D212—22no—
59C2×D212—22no—
60C2×D212—41no—
61C2×C4×D53—21no—
62C2×C4×D53—21no—
63C2×C4×D53—21no—
64C2×C4×D53—21no—
65C2×C4×D53—42no—
66C2×C4×D53—42no—
67C2×#88324—42no—
68C2×#88324—42no—
69#11719936—82no—
70D53⋊C8—82no—
71C4⋊C4×C53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C42×C53—21no—
75C42×C53—21no—
76#11719921—41no—
77#11719921—41no—
78#11719921—82no—
79#11719921—82no—
80C4⋊Dic53—41no—
81C4⋊Dic53—42no—
82C4×Dic53—24no—
83C23×C53C2×C22⋊C411yescharacteristic
84C22×D53—21no—
85C22×D53—21no—
86C22×D53—21no—
87C22×D53—21no—
88C22×D53—21no—
89C22×D53—41no—
90C22×D53—41no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53#10112yes—
94Q8×C53—84no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—21no—
99C2×C4×C53—41no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—42no—
103C2×C4×C53#9911yescharacteristic
104C2×C4×C53#9911yescharacteristic
105C2×C4×C53#13112yes—
106C2×Dic53—21no—
107C2×Dic53—21no—
108C2×Dic53—41no—
109C2×Dic53—41no—
110C2×Dic53—42no—
111C2×Dic53—42no—
112C2×Dic53—42no—
113D212—44no—
114C4×D53—42no—
115C4×D53—42no—
116C4×D53—81no—
117C4×D53—81no—
118#88324—82no—
119#88324—82no—
120#11569—531nomaximal, Sylow
121C22×C53—21no—
122C22×C53—21no—
123C22×C53#40811yescharacteristic
124C22×C53C2×#9911yescharacteristic
125C22×C53C2×#10111yescharacteristic
126C2×D53—22no—
127C2×D53—41no—
128C2×D53—41no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—41no—
132C2×D53—41no—
133C2×D53—41no—
134C2×D53—81no—
135C2×D53—81no—
136C4×C53—21no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—82no—
140C4×C53#35412yes—
141Dic53—41no—
142Dic53—41no—
143Dic53—82no—
144#2563—531no—
145#1465—531no—
146#1459—531no—
147#1324—534no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53#147211yescharacteristic
151C2×C53C2×#35411yescharacteristic
152C2×C53#213411yescharacteristic
153D53—42no—
154D53—81no—
155C2×C4⋊Q8#615711yescharacteristic
156C2×C4×D4—531no—
157#407—1061no—
158C2×C22⋊C8—1061no—
159#393—1061no—
160#393—1061no—
161#390—532no—
162#382—532no—
163#380—1061no—
164#380—1061no—
165C4×C4⋊C4—531no—
166#324—532no—
167#324—532no—
168#323—1061no—
169#323—1061no—
170C53#1156911yescharacteristic, complemented, Sylow
171C22×Q8—21no—
172C22×D4—531no—
173C23×C4—531no—
174C23×C4—531no—
175C2×M16—2121no—
176C22×C8—2121no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4×D4—2121no—
180C4×D4—2121no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—1061no—
184C2×C4⋊C4C2×#615712yes—
185C2×C22⋊C4—531no—
186C2×C22⋊C4—531no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1062no—
189C2×C42—532no—
190C2×C42C2×#615711yescharacteristic
191C22⋊C8—1062no—
192C22⋊C8—1062no—
193C22⋊C8—2121no—
194C22⋊C8—2121no—
195#95—1062no—
196#95—1062no—
197#95—1061no—
198C24—531no—
199C24—531no—
200C2×Q8—42no—
201C2×Q8—82no—
202C2×D4—1062no—
203C2×D4—1062no—
204C2×D4—2121no—
205C22×C4—22no—
206C22×C4—531no—
207C22×C4—531no—
208C22×C4—1061no—
209C22×C4—1061no—
210C22×C4—1061no—
211C22×C4—1061no—
212C22×C4—1062no—
213C22×C4—2122no—
214C22×C4—2122no—
215C22×C4C22⋊#615711yescharacteristic
216C22×C4C22⋊#615711yescharacteristic
217C22×C4C22×#615711yescharacteristic
218M16—4242no—
219C2×C8—4242no—
220C2×C8—2122no—
221C2×C8—2122no—
222C4⋊C4—42no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—2121no—
226C4⋊C4—2122no—
227C22⋊C4—4242no—
228C22⋊C4—4242no—
229C22⋊C4—2121no—
230C22⋊C4—2121no—
231C42—21no—
232C42—21no—
233C42—1064no—
234C23—532no—
235C23—1061no—
236C23—1061no—
237C23—1061no—
238C23—1061no—
239C23—1061no—
240C23—2121no—
241C23—2121no—
242C23—2121no—
243C23—2121no—
244C23C2×C22⋊#615711yescharacteristic
245Q8—84no—
246D4—2124no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—1061no—
256C2×C4—1061no—
257C2×C4—4241no—
258C2×C4—4241no—
259C2×C4—2122no—
260C2×C4—2122no—
261C2×C4—2122no—
262C2×C4—2122no—
263C2×C4—2122no—
264C2×C4—2121no—
265C2×C4—2121no—
266C2×C4#1313327511yescharacteristic
267C2×C4#1313327511yescharacteristic
268C2×C4#1313330512yes—
269C8—4242no—
270C8—4242no—
271C22—21no—
272C22—21no—
273C22—1062no—
274C22—4241no—
275C22—4241no—
276C22—2122no—
277C22—2122no—
278C22—2121no—
279C22—2121no—
280C22—2121no—
281C22—2121no—
282C22—2121no—
283C22#1681159811yescharacteristic
284C22C2×#1313327511yescharacteristic
285C22D53⋊#10111yescharacteristic
286C4—21no—
287C4—22no—
288C4—41no—
289C4—82no—
290C4—4242no—
291C4—2121no—
292C4—2121no—
293C4#1681152912yes—
294C2—21no—
295C2—21no—
296C2—4241no—
297C2—2122no—
298C2#2028999511yescharacteristic
299C2C2×#1681152911yescharacteristic
300C2#2029101411yescharacteristic
subgroupG/H
centerC22C2×#13133275
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socleC22×C53C2×#99
prime radicalC22×D212C2×C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#11569
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #20289656C2semidirect
170. C53#11569semidirect

Maximal subgroups

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

subgroupindex
#202899722
#202899522
#202896562
#202798892
#1156953