Finite Group Databasebeta

← #27293731

Subgroups — #27293731

Order 13568.

462 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 95 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#20273059C211yesmaximal, characteristic, complemented
2#20273039C211yesmaximal, characteristic, complemented
3#20272647C214yesmaximal, complemented
4C53×#2548C211yesmaximal, characteristic
5#16810680—21no—
6#16810676—21no—
7#16810349—21no—
8#16810349—21no—
9#16810349C2212yes—
10#16810349C2212yes—
11C2×#127×C53C411yescharacteristic
12C2×#124×C53C412yes—
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#394—21no—
15C53×#394—21no—
16C53×#390—21no—
17C53×#390—21no—
18C53×#390C2212yes—
19C4×C4⋊C4×C53C411yescharacteristic
20C4⋊#88322—41no—
21C4⋊#88322—41no—
22C22⋊#88322—22no—
23C22⋊#88322—22no—
24#13133128—41no—
25C4×#88322—41no—
26C22×Q8×C53C2×C411yescharacteristic
27C22×D4×C53—21no—
28C22×D4×C53—21no—
29C22×D4×C53C2×C412yes—
30#127×C53—41no—
31#127×C53—41no—
32#124×C53—44no—
33C4×Q8×C53—41no—
34C4×Q8×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53D411yescharacteristic
37C2×C4⋊C4×C53D411yescharacteristic
38C2×C4⋊C4×C53C2311yescharacteristic
39C2×C22⋊C4×C53—22no—
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—22no—
43C2×C42×C53C2×C412yes—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D411yescharacteristic
46C2×C42×C53D411yescharacteristic
47#95×C53—21no—
48C24×C53—22no—
49C2×Q8×C53—22no—
50C2×Q8×C53—22no—
51C2×Q8×C53—41no—
52C2×D4×C53—24no—
53C2×D4×C53—42no—
54C2×D4×C53—42no—
55C2×D4×C53—42no—
56C2×D4×C53—42no—
57C2×D4×C53—42no—
58C2×D4×C53—41no—
59C2×D4×C53—41no—
60C2×D4×C53—82no—
61C22×C4×C53—22no—
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C4⋊C4×C53—41no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C22⋊C4×C53—44no—
74C22⋊C4×C53—44no—
75C22⋊C4×C53—82no—
76C22⋊C4×C53—82no—
77C42×C53—21no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C42×C53—21no—
83C42×C53—42no—
84C2×#88322—42no—
85C2×#88322—42no—
86C23×C53—42no—
87C23×C53—42no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—42no—
91C23×C53—42no—
92C23×C53—42no—
93C23×C53C2×C22⋊C411yescharacteristic
94Q8×C53—44no—
95D4×C53—84no—
96D4×C53—82no—
97D4×C53—84no—
98D4×C53—82no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×C4×C53#10012yes—
119C2×C4×C53#13112yes—
120#88322—82no—
121#88322—82no—
122#11567—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—82no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53—82no—
129C22×C53—82no—
130C22×C53—82no—
131C22×C53—82no—
132C22×C53#40811yescharacteristic
133C22×C53C2×#9911yescharacteristic
134C22×C53C2×#10011yescharacteristic
135C4×C53—21no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—41no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—82no—
143C4×C53#35512yes—
144#2548D5311yescharacteristic
145#1467—531no—
146#1461—531no—
147#1318—534no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53—162no—
151C2×C53#147111yescharacteristic
152C2×C53#212811yescharacteristic
153C2×C53C2×#35511yescharacteristic
154C2×#127Dic5311yescharacteristic
155C2×#124Dic5312yes—
156C2×C4×Q8C2×D5311yescharacteristic
157#432—1061no—
158#431—1061no—
159#394—21no—
160#394—21no—
161#390—21no—
162#390—21no—
163#390C2×D5312yes—
164C4×C4⋊C4Dic5311yescharacteristic
165#324—532no—
166#324—532no—
167#324—1061no—
168#324—1061no—
169C53#1156711yescharacteristic, complemented, Sylow
170C22×Q8C2×Dic5311yescharacteristic
171C22×D4—21no—
172C22×D4—21no—
173C22×D4C2×Dic5312yes—
174#127—41no—
175#127—41no—
176#124—44no—
177C4×Q8—41no—
178C4×Q8—41no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4C22⋊D5311yescharacteristic
181C2×C4⋊C4C22⋊D5311yescharacteristic
182C2×C4⋊C4C22×D5311yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C42C2×Dic5311yescharacteristic
188C2×C42C2×Dic5312yes—
189C2×C42C22⋊D5311yescharacteristic
190C2×C42C22⋊D5311yescharacteristic
191C4⋊C8—2121no—
192C4⋊C8—2121no—
193C22⋊C8—1062no—
194C22⋊C8—1062no—
195#97—2121no—
196C4×C8—2121no—
197#95—21no—
198C24—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×D4—24no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—82no—
211C22×C4—22no—
212C22×C4C22⋊Dic5311yescharacteristic
213C22×C4C22⋊Dic5311yescharacteristic
214C22×C4C22⋊Dic5311yescharacteristic
215C22×C4C22⋊Dic5311yescharacteristic
216C22×C4C22×Dic5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218C22×C4C2×C22⋊D5311yescharacteristic
219C2×C8—2122no—
220C2×C8—2122no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—42no—
225C22⋊C4—44no—
226C22⋊C4—44no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23—42no—
242C23—42no—
243C23C2×C22⋊Dic5311yescharacteristic
244Q8—44no—
245D4—84no—
246D4—82no—
247D4—84no—
248D4—82no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#1313313211yescharacteristic
267C2×C4#1313313211yescharacteristic
268C2×C4#1313313412yes—
269C2×C4#1313322412yes—
270C8—4242no—
271C8—4242no—
272C22—21no—
273C22—21no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22—82no—
280C22—82no—
281C22#1681059011yescharacteristic
282C22C2×#1313313211yescharacteristic
283C22C2×#1313313411yescharacteristic
284C4—21no—
285C4—22no—
286C4—22no—
287C4—41no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—82no—
292C4#1681042012yes—
293C2—21no—
294C2—21no—
295C2—162no—
296C2#2027307511yescharacteristic
297C2#2027638711yescharacteristic
298C2C2×#1681042011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2548C2
socleC22×C53C2×#99
prime radicalC2×#127×C53C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2548D53
hypocenterC53#11567
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. #20273059C2semidirect
2. #20273039C2semidirect
3. #20272647C2semidirect
169. C53#11567semidirect

Maximal subgroups

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

subgroupindex
#202730592
#202730392
#202726472
C53×#25482
#1156753