Finite Group Databasebeta

← #27294648

Subgroups — #27294648

Order 13568.

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

normal subgroups only

This group has 482 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
1C4×#13133132C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic
3#20272715C214yesmaximal, complemented
4C53×#2486C211yesmaximal, characteristic
5#16810680—21no—
6C2×#13133132—21no—
7C4×C22⋊Dic53—21no—
8#16810390C2212yes—
9#16810347—21no—
10#16810347C2212yes—
11C2×#123×C53—21no—
12C2×C4×D4×C53C2211yescharacteristic
13C22×C4⋊C4×C53C411yescharacteristic, complemented
14C53×#398C412yescomplemented
15C53×#397C2212yes—
16C53×#389—21no—
17C53×#381—21no—
18C53×#380—21no—
19C53×#380C411yescharacteristic, complemented
20C2×C22⋊Dic53—22no—
21C2×C4×Dic53—41no—
22C4⋊#88322—41no—
23#13133132—24no—
24C22⋊#88322—22no—
25#13133128—41no—
26#13133124—41no—
27C22×D4×C53D411yescharacteristic
28C23×C4×C53—21no—
29C23×C4×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31#123×C53—44no—
32C4×D4×C53—41no—
33C4×D4×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—22no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53—41no—
39C2×C4⋊C4×C53C2×C412yes—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C2×C42×C53D411yescharacteristic
46#95×C53—22no—
47#95×C53—22no—
48#95×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C24×C53C2×D411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—42no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—42no—
73C4⋊C4×C53—44no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C22⋊C4×C53—41no—
79C22⋊C4×C53—42no—
80C22⋊C4×C53—44no—
81C42×C53—21no—
82C42×C53—21no—
83C22⋊Dic53—24no—
84C22⋊Dic53—42no—
85C4×Dic53—82no—
86C2×#88322—42no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—21no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53—41no—
95C23×C53#10412yes—
96C23×C53C2×C22⋊C411yescharacteristic
97C23×C53AGL(1,ℤ/8)12yes—
98D4×C53—44no—
99C2×C4×C53—22no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—42no—
113C2×C4×C53—42no—
114C2×C4×C53—42no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—81no—
119C2×C4×C53—84no—
120C2×C4×C53#9911yescharacteristic
121C2×C4×C53#9911yescharacteristic
122C2×Dic53—44no—
123C2×Dic53—82no—
124C2×Dic53—82no—
125#88322—82no—
126#11773—531nomaximal, Sylow
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—22no—
130C22×C53—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—81no—
139C22×C53C2×#9911yescharacteristic
140C22×C53#41811yescharacteristic
141C22×C53C2×#10411yescharacteristic
142C4×C53—21no—
143C4×C53—41no—
144C4×C53—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—82no—
148C4×C53—82no—
149C4×C53#35212yes—
150Dic53—84no—
151#2486D5311yescharacteristic
152C4×#99—531no—
153#1466—531no—
154#1344—534no—
155C2×C53—21no—
156C2×C53—21no—
157C2×C53—42no—
158C2×C53—42no—
159C2×C53#150711yescharacteristic
160C2×C53C2×#35211yescharacteristic
161C2×C53#212911yescharacteristic
162C2×#123—21no—
163C2×C4×D4C2×D5311yescharacteristic
164C22×C4⋊C4Dic5311yescharacteristic, complemented
165#432—1061no—
166C2×#99—1061no—
167#398Dic5312yescomplemented
168#397C2×D5312yes—
169#389—21no—
170#381—21no—
171#380—21no—
172#380Dic5311yescharacteristic, complemented
173C4×C22⋊C4—1061no—
174#342—532no—
175#323—532no—
176#323—1061no—
177C53#1177311yescharacteristic, complemented, Sylow
178C22×D4C22⋊D5311yescharacteristic
179C23×C4—21no—
180C23×C4C2×Dic5311yescharacteristic
181C23×C4C22⋊D5311yescharacteristic
182#123—44no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×Dic5312yes—
191C2×C4⋊C4C2×Dic5311yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—1062no—
195C2×C22⋊C4C22⋊D5311yescharacteristic
196C2×C22⋊C4C22×D5311yescharacteristic
197C2×C42—2121no—
198C2×C42C22⋊D5311yescharacteristic
199C4⋊C8—2121no—
200#99—1064no—
201C22⋊C8—1062no—
202#97—2121no—
203#95—22no—
204#95—22no—
205#95—2121no—
206#95C2×Dic5312yes—
207C24C22⋊Dic5311yescharacteristic
208C24C2×C22⋊D5311yescharacteristic
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2122no—
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22⋊Dic5311yescharacteristic
227C22×C4C22×Dic5311yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229C2×C8—2122no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—44no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—44no—
240C22⋊C4—1064no—
241C22⋊C4—2122no—
242C42—21no—
243C42—21no—
244C42—4242no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1313314412yes—
254C23C2×C22⋊Dic5311yescharacteristic
255C23#1313324312yes—
256D4—44no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—81no—
277C2×C4—84no—
278C2×C4—4242no—
279C2×C4—4242no—
280C2×C4—2124no—
281C2×C4#1313313211yescharacteristic
282C2×C4#1313313211yescharacteristic
283C8—4242no—
284C22—21no—
285C22—21no—
286C22—22no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22C2×#1313313211yescharacteristic
297C22#1681062811yescharacteristic
298C22C2×#1313314411yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—82no—
305C4—82no—
306C4—4244no—
307C4#1681041312yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027323111yescharacteristic
313C2C2×#1681041311yescharacteristic
314C2#2027639011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2486C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2486D53
hypocenterC53#11773
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
3. #20272715C2semidirect
13. C22×C4⋊C4×C53C4semidirect
14. C53×#398C4semidirect
19. C53×#380C4semidirect
164. C22×C4⋊C4Dic53semidirect
167. #398Dic53semidirect
172. #380Dic53semidirect
177. C53#11773semidirect

Maximal subgroups

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

subgroupindex
C4×#131331322
#202730562
#202727152
C53×#24862
#1177353