Finite Group Databasebeta

← #27294647

Subgroups — #27294647

Order 13568.

482 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 87 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#20279023C211yesmaximal, characteristic
2#20273056C211yesmaximal, characteristic
3#20272714C214yesmaximal, complemented
4C4×#99×C53C211yesmaximal, characteristic
5C2×#13133198—21no—
6C22×C4⋊Dic53C411yescharacteristic, complemented
7#16810680—21no—
8#16810553C412yescomplemented
9#16810548C2212yes—
10#16810521—21no—
11#16810484—21no—
12#16810483—21no—
13#16810483C411yescharacteristic, complemented
14#16810347—21no—
15#16810347C2212yes—
16C2×C4×D4×C53C2211yescharacteristic
17C2×#99×C53—21no—
18C4×C22⋊C4×C53—21no—
19C53×#342C2212yes—
20C23×Dic53—21no—
21#13133198—44no—
22C2×C4⋊Dic53—21no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C4⋊#88322—41no—
31C22⋊#88322—22no—
32#13133128—41no—
33#13133124—22no—
34#13133124—22no—
35#13133124C2×C412yes—
36C22×D4×C53D411yescharacteristic
37C23×C4×C53C2×C411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C4×D4×C53—41no—
40C4×D4×C53—41no—
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C22⋊C4×C53—22no—
43C2×C22⋊C4×C53D411yescharacteristic
44C2×C22⋊C4×C53C2311yescharacteristic
45C2×C42×C53—41no—
46C2×C42×C53D411yescharacteristic
47#99×C53—24no—
48#95×C53—41no—
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—42no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53—42no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—24no—
73C22⋊C4×C53—41no—
74C22⋊C4×C53—41no—
75C22⋊C4×C53—42no—
76C42×C53—21no—
77C42×C53—21no—
78C42×C53—82no—
79C22⋊Dic53—42no—
80C22⋊Dic53—44no—
81C4⋊Dic53—42no—
82C4⋊Dic53—41no—
83C4⋊Dic53—41no—
84C4⋊Dic53—42no—
85#11719919—44no—
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—44no—
113C2×C4×C53—81no—
114C2×C4×C53—82no—
115C2×C4×C53—82no—
116C2×C4×C53#9911yescharacteristic
117C2×C4×C53#9911yescharacteristic
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—84no—
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—84no—
148C4×C53#35212yes—
149Dic53—82no—
150Dic53—82no—
151#2486—531no—
152C4×#99D5311yescharacteristic
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—1061no—
163C2×C4×D4C2×D5311yescharacteristic
164C22×C4⋊C4—531no—
165#432—1061no—
166C2×#99—21no—
167#398—532no—
168#397—532no—
169#389—1061no—
170#381—1061no—
171#380—531no—
172#380—1061no—
173C4×C22⋊C4—21no—
174#342C2×D5312yes—
175#323—532no—
176#323—1061no—
177C53#1177311yescharacteristic, complemented, Sylow
178C22×D4C22⋊D5311yescharacteristic
179C23×C4—1061no—
180C23×C4C4×D5311yescharacteristic
181C23×C4C22⋊D5311yescharacteristic
182#123—2124no—
183C4×D4—41no—
184C4×D4—41no—
185C2×C4⋊C4—532no—
186C2×C4⋊C4—1062no—
187C2×C4⋊C4—1061no—
188C2×C4⋊C4—1061no—
189C2×C4⋊C4—2121no—
190C2×C4⋊C4—2121no—
191C2×C4⋊C4C4×D5311yescharacteristic
192C2×C22⋊C4—22no—
193C2×C22⋊C4—1062no—
194C2×C22⋊C4—1061no—
195C2×C22⋊C4D21211yescharacteristic
196C2×C22⋊C4C22×D5311yescharacteristic
197C2×C42—41no—
198C2×C42D21211yescharacteristic
199C4⋊C8—2121no—
200#99—24no—
201C22⋊C8—1062no—
202#97—2121no—
203#95—41no—
204#95—532no—
205#95—1062no—
206#95—1062no—
207C24#1171992111yescharacteristic
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—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1062no—
220C22×C4—1062no—
221C22×C4—2122no—
222C22×C4—2122no—
223C22×C4—2122no—
224C22×C4#1171992111yescharacteristic
225C22×C4#1171992111yescharacteristic
226C22×C4#1171992111yescharacteristic
227C22×C4C2×C4×D5311yescharacteristic
228C22×C4C2×D21211yescharacteristic
229C2×C8—2122no—
230C4⋊C4—41no—
231C4⋊C4—2122no—
232C4⋊C4—2124no—
233C4⋊C4—2122no—
234C4⋊C4—2121no—
235C4⋊C4—2121no—
236C22⋊C4—24no—
237C22⋊C4—41no—
238C22⋊C4—41no—
239C22⋊C4—42no—
240C22⋊C4—2122no—
241C22⋊C4—2124no—
242C42—21no—
243C42—21no—
244C42—82no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1313314512yes—
254C23C2×#1171992111yescharacteristic
255C23#1313324212yes—
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—44no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—4244no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313313111yescharacteristic
282C2×C4#1313313111yescharacteristic
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×#1313313111yescharacteristic
297C22#1681062711yescharacteristic
298C22C2×#1313314511yescharacteristic
299C4—21no—
300C4—41no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—84no—
305C4—4242no—
306C4—4242no—
307C4#1681041212yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#2027323011yescharacteristic
313C2C2×#1681041211yescharacteristic
314C2#2027638911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC4×#99×C53C2
socleC22×C53C2×#99
prime radicalC22×D4×C53D4
cosocleC2×C22⋊C4×C53C23
hypercenterC4×#99D53
hypocenterC53#11773
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #20272714C2semidirect
6. C22×C4⋊Dic53C4semidirect
8. #16810553C4semidirect
13. #16810483C4semidirect
177. C53#11773semidirect

Maximal subgroups

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

subgroupindex
#202790232
#202730562
#202727142
C4×#99×C532
#1177353