Finite Group Databasebeta

← #27274465

Subgroups — #27274465

Order 13568.

914 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 155 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#407×D53C212yesmaximal
2#20277095C211yesmaximal, characteristic
3#20272624C212yesmaximal, complemented
4#20272623C211yesmaximal, characteristic, complemented
5C53×#1306C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×M16×D53—22no—
8C2×#13133173C412yes—
9#16810587C2212yes—
10#16810586—22no—
11#16810585—22no—
12#16810585C414yescomplemented
13C22⋊C8×D53—22no—
14C22⋊C8×D53C414yescomplemented
15#16810486C2211yescharacteristic
16#16810482—22no—
17#16810478C412yes—
18#16810470—22no—
19C53×#407C2212yes—
20C53×#381C2211yescharacteristic
21C24×D53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C22×C4×D53—44no—
24C22×C4×D53C2×C412yes—
25C22×C4×D53C2×C412yes—
26C22×C4×D53D412yes—
27C22×C4×D53D412yes—
28C22×C4×D53D412yes—
29C22×C4×D53Q812yes—
30C2×#11719917—22no—
31C2×#11719912—24no—
32M16×D53—48no—
33C2×C8×D53—24no—
34#13133173—44no—
35C2×#11719919—22no—
36C2×C22⋊Dic53C2×C412yes—
37C2×#11719921—24no—
38C2×C4×Dic53—22no—
39C22⋊#88322—22no—
40C22⋊#88322C2×C414yes—
41#13133129—24no—
42#13133129—24no—
43#13133124—22no—
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C22⋊C8×C53—22no—
48C22⋊C8×C53C2×C414yes—
49#95×C53—22no—
50C24×C53C22×C411yescharacteristic
51C23×D53—44no—
52C23×D53—42no—
53C23×D53C4211yescharacteristic
54C23×D53C4211yescharacteristic
55C23×D53C22⋊C412yes—
56C23×D53C4⋊C412yes—
57C22×C4×C53—22no—
58C22×C4×C53—42no—
59C22×C4×C53C22×C412yes—
60C22×C4×C53C2×D412yes—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×Q811yescharacteristic
63C22×Dic53—24no—
64C22×Dic53—22no—
65C22×Dic53C4212yes—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C4⋊C412yes—
69C2×C4×D53—28no—
70C2×C4×D53—44no—
71C2×C4×D53—44no—
72C2×C4×D53—44no—
73C2×C4×D53—44no—
74C2×C4×D53—44no—
75C2×C4×D53C22⋊C412yes—
76C2×C4×D53C22⋊C412yes—
77C2×C4×D53C22⋊C412yes—
78C2×C4×D53C22⋊C412yes—
79C2×C4×D53C4⋊C412yes—
80C2×C4×D53C4⋊C412yes—
81C2×C4×D53C4⋊C412yes—
82C2×C4×D53C4⋊C412yes—
83M16×C53—44no—
84C2×C8×C53—24no—
85C22⋊C4×C53—42no—
86C22⋊Dic53—42no—
87#11719921—48no—
88#11719919—44no—
89C4×Dic53—44no—
90#11719917—44no—
91C2×#88322—24no—
92#11719912—48no—
93C8×D53—48no—
94C23×C53—22no—
95C23×C53—42no—
96C23×C53C2×C4211yescharacteristic
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53C2×C4⋊C411yescharacteristic
99C22×D53—24no—
100C22×D53—24no—
101C22×D53—24no—
102C22×D53—24no—
103C22×D53—41no—
104C22×D53—41no—
105C22×D53—42no—
106C22×D53—42no—
107C22×D53—42no—
108C22×D53—42no—
109C22×D53—416no—
110C22×D53#9512yes—
111C22×D53#9512yes—
112C2×C4×C53—24no—
113C2×C4×C53—42no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—44no—
117C2×C4×C53—42no—
118C2×C4×C53C2×C22⋊C412yes—
119C2×C4×C53C2×C4⋊C412yes—
120C2×Dic53—22no—
121C2×Dic53—24no—
122C2×Dic53—24no—
123C2×Dic53—44no—
124C2×Dic53—42no—
125C2×Dic53#9512yes—
126C2×Dic53#9512yes—
127C2×Dic53#9914yes—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—24no—
131C4×D53—24no—
132C4×D53—42no—
133C4×D53—42no—
134C4×D53—42no—
135C4×D53—42no—
136C8×C53—44no—
137#88322—44no—
138#8115—531nomaximal, Sylow
139C22×C53—22no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—44no—
145C22×C53C2×#9511yescharacteristic
146C22×C53C2×#9912yes—
147C2×D53—22no—
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—44no—
153C2×D53—48no—
154C2×D53—48no—
155C2×D53—44no—
156C2×D53#34414yes—
157C4×C53—22no—
158C4×C53—22no—
159C4×C53—82no—
160Dic53—22no—
161Dic53—82no—
162Dic53#34214yes—
163C2×#407—532no—
164#2280—531no—
165#1306—532no—
166#1306—531no—
167#1306D5311yescharacteristic, complemented
168C2×C53—22no—
169C2×C53—41no—
170C2×C53—41no—
171C2×C53C2×#34211yescharacteristic
172C2×C53C2×#34412yes—
173D53—24no—
174D53—42no—
175C24×C4—531no—
176C22×M16—1062no—
177C2×#117—532no—
178#407—532no—
179#407—534no—
180#407—1062no—
181#407—1062no—
182#407C2×D5312yes—
183C2×C22⋊C8—534no—
184C2×C22⋊C8—1062no—
185#381—531no—
186#381C2×D5311yescharacteristic
187#380—1062no—
188#379—532no—
189C4×C22⋊C4—1062no—
190C53#811511yescharacteristic, complemented, Sylow
191C25—531no—
192C23×C4—531no—
193C23×C4—532no—
194C23×C4—532no—
195C23×C4—532no—
196C23×C4—532no—
197C23×C4—532no—
198C23×C4—532no—
199C23×C4—2124no—
200C23×C4C22×D5311yescharacteristic
201C2×M16—22no—
202C2×M16—1064no—
203C2×M16—1062no—
204C2×M16—2128no—
205C22×C8—1064no—
206#117—2124no—
207C2×C4⋊C4—1062no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—1064no—
210C2×C22⋊C4C4×D5312yes—
211C2×C42—1062no—
212C22⋊C8—22no—
213C22⋊C8—534no—
214C22⋊C8—1064no—
215C22⋊C8—1064no—
216C22⋊C8—1062no—
217C22⋊C8C4×D5314yes—
218#95—22no—
219#95—1062no—
220C24—531no—
221C24—532no—
222C24—532no—
223C24—531no—
224C24—2122no—
225C24—2124no—
226C24C2×C4×D5311yescharacteristic
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—531no—
230C22×C4—532no—
231C22×C4—531no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—532no—
235C22×C4—532no—
236C22×C4—532no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—1068no—
242C22×C4—1064no—
243C22×C4—1062no—
244C22×C4—2124no—
245C22×C4—2124no—
246C22×C4—2124no—
247C22×C4—2124no—
248C22×C4—2124no—
249C22×C4C2×C4×D5312yes—
250C22×C4D4×D5312yes—
251C22×C4D4×D5311yescharacteristic
252C22×C4Q8×D5311yescharacteristic
253M16—44no—
254M16—2128no—
255M16—2124no—
256C2×C8—24no—
257C2×C8—1064no—
258C2×C8—2128no—
259C4⋊C4—2124no—
260C22⋊C4—42no—
261C22⋊C4—2128no—
262C22⋊C4—2122no—
263C42—2124no—
264C23—22no—
265C23—42no—
266C23—532no—
267C23—532no—
268C23—1064no—
269C23—1064no—
270C23—1064no—
271C23—1064no—
272C23—2121no—
273C23—2121no—
274C23—2122no—
275C23—2122no—
276C23—2122no—
277C23—2122no—
278C23—21216no—
279C23C42×D5311yescharacteristic
280C23C22⋊C4×D5311yescharacteristic
281C23C4⋊C4×D5311yescharacteristic
282C2×C4—24no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—534no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—1064no—
292C2×C4—1064no—
293C2×C4—1064no—
294C2×C4—1064no—
295C2×C4—1064no—
296C2×C4—1064no—
297C2×C4—1062no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4C22⋊C4×D5312yes—
305C2×C4C4⋊C4×D5312yes—
306C8—44no—
307C8—2124no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—22no—
312C22—22no—
313C22—44no—
314C22—534no—
315C22—1062no—
316C22—1062no—
317C22—1062no—
318C22—1062no—
319C22—1062no—
320C22—2128no—
321C22—2124no—
322C22—2128no—
323C22—2124no—
324C22#95×D5311yescharacteristic
325C22#99×D5312yes—
326C4—22no—
327C4—22no—
328C4—82no—
329C4—534no—
330C4—1062no—
331C4—4242no—
332C2—22no—
333C2—41no—
334C2—41no—
335C2—1064no—
336C2—2122no—
337C2#342×D5311yescharacteristic
338C2#344×D5312yes—
subgroupG/H
centerC22#95×D53
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC53×#1306C2
socleC22×C53C2×#95
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1306D53
hypocenterC53#8115
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1306

Complements

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

NH ≅ G/Ntype
3. #20272624C2semidirect
4. #20272623C2semidirect
5. C53×#1306C2semidirect
12. #16810585C4semidirect
14. C22⋊C8×D53C4semidirect
167. #1306D53semidirect
190. C53#8115semidirect

Maximal subgroups

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

subgroupindex
#407×D532
#202770952
#202726242
#202726232
C53×#13062
#811553