Finite Group Databasebeta

← #27274455

Subgroups — #27274455

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×D53D411yescharacteristic
28C22×C4×D53D411yescharacteristic
29C22×C4×D53D412yes—
30C22×C4×D53Q811yescharacteristic
31C22×C4×D53Q811yescharacteristic
32C2×#11719917—22no—
33C2×#11719912—24no—
34M16×D53—48no—
35C2×C8×D53—24no—
36#13133173—44no—
37C2×#11719919—22no—
38C2×C22⋊Dic53C2×C412yes—
39C2×#11719921—24no—
40C2×C4×Dic53—22no—
41C22⋊#88322—22no—
42C22⋊#88322C2×C414yes—
43#13133129—24no—
44#13133129—24no—
45#13133124—22no—
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C22⋊C8×C53—22no—
50C22⋊C8×C53C2×C414yes—
51#95×C53—22no—
52C24×C53C22×C411yescharacteristic
53C23×D53—44no—
54C23×D53—42no—
55C23×D53C4212yes—
56C23×D53C22⋊C412yes—
57C23×D53C4⋊C411yescharacteristic
58C23×D53C4⋊C411yescharacteristic
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53C22×C412yes—
62C22×C4×C53C2×D412yes—
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×Q811yescharacteristic
65C22×Dic53—24no—
66C22×Dic53—22no—
67C22×Dic53C4211yescharacteristic
68C22×Dic53C4211yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C4⋊C411yescharacteristic
72C22×Dic53C4⋊C411yescharacteristic
73C2×C4×D53—28no—
74C2×C4×D53—44no—
75C2×C4×D53—44no—
76C2×C4×D53—42no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53—44no—
81C2×C4×D53C22⋊C412yes—
82C2×C4×D53C22⋊C412yes—
83C2×C4×D53C22⋊C412yes—
84C2×C4×D53C22⋊C412yes—
85C2×C4×D53C4⋊C412yes—
86C2×C4×D53C4⋊C412yes—
87C2×C4×D53C4⋊C412yes—
88C2×C4×D53C4⋊C412yes—
89M16×C53—44no—
90C2×C8×C53—24no—
91C22⋊C4×C53—42no—
92C22⋊Dic53—42no—
93#11719921—48no—
94#11719919—44no—
95C4×Dic53—44no—
96#11719917—44no—
97C2×#88322—24no—
98#11719912—48no—
99C8×D53—48no—
100C23×C53—22no—
101C23×C53—42no—
102C23×C53C2×C4211yescharacteristic
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53C2×C4⋊C411yescharacteristic
105C22×D53—24no—
106C22×D53—24no—
107C22×D53—24no—
108C22×D53—24no—
109C22×D53—42no—
110C22×D53—41no—
111C22×D53—41no—
112C22×D53—42no—
113C22×D53—42no—
114C22×D53—42no—
115C22×D53—416no—
116C22×D53#9512yes—
117C22×D53#9512yes—
118C2×C4×C53—24no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—44no—
123C2×C4×C53—42no—
124C2×C4×C53C2×C22⋊C412yes—
125C2×C4×C53C2×C4⋊C412yes—
126C2×Dic53—22no—
127C2×Dic53—22no—
128C2×Dic53—22no—
129C2×Dic53—22no—
130C2×Dic53—22no—
131C2×Dic53—44no—
132C2×Dic53—42no—
133C2×Dic53#9511yescharacteristic
134C2×Dic53#9511yescharacteristic
135C2×Dic53#9511yescharacteristic
136C2×Dic53#9511yescharacteristic
137C2×Dic53#9914yes—
138C4×D53—24no—
139C4×D53—24no—
140C4×D53—24no—
141C4×D53—24no—
142C4×D53—42no—
143C4×D53—42no—
144C4×D53—42no—
145C4×D53—42no—
146C8×C53—44no—
147#88322—44no—
148#8112—531nomaximal, Sylow
149C22×C53—22no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—22no—
153C22×C53—22no—
154C22×C53—44no—
155C22×C53C2×#9511yescharacteristic
156C22×C53C2×#9912yes—
157C2×D53—22no—
158C2×D53—22no—
159C2×D53—22no—
160C2×D53—22no—
161C2×D53—22no—
162C2×D53—44no—
163C2×D53—48no—
164C2×D53—48no—
165C2×D53—44no—
166C2×D53#34414yes—
167C4×C53—22no—
168C4×C53—22no—
169C4×C53—82no—
170Dic53—21no—
171Dic53—21no—
172Dic53—82no—
173Dic53#34212yes—
174Dic53#34212yes—
175C2×#407—532no—
176#2280—531no—
177#1306—532no—
178#1306—531no—
179#1306D5311yescharacteristic, complemented
180C2×C53—22no—
181C2×C53—41no—
182C2×C53—41no—
183C2×C53C2×#34211yescharacteristic
184C2×C53C2×#34412yes—
185D53—24no—
186D53—42no—
187C24×C4—531no—
188C22×M16—1062no—
189C2×#117—532no—
190#407—532no—
191#407—534no—
192#407—1062no—
193#407—1062no—
194#407C2×D5312yes—
195C2×C22⋊C8—534no—
196C2×C22⋊C8—1062no—
197#381—531no—
198#381C2×D5311yescharacteristic
199#380—1062no—
200#379—532no—
201C4×C22⋊C4—1062no—
202C53#811211yescharacteristic, complemented, Sylow
203C25—531no—
204C23×C4—531no—
205C23×C4—532no—
206C23×C4—532no—
207C23×C4—532no—
208C23×C4—532no—
209C23×C4—531no—
210C23×C4—531no—
211C23×C4—531no—
212C23×C4—531no—
213C23×C4—2124no—
214C23×C4C22×D5311yescharacteristic
215C2×M16—22no—
216C2×M16—1064no—
217C2×M16—1062no—
218C2×M16—2128no—
219C22×C8—1064no—
220#117—2124no—
221C2×C4⋊C4—1062no—
222C2×C22⋊C4—532no—
223C2×C22⋊C4—1064no—
224C2×C22⋊C4C4×D5312yes—
225C2×C42—1062no—
226C22⋊C8—22no—
227C22⋊C8—534no—
228C22⋊C8—1064no—
229C22⋊C8—1064no—
230C22⋊C8—1062no—
231C22⋊C8C4×D5314yes—
232#95—22no—
233#95—1062no—
234C24—532no—
235C24—531no—
236C24—532no—
237C24—531no—
238C24—2122no—
239C24—2124no—
240C24C2×C4×D5311yescharacteristic
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—531no—
244C22×C4—531no—
245C22×C4—531no—
246C22×C4—531no—
247C22×C4—531no—
248C22×C4—531no—
249C22×C4—532no—
250C22×C4—532no—
251C22×C4—532no—
252C22×C4—532no—
253C22×C4—532no—
254C22×C4—532no—
255C22×C4—532no—
256C22×C4—532no—
257C22×C4—1068no—
258C22×C4—1064no—
259C22×C4—1062no—
260C22×C4—2124no—
261C22×C4—2124no—
262C22×C4—2124no—
263C22×C4—2122no—
264C22×C4—2122no—
265C22×C4—2122no—
266C22×C4—2122no—
267C22×C4C2×C4×D5312yes—
268C22×C4D4×D5312yes—
269C22×C4D4×D5311yescharacteristic
270C22×C4Q8×D5311yescharacteristic
271M16—44no—
272M16—2128no—
273M16—2124no—
274C2×C8—24no—
275C2×C8—1064no—
276C2×C8—2128no—
277C4⋊C4—2124no—
278C22⋊C4—42no—
279C22⋊C4—2128no—
280C22⋊C4—2122no—
281C42—2124no—
282C23—22no—
283C23—42no—
284C23—532no—
285C23—532no—
286C23—1064no—
287C23—1064no—
288C23—1064no—
289C23—1064no—
290C23—2122no—
291C23—2121no—
292C23—2121no—
293C23—2122no—
294C23—2122no—
295C23—2122no—
296C23—21216no—
297C23C42×D5311yescharacteristic
298C23C22⋊C4×D5311yescharacteristic
299C23C4⋊C4×D5311yescharacteristic
300C2×C4—24no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—534no—
307C2×C4—531no—
308C2×C4—531no—
309C2×C4—531no—
310C2×C4—531no—
311C2×C4—1064no—
312C2×C4—1064no—
313C2×C4—1064no—
314C2×C4—1064no—
315C2×C4—1062no—
316C2×C4—1062no—
317C2×C4—1062no—
318C2×C4—1062no—
319C2×C4—1062no—
320C2×C4—2122no—
321C2×C4—2122no—
322C2×C4—2122no—
323C2×C4—2122no—
324C2×C4—2124no—
325C2×C4—2122no—
326C2×C4C22⋊C4×D5312yes—
327C2×C4C4⋊C4×D5312yes—
328C8—44no—
329C8—2124no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—22no—
334C22—22no—
335C22—44no—
336C22—534no—
337C22—1062no—
338C22—1062no—
339C22—1062no—
340C22—1062no—
341C22—1062no—
342C22—2128no—
343C22—2124no—
344C22—2128no—
345C22—2124no—
346C22#95×D5311yescharacteristic
347C22#99×D5312yes—
348C4—22no—
349C4—22no—
350C4—82no—
351C4—532no—
352C4—532no—
353C4—1061no—
354C4—1061no—
355C4—4242no—
356C2—22no—
357C2—41no—
358C2—41no—
359C2—1064no—
360C2—2122no—
361C2#342×D5311yescharacteristic
362C2#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#8112
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
179. #1306D53semidirect
202. C53#8112semidirect

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
#811253