Finite Group Databasebeta

← #27274462

Subgroups — #27274462

Order 13568.

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

normal subgroups only

This group has 942 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#381×D53C211yesmaximal, characteristic
3#20272624C212yesmaximal, complemented
4#20272623C211yesmaximal, characteristic, complemented
5C53×#1306C211yesmaximal, characteristic, complemented
6C23×C4×D53C2211yescharacteristic
7C2×M16×D53—22no—
8C2×C22⋊C4×D53C412yes—
9#16810587C2212yes—
10#16810586—22no—
11#16810585—22no—
12#16810585C414yescomplemented
13C22⋊C8×D53—22no—
14C22⋊C8×D53C414yescomplemented
15#16810486C2211yescharacteristic
16#16810485—22no—
17#16810484C412yes—
18#95×D53—22no—
19C53×#407C2212yes—
20C53×#381C2211yescharacteristic
21C24×D53C2×C411yescharacteristic
22C23×Dic53C2×C411yescharacteristic
23C22×C4×D53—22no—
24C22×C4×D53—44no—
25C22×C4×D53C2×C412yes—
26C22×C4×D53C2×C412yes—
27C22×C4×D53D412yes—
28C22×C4×D53D412yes—
29C22×C4×D53D412yes—
30C22×C4×D53Q812yes—
31C2×#11719917—22no—
32C2×#11719912—24no—
33M16×D53—48no—
34C2×C8×D53—24no—
35C2×C22⋊Dic53C2×C412yes—
36C2×#11719921—22no—
37C22⋊C4×D53—44no—
38C22⋊#88322—22no—
39C22⋊#88322C2×C414yescomplemented
40#13133129—24no—
41#13133129—24no—
42#13133124—22no—
43#13133124—24no—
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C22⋊C8×C53—22no—
48C22⋊C8×C53C2×C414yescomplemented
49#95×C53—22no—
50C24×C53C22×C411yescharacteristic
51C23×D53—24no—
52C23×D53—44no—
53C23×D53C4212yes—
54C23×D53C22⋊C411yescharacteristic
55C23×D53C22⋊C411yescharacteristic
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—22no—
64C22×Dic53—42no—
65C22×Dic53C4211yescharacteristic
66C22×Dic53C4211yescharacteristic
67C22×Dic53C22⋊C412yes—
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×D53—44no—
76C2×C4×D53—48no—
77C2×C4×D53C22⋊C412yes—
78C2×C4×D53C22⋊C412yes—
79C2×C4×D53C22⋊C412yes—
80C2×C4×D53C22⋊C412yes—
81C2×C4×D53C4⋊C412yes—
82C2×C4×D53C4⋊C412yes—
83C2×C4×D53C4⋊C412yes—
84C2×C4×D53C4⋊C412yes—
85M16×C53—44no—
86C2×C8×C53—24no—
87C22⋊C4×C53—42no—
88C22⋊Dic53—42no—
89#11719921—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—22no—
100C22×D53—24no—
101C22×D53—24no—
102C22×D53—24no—
103C22×D53—24no—
104C22×D53—24no—
105C22×D53—42no—
106C22×D53—42no—
107C22×D53—42no—
108C22×D53—416no—
109C22×D53#9512yes—
110C22×D53#9512yes—
111C22×D53#9914yes—
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—24no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—42no—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53#9512yes—
127C2×Dic53#9512yes—
128C4×D53—24no—
129C4×D53—24no—
130C4×D53—24no—
131C4×D53—24no—
132C4×D53—42no—
133C4×D53—42no—
134C4×D53—42no—
135C4×D53—42no—
136C4×D53—88no—
137C8×C53—44no—
138#88322—44no—
139#8114—531nomaximal, Sylow
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—44no—
146C22×C53C2×#9511yescharacteristic
147C22×C53C2×#9912yes—
148C2×D53—28no—
149C2×D53—22no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—22no—
153C2×D53—44no—
154C2×D53—48no—
155C2×D53—48no—
156C2×D53#34214yes—
157C2×D53#34414yes—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—82no—
161Dic53—22no—
162Dic53—22no—
163Dic53—82no—
164C2×#407—532no—
165C2×#381—531no—
166#1306—532no—
167#1306—531no—
168#1306D5311yescharacteristic, complemented
169C2×C53—22no—
170C2×C53—41no—
171C2×C53—41no—
172C2×C53C2×#34211yescharacteristic
173C2×C53C2×#34412yes—
174D53—24no—
175D53—24no—
176C24×C4—531no—
177C22×M16—1062no—
178C22×C22⋊C4—532no—
179#407—532no—
180#407—534no—
181#407—1062no—
182#407—1062no—
183#407C2×D5312yes—
184C2×C22⋊C8—534no—
185C2×C22⋊C8—1062no—
186#381—531no—
187#381—532no—
188#381—1062no—
189#381C2×D5311yescharacteristic
190C2×#95—1062no—
191C53#811411yescharacteristic, complemented, Sylow
192C25—531no—
193C23×C4—531no—
194C23×C4—532no—
195C23×C4—532no—
196C23×C4—532no—
197C23×C4—532no—
198C23×C4—532no—
199C23×C4—532no—
200C23×C4—1062no—
201C23×C4—2124no—
202C23×C4C22×D5311yescharacteristic
203C2×M16—22no—
204C2×M16—1064no—
205C2×M16—1062no—
206C2×M16—2128no—
207C22×C8—1064no—
208C2×C22⋊C4—532no—
209C2×C22⋊C4—1062no—
210C2×C22⋊C4—2124no—
211C2×C22⋊C4C4×D5312yes—
212C22⋊C8—22no—
213C22⋊C8—534no—
214C22⋊C8—1064no—
215C22⋊C8—1064no—
216C22⋊C8—1062no—
217C22⋊C8C4×D5314yescomplemented
218#95—22no—
219#95—1064no—
220#95—1062no—
221C24—531no—
222C24—531no—
223C24—532no—
224C24—532no—
225C24—1064no—
226C24—2124no—
227C24C2×C4×D5311yescharacteristic
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—532no—
231C22×C4—531no—
232C22×C4—531no—
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—532no—
242C22×C4—1068no—
243C22×C4—1062no—
244C22×C4—2124no—
245C22×C4—2124no—
246C22×C4—2124no—
247C22×C4—2124no—
248C22×C4—2124no—
249C22×C4—2124no—
250C22×C4—2128no—
251C22×C4—2122no—
252C22×C4C2×C4×D5312yes—
253C22×C4D4×D5312yes—
254C22×C4D4×D5311yescharacteristic
255C22×C4Q8×D5311yescharacteristic
256M16—44no—
257M16—2128no—
258M16—2124no—
259C2×C8—24no—
260C2×C8—1064no—
261C2×C8—2128no—
262C22⋊C4—42no—
263C22⋊C4—2124no—
264C22⋊C4—2122no—
265C23—22no—
266C23—42no—
267C23—532no—
268C23—532no—
269C23—534no—
270C23—1064no—
271C23—1064no—
272C23—1064no—
273C23—1064no—
274C23—1064no—
275C23—1062no—
276C23—2122no—
277C23—2122no—
278C23—2122no—
279C23—21216no—
280C23C42×D5311yescharacteristic
281C23C22⋊C4×D5311yescharacteristic
282C23C4⋊C4×D5311yescharacteristic
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—42no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—1064no—
292C2×C4—1064no—
293C2×C4—1064no—
294C2×C4—1064no—
295C2×C4—1064no—
296C2×C4—4248no—
297C2×C4—2122no—
298C2×C4—2122no—
299C2×C4—2122no—
300C2×C4—2122no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2122no—
304C2×C4—2124no—
305C2×C4—2122no—
306C2×C4C22⋊C4×D5312yes—
307C2×C4C4⋊C4×D5312yes—
308C8—44no—
309C8—2124no—
310C22—22no—
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—44no—
316C22—534no—
317C22—534no—
318C22—1068no—
319C22—1062no—
320C22—1062no—
321C22—1062no—
322C22—1062no—
323C22—2124no—
324C22—2128no—
325C22—2128no—
326C22#95×D5311yescharacteristic
327C22#99×D5312yes—
328C4—22no—
329C4—22no—
330C4—82no—
331C4—1062no—
332C4—1062no—
333C4—4242no—
334C2—22no—
335C2—41no—
336C2—41no—
337C2—1064no—
338C2—1064no—
339C2#342×D5311yescharacteristic
340C2#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#8114
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
39. C22⋊#88322C2×C4semidirect
48. C22⋊C8×C53C2×C4semidirect
168. #1306D53semidirect
191. C53#8114semidirect
217. C22⋊C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
#407×D532
#381×D532
#202726242
#202726232
C53×#13062
#811453