Finite Group Databasebeta

← #28301022

Subgroups — #28301022

Order 13568.

954 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 954 conjugacy classes of subgroups — too many to draw. Only its 148 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#20291159C212yesmaximal
2#20289595C214yesmaximal, complemented
3#407×D53C211yesmaximal, characteristic
4C2×#13133296—22no—
5#16811566C414yescomplemented
6#16811565—22no—
7#16811565C414yescomplemented
8#16811564—22no—
9C4×C22⋊#6157—22no—
10C4×C22⋊#6157C2214yes—
11C23×C4×D53C2211yescharacteristic
12C2×M16×D53C412yescomplemented
13#16810587C411yescharacteristic, complemented
14#16810586C2212yes—
15#16810585—22no—
16C22⋊C8×D53—22no—
17C53×#407C411yescharacteristic, complemented
18#13133296—48no—
19C2×C4⋊#6157—24no—
20C2×C22⋊#6157—24no—
21C2×C22⋊#6157—24no—
22C2×C22⋊#6157—24no—
23C2×C4×#6157—24no—
24(C2×#6157)⋊C4—24no—
25(C2×#6157)⋊C4C2×C418yes—
26C24×D53D411yescharacteristic
27C23×Dic53Q811yescharacteristic
28C22×C4×D53—44no—
29C22×C4×D53C2×C411yescharacteristic
30C22×C4×D53C2×C412yes—
31C22×C4×D53C2×C412yes—
32C22×C4×D53D412yes—
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D412yes—
35C22×C4×D53Q811yescharacteristic
36C22×C4×D53C2311yescharacteristic
37C2×#11719917C2×C412yes—
38C2×#11719912—22no—
39M16×D53—44no—
40C2×C8×D53—22no—
41C22⋊#88322—22no—
42#13133129—24no—
43C23×C4×C53C2×C411yescharacteristic
44C2×M16×C53C2×C412yes—
45C22⋊C8×C53—22no—
46C24×C53C4⋊C411yescharacteristic
47C23×D53—42no—
48C23×D53—44no—
49C23×D53C22⋊C412yes—
50C23×D53C22⋊C412yes—
51C23×D53C4⋊C411yescharacteristic
52C23×D53C2×D411yescharacteristic
53C22×#6157—28no—
54C22×C4×C53—42no—
55C22×C4×C53C4212yes—
56C22×C4×C53C22⋊C412yes—
57C22×C4×C53C4⋊C411yescharacteristic
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—42no—
60C22×Dic53C4⋊C411yescharacteristic
61C22×Dic53C4⋊C412yes—
62C22×Dic53C4⋊C412yes—
63C22×Dic53C2×Q811yescharacteristic
64C2×C4×D53—24no—
65C2×C4×D53—28no—
66C2×C4×D53—44no—
67C2×C4×D53—42no—
68C2×C4×D53—44no—
69C2×C4×D53—42no—
70C2×C4×D53—44no—
71C2×C4×D53—42no—
72C2×C4×D53C4211yescharacteristic
73C2×C4×D53C4211yescharacteristic
74C2×C4×D53C22⋊C412yes—
75C2×C4×D53C22⋊C411yescharacteristic
76C2×C4×D53C22⋊C412yes—
77C2×C4×D53C22⋊C411yescharacteristic
78C2×C4×D53C4⋊C412yes—
79C2×C4×D53C4⋊C412yes—
80C2×C4×D53C22×C412yes—
81C2×C4×D53C2×D412yes—
82C22⋊#6157—48no—
83C22⋊#6157—48no—
84C22⋊#6157—48no—
85C4⋊#6157—48no—
86C4×#6157—48no—
87M16×C53—42no—
88C2×C8×C53—22no—
89#11719917—42no—
90C2×#88322—22no—
91#11719912—44no—
92C8×D53—44no—
93C23×C53—41no—
94C23×C53—42no—
95C23×C53#9512yes—
96C23×C53C2×C4⋊C411yescharacteristic
97C22×D53—24no—
98C22×D53—22no—
99C22×D53—24no—
100C22×D53—24no—
101C22×D53—24no—
102C22×D53—42no—
103C22×D53—41no—
104C22×D53—42no—
105C22×D53—42no—
106C22×D53—42no—
107C22×D53—416no—
108C22×D53#9511yescharacteristic
109C22×D53#9511yescharacteristic
110C22×D53C2×C22⋊C412yes—
111C2×#6157—48no—
112C2×#6157—48no—
113C2×#6157—48no—
114C2×C4×C53—24no—
115C2×C4×C53—22no—
116C2×C4×C53—42no—
117C2×C4×C53—41no—
118C2×C4×C53#9512yes—
119C2×C4×C53C2×C4211yescharacteristic
120C2×C4×C53C2×C22⋊C411yescharacteristic
121C2×Dic53—22no—
122C2×Dic53—24no—
123C2×Dic53—42no—
124C2×Dic53—41no—
125C2×Dic53#9511yescharacteristic
126C2×Dic53#9511yescharacteristic
127C2×Dic53C2×C4⋊C412yes—
128C4×D53—22no—
129C4×D53—24no—
130C4×D53—24no—
131C4×D53—22no—
132C4×D53—24no—
133C4×D53—22no—
134C4×D53—42no—
135C4×D53—42no—
136C4×D53—42no—
137C4×D53#9914yes—
138C8×C53—42no—
139#88322—42no—
140#8049—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—21no—
146C22×C53—44no—
147C22×C53C2×#9511yescharacteristic
148C2×D53—22no—
149C2×D53—22no—
150C2×D53—22no—
151C2×D53—22no—
152C2×D53—21no—
153C2×D53—44no—
154C2×D53—48no—
155C2×D53—48no—
156C2×D53—44no—
157C2×D53#34212yes—
158C2×D53#42114yes—
159#6157—88no—
160C4×C53—22no—
161C4×C53—21no—
162C4×C53C2×#9912yes—
163Dic53—21no—
164Dic53—22no—
165Dic53#34212yes—
166C2×#407—531no—
167#2280—532no—
168#1297—534no—
169C2×C53—42no—
170C2×C53—41no—
171C2×C53#173312yes—
172C2×C53C2×#34211yescharacteristic
173D53—24no—
174D53—22no—
175D53—41no—
176C24×C4—531no—
177C22×M16—532no—
178C2×#117—1062no—
179#407—531no—
180#407—532no—
181#407—1062no—
182#407#615711yescharacteristic, complemented
183C2×C22⋊C8—1062no—
184#381—534no—
185#380—534no—
186#380—1062no—
187#379—1062no—
188C4×C22⋊C4—534no—
189C4×C22⋊C4—1062no—
190C53#804911yescharacteristic, complemented, Sylow
191C25—531no—
192C23×C4—531no—
193C23×C4—532no—
194C23×C4—531no—
195C23×C4—532no—
196C23×C4—532no—
197C23×C4—531no—
198C23×C4—531no—
199C23×C4—532no—
200C23×C4—531no—
201C23×C4—2124no—
202C23×C4C2×#615711yescharacteristic
203C2×M16—532no—
204C2×M16—1062no—
205C2×M16—2124no—
206C2×M16C2×#615712yes—
207C22×C8—1062no—
208#117—2128no—
209C2×C4⋊C4—1064no—
210C2×C22⋊C4—1064no—
211C2×C22⋊C4—1064no—
212C2×C22⋊C4—1064no—
213C2×C42—1064no—
214C22⋊C8—22no—
215C22⋊C8—1064no—
216C22⋊C8—1062no—
217#95—538no—
218#95—1064no—
219C24—532no—
220C24—531no—
221C24—532no—
222C24—531no—
223C24—2122no—
224C24—2124no—
225C24C4⋊#615711yescharacteristic
226C22×C4—42no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—531no—
230C22×C4—532no—
231C22×C4—531no—
232C22×C4—532no—
233C22×C4—532no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—532no—
240C22×C4—532no—
241C22×C4—1068no—
242C22×C4—1064no—
243C22×C4—1068no—
244C22×C4—2122no—
245C22×C4—2124no—
246C22×C4—2124no—
247C22×C4—2122no—
248C22×C4—2124no—
249C22×C4—2122no—
250C22×C4—2122no—
251C22×C4C4×#615712yes—
252C22×C4C4⋊#615711yescharacteristic
253C22×C4C22⋊#615712yes—
254C22×C4C22×#615711yescharacteristic
255M16—42no—
256M16—2124no—
257M16—2122no—
258C2×C8—22no—
259C2×C8—1062no—
260C2×C8—2124no—
261C4⋊C4—2128no—
262C22⋊C4—2128no—
263C22⋊C4—2128no—
264C22⋊C4—2128no—
265C42—2128no—
266C23—41no—
267C23—42no—
268C23—531no—
269C23—532no—
270C23—531no—
271C23—1064no—
272C23—1064no—
273C23—1064no—
274C23—1062no—
275C23—1064no—
276C23—2122no—
277C23—2121no—
278C23—2122no—
279C23—2122no—
280C23—2122no—
281C23—21216no—
282C23(C2×#6157)⋊C412yes—
283C23C2×C4⋊#615711yescharacteristic
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—531no—
289C2×C4—534no—
290C2×C4—531no—
291C2×C4—532no—
292C2×C4—1064no—
293C2×C4—1062no—
294C2×C4—1064no—
295C2×C4—1062no—
296C2×C4—1064no—
297C2×C4—1062no—
298C2×C4—1064no—
299C2×C4—1062no—
300C2×C4—2122no—
301C2×C4—2122no—
302C2×C4—2122no—
303C2×C4—2122no—
304C2×C4—2121no—
305C2×C4—2128no—
306C2×C4—2128no—
307C2×C4—2128no—
308C2×C4(C2×#6157)⋊C412yes—
309C2×C4C2×C4×#615711yescharacteristic
310C2×C4C2×C22⋊#615711yescharacteristic
311C8—42no—
312C8—2122no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—21no—
318C22—44no—
319C22—534no—
320C22—532no—
321C22—1062no—
322C22—1062no—
323C22—1062no—
324C22—1062no—
325C22—1061no—
326C22—2124no—
327C22—2128no—
328C22—2128no—
329C22—2124no—
330C22C2×(C2×#6157)⋊C411yescharacteristic
331C4—22no—
332C4—21no—
333C4—532no—
334C4—1061no—
335C4—1062no—
336C4—4248no—
337C4D53⋊#9912yes—
338C2—42no—
339C2—41no—
340C2—1064no—
341C2—1062no—
342C2—2121no—
343C2#2029039712yes—
344C2#2029041211yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#407C4
socleC22×C53C2×#95
prime radicalC24×D53D4
cosocleC22×C4×D53C23
hypercenter#407#6157
hypocenterC53#8049
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. #20289595C2semidirect
5. #16811566C4semidirect
7. #16811565C4semidirect
12. C2×M16×D53C4semidirect
13. #16810587C4semidirect
17. C53×#407C4semidirect
182. #407#6157semidirect
190. C53#8049semidirect

Maximal subgroups

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

subgroupindex
#202911592
#202895952
#407×D532
#804953