Finite Group Databasebeta

← #28301014

Subgroups — #28301014

Order 13568.

1226 conjugacy classes of subgroups; only the 172 normal subgroups are shown.

normal subgroups only

This group has 1226 conjugacy classes of subgroups — too many to draw. Only its 172 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
1C2×C4×C22⋊#6157C212yesmaximal
2#20289595C214yesmaximal, complemented
3#407×D53C211yesmaximal, characteristic
4C22×C22⋊#6157—22no—
5C22×C4×#6157—22no—
6C4×C22⋊#6157—22no—
7C4×C22⋊#6157—22no—
8C4×C22⋊#6157C414yescomplemented
9C4×C22⋊#6157C2214yes—
10C2×(C2×#6157)⋊C4C414yescomplemented
11C23×C4×D53C2211yescharacteristic
12C2×M16×D53C412yescomplemented
13#16810587C411yescharacteristic, complemented
14#16810586C2212yes—
15#16810585—22no—
16C22⋊C8×D53—22no—
17C53×#407C411yescharacteristic, complemented
18C23×#6157—24no—
19C2×C22⋊#6157—216no—
20C2×C22⋊#6157—24no—
21C2×C22⋊#6157—24no—
22C2×C4×#6157—24no—
23C2×C4×#6157—24no—
24C2×C4×#6157—48no—
25(C2×#6157)⋊C4—24no—
26(C2×#6157)⋊C4C2×C418yescomplemented
27C24×D53D411yescharacteristic
28C23×Dic53Q811yescharacteristic
29C22×C4×D53—44no—
30C22×C4×D53C2×C411yescharacteristic
31C22×C4×D53C2×C412yes—
32C22×C4×D53C2×C412yes—
33C22×C4×D53D412yes—
34C22×C4×D53D411yescharacteristic
35C22×C4×D53D412yes—
36C22×C4×D53Q811yescharacteristic
37C22×C4×D53C2311yescharacteristic
38C2×#11719917C2×C412yescomplemented
39C2×#11719912—22no—
40M16×D53—44no—
41C2×C8×D53—22no—
42C22⋊#88322—22no—
43#13133129—24no—
44C23×C4×C53C2×C411yescharacteristic
45C2×M16×C53C2×C412yescomplemented
46C22⋊C8×C53—22no—
47C24×C53C4⋊C411yescharacteristic
48C23×D53—28no—
49C23×D53—42no—
50C23×D53C22⋊C412yes—
51C23×D53C22⋊C412yes—
52C23×D53C4⋊C411yescharacteristic
53C23×D53C2×D411yescharacteristic
54C22×#6157—216no—
55C22×#6157—28no—
56C22×#6157—48no—
57C22×#6157—48no—
58C22×C4×C53—42no—
59C22×C4×C53C4212yes—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C4⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—42no—
64C22×Dic53C4⋊C411yescharacteristic
65C22×Dic53C4⋊C412yes—
66C22×Dic53C4⋊C412yes—
67C22×Dic53C2×Q811yescharacteristic
68C2×C4×D53—24no—
69C2×C4×D53—28no—
70C2×C4×D53—44no—
71C2×C4×D53—42no—
72C2×C4×D53—44no—
73C2×C4×D53—42no—
74C2×C4×D53—44no—
75C2×C4×D53—42no—
76C2×C4×D53C4211yescharacteristic
77C2×C4×D53C4211yescharacteristic
78C2×C4×D53C22⋊C412yes—
79C2×C4×D53C22⋊C411yescharacteristic
80C2×C4×D53C22⋊C412yes—
81C2×C4×D53C22⋊C411yescharacteristic
82C2×C4×D53C4⋊C412yes—
83C2×C4×D53C4⋊C412yes—
84C2×C4×D53C22×C412yes—
85C2×C4×D53C2×D412yes—
86C22⋊#6157—216no—
87C22⋊#6157—216no—
88C4×#6157—48no—
89C4×#6157—48no—
90M16×C53—42no—
91C2×C8×C53—22no—
92#11719917—42no—
93C2×#88322—22no—
94#11719912—44no—
95C8×D53—44no—
96C23×C53—24no—
97C23×C53—41no—
98C23×C53#9512yes—
99C23×C53C2×C4⋊C411yescharacteristic
100C22×D53—24no—
101C22×D53—24no—
102C22×D53—24no—
103C22×D53—24no—
104C22×D53—22no—
105C22×D53—24no—
106C22×D53—24no—
107C22×D53—42no—
108C22×D53—41no—
109C22×D53—416no—
110C22×D53#9511yescharacteristic
111C22×D53#9511yescharacteristic
112C22×D53#10418yes—
113C22×D53C2×C22⋊C412yes—
114C2×#6157—216no—
115C2×#6157—48no—
116C2×#6157—432no—
117C2×#6157—48no—
118C2×C4×C53—24no—
119C2×C4×C53—22no—
120C2×C4×C53—42no—
121C2×C4×C53—41no—
122C2×C4×C53#9512yes—
123C2×C4×C53C2×C4211yescharacteristic
124C2×C4×C53C2×C22⋊C411yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—24no—
127C2×Dic53—42no—
128C2×Dic53—41no—
129C2×Dic53#9511yescharacteristic
130C2×Dic53#9511yescharacteristic
131C2×Dic53C2×C4⋊C412yes—
132C4×D53—22no—
133C4×D53—24no—
134C4×D53—24no—
135C4×D53—22no—
136C4×D53—24no—
137C4×D53—22no—
138C4×D53—42no—
139C4×D53—42no—
140C4×D53—42no—
141C4×D53#9914yes—
142C8×C53—42no—
143#88322—42no—
144C2×#1297—531nomaximal, Sylow
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—21no—
149C22×C53—44no—
150C22×C53#33914yes—
151C22×C53C2×#9511yescharacteristic
152C2×D53—22no—
153C2×D53—22no—
154C2×D53—216no—
155C2×D53—22no—
156C2×D53—21no—
157C2×D53—44no—
158C2×D53—48no—
159C2×D53—44no—
160C2×D53#33914yes—
161C2×D53#34212yes—
162C2×D53C2×#10414yes—
163#6157—416no—
164C4×C53—22no—
165C4×C53—21no—
166C4×C53C2×#9912yes—
167Dic53—21no—
168Dic53—22no—
169Dic53#34212yes—
170C2×#407—531no—
171C2×C4×C22⋊C4—532no—
172#1297—534no—
173C2×C53—24no—
174C2×C53—41no—
175C2×C53C2×#33912yes—
176C2×C53C2×#34211yescharacteristic
177D53—24no—
178D53—41no—
179D53#129714yescomplemented
180C24×C4—531no—
181C22×M16—532no—
182C22×C22⋊C4—1062no—
183C22×C42—1062no—
184#407—531no—
185#407—532no—
186#407—1062no—
187#407#615711yescharacteristic, complemented
188C2×C22⋊C8—1062no—
189C4×C22⋊C4—534no—
190C4×C22⋊C4—534no—
191C4×C22⋊C4—1062no—
192C4×C22⋊C4—1062no—
193C2×#95—534no—
194C53C2×#129711yescharacteristic, complemented, Sylow
195C25—531no—
196C23×C4—531no—
197C23×C4—532no—
198C23×C4—531no—
199C23×C4—532no—
200C23×C4—532no—
201C23×C4—531no—
202C23×C4—531no—
203C23×C4—532no—
204C23×C4—531no—
205C23×C4—1064no—
206C23×C4—2124no—
207C23×C4C2×#615711yescharacteristic
208C2×M16—532no—
209C2×M16—1062no—
210C2×M16—2124no—
211C2×M16C2×#615712yescomplemented
212C22×C8—1062no—
213C2×C22⋊C4—1064no—
214C2×C22⋊C4—1064no—
215C2×C22⋊C4—10616no—
216C2×C42—1064no—
217C2×C42—1064no—
218C2×C42—2128no—
219C22⋊C8—22no—
220C22⋊C8—1064no—
221C22⋊C8—1062no—
222#95—538no—
223#95—1064no—
224C24—532no—
225C24—531no—
226C24—532no—
227C24—531no—
228C24—1068no—
229C24—2122no—
230C24C4⋊#615711yescharacteristic
231C22×C4—42no—
232C22×C4—531no—
233C22×C4—532no—
234C22×C4—531no—
235C22×C4—532no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—532no—
243C22×C4—532no—
244C22×C4—532no—
245C22×C4—532no—
246C22×C4—1068no—
247C22×C4—1064no—
248C22×C4—1068no—
249C22×C4—10616no—
250C22×C4—2122no—
251C22×C4—2124no—
252C22×C4—2124no—
253C22×C4—2122no—
254C22×C4—2124no—
255C22×C4—2122no—
256C22×C4—2122no—
257C22×C4—2128no—
258C22×C4—2128no—
259C22×C4C4×#615712yes—
260C22×C4C4⋊#615711yescharacteristic
261C22×C4C22⋊#615712yes—
262C22×C4C22×#615711yescharacteristic
263M16—42no—
264M16—2124no—
265M16—2122no—
266C2×C8—22no—
267C2×C8—1062no—
268C2×C8—2124no—
269C22⋊C4—10616no—
270C22⋊C4—10616no—
271C42—2128no—
272C42—2128no—
273C23—24no—
274C23—41no—
275C23—531no—
276C23—532no—
277C23—538no—
278C23—531no—
279C23—1064no—
280C23—1064no—
281C23—1064no—
282C23—1062no—
283C23—1064no—
284C23—1064no—
285C23—1064no—
286C23—2122no—
287C23—2121no—
288C23—21216no—
289C23(C2×#6157)⋊C412yes—
290C23C2×C4⋊#615711yescharacteristic
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—531no—
296C2×C4—534no—
297C2×C4—531no—
298C2×C4—532no—
299C2×C4—1064no—
300C2×C4—1062no—
301C2×C4—1064no—
302C2×C4—1062no—
303C2×C4—1064no—
304C2×C4—1062no—
305C2×C4—1064no—
306C2×C4—1062no—
307C2×C4—10616no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2121no—
313C2×C4—2128no—
314C2×C4—2128no—
315C2×C4—21232no—
316C2×C4(C2×#6157)⋊C412yes—
317C2×C4C2×C4×#615711yescharacteristic
318C2×C4C2×C22⋊#615711yescharacteristic
319C8—42no—
320C8—2122no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—21no—
325C22—44no—
326C22—534no—
327C22—532no—
328C22—534no—
329C22—10616no—
330C22—1062no—
331C22—1062no—
332C22—1062no—
333C22—1061no—
334C22—2128no—
335C22—2124no—
336C22—2124no—
337C22#1681149514yes—
338C22C2×(C2×#6157)⋊C411yescharacteristic
339C4—22no—
340C4—21no—
341C4—532no—
342C4—1061no—
343C4—1062no—
344C4—21216no—
345C4D53⋊#9912yes—
346C2—24no—
347C2—41no—
348C2—534no—
349C2—1064no—
350C2—2121no—
351C2C2×#1681149512yes—
352C2#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
hypocenterC53C2×#1297
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
8. C4×C22⋊#6157C4semidirect
10. C2×(C2×#6157)⋊C4C4semidirect
12. C2×M16×D53C4semidirect
13. #16810587C4semidirect
17. C53×#407C4semidirect
26. (C2×#6157)⋊C4C2×C4semidirect
38. C2×#11719917C2×C4semidirect
45. C2×M16×C53C2×C4semidirect
179. D53#1297semidirect
187. #407#6157semidirect
194. C53C2×#1297semidirect
211. C2×M16C2×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊#61572
#202895952
#407×D532
C2×#129753