Finite Group Databasebeta

← #28306443

Subgroups — #28306443

Order 13568.

1374 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 216 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
2C2×#16811491C214yesmaximal, complemented
3C22×M16×D53C211yesmaximal, characteristic
4C2×#13133296C2214yes—
5#16811566—22no—
6#16811565—24no—
7#16811564—22no—
8C4×C22⋊#6157C418yescomplemented
9#16811491—28no—
10C23×C4×D53C2211yescharacteristic
11C22×#11719917C411yescharacteristic, complemented
12C22×#11719912C2212yes—
13C2×M16×D53—24no—
14C2×M16×D53—28no—
15C22×C8×D53C412yescomplemented
16C22×M16×C53C411yescharacteristic, complemented
17#13133296—28no—
18#13133296—28no—
19C2×C4⋊#6157—24no—
20C2×C22⋊#6157—28no—
21C2×C22⋊#6157—24no—
22C2×C4×#6157C2×C418yes—
23(C2×#6157)⋊C4—28no—
24C24×D53D411yescharacteristic
25C23×Dic53Q811yescharacteristic
26C22×C4×D53—24no—
27C22×C4×D53—24no—
28C22×C4×D53C2×C411yescharacteristic
29C22×C4×D53C2×C414yes—
30C22×C4×D53D411yescharacteristic
31C22×C4×D53D414yes—
32C22×C4×D53Q811yescharacteristic
33C22×C4×D53C2311yescharacteristic
34C2×#11719917—24no—
35C2×#11719917—22no—
36C2×#11719912—24no—
37C2×#11719912—24no—
38C2×#11719912D418yes—
39M16×D53—232no—
40C22×#88322C2×C412yes—
41C2×C8×D53—24no—
42C2×C8×D53—24no—
43C2×C8×D53—24no—
44C23×C4×C53C2×C411yescharacteristic
45C2×M16×C53—24no—
46C2×M16×C53—22no—
47C22×C8×C53C2×C412yes—
48C24×C53C4⋊C411yescharacteristic
49C23×D53—24no—
50C23×D53—44no—
51C23×D53C22⋊C414yes—
52C23×D53C4⋊C411yescharacteristic
53C23×D53C2×D411yescharacteristic
54C22×#6157—28no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53C4212yes—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C4⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—24no—
62C22×Dic53C4⋊C411yescharacteristic
63C22×Dic53C4⋊C414yes—
64C22×Dic53C2×Q811yescharacteristic
65C2×C4×D53—24no—
66C2×C4×D53—22no—
67C2×C4×D53—22no—
68C2×C4×D53—24no—
69C2×C4×D53—216no—
70C2×C4×D53—28no—
71C2×C4×D53—28no—
72C2×C4×D53C4212yes—
73C2×C4×D53C22⋊C414yes—
74C2×C4×D53C22⋊C412yes—
75C2×C4×D53C4⋊C414yes—
76C2×C4×D53C22×C412yes—
77C2×C4×D53C2×D414yes—
78C2×C4×D53C2×D412yes—
79C2×C4×D53#4114yes—
80C22⋊#6157—416no—
81C22⋊#6157—48no—
82C4⋊#6157—44no—
83C4⋊#6157—44no—
84C4×#6157—216no—
85M16×C53—28no—
86C2×C8×C53—22no—
87C2×C8×C53—22no—
88C2×C8×C53C22⋊C414yescomplemented
89#11719917—28no—
90C2×#88322—22no—
91C2×#88322—22no—
92C2×#88322C22⋊C414yescomplemented
93#11719912—216no—
94C8×D53—216no—
95C23×C53—22no—
96C23×C53—42no—
97C23×C53#9512yes—
98C23×C53C2×C4⋊C411yescharacteristic
99C22×D53—22no—
100C22×D53—24no—
101C22×D53—28no—
102C22×D53—28no—
103C22×D53—44no—
104C22×D53—42no—
105C22×D53—416no—
106C22×D53#9512yes—
107C22×D53C2×C22⋊C412yes—
108C22×D53#12114yes—
109C2×#6157—416no—
110C2×#6157—48no—
111C2×C4×C53—22no—
112C2×C4×C53—24no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53#9512yes—
116C2×C4×C53C2×C4211yescharacteristic
117C2×C4×C53C2×C22⋊C412yes—
118C2×C4×C53C2×C22⋊C411yescharacteristic
119C2×C4×C53#11712yes—
120C2×Dic53—22no—
121C2×Dic53—28no—
122C2×Dic53#9512yes—
123C2×Dic53C2×C4⋊C412yes—
124C2×Dic53C22⋊Q814yes—
125C4×D53—28no—
126C4×D53—24no—
127C4×D53—22no—
128C4×D53—22no—
129C4×D53—28no—
130C4×D53C4×D418yes—
131C4×D53#12014yes—
132C4×D53#12314yes—
133C8×C53—24no—
134#88322—24no—
135#10717—531nomaximal, Sylow
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—24no—
139C22×C53—44no—
140C22×C53C2×#9511yescharacteristic
141C22×C53#38012yes—
142C2×D53—28no—
143C2×D53—22no—
144C2×D53—24no—
145C2×D53—22no—
146C2×D53—44no—
147C2×D53—416no—
148C2×D53H(3,ℤ/4)14yescomplemented
149C2×D53#38614yes—
150#6157—88no—
151C4×C53—22no—
152C4×C53C4×C22⋊C412yes—
153C4×C53#37911yescharacteristic
154C4×C53#38111yescharacteristic
155Dic53—22no—
156Dic53#38514yes—
157C23×M16—531no—
158#2280—532no—
159C2×H(3,ℤ/4)—534no—
160C2×C53—22no—
161C2×C53—42no—
162C2×C53C22⋊#9511yescharacteristic
163C2×C53C2×H(3,ℤ/4)12yes—
164D53—24no—
165D53—24no—
166C24×C4—531no—
167C22×M16—531no—
168C22×M16—532no—
169C22×M16—1068no—
170C22×M16—1064no—
171C22×M16#615711yescharacteristic, complemented
172C23×C8—532no—
173C2×#117—534no—
174#381—1062no—
175#380—1064no—
176#379—1062no—
177C4×C22⋊C4—538no—
178H(3,ℤ/4)—1068no—
179C53#1071711yescharacteristic, complemented, Sylow
180C25—531no—
181C23×C4—531no—
182C23×C4—531no—
183C23×C4—534no—
184C23×C4—531no—
185C23×C4—531no—
186C23×C4—534no—
187C23×C4—531no—
188C23×C4—1064no—
189C23×C4—1064no—
190C23×C4C2×#615711yescharacteristic
191C2×M16—22no—
192C2×M16—24no—
193C2×M16—538no—
194C2×M16—1064no—
195C2×M16—1064no—
196C2×M16—1064no—
197C2×M16—10632no—
198C2×M16—1062no—
199C22×C8—532no—
200C22×C8—1064no—
201C22×C8—1064no—
202C22×C8—1064no—
203C22×C8C2×#615712yes—
204#117—1068no—
205#117—1068no—
206C2×C4⋊C4—1064no—
207C2×C22⋊C4—1064no—
208C2×C22⋊C4—1068no—
209C2×C42—538no—
210#95—1068no—
211C24—531no—
212C24—531no—
213C24—534no—
214C24—1064no—
215C24—2124no—
216C24C4⋊#615711yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—531no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—534no—
223C22×C4—534no—
224C22×C4—534no—
225C22×C4—531no—
226C22×C4—534no—
227C22×C4—534no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—1068no—
231C22×C4—1064no—
232C22×C4—10616no—
233C22×C4—1068no—
234C22×C4—1064no—
235C22×C4—1064no—
236C22×C4—1062no—
237C22×C4—1062no—
238C22×C4—1068no—
239C22×C4C4×#615712yes—
240C22×C4C4⋊#615711yescharacteristic
241C22×C4C22⋊#615712yes—
242C22×C4C22×#615711yescharacteristic
243M16—28no—
244M16—10616no—
245M16—1068no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8—534no—
249C2×C8—10616no—
250C2×C8—1062no—
251C2×C8—1062no—
252C2×C8C22⋊#615714yescomplemented
253C4⋊C4—2124no—
254C4⋊C4—2124no—
255C22⋊C4—2128no—
256C22⋊C4—21216no—
257C42—10616no—
258C23—22no—
259C23—42no—
260C23—532no—
261C23—532no—
262C23—534no—
263C23—1064no—
264C23—1068no—
265C23—1068no—
266C23—1062no—
267C23—2122no—
268C23—2124no—
269C23—21216no—
270C23(C2×#6157)⋊C412yes—
271C23C2×C4⋊#615711yescharacteristic
272C2×C4—22no—
273C2×C4—24no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—532no—
277C2×C4—534no—
278C2×C4—538no—
279C2×C4—534no—
280C2×C4—534no—
281C2×C4—532no—
282C2×C4—1068no—
283C2×C4—1068no—
284C2×C4—1068no—
285C2×C4—1062no—
286C2×C4—1064no—
287C2×C4—1062no—
288C2×C4—1062no—
289C2×C4—2128no—
290C2×C4—21216no—
291C2×C4(C2×#6157)⋊C412yes—
292C2×C4C2×C4×#615711yescharacteristic
293C2×C4C2×C22⋊#615712yes—
294C2×C4C2×C22⋊#615711yescharacteristic
295C2×C4#1313329612yes—
296C8—24no—
297C8—1064no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—44no—
302C22—534no—
303C22—534no—
304C22—1068no—
305C22—1064no—
306C22—1062no—
307C22—1062no—
308C22—2124no—
309C22—21216no—
310C22C2×(C2×#6157)⋊C411yescharacteristic
311C22#1681156512yes—
312C4—22no—
313C4—534no—
314C4—1062no—
315C4—4248no—
316C4C4×C22⋊#615712yes—
317C4#1681156411yescharacteristic
318C4#1681156611yescharacteristic
319C2—22no—
320C2—42no—
321C2—1064no—
322C2—1064no—
323C2#2028988911yescharacteristic
324C2#2029038712yes—
subgroupG/H
centerC2×C4C2×C4×#6157
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC22×M16×C53C4
socleC22×C53C2×#95
prime radicalC24×D53D4
cosocleC22×C4×D53C23
hypercenterC22×M16#6157
hypocenterC53#10717
LCSG ▷ C2×C4×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×M16

Complements

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

NH ≅ G/Ntype
2. C2×#16811491C2semidirect
8. C4×C22⋊#6157C4semidirect
11. C22×#11719917C4semidirect
15. C22×C8×D53C4semidirect
16. C22×M16×C53C4semidirect
88. C2×C8×C53C22⋊C4semidirect
92. C2×#88322C22⋊C4semidirect
148. C2×D53H(3,ℤ/4)semidirect
171. C22×M16#6157semidirect
179. C53#10717semidirect
252. C2×C8C22⋊#6157semidirect

Maximal subgroups

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

subgroupindex
#202911592
C2×#168114912
C22×M16×D532
#1071753