Finite Group Databasebeta

← #28307732

Subgroups — #28307732

Order 13568.

1050 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 1050 conjugacy classes of subgroups — too many to draw. Only its 176 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×#16811493C214yesmaximal
2#20290379C211yesmaximal, characteristic
3C4×(C2×#6157)⋊C4C211yesmaximal, characteristic
4C2×#16811179C211yesmaximal, characteristic
5C22×#11719936C2212yes—
6C22×C4×#6157C2212yes—
7C2×(C2×#6157)⋊C4—22no—
8#16811493—28no—
9#16811489—22no—
10#16811179—28no—
11C2×#13133216C412yescomplemented
12C2×#13133209C411yescharacteristic, complemented
13C2×#13133195—22no—
14C2×#13133182—22no—
15C2×#13133175C2212yes—
16C2×C42×D53C2211yescharacteristic
17C2×#125×C53C411yescharacteristic, complemented
18C23×#6157—22no—
19C2×#11719936—24no—
20C2×#11719936—24no—
21C2×#11719936—22no—
22C2×#11719936Q814yescomplemented
23C22×#88324—22no—
24C2×C4×#6157—24no—
25C2×C4×#6157—24no—
26C2×C4×#6157—22no—
27C2×C4×#6157Q814yescomplemented
28(C2×#6157)⋊C4—28no—
29C22×Dic106—22no—
30C22×C4×D53D412yes—
31C22×C4×D53C2311yescharacteristic
32#13133216—24no—
33#13133216—24no—
34#13133209—24no—
35#13133195—216no—
36#13133182—216no—
37#13133175—24no—
38#13133175D418yes—
39C2×C4⋊Dic53C2×C412yes—
40C2×#11719919—22no—
41C2×#11719919—22no—
42C2×#11719921—22no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53C2×C411yescharacteristic
45C2×C4×Dic53D412yes—
46C42×D53—24no—
47#125×C53—24no—
48C2×C4⋊C4×C53—22no—
49C2×C4⋊C4×C53C2×C412yes—
50C2×C42×C53C2×C411yescharacteristic
51C23×D53C2×D411yescharacteristic
52C22×#6157—28no—
53C22×#6157—24no—
54C22×#6157—44no—
55C22×#6157—42no—
56C22×#6157—42no—
57C22×C4×C53—22no—
58C22×C4×C53C22⋊C412yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—22no—
61C22×Dic53C22⋊C412yes—
62C22×Dic53C2×D411yescharacteristic
63C2×C4×D53—24no—
64C2×C4×D53—24no—
65C2×C4×D53—42no—
66C2×C4×D53C2×D414yes—
67C2×C4×D53C2×Q812yes—
68C2×C4×D53C2×Q812yes—
69C2×C4×D53#4112yes—
70C2×C4×D53#4112yes—
71C2×Dic106—28no—
72C2×Dic106—44no—
73C2×Dic106—44no—
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C4×#6157—28no—
78C4×#6157—44no—
79#11719936—28no—
80#11719936—44no—
81C4⋊C4×C53—28no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53C22⋊C414yes—
84C42×C53—22no—
85#11719921—28no—
86C4⋊Dic53—22no—
87C4⋊Dic53C22⋊C414yes—
88#11719919—28no—
89#11719919—28no—
90C4×Dic53—22no—
91C4×Dic53—24no—
92C4×Dic53—28no—
93C23×C53C2×C22⋊C411yescharacteristic
94C22×D53—22no—
95C22×D53—22no—
96C22×D53#12012yes—
97C22×D53C22⋊Q812yes—
98C22×D53#12411yescharacteristic
99C22×D53C4⋊Q811yescharacteristic
100C2×#6157—28no—
101C2×#6157—48no—
102C2×#6157—44no—
103C2×#6157—48no—
104C2×#6157—44no—
105C2×C4×C53—24no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53C2×C22⋊C412yes—
112C2×C4×C53C4×Q812yes—
113C2×C4×C53C4×Q812yes—
114C2×Dic53—22no—
115C2×Dic53—22no—
116C2×Dic53—24no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53#10414yes—
120C2×Dic53#12012yes—
121C2×Dic53C22⋊Q812yes—
122C2×Dic53#12411yescharacteristic
123C2×Dic53C4⋊Q811yescharacteristic
124C4×D53—48no—
125C4×D53C22⋊Q814yes—
126C4×D53C22⋊Q814yes—
127C4×D53#12414yes—
128C4×D53C4⋊Q814yes—
129Dic106—416no—
130#88324—44no—
131#11386—531nomaximal, Sylow
132C22×C53#37911yescharacteristic
133C22×C53#39111yescharacteristic
134C22×C53#39111yescharacteristic
135C22×C53C2×#10412yes—
136C22×C53#42112yes—
137C2×D53—22no—
138C2×D53—22no—
139C2×D53—22no—
140C2×D53—22no—
141C2×D53—24no—
142C2×D53#39312yes—
143C2×D53#39512yes—
144#6157—48no—
145C4×C53—44no—
146C4×C53—42no—
147C4×C53#39114yes—
148Dic53—24no—
149Dic53—44no—
150Dic53#39312yes—
151Dic53#39512yes—
152C2×#533—531no—
153C2×#339—534no—
154#1728—531no—
155C4×#95—531no—
156C2×C53#144611yescharacteristic
157C2×C53#178912yes—
158C2×C53#198612yes—
159C2×C53#199411yescharacteristic
160C2×C53#199511yescharacteristic
161D53—22no—
162D53—22no—
163C22×M16—532no—
164#533—1068no—
165C2×C4⋊Q8—532no—
166C2×#125—531no—
167C2×#125#615711yescharacteristic, complemented
168C2×C22⋊Q8—1062no—
169C2×C4×Q8—1062no—
170C2×#117—532no—
171C22×C42—531no—
172C22×C42—532no—
173C2×#95—1062no—
174#339—1068no—
175#336—1062no—
176C53#1138611yescharacteristic, complemented, Sylow
177C22×Q8—1062no—
178C23×C4—531no—
179C23×C4—532no—
180C23×C4—1062no—
181C2×M16—534no—
182C2×M16—1062no—
183C2×M16—1064no—
184C2×M16—1064no—
185C22×C8—1062no—
186C4⋊Q8—1064no—
187C4⋊Q8—1064no—
188#125—24no—
189#125—1064no—
190C22⋊Q8—10616no—
191C4×Q8—10616no—
192#117—538no—
193#117—1064no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—532no—
196C2×C4⋊C4—1062no—
197C2×C4⋊C4—1062no—
198C2×C4⋊C4C2×#615712yes—
199C2×C22⋊C4—1062no—
200C2×C42—532no—
201C2×C42—531no—
202C2×C42—534no—
203C2×C42—1064no—
204C2×C42—1062no—
205C2×C42—1064no—
206C2×C42—1064no—
207C2×C42—1062no—
208C2×C42C2×#615711yescharacteristic
209#95—1068no—
210C24—531no—
211C2×Q8—1068no—
212C2×Q8—2124no—
213C2×Q8—2124no—
214C22×C4—22no—
215C22×C4—531no—
216C22×C4—534no—
217C22×C4—532no—
218C22×C4—532no—
219C22×C4—532no—
220C22×C4—532no—
221C22×C4—532no—
222C22×C4—1064no—
223C22×C4—1064no—
224C22×C4—1064no—
225C22×C4—1068no—
226C22×C4—1062no—
227C22×C4—2122no—
228C22×C4—2122no—
229C22×C4—2122no—
230C22×C4—2124no—
231C22×C4C22⋊#615712yes—
232C22×C4C22×#615711yescharacteristic
233M16—1068no—
234M16—2124no—
235C2×C8—1064no—
236C2×C8—2122no—
237C2×C8—2122no—
238C4⋊C4—28no—
239C4⋊C4—22no—
240C4⋊C4—534no—
241C4⋊C4—1068no—
242C4⋊C4—1068no—
243C4⋊C4—1062no—
244C4⋊C4C22⋊#615714yes—
245C22⋊C4—1068no—
246C42—22no—
247C42—1064no—
248C42—1068no—
249C42—1068no—
250C42—1062no—
251C42—2124no—
252C23—532no—
253C23—532no—
254C23—531no—
255C23—531no—
256C23—1062no—
257C23—1062no—
258C23C2×C22⋊#615711yescharacteristic
259Q8—21216no—
260C2×C4—22no—
261C2×C4—24no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—532no—
267C2×C4—532no—
268C2×C4—531no—
269C2×C4—531no—
270C2×C4—534no—
271C2×C4—534no—
272C2×C4—534no—
273C2×C4—534no—
274C2×C4—534no—
275C2×C4—1062no—
276C2×C4—1062no—
277C2×C4—1068no—
278C2×C4—1064no—
279C2×C4—2128no—
280C2×C4—2124no—
281C2×C4—2128no—
282C2×C4—2124no—
283C2×C4—2128no—
284C2×C4—2122no—
285C2×C4—2122no—
286C2×C4C2×C22⋊#615712yes—
287C2×C4Q8×#615712yes—
288C2×C4Q8×#615712yes—
289C8—2124no—
290C22—532no—
291C22—532no—
292C22—1062no—
293C22—1062no—
294C22—1062no—
295C22—1062no—
296C22—1064no—
297C22#1681156411yescharacteristic
298C22(C2×#6157)⋊Q811yescharacteristic
299C22(C2×#6157)⋊Q811yescharacteristic
300C22D53⋊#10412yes—
301C22#1681162012yes—
302C4—42no—
303C4—44no—
304C4—532no—
305C4—532no—
306C4—1064no—
307C4—2128no—
308C4—2124no—
309C4(C2×#6157)⋊Q814yes—
310C2—1062no—
311C2—1062no—
312C2#2028990511yescharacteristic
313C2#2029055712yes—
314C2#2029089212yes—
315C2#2029090111yescharacteristic
316C2#2029090211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×#125×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C2×D4
cosocleC22×C4×D53C23
hypercenterC2×#125#6157
hypocenterC53#11386
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
11. C2×#13133216C4semidirect
12. C2×#13133209C4semidirect
17. C2×#125×C53C4semidirect
22. C2×#11719936Q8semidirect
27. C2×C4×#6157Q8semidirect
167. C2×#125#6157semidirect
176. C53#11386semidirect

Maximal subgroups

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

subgroupindex
C2×#168114932
#202903792
C4×(C2×#6157)⋊C42
C2×#168111792
#1138653