Finite Group Databasebeta

← #27281899

Subgroups — #27281899

Order 13568.

734 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 107 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#20278989C211yesmaximal, characteristic
2#20277068C211yesmaximal, characteristic
3#20274812C212yesmaximal, complemented
4#20272569C212yesmaximal, complemented
5C53×#1462C211yesmaximal, characteristic
6C2×#13133176C2211yescharacteristic
7C22×C22⋊Dic53—21no—
8#16810559C412yescomplemented
9#16810548C412yescomplemented
10#16810511—22no—
11#16810496—21no—
12#16810486C2211yescharacteristic
13#16810479—21no—
14C22⋊C4×Dic53—24no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C2212yes—
17#16810384—24no—
18C22×C22⋊C4×C53C2211yescharacteristic
19C53×#407C2212yes—
20C53×#323—22no—
21C23×Dic53—22no—
22#13133176—22no—
23#13133176—22no—
24C2×C4⋊Dic53C2×C411yescharacteristic, complemented
25C2×C4⋊Dic53C2×C411yescharacteristic, complemented
26C2×#11719919—22no—
27C2×C22⋊Dic53—28no—
28C2×C22⋊Dic53—22no—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C22⋊Dic53—22no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C4×Dic53—22no—
35C2×C4×Dic53—21no—
36C2×C4×Dic53D414yes—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40#13133124—22no—
41#13133124C2×C412yescomplemented
42C25×C53D411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53C2×C412yes—
49C2×C22⋊C4×C53D412yes—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C24×C53—21no—
53C24×C53—24no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—81no—
57C24×C53C22⋊C412yes—
58C24×C53C2×D411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—81no—
63C22×C4×C53C22⋊C412yes—
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D412yes—
66C22×C4×C53C2×D411yescharacteristic
67C22×C4×C53#4112yes—
68C22×Dic53—24no—
69C22×Dic53—28no—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C22×Dic53—82no—
74C22×Dic53—82no—
75M16×C53—44no—
76C2×C8×C53—42no—
77C22⋊C4×C53—24no—
78C22⋊C4×C53—24no—
79C22⋊Dic53—28no—
80C22⋊Dic53—28no—
81C22⋊Dic53—42no—
82C22⋊Dic53—44no—
83C22⋊Dic53—41no—
84C22⋊Dic53—41no—
85C22⋊Dic53—82no—
86C4⋊Dic53—22no—
87C4⋊Dic53—22no—
88#11719919—82no—
89C4×Dic53—42no—
90C4×Dic53—44no—
91C4×Dic53—82no—
92C23×C53—22no—
93C23×C53—22no—
94C23×C53—24no—
95C23×C53—22no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—41no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—81no—
105C23×C53—81no—
106C23×C53—88no—
107C23×C53#10414yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C23×C53#12011yescharacteristic
110C23×C53#12311yescharacteristic
111C2×C4×C53—24no—
112C2×C4×C53—22no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—81no—
119C2×C4×C53C4×D412yes—
120C2×C4×C53#12112yes—
121C2×Dic53—28no—
122C2×Dic53—44no—
123C2×Dic53—44no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—88no—
132C8×C53—82no—
133#9477—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—24no—
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—44no—
143C22×C53—44no—
144C22×C53—44no—
145C22×C53—44no—
146C22×C53—42no—
147C22×C53—42no—
148C22×C53—82no—
149C22×C53—84no—
150C22×C53#38611yescharacteristic
151C22×C53C2×#10412yes—
152C22×C53#45812yes—
153C4×C53—22no—
154C4×C53—22no—
155C4×C53—42no—
156Dic53—48no—
157Dic53—82no—
158Dic53—81no—
159Dic53—81no—
160#2482—531no—
161#2273—531no—
162#1826—532no—
163#1462D5311yescharacteristic
164#1284—532no—
165C2×C53—22no—
166C2×C53—22no—
167C2×C53—41no—
168C2×C53—44no—
169C2×C53#189011yescharacteristic
170C2×C53#189611yescharacteristic
171C2×C53#191111yescharacteristic
172C2×#117—531no—
173C22×C22⋊C4—1061no—
174C22×C22⋊C4C2×D5311yescharacteristic
175#407C2×D5312yes—
176#399—532no—
177#397—532no—
178#388—1062no—
179#385—1061no—
180#381—531no—
181#379—1061no—
182C4×C22⋊C4—532no—
183C4×C22⋊C4—1064no—
184C4×C22⋊C4—1061no—
185#339—1064no—
186#323—22no—
187C53#947711yescharacteristic, complemented, Sylow
188C25D21211yescharacteristic
189C23×C4—1062no—
190C23×C4D21211yescharacteristic
191C23×C4C22×D5311yescharacteristic
192C2×M16—22no—
193#117—1062no—
194#117—1062no—
195C2×C4⋊C4—531no—
196C2×C4⋊C4—531no—
197C2×C4⋊C4—1062no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—1068no—
201C2×C22⋊C4—1062no—
202C2×C22⋊C4—1062no—
203C2×C22⋊C4—1062no—
204C2×C22⋊C4—1061no—
205C2×C22⋊C4—1061no—
206C2×C22⋊C4—1062no—
207C2×C22⋊C4C4×D5312yes—
208C2×C22⋊C4C22⋊D5312yes—
209C2×C42—534no—
210C2×C42—1062no—
211C2×C42—1061no—
212C22⋊C8—22no—
213C22⋊C8—22no—
214#95—532no—
215#95—1062no—
216#95—1062no—
217#95—1061no—
218#95—1062no—
219C24—21no—
220C24—24no—
221C24—21no—
222C24—42no—
223C24—81no—
224C24#1171992112yes—
225C24C2×D21211yescharacteristic
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—42no—
229C22×C4—81no—
230C22×C4—1068no—
231C22×C4—1064no—
232C22×C4—1062no—
233C22×C4—1061no—
234C22×C4—1061no—
235C22×C4—4242no—
236C22×C4—4242no—
237C22×C4#1171992112yes—
238C22×C4C2×C4×D5311yescharacteristic
239C22×C4D4×D5312yes—
240C22×C4#1171994712yes—
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—44no—
243C2×C8—42no—
244C4⋊C4—1062no—
245C4⋊C4—1062no—
246C4⋊C4—4242no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—1068no—
250C22⋊C4—1068no—
251C22⋊C4—4242no—
252C22⋊C4—2122no—
253C22⋊C4—2124no—
254C22⋊C4—2121no—
255C22⋊C4—2121no—
256C42—4242no—
257C42—2124no—
258C42—2122no—
259C23—22no—
260C23—22no—
261C23—24no—
262C23—22no—
263C23—22no—
264C23—22no—
265C23—22no—
266C23—41no—
267C23—42no—
268C23—42no—
269C23—42no—
270C23—42no—
271C23—81no—
272C23—81no—
273C23—88no—
274C23#1313314314yes—
275C23C2×#1171992111yescharacteristic
276C23#1313318711yescharacteristic
277C23#1313320111yescharacteristic
278C2×C4—24no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—81no—
286C2×C4—1068no—
287C2×C4—4248no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2122no—
291C2×C4—2121no—
292C2×C4—2121no—
293C2×C4—2122no—
294C2×C4—2122no—
295C2×C4—2124no—
296C2×C4—2124no—
297C2×C4#1313317712yes—
298C2×C4#1313318912yes—
299C8—82no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—24no—
307C22—21no—
308C22—44no—
309C22—44no—
310C22—44no—
311C22—44no—
312C22—42no—
313C22—42no—
314C22—82no—
315C22—84no—
316C22#1681050011yescharacteristic
317C22C2×#1313314312yes—
318C22#1681081412yes—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—4242no—
323C4—4241no—
324C4—4241no—
325C4—2128no—
326C2—22no—
327C2—22no—
328C2—41no—
329C2—44no—
330C2#2027512611yescharacteristic
331C2#2027515611yescharacteristic
332C2#2027524511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1462C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#1462D53
hypocenterC53#9477
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
3. #20274812C2semidirect
4. #20272569C2semidirect
8. #16810559C4semidirect
9. #16810548C4semidirect
24. C2×C4⋊Dic53C2×C4semidirect
25. C2×C4⋊Dic53C2×C4semidirect
41. #13133124C2×C4semidirect
187. C53#9477semidirect

Maximal subgroups

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

subgroupindex
#202789892
#202770682
#202748122
#202725692
C53×#14622
#947753