Finite Group Databasebeta

← #27281528

Subgroups — #27281528

Order 13568.

726 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 726 conjugacy classes of subgroups — too many to draw. Only its 115 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#20277068C211yesmaximal, characteristic
2#20273235C212yesmaximal, complemented
3#20273042C211yesmaximal, characteristic
4#20272601C212yesmaximal, complemented
5C53×#2561C211yesmaximal, characteristic
6C22×C22⋊Dic53—21no—
7#16810587C2212yes—
8C22⋊C4×Dic53—24no—
9C4×C22⋊Dic53—21no—
10C4×C22⋊Dic53C2212yes—
11#16810360—24no—
12#16810347—22no—
13C2×C22⋊Q8×C53C2211yescharacteristic
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#397C412yescomplemented
16C53×#393C412yescomplemented
17C53×#388—22no—
18C53×#381—21no—
19C53×#379—21no—
20C4×C22⋊C4×C53C2211yescharacteristic
21C23×Dic53—22no—
22C2×#11719917—22no—
23C2×C22⋊Dic53—28no—
24C2×C22⋊Dic53—22no—
25C2×C22⋊Dic53—22no—
26C2×C4×Dic53—22no—
27C2×C4×Dic53—22no—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30#13133124—22no—
31#13133124D414yes—
32C25×C53D411yescharacteristic
33C22×Q8×C53C2×C411yescharacteristic, complemented
34C23×C4×C53D411yescharacteristic
35C23×C4×C53C2311yescharacteristic
36C22⋊Q8×C53—22no—
37C22⋊Q8×C53—22no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
40C2×C22⋊C4×C53—22no—
41C2×C22⋊C4×C53—22no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—22no—
46C2×C22⋊C4×C53C2×C412yes—
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53—21no—
49#95×C53—22no—
50#95×C53—22no—
51#95×C53C2×C412yescomplemented
52C24×C53—21no—
53C24×C53—24no—
54C24×C53—21no—
55C24×C53—42no—
56C24×C53—81no—
57C24×C53C22⋊C412yes—
58C24×C53C2×D411yescharacteristic
59C2×Q8×C53—22no—
60C2×Q8×C53—81no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—42no—
67C22×C4×C53—81no—
68C22×C4×C53C22⋊C412yes—
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4112yes—
73C22×Dic53—24no—
74C22×Dic53—28no—
75C22×Dic53—82no—
76C22×Dic53—82no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—24no—
80C22⋊C4×C53—24no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—44no—
83C22⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—82no—
86C42×C53—81no—
87C22⋊Dic53—28no—
88C22⋊Dic53—28no—
89C4×Dic53—82no—
90C4×Dic53—82no—
91#11719917—82no—
92C2×#88322—42no—
93C23×C53—22no—
94C23×C53—22no—
95C23×C53—24no—
96C23×C53—22no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—41no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53—81no—
106C23×C53—81no—
107C23×C53—88no—
108C23×C53#10414yes—
109C23×C53C2×C22⋊C411yescharacteristic
110C23×C53#12011yescharacteristic
111C23×C53#12311yescharacteristic
112Q8×C53—82no—
113C2×C4×C53—22no—
114C2×C4×C53—41no—
115C2×C4×C53—42no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—81no—
128C2×C4×C53C4×D412yes—
129C2×C4×C53#12112yes—
130C2×Dic53—28no—
131C2×Dic53—44no—
132C2×Dic53—44no—
133C2×Dic53—88no—
134#88322—82no—
135#9407—531nomaximal, Sylow
136C22×C53—21no—
137C22×C53—21no—
138C22×C53—21no—
139C22×C53—21no—
140C22×C53—21no—
141C22×C53—24no—
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—44no—
145C22×C53—44no—
146C22×C53—44no—
147C22×C53—44no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—82no—
151C22×C53—84no—
152C22×C53#38611yescharacteristic
153C22×C53C2×#10412yes—
154C22×C53#45512yes—
155C4×C53—22no—
156C4×C53—22no—
157C4×C53—42no—
158C4×C53—82no—
159C4×C53—81no—
160C4×C53—81no—
161Dic53—48no—
162#2561D5311yescharacteristic
163#2273—531no—
164#1508—532no—
165#1462—531no—
166#1297—532no—
167C2×C53—22no—
168C2×C53—22no—
169C2×C53—41no—
170C2×C53—44no—
171C2×C53#188511yescharacteristic
172C2×C53#189611yescharacteristic
173C2×C53#191311yescharacteristic
174C2×C22⋊Q8C2×D5311yescharacteristic
175C22×C22⋊C4—1061no—
176C22×C22⋊C4C2×D5311yescharacteristic
177#407—532no—
178#397Dic5312yescomplemented
179#393Dic5312yescomplemented
180#388—22no—
181#381—21no—
182#379—21no—
183C4×C22⋊C4—532no—
184C4×C22⋊C4—1064no—
185C4×C22⋊C4—1061no—
186C4×C22⋊C4C2×D5311yescharacteristic
187#328—1064no—
188#323—1062no—
189C53#940711yescharacteristic, complemented, Sylow
190C25C22⋊D5311yescharacteristic
191C22×Q8C2×Dic5311yescharacteristic, complemented
192C23×C4—1062no—
193C23×C4C22⋊D5311yescharacteristic
194C23×C4C22×D5311yescharacteristic
195C2×M16—1062no—
196C22⋊Q8—22no—
197C22⋊Q8—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×Dic5311yescharacteristic, complemented
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—1068no—
207C2×C22⋊C4—1062no—
208C2×C22⋊C4—1062no—
209C2×C22⋊C4C2×Dic5312yes—
210C2×C22⋊C4C22⋊D5312yes—
211C2×C42—21no—
212C2×C42—1062no—
213C2×C42—1062no—
214C22⋊C8—1062no—
215C22⋊C8—1062no—
216#95—22no—
217#95—22no—
218#95—534no—
219#95—1062no—
220#95C2×Dic5312yescomplemented
221C24—21no—
222C24—24no—
223C24—21no—
224C24—42no—
225C24—81no—
226C24C22⋊Dic5312yes—
227C24C2×C22⋊D5311yescharacteristic
228C2×Q8—22no—
229C2×Q8—81no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—1068no—
238C22×C4—1064no—
239C22×C4—4242no—
240C22×C4—4242no—
241C22×C4C22⋊Dic5312yes—
242C22×C4D4×D5312yes—
243C22×C4#1171994712yes—
244C22×C4C22×Dic5311yescharacteristic
245C22×C4C2×C22⋊D5311yescharacteristic
246M16—4242no—
247C2×C8—2122no—
248C4⋊C4—22no—
249C4⋊C4—42no—
250C22⋊C4—24no—
251C22⋊C4—24no—
252C22⋊C4—42no—
253C22⋊C4—44no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—82no—
257C22⋊C4—1068no—
258C22⋊C4—1068no—
259C42—81no—
260C42—4242no—
261C42—4242no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23—22no—
269C23—41no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—81no—
275C23—81no—
276C23—88no—
277C23#1313314414yes—
278C23C2×C22⋊Dic5311yescharacteristic
279C23#1313318511yescharacteristic
280C23#1313319811yescharacteristic
281Q8—82no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—1068no—
298C2×C4—4248no—
299C2×C4—2124no—
300C2×C4—2124no—
301C2×C4D4×Dic5312yes—
302C2×C4#1313318912yes—
303C8—4242no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—24no—
311C22—21no—
312C22—44no—
313C22—44no—
314C22—44no—
315C22—44no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22#1681050311yescharacteristic
321C22C2×#1313314412yes—
322C22#1681080012yes—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—82no—
327C4—81no—
328C4—81no—
329C4—2128no—
330C2—22no—
331C2—22no—
332C2—41no—
333C2—44no—
334C2#2027510411yescharacteristic
335C2#2027515911yescharacteristic
336C2#2027525811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2561C2
socleC22×C53#386
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenter#2561D53
hypocenterC53#9407
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
2. #20273235C2semidirect
4. #20272601C2semidirect
15. C53×#397C4semidirect
16. C53×#393C4semidirect
33. C22×Q8×C53C2×C4semidirect
39. C2×C4⋊C4×C53C2×C4semidirect
51. #95×C53C2×C4semidirect
178. #397Dic53semidirect
179. #393Dic53semidirect
189. C53#9407semidirect
191. C22×Q8C2×Dic53semidirect
199. C2×C4⋊C4C2×Dic53semidirect
220. #95C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202770682
#202732352
#202730422
#202726012
C53×#25612
#940753