Finite Group Databasebeta

← #27281432

Subgroups — #27281432

Order 13568.

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

normal subgroups only

This group has 532 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#20280413C211yesmaximal, characteristic
2#20273316C212yesmaximal, complemented
3#20273049C211yesmaximal, characteristic
4#20272609C212yesmaximal, complemented
5C53×#2873C211yesmaximal, characteristic
6C2×#13133209C2211yescharacteristic
7#16810680C2212yes—
8#16810562C411yescharacteristic
9#16810559C411yescharacteristic
10#16810553C411yescharacteristic
11#16810548C411yescharacteristic
12#16810518—21no—
13#16810517—21no—
14#16810507—21no—
15#16810496—21no—
16#16810469C2211yescharacteristic
17#16810364—24no—
18#16810349—21no—
19#16810347—21no—
20C22×M16×C53—21no—
21C2×#117×C53C2211yescharacteristic
22C53×#405—24no—
23C4×M16×C53—21no—
24C4×M16×C53C2212yes—
25#13133209—22no—
26#13133209—22no—
27C2×C4⋊Dic53C2×C411yescharacteristic
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—21no—
35C2×C4×Dic53—21no—
36C4⋊#88322—22no—
37C22⋊#88322—22no—
38#13133128—22no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124C2×C411yescharacteristic
45#13133124C2×C411yescharacteristic
46C23×C4×C53D411yescharacteristic
47C2×M16×C53—24no—
48C2×M16×C53—24no—
49C2×M16×C53—22no—
50C2×M16×C53—22no—
51C22×C8×C53—22no—
52#117×C53—22no—
53#117×C53—22no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53D411yescharacteristic
56C2×C22⋊C4×C53C2×C411yescharacteristic
57C2×C22⋊C4×C53D411yescharacteristic
58C2×C42×C53D411yescharacteristic
59C2×C42×C53C2311yescharacteristic
60#97×C53—22no—
61#97×C53D414yes—
62C4×C8×C53—22no—
63C4×C8×C53—22no—
64C24×C53C22⋊C411yescharacteristic
65C22×C4×C53—22no—
66C22×C4×C53—22no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53C2×D411yescharacteristic
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—21no—
78C22×Dic53—21no—
79M16×C53—28no—
80M16×C53—24no—
81M16×C53—24no—
82C2×C8×C53—28no—
83C2×C8×C53—24no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53C2×D412yes—
91C42×C53#4112yes—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—22no—
95C4⋊Dic53—22no—
96#11719919—42no—
97#11719919—42no—
98#11719919—41no—
99#11719919—42no—
100#11719919—41no—
101C4×Dic53—42no—
102C2×#88322—42no—
103C23×C53—21no—
104C23×C53—21no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53#10412yes—
108C23×C53C2×C22⋊C411yescharacteristic
109C2×C4×C53—24no—
110C2×C4×C53—22no—
111C2×C4×C53—22no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53#10412yes—
123C2×C4×C53C4×D412yes—
124C2×C4×C53#12011yescharacteristic
125C2×C4×C53#12112yes—
126C2×C4×C53#12311yescharacteristic
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C8×C53—28no—
140#88322—82no—
141#9388—531nomaximal, Sylow
142C22×C53—21no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—81no—
149C22×C53#38611yescharacteristic
150C22×C53C2×#10411yescharacteristic
151C22×C53C2×#10411yescharacteristic
152C4×C53—22no—
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—41no—
158C4×C53—41no—
159Dic53—42no—
160Dic53—81no—
161Dic53—81no—
162Dic53—81no—
163#2873D5311yescharacteristic
164#2611—531no—
165#1528—532no—
166#1464—531no—
167#1300—532no—
168C2×C53—21no—
169C2×C53—21no—
170C2×C53—42no—
171C2×C53#188411yescharacteristic
172C2×C53#189611yescharacteristic
173C2×C53#191511yescharacteristic
174C22×M16—21no—
175C2×#125—531no—
176C2×#117C2×D5311yescharacteristic
177#432—532no—
178#405—24no—
179C4×M16—21no—
180C4×M16C2×D5312yes—
181#400—531no—
182#399—531no—
183#398—531no—
184#397—531no—
185#389—1061no—
186#388—1061no—
187#387—1061no—
188#385—1061no—
189#376—531no—
190#330—1064no—
191#324—1061no—
192#323—1061no—
193C53#938811yescharacteristic, complemented, Sylow
194C23×C4C22⋊D5311yescharacteristic
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—22no—
199C22×C8—22no—
200#125—1062no—
201#125—1062no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—531no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C4⋊C4D21211yescharacteristic
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4C4×D5311yescharacteristic
215C2×C22⋊C4D21211yescharacteristic
216C2×C42—1061no—
217C2×C42C22⋊D5311yescharacteristic
218C2×C42C22×D5311yescharacteristic
219C4⋊C8—1062no—
220C22⋊C8—1062no—
221#97—22no—
222#97—1062no—
223#97C22⋊D5314yes—
224C4×C8—22no—
225C4×C8—22no—
226#95—531no—
227#95—531no—
228#95—1061no—
229#95—1061no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233C24#1171992111yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4#1171992111yescharacteristic
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4C2×C4×D5311yescharacteristic
246C22×C4C2×D21211yescharacteristic
247C22×C4C2×C22⋊D5311yescharacteristic
248M16—28no—
249M16—24no—
250M16—24no—
251C2×C8—28no—
252C2×C8—24no—
253C2×C8—2122no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—1062no—
257C4⋊C4—1062no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C4⋊C4—2121no—
261C4⋊C4—2122no—
262C4⋊C4—2121no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—21no—
269C42—2122no—
270C42#1171994512yes—
271C42C2×C22⋊D5312yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1313314512yes—
277C23C2×#1171992111yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—1062no—
292C2×C4—2121no—
293C2×C4—2121no—
294C2×C4—2121no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4#1313314512yes—
304C2×C4C4×C22⋊D5312yes—
305C2×C4#1313318611yescharacteristic
306C2×C4#1313319112yes—
307C2×C4#1313320011yescharacteristic
308C8—28no—
309C8—4242no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1681050211yescharacteristic
318C22C2×#1313314511yescharacteristic
319C22C2×#1313314511yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—4241no—
328C4—4241no—
329C4—4241no—
330C4—2122no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2027509811yescharacteristic
335C2#2027515811yescharacteristic
336C2#2027526711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2873C2
socleC22×C53#386
prime radicalC24×C53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#2873D53
hypocenterC53#9388
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. #20273316C2semidirect
4. #20272609C2semidirect
193. C53#9388semidirect

Maximal subgroups

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

subgroupindex
#202804132
#202733162
#202730492
#202726092
C53×#28732
#938853