Finite Group Databasebeta

← #27282064

Subgroups — #27282064

Order 13568.

522 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283756C211yesmaximal, characteristic
2#20279032C211yesmaximal, characteristic
3#20274793C212yesmaximal, complemented
4#20272622C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7#16810701—24no—
8#16810676—21no—
9#16810587—21no—
10#16810587C2212yes—
11#16810562—21no—
12#16810559—21no—
13#16810521—21no—
14#16810503C412yes—
15#16810495C412yes—
16#16810483—21no—
17#16810483C2211yescharacteristic
18#16810391—24no—
19C2×C4×D4×C53C2211yescharacteristic
20C53×#407C2212yes—
21C53×#324—21no—
22C53×#323—21no—
23C23×Dic53C2×C412yes—
24C2×#11719917—22no—
25C2×#11719917D414yes—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53C2×C412yes—
34C4⋊#88322—22no—
35C22⋊#88322—22no—
36C22⋊#88322—22no—
37C22⋊#88322—22no—
38C4×#88322—22no—
39#13133124—22no—
40#13133124—22no—
41#13133124—21no—
42C22×D4×C53C2×C411yescharacteristic
43C23×C4×C53D411yescharacteristic
44C23×C4×C53C2311yescharacteristic
45C2×M16×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C24×C53C22⋊C411yescharacteristic
55C24×C53C22×C411yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—42no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C2×D412yes—
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53#4112yes—
70C22×Dic53—22no—
71C22×Dic53—22no—
72C22×Dic53—24no—
73C22×Dic53—82no—
74M16×C53—44no—
75C2×C8×C53—42no—
76C4⋊C4×C53—22no—
77C22⋊C4×C53—22no—
78C22⋊C4×C53—22no—
79C42×C53—21no—
80C42×C53—21no—
81C22⋊Dic53—42no—
82C22⋊Dic53—82no—
83C4⋊Dic53—42no—
84C4⋊Dic53—41no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87#11719919—82no—
88#11719917—44no—
89#11719917—82no—
90C2×#88322—24no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53—41no—
100C23×C53C2×C22⋊C411yescharacteristic
101C23×C53C4×D412yes—
102D4×C53—44no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—24no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—41no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—81no—
115C2×C4×C53#10112yes—
116C2×C4×C53#12011yescharacteristic
117C2×C4×C53#12112yes—
118C2×C4×C53#12311yescharacteristic
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—42no—
125C2×Dic53—84no—
126C8×C53—82no—
127#88322—44no—
128#9510—531nomaximal, Sylow
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—22no—
132C22×C53—22no—
133C22×C53—22no—
134C22×C53—42no—
135C22×C53—42no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53—81no—
141C22×C53#38611yescharacteristic
142C22×C53C2×#10111yescharacteristic
143C22×C53#41311yescharacteristic
144C4×C53—22no—
145C4×C53—21no—
146C4×C53—22no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53#45812yes—
150Dic53—82no—
151Dic53—82no—
152#2975—531no—
153#2487—531no—
154#1821—532no—
155#1465D5311yescharacteristic
156#1305—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—42no—
160C2×C53—81no—
161C2×C53#189011yescharacteristic
162C2×C53#190111yescharacteristic
163C2×C53#191411yescharacteristic
164C2×C4×D4C2×D5311yescharacteristic
165C22×C4⋊C4—531no—
166#436—1064no—
167#431—1061no—
168#407—532no—
169#407—1061no—
170#407C2×D5312yes—
171#400—1061no—
172#399—1061no—
173#389—1061no—
174#386—532no—
175#385—532no—
176#380—531no—
177#380—1061no—
178#343—1064no—
179#324—21no—
180#323—21no—
181C53#951011yescharacteristic, complemented, Sylow
182C22×D4C4×D5311yescharacteristic
183C23×C4—532no—
184C23×C4D21211yescharacteristic
185C23×C4C22×D5311yescharacteristic
186C2×M16—22no—
187C2×M16—534no—
188C2×M16—1062no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—1062no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4—1062no—
195C2×C4⋊C4—1062no—
196C2×C4⋊C4—1061no—
197C2×C4⋊C4C22⋊D5311yescharacteristic
198C2×C22⋊C4—532no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4C4×D5311yescharacteristic
201C2×C22⋊C4C22⋊D5311yescharacteristic
202C2×C42D21211yescharacteristic
203C4⋊C8—1062no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—1062no—
207C22⋊C8—1062no—
208C22⋊C8—1062no—
209C4×C8—1062no—
210#95—1061no—
211#95—1062no—
212#95—1062no—
213C24#1171992111yescharacteristic
214C24C2×C4×D5311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—41no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—1064no—
223C22×C4—1062no—
224C22×C4—1062no—
225C22×C4—4242no—
226C22×C4#1171992111yescharacteristic
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4C2×D21211yescharacteristic
230C22×C4D4×D5312yes—
231C22×C4#1171994712yes—
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—44no—
234M16—4242no—
235M16—2124no—
236C2×C8—42no—
237C2×C8—1064no—
238C4⋊C4—22no—
239C4⋊C4—4242no—
240C4⋊C4—2122no—
241C4⋊C4—2121no—
242C4⋊C4—2122no—
243C4⋊C4—2121no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—4242no—
247C22⋊C4—2122no—
248C42—21no—
249C42—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C2×#1171992111yescharacteristic
260C23#1313317712yes—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—24no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—81no—
274C2×C4—4244no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4—2122no—
280C2×C4—2122no—
281C2×C4#1313313512yes—
282C2×C4#1313318711yescharacteristic
283C2×C4#1313318912yes—
284C2×C4#1313320111yescharacteristic
285C8—82no—
286C8—2124no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22#1681050011yescharacteristic
300C22C2×#1313313511yescharacteristic
301C22#1681060511yescharacteristic
302C4—22no—
303C4—21no—
304C4—22no—
305C4—41no—
306C4—41no—
307C4—4242no—
308C4—4242no—
309C4#1681081412yes—
310C2—21no—
311C2—21no—
312C2—42no—
313C2—81no—
314C2#2027512611yescharacteristic
315C2#2027518111yescharacteristic
316C2#2027526011yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC24×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1465C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×C4×C53C23
hypercenter#1465D53
hypocenterC53#9510
LCSG ▷ C24×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1465

Complements

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

NH ≅ G/Ntype
3. #20274793C2semidirect
4. #20272622C2semidirect
181. C53#9510semidirect

Maximal subgroups

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

subgroupindex
#202837562
#202790322
#202747932
#202726222
C53×#14652
#951053