Finite Group Databasebeta

← #27294985

Subgroups — #27294985

Order 13568.

382 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 87 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#20274792C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272739C212yesmaximal
4#20272715C212yesmaximal
5C53×#2611C211yesmaximal, characteristic
6#16810676—21no—
7#16810486—21no—
8#16810390—21no—
9#16810390C2212yes—
10#16810349—21no—
11#16810347C2212yes—
12C2×#125×C53C411yescharacteristic, complemented
13C2×#117×C53C2211yescharacteristic
14C53×#400C411yescharacteristic, complemented
15C53×#399C2211yescharacteristic
16C53×#398C411yescharacteristic, complemented
17C53×#397C2211yescharacteristic
18C53×#389—21no—
19C53×#388—21no—
20C53×#387—21no—
21C53×#385—21no—
22C53×#376C411yescharacteristic, complemented
23C2×C22⋊Dic53—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C4×#88322—41no—
27#13133124—41no—
28#13133124—41no—
29C23×C4×C53D411yescharacteristic
30#125×C53—41no—
31#125×C53—41no—
32#117×C53—41no—
33#117×C53—41no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53D411yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C22⋊C4×C53C2311yescharacteristic
46C2×C42×C53—21no—
47C2×C42×C53C2×C411yescharacteristic
48C2×C42×C53D411yescharacteristic
49#95×C53—21no—
50#95×C53—21no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53C2×C411yescharacteristic
55#95×C53C2×C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—41no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C22×Dic53—42no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—41no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—42no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—41no—
83C22⋊C4×C53—42no—
84C22⋊C4×C53—42no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53#13512yes—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—22no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—42no—
110C2×C4×C53—42no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#9911yescharacteristic
128C2×Dic53—82no—
129C2×Dic53—82no—
130C2×Dic53—82no—
131#88322—82no—
132#11846—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—22no—
136C22×C53—42no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—81no—
140C22×C53C2×#9911yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—21no—
144C4×C53—21no—
145C4×C53—42no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—81no—
152C4×C53—81no—
153C4×C53—81no—
154Dic53—162no—
155#2611D5311yescharacteristic
156#1820—531no—
157#1458—531no—
158#1352—532no—
159#1344—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53#151611yescharacteristic
164C2×C53#212311yescharacteristic
165C2×C53#212611yescharacteristic
166C2×#125Dic5311yescharacteristic, complemented
167C2×#117C2×D5311yescharacteristic
168#431—1061no—
169#400Dic5311yescharacteristic, complemented
170#399C2×D5311yescharacteristic
171#398Dic5311yescharacteristic, complemented
172#397C2×D5311yescharacteristic
173#389—21no—
174#388—21no—
175#387—21no—
176#385—21no—
177#381—1061no—
178#376Dic5311yescharacteristic, complemented
179#342—532no—
180#342—1061no—
181#324—1061no—
182#323—532no—
183C53#1184611yescharacteristic, complemented, Sylow
184C23×C4C22⋊D5311yescharacteristic
185#125—41no—
186#125—41no—
187#117—41no—
188#117—41no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5311yescharacteristic
194C2×C4⋊C4C2×Dic5311yescharacteristic
195C2×C4⋊C4C2×Dic5311yescharacteristic
196C2×C4⋊C4C22⋊D5311yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—1062no—
200C2×C22⋊C4C22⋊D5311yescharacteristic
201C2×C22⋊C4C22×D5311yescharacteristic
202C2×C42—21no—
203C2×C42C2×Dic5311yescharacteristic
204C2×C42C22⋊D5311yescharacteristic
205C4⋊C8—2121no—
206C22⋊C8—1062no—
207C4×C8—2121no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—2121no—
214#95—2121no—
215#95C2×Dic5311yescharacteristic
216#95C2×Dic5311yescharacteristic
217C24C2×C22⋊D5311yescharacteristic
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—2122no—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22⋊Dic5311yescharacteristic
231C22×C4C22×Dic5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233C2×C8—2122no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—2122no—
248C22⋊C4—2122no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—42no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic5311yescharacteristic
259C23#1313323712yes—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—4242no—
288C2×C4—4242no—
289C2×C4—4242no—
290C2×C4#1313313211yescharacteristic
291C2×C4#1313313211yescharacteristic
292C8—4242no—
293C22—21no—
294C22—21no—
295C22—22no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22C2×#1313313211yescharacteristic
301C22#1681061911yescharacteristic
302C22#1681064011yescharacteristic
303C4—21no—
304C4—21no—
305C4—42no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4—81no—
313C4—81no—
314C4—8482no—
315C2—21no—
316C2—21no—
317C2—42no—
318C2#2027327111yescharacteristic
319C2#2027636911yescharacteristic
320C2#2027638011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2611C2
socleC22×C53C2×#99
prime radicalC24×C53C2×D4
cosocleC2×C22⋊C4×C53C23
hypercenter#2611D53
hypocenterC53#11846
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2611

Complements

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

NH ≅ G/Ntype
12. C2×#125×C53C4semidirect
14. C53×#400C4semidirect
16. C53×#398C4semidirect
22. C53×#376C4semidirect
166. C2×#125Dic53semidirect
169. #400Dic53semidirect
171. #398Dic53semidirect
178. #376Dic53semidirect
183. C53#11846semidirect

Maximal subgroups

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

subgroupindex
#202747922
#202730292
#202727392
#202727152
C53×#26112
#1184653