Finite Group Databasebeta

← #27295020

Subgroups — #27295020

Order 13568.

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

normal subgroups only

This group has 368 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#20274798C211yesmaximal, characteristic
2#20273029C211yesmaximal, characteristic
3#20272742C212yesmaximal, complemented
4#20272721C212yesmaximal, complemented
5C53×#2611C211yesmaximal, characteristic
6#16810676—21no—
7#16810587—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×#125×C53C411yescharacteristic
13C2×#117×C53C2211yescharacteristic
14C53×#400C2211yescharacteristic
15C53×#399C411yescharacteristic
16C53×#398C2211yescharacteristic
17C53×#397C411yescharacteristic
18C53×#389—21no—
19C53×#388—21no—
20C53×#387—21no—
21C53×#385—21no—
22C53×#376C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C4×#88322—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×C53D411yescharacteristic
41C2×C4⋊C4×C53C2311yescharacteristic
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53D411yescharacteristic
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×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53—22no—
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×C53C2×D411yescharacteristic
70C22×C4×C53C2×D411yescharacteristic
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53—42no—
74C4⋊C4×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—42no—
80C22⋊C4×C53—41no—
81C22⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C22⋊C4×C53—42no—
84C42×C53—21no—
85C42×C53—21no—
86C42×C53—21no—
87C42×C53—21no—
88C42×C53—42no—
89#11719917—42no—
90#11719917—42no—
91C2×#88322—42no—
92C2×#88322—42no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—22no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
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#9911yescharacteristic
126C2×C4×C53#9911yescharacteristic
127C2×C4×C53#13512yes—
128#88322—82no—
129#88322—82no—
130#11853—531nomaximal, Sylow
131C22×C53—21no—
132C22×C53—21no—
133C22×C53—42no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—81no—
138C22×C53C2×#9911yescharacteristic
139C22×C53#41611yescharacteristic
140C22×C53#42111yescharacteristic
141C4×C53—22no—
142C4×C53—21no—
143C4×C53—21no—
144C4×C53—42no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53—41no—
149C4×C53—81no—
150C4×C53—81no—
151C4×C53—81no—
152#2611D5311yescharacteristic
153#1822—531no—
154#1458—531no—
155#1353—532no—
156#1346—532no—
157C2×C53—21no—
158C2×C53—21no—
159C2×C53—81no—
160C2×C53#151611yescharacteristic
161C2×C53#212311yescharacteristic
162C2×C53#213211yescharacteristic
163C2×#125Dic5311yescharacteristic
164C2×#117C2×D5311yescharacteristic
165#431—1061no—
166#407—1061no—
167#400C2×D5311yescharacteristic
168#399Dic5311yescharacteristic
169#398C2×D5311yescharacteristic
170#397Dic5311yescharacteristic
171#389—21no—
172#388—21no—
173#387—21no—
174#385—21no—
175#376Dic5311yescharacteristic
176#343—532no—
177#343—1061no—
178#324—532no—
179#323—1061no—
180C53#1185311yescharacteristic, complemented, Sylow
181C23×C4C22⋊D5311yescharacteristic
182C2×M16—1062no—
183#125—41no—
184#125—41no—
185#117—41no—
186#117—41no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C2×Dic5311yescharacteristic
192C2×C4⋊C4C2×Dic5311yescharacteristic
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4C2×Dic5311yescharacteristic
198C2×C22⋊C4C22⋊D5311yescharacteristic
199C2×C42—21no—
200C2×C42C2×Dic5311yescharacteristic
201C2×C42C22⋊D5311yescharacteristic
202C4⋊C8—2121no—
203C22⋊C8—1062no—
204C22⋊C8—2121no—
205C22⋊C8—2121no—
206C4×C8—2121no—
207#95—21no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95C2×Dic5311yescharacteristic
213#95C2×Dic5311yescharacteristic
214C24C22⋊Dic5311yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—41no—
223C22×C4C22⋊Dic5311yescharacteristic
224C22×C4C22⋊Dic5311yescharacteristic
225C22×C4C22⋊Dic5311yescharacteristic
226C22×C4C22×Dic5311yescharacteristic
227C22×C4C2×C22⋊D5311yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229M16—2122no—
230M16—2122no—
231C2×C8—2122no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—42no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—42no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×C22⋊Dic5311yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
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#1313313211yescharacteristic
284C2×C4#1313313211yescharacteristic
285C2×C4#1313323712yes—
286C8—4242no—
287C8—4242no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1313313211yescharacteristic
296C22#1681061911yescharacteristic
297C22#1681064011yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—42no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—41no—
306C4—81no—
307C4—81no—
308C4—81no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#2027327111yescharacteristic
313C2#2027636911yescharacteristic
314C2#2027640011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2611C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2611D53
hypocenterC53#11853
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
3. #20272742C2semidirect
4. #20272721C2semidirect
180. C53#11853semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730292
#202727422
#202727212
C53×#26112
#1185353