Finite Group Databasebeta

← #27295018

Subgroups — #27295018

Order 13568.

368 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 79 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#20273029C211yesmaximal, characteristic
3#20272741C212yesmaximal, complemented
4#20272720C212yesmaximal, complemented
5C53×#1822C211yesmaximal, characteristic
6C2×#13133209C411yescharacteristic
7#16810676—21no—
8#16810562C2211yescharacteristic
9#16810559C411yescharacteristic
10#16810553C2211yescharacteristic
11#16810548C411yescharacteristic
12#16810518—21no—
13#16810517—21no—
14#16810507—21no—
15#16810496—21no—
16#16810469C411yescharacteristic
17#16810349C2212yes—
18#16810347—21no—
19C2×#117×C53C2211yescharacteristic
20C53×#407—21no—
21C53×#343—21no—
22C53×#343C2212yes—
23#13133209—41no—
24#13133209—41no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C411yescharacteristic
27C2×#11719919—21no—
28C2×#11719919—21no—
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×C22⋊Dic53—21no—
32C2×C22⋊Dic53—21no—
33C2×C4×Dic53—21no—
34C4⋊#88322—41no—
35C22⋊#88322—22no—
36C4×#88322—41no—
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124C2×C411yescharacteristic
43#13133124C2×C411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C2×M16×C53—22no—
46#117×C53—41no—
47#117×C53—41no—
48C2×C4⋊C4×C53D411yescharacteristic
49C2×C4⋊C4×C53C2311yescharacteristic
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C22⋊C8×C53—41no—
55C22⋊C8×C53—41no—
56C24×C53C22⋊C411yescharacteristic
57C22×C4×C53—21no—
58C22×C4×C53—21no—
59C22×C4×C53—22no—
60C22×C4×C53—41no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×D411yescharacteristic
67C22×Dic53—21no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53—21no—
71M16×C53—42no—
72M16×C53—42no—
73C2×C8×C53—42no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C22⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78C42×C53—21no—
79C42×C53—21no—
80C42×C53—21no—
81C42×C53—21no—
82C22⋊Dic53—42no—
83C22⋊Dic53—42no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86#11719919—42no—
87#11719919—42no—
88#11719919—41no—
89#11719919—41no—
90#11719919—42no—
91C4×Dic53—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—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—42no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53#9911yescharacteristic
114C2×C4×C53#9911yescharacteristic
115C2×C4×C53#13512yes—
116C2×Dic53—22no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—41no—
120C2×Dic53—41no—
121C2×Dic53—41no—
122C2×Dic53—41no—
123C2×Dic53—41no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C8×C53—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—41no—
145C4×C53—41no—
146C4×C53—41no—
147C4×C53—41no—
148Dic53—42no—
149Dic53—81no—
150Dic53—81no—
151Dic53—81no—
152#2611—531no—
153#1822D5311yescharacteristic
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×#125—531no—
164C2×#117C2×D5311yescharacteristic
165#431—1061no—
166#407—21no—
167#400—531no—
168#399—531no—
169#398—531no—
170#397—531no—
171#389—1061no—
172#388—1061no—
173#387—1061no—
174#385—1061no—
175#376—531no—
176#343—21no—
177#343C2×D5312yes—
178#324—532no—
179#323—1061no—
180C53#1185311yescharacteristic, complemented, Sylow
181C23×C4C22⋊D5311yescharacteristic
182C2×M16—22no—
183#125—2121no—
184#125—2121no—
185#117—41no—
186#117—41no—
187C2×C4⋊C4—531no—
188C2×C4⋊C4—531no—
189C2×C4⋊C4—1061no—
190C2×C4⋊C4—1061no—
191C2×C4⋊C4—1061no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4C22⋊D5311yescharacteristic
194C2×C4⋊C4C22×D5311yescharacteristic
195C2×C22⋊C4—1061no—
196C2×C22⋊C4—1061no—
197C2×C22⋊C4C4×D5311yescharacteristic
198C2×C22⋊C4D21211yescharacteristic
199C2×C42—1061no—
200C2×C42C4×D5311yescharacteristic
201C2×C42D21211yescharacteristic
202C4⋊C8—2121no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—1062no—
206C4×C8—2121no—
207#95—531no—
208#95—531no—
209#95—1061no—
210#95—1061no—
211#95—1061no—
212#95—1061no—
213#95—1061no—
214C24#1171992111yescharacteristic
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—41no—
219C22×C4—1061no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4—1061no—
223C22×C4#1171992111yescharacteristic
224C22×C4#1171992111yescharacteristic
225C22×C4#1171992111yescharacteristic
226C22×C4C2×C4×D5311yescharacteristic
227C22×C4C2×D21211yescharacteristic
228C22×C4C2×C22⋊D5311yescharacteristic
229M16—42no—
230M16—42no—
231C2×C8—42no—
232C2×C8—2122no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—2121no—
236C4⋊C4—2121no—
237C4⋊C4—2122no—
238C4⋊C4—2122no—
239C4⋊C4—2121no—
240C4⋊C4—2121no—
241C4⋊C4—2122no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—2122no—
245C22⋊C4—2122no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—2122no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23C2×#1171992111yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—1062no—
272C2×C4—2121no—
273C2×C4—2121no—
274C2×C4—2121no—
275C2×C4—2121no—
276C2×C4—2121no—
277C2×C4—2121no—
278C2×C4—2121no—
279C2×C4—2121no—
280C2×C4—2121no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4#1313313111yescharacteristic
284C2×C4#1313313111yescharacteristic
285C2×C4#1313323712yes—
286C8—82no—
287C8—4242no—
288C22—21no—
289C22—21no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22C2×#1313313111yescharacteristic
296C22#1681061711yescharacteristic
297C22#1681064111yescharacteristic
298C4—22no—
299C4—21no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—4241no—
306C4—4241no—
307C4—4241no—
308C4—2122no—
309C2—21no—
310C2—21no—
311C2—81no—
312C2#2027326911yescharacteristic
313C2#2027637011yescharacteristic
314C2#2027639811yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1822C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1822D53
hypocenterC53#11853
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
3. #20272741C2semidirect
4. #20272720C2semidirect
180. C53#11853semidirect

Maximal subgroups

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

subgroupindex
#202804132
#202730292
#202727412
#202727202
C53×#18222
#1185353