Finite Group Databasebeta

← #27295000

Subgroups — #27295000

Order 13568.

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

normal subgroups only

This group has 376 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#20272745C212yesmaximal, complemented
4#20272721C212yesmaximal, complemented
5C53×#2599C211yesmaximal, characteristic
6#16810676—21no—
7#16810587—21no—
8#16810392—21no—
9#16810392C2212yes—
10#16810349C2212yes—
11#16810347—21no—
12C2×C4×Q8×C53C411yescharacteristic
13C2×#117×C53C2211yescharacteristic
14C53×#398C2211yescharacteristic
15C53×#397C411yescharacteristic
16C53×#395C2211yescharacteristic
17C53×#393C411yescharacteristic
18C53×#388—21no—
19C53×#387—21no—
20C53×#387—21no—
21C53×#385—21no—
22C53×#383C411yescharacteristic
23C2×#11719917—22no—
24C4⋊#88322—41no—
25C22⋊#88322—22no—
26C22⋊#88322—41no—
27C22⋊#88322—41no—
28C4×#88322—41no—
29C22×Q8×C53C2×C411yescharacteristic
30C23×C4×C53D411yescharacteristic
31C4×Q8×C53—41no—
32C4×Q8×C53—41no—
33#117×C53—41no—
34#117×C53—41no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53D411yescharacteristic
40C2×C4⋊C4×C53C2311yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53C2×C411yescharacteristic
44C2×C22⋊C4×C53D411yescharacteristic
45C2×C42×C53—21no—
46C2×C42×C53C2×C411yescharacteristic
47C2×C42×C53D411yescharacteristic
48#95×C53—21no—
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
57C2×Q8×C53—42no—
58C2×Q8×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—22no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C4⋊C4×C53—41no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—42no—
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
99Q8×C53—42no—
100Q8×C53—42no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—22no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—42no—
111C2×C4×C53—42no—
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—41no—
127C2×C4×C53#9911yescharacteristic
128C2×C4×C53#9911yescharacteristic
129C2×C4×C53#13512yes—
130#88322—82no—
131#88322—82no—
132#11849—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—42no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—41no—
139C22×C53—81no—
140C22×C53C2×#9911yescharacteristic
141C22×C53#41611yescharacteristic
142C22×C53#42111yescharacteristic
143C4×C53—22no—
144C4×C53—21no—
145C4×C53—21no—
146C4×C53—41no—
147C4×C53—42no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—41no—
151C4×C53—81no—
152C4×C53—81no—
153C4×C53—81no—
154#2599D5311yescharacteristic
155#1822—531no—
156#1458—531no—
157#1354—532no—
158#1346—532no—
159C2×C53—21no—
160C2×C53—21no—
161C2×C53—81no—
162C2×C53#151611yescharacteristic
163C2×C53#212011yescharacteristic
164C2×C53#213511yescharacteristic
165C2×C4×Q8Dic5311yescharacteristic
166C2×#117C2×D5311yescharacteristic
167#431—1061no—
168#407—1061no—
169#398C2×D5311yescharacteristic
170#397Dic5311yescharacteristic
171#395C2×D5311yescharacteristic
172#393Dic5311yescharacteristic
173#388—21no—
174#387—21no—
175#387—21no—
176#385—21no—
177#383Dic5311yescharacteristic
178#343—532no—
179#343—1061no—
180#324—532no—
181#323—1061no—
182C53#1184911yescharacteristic, complemented, Sylow
183C22×Q8C2×Dic5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C2×M16—1062no—
186C4×Q8—41no—
187C4×Q8—41no—
188#117—41no—
189#117—41no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic5311yescharacteristic
194C2×C4⋊C4C22⋊D5311yescharacteristic
195C2×C4⋊C4C22×D5311yescharacteristic
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4C2×Dic5311yescharacteristic
199C2×C22⋊C4C22⋊D5311yescharacteristic
200C2×C42—21no—
201C2×C42C2×Dic5311yescharacteristic
202C2×C42C22⋊D5311yescharacteristic
203C4⋊C8—2121no—
204C22⋊C8—1062no—
205C22⋊C8—2121no—
206C22⋊C8—2121no—
207C4×C8—2121no—
208#95—21no—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95C2×Dic5311yescharacteristic
215#95C2×Dic5311yescharacteristic
216C24C22⋊Dic5311yescharacteristic
217C2×Q8—42no—
218C2×Q8—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—41no—
227C22×C4C22⋊Dic5311yescharacteristic
228C22×C4C22⋊Dic5311yescharacteristic
229C22×C4C22⋊Dic5311yescharacteristic
230C22×C4C22×Dic5311yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232C22×C4C2×C22⋊D5311yescharacteristic
233M16—2122no—
234M16—2122no—
235C2×C8—2122no—
236C2×C8—2122no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—42no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊Dic5311yescharacteristic
259Q8—42no—
260Q8—42no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
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#1313313211yescharacteristic
288C2×C4#1313313211yescharacteristic
289C2×C4#1313323712yes—
290C8—4242no—
291C8—4242no—
292C22—21no—
293C22—21no—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22C2×#1313313211yescharacteristic
300C22#1681061911yescharacteristic
301C22#1681064011yescharacteristic
302C4—22no—
303C4—21no—
304C4—21no—
305C4—41no—
306C4—42no—
307C4—41no—
308C4—41no—
309C4—41no—
310C4—81no—
311C4—81no—
312C4—81no—
313C2—21no—
314C2—21no—
315C2—81no—
316C2#2027327111yescharacteristic
317C2#2027635611yescharacteristic
318C2#2027641011yescharacteristic
subgroupG/H
centerC22C2×#13133132
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#2599C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#2599D53
hypocenterC53#11849
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2599

Complements

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

NH ≅ G/Ntype
3. #20272745C2semidirect
4. #20272721C2semidirect
182. C53#11849semidirect

Maximal subgroups

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

subgroupindex
#202747982
#202730292
#202727452
#202727212
C53×#25992
#1184953