Finite Group Databasebeta

← #27294618

Subgroups — #27294618

Order 13568.

420 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 91 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#20274454C211yesmaximal, characteristic
2#20273049C211yesmaximal, characteristic
3#20272742C212yesmaximal, complemented
4#20272721C212yesmaximal, complemented
5C53×#2611C211yesmaximal, characteristic
6C22×#11719917—21no—
7#16810680—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—24no—
24C2×#11719917—22no—
25C2×#11719917—41no—
26C2×#11719917—42no—
27C22×#88322—41no—
28C4⋊#88322—41no—
29C22⋊#88322—22no—
30#13133128—41no—
31C23×C4×C53D411yescharacteristic
32#125×C53—41no—
33#125×C53—41no—
34#117×C53—41no—
35#117×C53—41no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53—21no—
39C2×C4⋊C4×C53—21no—
40C2×C4⋊C4×C53C2×C411yescharacteristic
41C2×C4⋊C4×C53C2×C411yescharacteristic
42C2×C4⋊C4×C53D411yescharacteristic
43C2×C4⋊C4×C53C2311yescharacteristic
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53—21no—
49C2×C42×C53C2×C411yescharacteristic
50C2×C42×C53D411yescharacteristic
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C22×C4×C53—22no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—42no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53—41no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—41no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53—21no—
92C42×C53—42no—
93#11719917—24no—
94#11719917—44no—
95#11719917—42no—
96C2×#88322—44no—
97C2×#88322—42no—
98C2×#88322—42no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53#10412yes—
104C23×C53C2×C22⋊C411yescharacteristic
105C2×C4×C53—22no—
106C2×C4×C53—22no—
107C2×C4×C53—22no—
108C2×C4×C53—22no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
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—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53#9911yescharacteristic
134C2×C4×C53#9911yescharacteristic
135C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C53#13711yescharacteristic
137#88322—44no—
138#88322—82no—
139#11767—531nomaximal, Sylow
140C22×C53—21no—
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—42no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—81no—
147C22×C53C2×#9911yescharacteristic
148C22×C53#41711yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—21no—
154C4×C53—42no—
155C4×C53—41no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—81no—
160C4×C53—81no—
161C4×C53—81no—
162#2611D5311yescharacteristic
163#1742—531no—
164#1464—531no—
165#1353—532no—
166#1346—532no—
167C2×C53—21no—
168C2×C53—21no—
169C2×C53—42no—
170C2×C53#150411yescharacteristic
171C2×C53#212311yescharacteristic
172C2×C53#213211yescharacteristic
173C22×M16—1061no—
174C2×#125Dic5311yescharacteristic
175C2×#117C2×D5311yescharacteristic
176#432—1061no—
177#400C2×D5311yescharacteristic
178#399Dic5311yescharacteristic
179#398C2×D5311yescharacteristic
180#397Dic5311yescharacteristic
181#389—21no—
182#388—21no—
183#387—21no—
184#385—21no—
185#376Dic5311yescharacteristic
186#343—532no—
187#343—1061no—
188#324—532no—
189#323—1061no—
190C53#1176711yescharacteristic, complemented, Sylow
191C23×C4C22⋊D5311yescharacteristic
192C2×M16—1064no—
193C2×M16—1062no—
194C2×M16—2121no—
195C2×M16—2122no—
196C22×C8—2121no—
197#125—41no—
198#125—41no—
199#117—41no—
200#117—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C2×Dic5311yescharacteristic
206C2×C4⋊C4C2×Dic5311yescharacteristic
207C2×C4⋊C4C22⋊D5311yescharacteristic
208C2×C4⋊C4C22×D5311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4C2×Dic5311yescharacteristic
212C2×C22⋊C4C22⋊D5311yescharacteristic
213C2×C42—21no—
214C2×C42C2×Dic5311yescharacteristic
215C2×C42C22⋊D5311yescharacteristic
216C4⋊C8—2121no—
217C22⋊C8—1062no—
218#97—2121no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95C2×Dic5311yescharacteristic
225#95C2×Dic5311yescharacteristic
226C24C22⋊Dic5311yescharacteristic
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4C22⋊Dic5311yescharacteristic
237C22×C4C22⋊Dic5311yescharacteristic
238C22×C4C22⋊Dic5311yescharacteristic
239C22×C4C22×Dic5311yescharacteristic
240C22×C4C2×C22⋊D5311yescharacteristic
241C22×C4C2×C22⋊D5311yescharacteristic
242M16—1064no—
243M16—2124no—
244M16—2122no—
245C2×C8—2122no—
246C2×C8—2124no—
247C2×C8—2122no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—42no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23#1313314412yes—
272C23C2×C22⋊Dic5311yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4#1313313211yescharacteristic
302C2×C4#1313313211yescharacteristic
303C2×C4#1313324211yescharacteristic
304C2×C4#1313324711yescharacteristic
305C8—4242no—
306C8—2124no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—81no—
314C22C2×#1313313211yescharacteristic
315C22#1681062411yescharacteristic
316C22C2×#1313314411yescharacteristic
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—42no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—81no—
327C4—81no—
328C4—81no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2#2027321911yescharacteristic
333C2#2027636911yescharacteristic
334C2#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#11767
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
190. C53#11767semidirect

Maximal subgroups

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

subgroupindex
#202744542
#202730492
#202727422
#202727212
C53×#26112
#1176753