Finite Group Databasebeta

← #28300017

Subgroups — #28300017

Order 13568.

450 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 90 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#20290605C211yesmaximal, characteristic
2#20290571C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#20283023C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133208C411yescharacteristic
13C22×C4×Dic53C2211yescharacteristic
14#16810522C2211yescharacteristic
15#16810519C2211yescharacteristic
16#16810510—21no—
17#16810506—21no—
18#16810498—21no—
19#16810495—21no—
20#16810488C411yescharacteristic
21#16810483C411yescharacteristic
22C53×#380C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C23×Dic53D411yescharacteristic
29C23×Dic53D411yescharacteristic
30#13133208—42no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53C2×C411yescharacteristic
34C2×C4⋊Dic53C2×C411yescharacteristic
35C2×#11719919—21no—
36C2×#11719919—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53C2311yescharacteristic
43#13133124—21no—
44#13133124—21no—
45#13133124—21no—
46#13133124—21no—
47#13133124—21no—
48C23×C4×C53C2×C411yescharacteristic
49C2×C4⋊C4×C53C2×C411yescharacteristic
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—21no—
52#95×C53—21no—
53C24×C53C22⋊C411yescharacteristic
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22×C411yescharacteristic
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—41no—
68C22×Dic53—41no—
69C22×Dic53—41no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×#88324—42no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—41no—
82C22⋊C4×C53—42no—
83C22⋊Dic53—42no—
84C4⋊Dic53—22no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4⋊Dic53—41no—
89#11719919—42no—
90#11719919—42no—
91C4×Dic53—22no—
92C4×Dic53—22no—
93C4×Dic53—41no—
94C4×Dic53—41no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53#10112yes—
102C23×C53C2×C22⋊C411yescharacteristic
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—44no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—41no—
113C2×C4×C53—41no—
114C2×C4×C53—41no—
115C2×C4×C53#13112yes—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—44no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53#9912yes—
135C2×Dic53#13512yes—
136C2×Dic53AGL(1,ℤ/8)11yescharacteristic
137C2×Dic53#13711yescharacteristic
138#88324—82no—
139#88324—82no—
140#7731—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53—22no—
143C22×C53—22no—
144C22×C53—41no—
145C22×C53—41no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53#40811yescharacteristic
149C22×C53C2×#9911yescharacteristic
150C22×C53C2×#10111yescharacteristic
151C22×C53#41611yescharacteristic
152C22×C53#41711yescharacteristic
153C22×C53#42111yescharacteristic
154C22×C53#42111yescharacteristic
155C4×C53—22no—
156C4×C53—41no—
157C4×C53—81no—
158C4×C53—81no—
159Dic53—21no—
160Dic53—21no—
161Dic53—22no—
162Dic53—22no—
163Dic53—41no—
164Dic53—81no—
165Dic53—81no—
166#2851—531no—
167#1817—531no—
168#1797—531no—
169#1271—532no—
170#1271—532no—
171C2×C53—42no—
172C2×C53#147211yescharacteristic
173C2×C53#148211yescharacteristic
174C2×C53#149511yescharacteristic
175C2×C53#151311yescharacteristic
176C2×C53#151611yescharacteristic
177C2×C53#153411yescharacteristic
178C2×C53#155211yescharacteristic
179C2×#125—531no—
180C22×C42—531no—
181C2×C4⋊C8—1061no—
182C2×C22⋊C8—1061no—
183C2×C22⋊C8—1061no—
184#389—531no—
185#389—531no—
186#387—1061no—
187#387—1061no—
188#385—1061no—
189#385—1061no—
190#382—531no—
191#380—531no—
192#380#615711yescharacteristic
193#336—532no—
194#336—532no—
195#336—1061no—
196C53#773111yescharacteristic, complemented, Sylow
197C23×C4—531no—
198C23×C4—531no—
199C23×C4C2×#615711yescharacteristic
200C22×C8—1062no—
201C22×C8—1062no—
202#125—2122no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—531no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4—1061no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4C2×#615711yescharacteristic
210C2×C4⋊C4C2×#615711yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—1061no—
213C2×C42—531no—
214C2×C42—531no—
215C2×C42—531no—
216C2×C42—531no—
217C2×C42—2122no—
218C4⋊C8—2122no—
219C22⋊C8—2122no—
220C22⋊C8—2122no—
221#95—21no—
222#95—1061no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227C24C22⋊#615711yescharacteristic
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—531no—
237C22×C4—1061no—
238C22×C4—1062no—
239C22×C4—1062no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—1061no—
243C22×C4—2121no—
244C22×C4—2121no—
245C22×C4—2121no—
246C22×C4C22⋊#615711yescharacteristic
247C22×C4C22×#615711yescharacteristic
248C2×C8—2122no—
249C2×C8—2122no—
250C2×C8—2122no—
251C2×C8—2122no—
252C2×C8—2122no—
253C2×C8—2122no—
254C4⋊C4—22no—
255C4⋊C4—41no—
256C4⋊C4—1062no—
257C4⋊C4—2121no—
258C4⋊C4—2122no—
259C4⋊C4—2122no—
260C4⋊C4—2121no—
261C4⋊C4—2121no—
262C4⋊C4—2122no—
263C22⋊C4—42no—
264C22⋊C4—2122no—
265C42—1062no—
266C42—1062no—
267C42—2121no—
268C42—2121no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23#1313327812yes—
276C23C2×C22⋊#615711yescharacteristic
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—44no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—532no—
290C2×C4—532no—
291C2×C4—531no—
292C2×C4—531no—
293C2×C4—1061no—
294C2×C4—1061no—
295C2×C4—1061no—
296C2×C4—1061no—
297C2×C4—1061no—
298C2×C4—1061no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—2124no—
303C2×C4—2122no—
304C2×C4—2122no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4#1313330412yes—
312C8—4242no—
313C8—4242no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22#1681159611yescharacteristic
322C22D53⋊#9911yescharacteristic
323C22C2×#1313327811yescharacteristic
324C22#1681161311yescharacteristic
325C22#1681161411yescharacteristic
326C22#1681162111yescharacteristic
327C22#1681162211yescharacteristic
328C4—22no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—1061no—
333C4—1061no—
334C4—1062no—
335C4—1062no—
336C4—4241no—
337C4—4241no—
338C4—2121no—
339C2—42no—
340C2#2028999411yescharacteristic
341C2#2029002811yescharacteristic
342C2#2029007011yescharacteristic
343C2#2029013311yescharacteristic
344C2#2029014011yescharacteristic
345C2#2029019711yescharacteristic
346C2#2029023611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#380C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#380#6157
hypocenterC53#7731
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
196. C53#7731semidirect

Maximal subgroups

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

subgroupindex
#202906052
#202905712
#202895342
#202895332
#202830232
#773153