Finite Group Databasebeta

← #28312300

Subgroups — #28312300

Order 13568.

420 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 80 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#20290582C211yesmaximal, characteristic
2#20290468C211yesmaximal, characteristic
3#20289813C212yesmaximal
4#20289812C212yesmaximal
5#20279921C211yesmaximal, characteristic
6#16811626—21no—
7C2×#13133278C2212yes—
8C2×#13133278C2212yes—
9C4×#11719940—21no—
10#16811503—21no—
11#16811503—21no—
12C2×#13133216C411yescharacteristic
13C2×#13133182C2211yescharacteristic
14#16810543—21no—
15#16810543—21no—
16#16810529C2211yescharacteristic
17#16810528C2211yescharacteristic
18#16810518—21no—
19#16810518—21no—
20#16810492C411yescharacteristic
21C4×#11719919C411yescharacteristic
22C53×#384C411yescharacteristic
23C2×#11719940—22no—
24C2×#11719940—22no—
25#13133284—41no—
26C4⋊#88324—41no—
27#13133278—22no—
28#13133278—24no—
29#13133278Q814yescomplemented
30#13133272—41no—
31C4×#88324—41no—
32C22×Dic106—21no—
33C22×Dic106C2311yescharacteristic
34#13133216—41no—
35#13133216—41no—
36#13133182—41no—
37#13133182—41no—
38C2×C4⋊Dic53C2×C411yescharacteristic
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919—21no—
43C2×#11719919D411yescharacteristic
44C2×#11719919D411yescharacteristic
45C2×C4×Dic53C2×C411yescharacteristic
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53D411yescharacteristic
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53#13133124—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C42×C53C2×C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C22⋊C411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C22×Dic53C2×D411yescharacteristic
70C2×Dic106—41no—
71C2×Dic106—41no—
72C2×Dic106—42no—
73C2×Dic106—81no—
74C2×Dic106—81no—
75C2×Dic106C2×Q812yes—
76C2×Dic106#4112yes—
77#11719940—24no—
78#11719940—42no—
79#11719940—42no—
80#11719940—42no—
81C2×#88324—42no—
82C2×#88324—42no—
83C4⋊C4×C53—41no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C42×C53—41no—
88C4⋊Dic53—41no—
89#11719919—24no—
90#11719919—41no—
91#11719919—41no—
92#11719919—41no—
93#11719919—41no—
94#11719919—81no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C4×Dic53—21no—
101C4×Dic53—21no—
102C4×Dic53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—22no—
107C2×C4×C53—21no—
108C2×C4×C53—21no—
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—41no—
116C2×C4×C53C4×Q812yes—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—21no—
124C2×Dic53—41no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53#9911yescharacteristic
131C2×Dic53#9911yescharacteristic
132C2×Dic53C22⋊Q812yes—
133C2×Dic53#12411yescharacteristic
134C2×Dic53C4⋊Q811yescharacteristic
135Dic106—42no—
136Dic106—42no—
137Dic106—84no—
138#88324—44no—
139#88324—82no—
140#13302—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53#39111yescharacteristic
144C22×C53C2×#9911yescharacteristic
145C22×C53#41011yescharacteristic
146C4×C53—22no—
147C4×C53—42no—
148C4×C53—41no—
149C4×C53—41no—
150C4×C53—81no—
151Dic53—21no—
152Dic53—22no—
153Dic53—21no—
154Dic53—21no—
155Dic53—42no—
156Dic53—41no—
157Dic53—81no—
158Dic53#35612yes—
159#2567—531no—
160#1801—531no—
161C4×#101—531no—
162#1394—532no—
163#1394—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53#199611yescharacteristic
167C2×C53C2×#35611yescharacteristic
168C2×C53#213411yescharacteristic
169C2×C4⋊Q8—531no—
170C2×C4×Q8—531no—
171#423—1061no—
172C2×#101—532no—
173C2×#101—532no—
174C4×M16—1061no—
175#395—1061no—
176#395—1061no—
177#391—531no—
178#391—531no—
179#389—1061no—
180#389—1061no—
181#384—531no—
182#384#615711yescharacteristic
183C4×C4⋊C4—531no—
184#343—1061no—
185#343—1061no—
186C53#1330211yescharacteristic, complemented, Sylow
187C22×Q8—531no—
188C22×Q8—1061no—
189C2×M16—1062no—
190C2×M16—1062no—
191C4⋊Q8—2121no—
192C4⋊Q8—2121no—
193C4×Q8—2121no—
194C4×Q8—2121no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—531no—
198C2×C4⋊C4—531no—
199C2×C4⋊C4—531no—
200C2×C4⋊C4—1061no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4C2×#615711yescharacteristic
205C2×C4⋊C4C2×#615711yescharacteristic
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42C2×#615711yescharacteristic
211C4⋊C8—2121no—
212C4⋊C8—2121no—
213#101—534no—
214#101—1062no—
215#101—1064no—
216#97—2121no—
217C4×C8—2121no—
218#95—1061no—
219#95—1061no—
220#95—1061no—
221#95—1061no—
222#95—1061no—
223C2×Q8—532no—
224C2×Q8—532no—
225C2×Q8—4241no—
226C2×Q8—4241no—
227C2×Q8—2121no—
228C2×Q8—2121no—
229C2×Q8—2122no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—531no—
233C22×C4—531no—
234C22×C4—531no—
235C22×C4—531no—
236C22×C4—1061no—
237C22×C4—1061no—
238C22×C4C22⋊#615711yescharacteristic
239C22×C4C22⋊#615711yescharacteristic
240C22×C4C22×#615711yescharacteristic
241M16—1064no—
242M16—2122no—
243M16—2122no—
244M16—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—81no—
250C4⋊C4—81no—
251C4⋊C4—1064no—
252C4⋊C4—4241no—
253C4⋊C4—4241no—
254C4⋊C4—2121no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C42—41no—
260C42—1061no—
261C42—1061no—
262C42—1061no—
263C42—1061no—
264C42—1061no—
265C42—1061no—
266C42—2121no—
267C23C2×C22⋊#615711yescharacteristic
268Q8—4244no—
269Q8—2122no—
270Q8—2122no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—531no—
284C2×C4—532no—
285C2×C4—531no—
286C2×C4—531no—
287C2×C4—531no—
288C2×C4—1062no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—1061no—
293C2×C4—1062no—
294C2×C4—1061no—
295C2×C4—2121no—
296C2×C4—2121no—
297C2×C4—2121no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4Q8×#615712yes—
302C8—4242no—
303C8—2124no—
304C22—21no—
305C22—21no—
306C22(C2×#6157)⋊Q811yescharacteristic
307C22D53⋊#9911yescharacteristic
308C22#1681160211yescharacteristic
309C4—22no—
310C4—42no—
311C4—41no—
312C4—41no—
313C4—81no—
314C4—532no—
315C4—1061no—
316C4—1062no—
317C4—1061no—
318C4—1061no—
319C4—4241no—
320C4—2122no—
321C4—2121no—
322C2—21no—
323C2—21no—
324C2#2029090611yescharacteristic
325C2D53⋊#35611yescharacteristic
326C2#2029101511yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#384C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenter#384#6157
hypocenterC53#13302
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #384

Complements

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

NH ≅ G/Ntype
29. #13133278Q8semidirect
186. C53#13302semidirect

Maximal subgroups

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

subgroupindex
#202905822
#202904682
#202898132
#202898122
#202799212
#1330253