Finite Group Databasebeta

← #28308923

Subgroups — #28308923

Order 13568.

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

normal subgroups only

This group has 432 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#20290421C211yesmaximal, characteristic
2#20289947C211yesmaximal, characteristic
3#20289716C212yesmaximal, complemented
4#20289715C212yesmaximal, complemented
5#20280535C211yesmaximal, characteristic
6C22×#11719940—21no—
7#16811640—21no—
8#16811503—21no—
9#16811503C2212yes—
10#16811469C2212yes—
11#16811466—21no—
12C2×#13133215C411yescharacteristic
13C2×#13133173C2211yescharacteristic
14#16810564—21no—
15#16810559—21no—
16#16810544C2211yescharacteristic
17#16810542C2211yescharacteristic
18#16810535C411yescharacteristic
19#16810520—21no—
20#16810487C411yescharacteristic
21#16810481—21no—
22C53×#393C411yescharacteristic
23C2×#11719940—24no—
24C2×#11719940—22no—
25C2×#11719940—41no—
26C2×#11719940—42no—
27C22×#88324—41no—
28#13133284—41no—
29C22⋊#88324—22no—
30C4×#88324—41no—
31C22×Dic106C2×C411yescharacteristic
32C23×Dic53D411yescharacteristic
33#13133215—41no—
34#13133215—41no—
35#13133173—41no—
36#13133173—41no—
37C2×C4⋊Dic53—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C2×#11719919—21no—
42C2×#11719919D411yescharacteristic
43C2×#11719919C2311yescharacteristic
44C2×C22⋊Dic53—21no—
45C2×C22⋊Dic53D411yescharacteristic
46C2×C4×Dic53C2×C411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52#13133124C2×C411yescharacteristic
53C22×Q8×C53C2×C411yescharacteristic
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53C2×C411yescharacteristic
56#95×C53—21no—
57#95×C53C2×C411yescharacteristic
58C24×C53C22⋊C411yescharacteristic
59C2×Q8×C53—41no—
60C2×Q8×C53—81no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53C22⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×Dic53—21no—
66C22×Dic53—21no—
67C22×Dic53—22no—
68C22×Dic53—21no—
69C22×Dic53—21no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53SD1612yes—
73C22×Dic53Q1612yes—
74C22×Dic53C2×D411yescharacteristic
75C22×Dic53C2×D411yescharacteristic
76C2×Dic106—41no—
77C2×Dic106—81no—
78#11719940—24no—
79#11719940—44no—
80#11719940—42no—
81C2×#88324—42no—
82C2×#88324—44no—
83C2×#88324—42no—
84C22⋊C4×C53—41no—
85C22⋊C4×C53—81no—
86C22⋊Dic53—41no—
87C22⋊Dic53—81no—
88C4⋊Dic53—81no—
89#11719919—41no—
90#11719919—41no—
91#11719919—42no—
92#11719919—41no—
93#11719919—41no—
94#11719919—42no—
95#11719919—81no—
96C4×Dic53—21no—
97C4×Dic53—21no—
98C4×Dic53—21no—
99C4×Dic53—21no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53#10312yes—
105C23×C53C2×C22⋊C411yescharacteristic
106Q8×C53—82no—
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×C53—41no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—22no—
120C2×Dic53—22no—
121C2×Dic53—22no—
122C2×Dic53—22no—
123C2×Dic53—21no—
124C2×Dic53—21no—
125C2×Dic53—41no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53#9911yescharacteristic
134C2×Dic53#9911yescharacteristic
135C2×Dic53#10312yes—
136C2×Dic53#10412yes—
137C2×Dic53C2×SD1611yescharacteristic
138C2×Dic53C2×Q1611yescharacteristic
139Dic106—82no—
140#88324—44no—
141#88324—82no—
142#11724—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—81no—
150C22×C53C2×#9911yescharacteristic
151C22×C53C2×#10311yescharacteristic
152C22×C53C2×#10411yescharacteristic
153C4×C53—41no—
154C4×C53—41no—
155C4×C53—81no—
156C4×C53—81no—
157Dic53—21no—
158Dic53—21no—
159Dic53—41no—
160Dic53—41no—
161Dic53—81no—
162Dic53—81no—
163Dic53#32812yes—
164Dic53#32812yes—
165#2628—531no—
166#1742—531no—
167#1458—531no—
168#1354—532no—
169#1354—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53C2×#32811yescharacteristic
174C2×C53#212211yescharacteristic
175C2×C53#213511yescharacteristic
176C22×M16—1061no—
177C2×C4⋊Q8—531no—
178C2×#117—531no—
179#431—1061no—
180#400—1061no—
181#399—1061no—
182#395—531no—
183#395—531no—
184#393—531no—
185#393#615711yescharacteristic
186#389—1061no—
187#382—531no—
188#380—1061no—
189#343—532no—
190#343—1061no—
191#324—532no—
192#323—1061no—
193C53#1172411yescharacteristic, complemented, Sylow
194C22×Q8—531no—
195C22×Q8C2×#615711yescharacteristic
196C23×C4—531no—
197C2×M16—1062no—
198C2×M16—1064no—
199C2×M16—2122no—
200C2×M16—2121no—
201C22×C8—2121no—
202C4⋊Q8—2121no—
203C4⋊Q8—2121no—
204#117—2121no—
205#117—2121no—
206C2×C4⋊C4—531no—
207C2×C4⋊C4—531no—
208C2×C4⋊C4—1061no—
209C2×C4⋊C4—1061no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1061no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—531no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4C2×#615711yescharacteristic
217C2×C42—531no—
218C2×C42—531no—
219C4⋊C8—2121no—
220C22⋊C8—1062no—
221C4×C8—2121no—
222#95—21no—
223#95—531no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95C2×#615711yescharacteristic
229C24C22⋊#615711yescharacteristic
230C2×Q8—41no—
231C2×Q8—81no—
232C2×Q8—4241no—
233C2×Q8—2121no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—531no—
237C22×C4—532no—
238C22×C4—532no—
239C22×C4—531no—
240C22×C4—531no—
241C22×C4—531no—
242C22×C4—1061no—
243C22×C4—1061no—
244C22×C4—1062no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4C22⋊#615711yescharacteristic
248C22×C4C22×#615711yescharacteristic
249M16—1064no—
250M16—2122no—
251M16—2124no—
252C2×C8—2122no—
253C2×C8—2124no—
254C2×C8—2122no—
255C4⋊C4—4241no—
256C4⋊C4—4241no—
257C4⋊C4—2121no—
258C4⋊C4—2121no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C4⋊C4—2122no—
262C4⋊C4—2122no—
263C22⋊C4—41no—
264C22⋊C4—81no—
265C22⋊C4—4241no—
266C22⋊C4—2121no—
267C42—1061no—
268C42—1061no—
269C42—1061no—
270C42—1061no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1313328012yes—
276C23C2×C22⋊#615711yescharacteristic
277Q8—82no—
278Q8—4242no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—531no—
290C2×C4—531no—
291C2×C4—532no—
292C2×C4—532no—
293C2×C4—531no—
294C2×C4—531no—
295C2×C4—1062no—
296C2×C4—1062no—
297C2×C4—1062no—
298C2×C4—1062no—
299C2×C4—1061no—
300C2×C4—1061no—
301C2×C4—1061no—
302C2×C4—1061no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C8—4242no—
312C8—2124no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22D53⋊#9911yescharacteristic
321C22C2×#1313328011yescharacteristic
322C22D53⋊#10411yescharacteristic
323C4—41no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4—532no—
328C4—532no—
329C4—1061no—
330C4—1061no—
331C4—4241no—
332C4—4241no—
333C4—2121no—
334C4—2121no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2D53⋊#32811yescharacteristic
339C2#2029097511yescharacteristic
340C2#2029101811yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#393C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×#11719919C23
hypercenter#393#6157
hypocenterC53#11724
LCSG ▷ C22×C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #393

Complements

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

NH ≅ G/Ntype
3. #20289716C2semidirect
4. #20289715C2semidirect
193. C53#11724semidirect

Maximal subgroups

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

subgroupindex
#202904212
#202899472
#202897162
#202897152
#202805352
#1172453