Finite Group Databasebeta

← #27277654

Subgroups — #27277654

Order 13568.

472 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 107 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#20277655C211yesmaximal, characteristic
2#20274703C211yesmaximal, characteristic
3#20272595C212yesmaximal, complemented
4#20272583C212yesmaximal, complemented
5C53×#1459C211yesmaximal, characteristic
6C2×#13133132—21no—
7#16810516—21no—
8#16810509—21no—
9#16810503—21no—
10#16810491C411yescharacteristic, complemented
11#16810486C411yescharacteristic, complemented
12#16810483C2212yes—
13#16810483C2211yescharacteristic
14C22⋊C4×Dic53—21no—
15C4×C22⋊Dic53C411yescharacteristic, complemented
16#16810469C411yescharacteristic, complemented
17C2×#13133124C2211yescharacteristic
18#16810390—21no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×C22⋊C8×C53C2212yes—
21C53×#324—21no—
22C53×#323—21no—
23C23×Dic53—21no—
24C2×C4⋊Dic53—21no—
25C2×C4⋊Dic53—22no—
26C2×C22⋊Dic53—21no—
27C2×C22⋊Dic53—21no—
28C2×C22⋊Dic53—21no—
29C2×C22⋊Dic53—22no—
30C2×C4×Dic53—21no—
31C2×C4×Dic53—21no—
32#13133132—42no—
33#13133124—21no—
34#13133124—21no—
35#13133124—21no—
36#13133124—21no—
37#13133124—21no—
38#13133124—22no—
39#13133124C2×C411yescharacteristic
40#13133124C2×C411yescharacteristic
41#13133124C2×C411yescharacteristic
42#13133124C2×C411yescharacteristic
43C22×D4×C53D411yescharacteristic
44C23×C4×C53C2×C411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C22×C8×C53—22no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53C2×C411yescharacteristic
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—21no—
60C2×D4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—81no—
64C22×C4×C53—81no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53#4112yes—
71C22×C4×C53#4112yes—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—42no—
77C22×Dic53—42no—
78C22×Dic53—42no—
79C22×Dic53—81no—
80C2×C8×C53—42no—
81C2×C8×C53—82no—
82C4⋊C4×C53—22no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C42×C53—21no—
86C42×C53—21no—
87C22⋊Dic53—42no—
88C22⋊Dic53—41no—
89C22⋊Dic53—41no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C22⋊Dic53—81no—
93C4⋊Dic53—81no—
94C4⋊Dic53—82no—
95C4×Dic53—42no—
96C4×Dic53—81no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53C2×C22⋊C411yescharacteristic
105C23×C53#12312yes—
106C23×C53#13512yes—
107C23×C53AGL(1,ℤ/8)11yescharacteristic
108C23×C53#13711yescharacteristic
109D4×C53—42no—
110D4×C53—42no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53#11711yescharacteristic
125C2×C4×C53#11711yescharacteristic
126C2×C4×C53#12312yes—
127C2×Dic53—22no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—84no—
140C2×Dic53—82no—
141C2×Dic53—82no—
142C2×Dic53—82no—
143C8×C53—82no—
144#8763—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—21no—
148C22×C53—22no—
149C22×C53—42no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—81no—
156C22×C53—81no—
157C22×C53#38111yescharacteristic
158C22×C53#41611yescharacteristic
159C22×C53#41711yescharacteristic
160C22×C53#48212yes—
161C22×C53#48512yes—
162C4×C53—22no—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—41no—
166C4×C53—41no—
167C4×C53—41no—
168Dic53—42no—
169Dic53—81no—
170Dic53—81no—
171Dic53—81no—
172Dic53—162no—
173#2353—531no—
174#1799—531no—
175#1459D5311yescharacteristic
176#1294—532no—
177#1289—532no—
178C2×C53—21no—
179C2×C53—21no—
180C2×C53—22no—
181C2×C53—22no—
182C2×C53—81no—
183C2×C53#182511yescharacteristic
184C2×C53#182711yescharacteristic
185C2×C53#182811yescharacteristic
186C2×C4×D4C2×D5311yescharacteristic
187C2×#99—1061no—
188C2×C22⋊C8C2×D5312yes—
189#388—1061no—
190#387—1061no—
191#386—1061no—
192#383—531no—
193#381—531no—
194#380—531no—
195#380—532no—
196C4×C22⋊C4—531no—
197C4×C22⋊C4—1061no—
198#376—531no—
199C2×#95—531no—
200#342—1061no—
201#324—21no—
202#323—21no—
203C53#876311yescharacteristic, complemented, Sylow
204C22×D4D21211yescharacteristic
205C23×C4—1061no—
206C23×C4C4×D5311yescharacteristic
207C23×C4C22×D5311yescharacteristic
208C22×C8—22no—
209C4×D4—22no—
210C4×D4—22no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4C22⋊D5311yescharacteristic
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4D21211yescharacteristic
219C2×C22⋊C4C22⋊D5311yescharacteristic
220C2×C42—1061no—
221C2×C42—1061no—
222C2×C42C4×D5311yescharacteristic
223#99—2122no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#95—531no—
227#95—531no—
228#95—531no—
229#95—531no—
230#95—1061no—
231#95—1061no—
232#95—1061no—
233#95—1061no—
234#95—1061no—
235#95—1062no—
236C24#1171992111yescharacteristic
237C24C2×D21211yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—21no—
241C2×D4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—1061no—
247C22×C4—1061no—
248C22×C4—1061no—
249C22×C4—1061no—
250C22×C4—4241no—
251C22×C4—2122no—
252C22×C4—2122no—
253C22×C4—2122no—
254C22×C4#1171992111yescharacteristic
255C22×C4#1171992111yescharacteristic
256C22×C4#1171992111yescharacteristic
257C22×C4C2×C4×D5311yescharacteristic
258C22×C4#1171994712yes—
259C22×C4#1171994712yes—
260C22×C4C2×C22⋊D5311yescharacteristic
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—4242no—
265C4⋊C4—4241no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—4241no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C22⋊C4—2121no—
273C22⋊C4—2121no—
274C42—21no—
275C42—21no—
276C42—4241no—
277C42—2122no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23C2×#1171992111yescharacteristic
286C23#1313320112yes—
287C23#1313323912yes—
288C23#1313324511yescharacteristic
289C23#1313325011yescharacteristic
290D4—42no—
291D4—42no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—1062no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—4242no—
309C2×C4—4244no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4—2121no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4—2121no—
321C2×C4#1313317311yescharacteristic
322C2×C4#1313317311yescharacteristic
323C2×C4#1313319812yes—
324C8—82no—
325C22—21no—
326C22—22no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22—81no—
337C22#1681048411yescharacteristic
338C22#1681062011yescharacteristic
339C22#1681062511yescharacteristic
340C22#1681094812yes—
341C22#1681096512yes—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—8482no—
349C4—4241no—
350C4—4241no—
351C4—4241no—
352C4—2122no—
353C2—22no—
354C2—21no—
355C2—21no—
356C2—22no—
357C2—81no—
358C2#2027481111yescharacteristic
359C2#2027482011yescharacteristic
360C2#2027482111yescharacteristic
subgroupG/H
centerC22#16810484
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1459C2
socleC22×C53#381
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1459D53
hypocenterC53#8763
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #20272595C2semidirect
4. #20272583C2semidirect
10. #16810491C4semidirect
11. #16810486C4semidirect
15. C4×C22⋊Dic53C4semidirect
16. #16810469C4semidirect
203. C53#8763semidirect

Maximal subgroups

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

subgroupindex
#202776552
#202747032
#202725952
#202725832
C53×#14592
#876353