Finite Group Databasebeta

← #27293344

Subgroups — #27293344

Order 13568.

488 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 89 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#20277232C211yesmaximal, characteristic
2#20273034C211yesmaximal, characteristic
3#20273030C211yesmaximal, characteristic
4#20272664C211yesmaximal, characteristic, complemented
5#20272662C211yesmaximal, characteristic, complemented
6#20272645C211yesmaximal, characteristic, complemented
7C53×#1317C211yesmaximal, characteristic, complemented
8C2×#13133200C411yescharacteristic
9C2×#13133177C2211yescharacteristic
10C22×C4×Dic53C411yescharacteristic
11#16810673—21no—
12C22⋊C8×D53—21no—
13#16810512—21no—
14#16810505—21no—
15#16810500C411yescharacteristic
16#16810495C2211yescharacteristic
17#16810474—21no—
18C22⋊C4×Dic53C411yescharacteristic
19#16810470—21no—
20#16810348—21no—
21#16810347C2211yescharacteristic
22#16810347C2211yescharacteristic
23#16810346—21no—
24C53×#381C2211yescharacteristic
25C53×#323C2211yescharacteristic
26C53×#323C2211yescharacteristic
27C22×C22⋊D53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C411yescharacteristic
30C22×C4×D53D411yescharacteristic
31C2×C8×D53—41no—
32#13133200—22no—
33#13133200C814yescomplemented
34#13133177—41no—
35#13133177—41no—
36C2×#11719919—21no—
37C2×#11719919—21no—
38C2×#11719919D411yescharacteristic
39C2×C22⋊Dic53C2×C411yescharacteristic
40C2×#11719921—21no—
41C2×#11719921—21no—
42C2×#11719921D411yescharacteristic
43C2×C4×Dic53—21no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C2×C4×Dic53—41no—
47C2×C4×Dic53—41no—
48C2×C4×Dic53D411yescharacteristic
49#13133146—41no—
50C22⋊#88322—21no—
51C22⋊#88322—21no—
52#13133129—41no—
53C8×Dic53—41no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C23×C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53C2311yescharacteristic
60C22⋊C8×C53—21no—
61C22⋊C8×C53—21no—
62#95×C53—21no—
63#95×C53—21no—
64C24×C53C22×C411yescharacteristic
65C23×D53C22⋊C411yescharacteristic
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C2×C812yescomplemented
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C2×C22⋊D53—41no—
75C2×C22⋊D53—41no—
76C2×C22⋊D53C2×C812yes—
77C2×C22⋊D53M1612yes—
78C22×Dic53—21no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53—41no—
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C22⋊C411yescharacteristic
88C2×C4×D53—21no—
89C2×C4×D53—21no—
90C2×C4×D53—42no—
91C2×C8×C53—41no—
92C2×C8×C53—41no—
93C22⋊C4×C53—41no—
94C22⋊C4×C53—41no—
95C22⋊Dic53—21no—
96C22⋊Dic53C2×C812yescomplemented
97#11719921—24no—
98#11719921—41no—
99#11719921—42no—
100#11719919—24no—
101#11719919—41no—
102#11719919—42no—
103C4×Dic53—21no—
104C4×Dic53—21no—
105C4×Dic53—42no—
106C4×Dic53—42no—
107C4×Dic53—41no—
108C4×Dic53—41no—
109C2×#88322—41no—
110C2×#88322—41no—
111C8×D53—82no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53C22×C811yescharacteristic
118C23×C53C2×M1611yescharacteristic
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—41no—
122C22×D53—41no—
123C22×D53C22⋊C812yes—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—42no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—42no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C22⋊D53—42no—
137C22⋊D53—42no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—21no—
144C2×Dic53—42no—
145C2×Dic53—41no—
146C2×Dic53—41no—
147C2×Dic53—41no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—42no—
151C2×Dic53—41no—
152C2×Dic53—42no—
153C2×Dic53C22⋊C812yes—
154C2×Dic53#9911yescharacteristic
155C2×Dic53#9911yescharacteristic
156C4×D53—42no—
157C4×D53—81no—
158C8×C53—81no—
159C8×C53—81no—
160#88322—81no—
161#88322—81no—
162#11472—531nomaximal, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53C2×C22⋊C811yescharacteristic
173C22×C53C2×#9911yescharacteristic
174C22×C53#41311yescharacteristic
175C2×D53—22no—
176C2×D53—41no—
177C2×D53—41no—
178C2×D53—42no—
179C2×D53—81no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—42no—
183C4×C53—81no—
184Dic53—21no—
185Dic53—22no—
186Dic53—42no—
187Dic53—41no—
188Dic53—42no—
189Dic53#35212yes—
190C4×#123—531no—
191#1460—531no—
192#1459—531no—
193#1326—531no—
194#1326—531no—
195#1317—531no—
196#1317D5311yescharacteristic, complemented
197C2×C53—21no—
198C2×C53—21no—
199C2×C53—41no—
200C2×C53—41no—
201C2×C53#145911yescharacteristic
202C2×C53C2×#35211yescharacteristic
203C2×C53#212011yescharacteristic
204D53—42no—
205C2×#123—531no—
206C2×C4×D4—531no—
207C22×C42—531no—
208#431—1061no—
209C2×C22⋊C8—1061no—
210#388—1061no—
211#387—1061no—
212#386—531no—
213#385—531no—
214#381C2×D5311yescharacteristic
215C4×C4⋊C4—1061no—
216C4×C22⋊C4—531no—
217C4×C22⋊C4—1061no—
218#324—1061no—
219#323—531no—
220#323—531no—
221#323—1061no—
222#323C2×D5311yescharacteristic
223#323C2×D5311yescharacteristic
224C53#1147211yescharacteristic, complemented, Sylow
225C22×D4—531no—
226C23×C4—531no—
227C23×C4—531no—
228C23×C4—531no—
229C23×C4C4×D5311yescharacteristic
230C22×C8—2121no—
231#123—534no—
232#123—1062no—
233C4×D4—2121no—
234C4×D4—2121no—
235C2×C4⋊C4—531no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C22⋊C4C4×D5311yescharacteristic
243C2×C22⋊C4C22×D5311yescharacteristic
244C2×C42—531no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42—1061no—
248C2×C42—2121no—
249C2×C42—2121no—
250C4⋊C8—2121no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—1061no—
254C22⋊C8—1061no—
255C22⋊C8—2121no—
256C4×C8—2121no—
257#95—21no—
258#95—21no—
259#95—1061no—
260#95—1061no—
261#95—1061no—
262C24—531no—
263C24C2×C4×D5311yescharacteristic
264C2×D4—532no—
265C2×D4—532no—
266C2×D4—2121no—
267C2×D4—2121no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—1061no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—2122no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4—2121no—
286C22×C4C8×D5312yescomplemented
287C22×C4D4×D5311yescharacteristic
288C22×C4D4×D5311yescharacteristic
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—4242no—
292C2×C8—2121no—
293C2×C8—2121no—
294C4⋊C4—1064no—
295C4⋊C4—2121no—
296C4⋊C4—2122no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—532no—
300C22⋊C4—1064no—
301C22⋊C4—1061no—
302C22⋊C4—2121no—
303C22⋊C4—2122no—
304C42—1061no—
305C42—1061no—
306C42—2121no—
307C42—2121no—
308C42—2122no—
309C42—2122no—
310C23—21no—
311C23—21no—
312C23—41no—
313C23—41no—
314C23—532no—
315C23—1061no—
316C23—1061no—
317C23—2121no—
318C23—2121no—
319C23C22⋊C4×D5311yescharacteristic
320C23C2×C8×D5311yescharacteristic
321C23M16×D5311yescharacteristic
322D4—2122no—
323D4—2122no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—531no—
337C2×C4—532no—
338C2×C4—531no—
339C2×C4—1062no—
340C2×C4—1061no—
341C2×C4—1061no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—1061no—
345C2×C4—4241no—
346C2×C4—2122no—
347C2×C4—2122no—
348C2×C4—2122no—
349C2×C4—2121no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2122no—
353C2×C4—2121no—
354C2×C4—2121no—
355C2×C4—2122no—
356C8—81no—
357C8—81no—
358C8—4241no—
359C8—4241no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—41no—
365C22—41no—
366C22—42no—
367C22—41no—
368C22—41no—
369C22—1062no—
370C22—4241no—
371C22—2122no—
372C22—2121no—
373C22—2121no—
374C22C22⋊C8×D5311yescharacteristic
375C22#99×D5311yescharacteristic
376C22#1681060411yescharacteristic
377C4—41no—
378C4—41no—
379C4—42no—
380C4—81no—
381C4—532no—
382C4—1061no—
383C4—1062no—
384C4—2122no—
385C4—2121no—
386C4—2122no—
387C2—21no—
388C2—21no—
389C2—41no—
390C2—41no—
391C2—2122no—
392C2#2027303011yescharacteristic
393C2#352×D5311yescharacteristic
394C2#2027635511yescharacteristic
subgroupG/H
centerC22#99×D53
commutatorC23×C53C22×C8
FrattiniC2×C22⋊C4C22×D53
FittingC53×#1317C2
socleC22×C53C2×#99
prime radicalC22×C22⋊D53C2×C4
cosocleC2×C22⋊C4×C53C23
hypercenter#1317D53
hypocenterC53#11472
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #1317

Complements

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

NH ≅ G/Ntype
4. #20272664C2semidirect
5. #20272662C2semidirect
6. #20272645C2semidirect
7. C53×#1317C2semidirect
33. #13133200C8semidirect
71. C22×C4×C53C2×C8semidirect
96. C22⋊Dic53C2×C8semidirect
196. #1317D53semidirect
224. C53#11472semidirect
286. C22×C4C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202772322
#202730342
#202730302
#202726642
#202726622
#202726452
C53×#13172
#1147253