Finite Group Databasebeta

← #27281539

Subgroups — #27281539

Order 13568.

644 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 97 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#20280196C211yesmaximal, characteristic
2#20277070C211yesmaximal, characteristic, complemented
3#20273231C211yesmaximal, characteristic
4#20273044C211yesmaximal, characteristic, complemented
5#20272601C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#1507C211yesmaximal, characteristic
8C2×#13133173—21no—
9C22×#11719919C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810561C412yescomplemented
12#16810537C412yescomplemented
13#16810517—22no—
14#16810503—21no—
15#16810503—21no—
16C4×#11719919—22no—
17#16810474—22no—
18C22⋊C4×Dic53—21no—
19C4×C22⋊Dic53C2211yescharacteristic
20C4×C22⋊Dic53C2211yescharacteristic
21#16810357—22no—
22#16810348—22no—
23C2×#121×C53C2211yescharacteristic
24C53×#407C2211yescharacteristic
25C4×C22⋊C4×C53C2211yescharacteristic
26C53×#327—22no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×#11719917—21no—
30#13133173—28no—
31C2×#11719919—22no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—21no—
36C2×#11719919C2×C412yes—
37C2×#11719919D412yes—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—22no—
41C2×C22⋊Dic53—22no—
42C2×C4×Dic53—22no—
43C2×C4×Dic53—22no—
44C2×C4×Dic53—21no—
45C2×C4×Dic53—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48#13133124—22no—
49#13133124—22no—
50#13133124—22no—
51#13133124C2×C412yes—
52#13133124D412yes—
53C22×D4×C53—21no—
54C22×D4×C53C2×C411yescharacteristic
55C23×C4×C53C2311yescharacteristic
56C2×M16×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C2×C4⋊C4×C53C2×C411yescharacteristic
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C42×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67#95×C53D412yes—
68C24×C53—21no—
69C24×C53C2×D411yescharacteristic
70C2×D4×C53—21no—
71C2×D4×C53—21no—
72C2×D4×C53—44no—
73C2×D4×C53—41no—
74C2×D4×C53—41no—
75C2×D4×C53—81no—
76C22×C4×C53—22no—
77C22×C4×C53—21no—
78C22×C4×C53—81no—
79C22×C4×C53C22×C411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53#4111yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—24no—
88C22×Dic53—21no—
89C22×Dic53—22no—
90C22×Dic53—21no—
91C22×Dic53—22no—
92C22×Dic53—81no—
93C22×Dic53C22⋊C412yes—
94C22×Dic53C22⋊C412yes—
95M16×C53—81no—
96C2×C8×C53—41no—
97C4⋊C4×C53—21no—
98C4⋊C4×C53—21no—
99C22⋊C4×C53—24no—
100C22⋊C4×C53—41no—
101C22⋊C4×C53—41no—
102C42×C53—81no—
103C22⋊Dic53—24no—
104C22⋊Dic53—42no—
105C22⋊Dic53—44no—
106C22⋊Dic53—82no—
107#11719919—28no—
108#11719919—22no—
109#11719919—22no—
110#11719919—22no—
111#11719919—22no—
112C4×Dic53—28no—
113C4×Dic53—82no—
114C4×Dic53—81no—
115C4×Dic53—81no—
116#11719917—81no—
117C2×#88322—41no—
118C23×C53—22no—
119C23×C53—22no—
120C23×C53—41no—
121C23×C53—42no—
122C23×C53—42no—
123C23×C53—41no—
124C23×C53—42no—
125C23×C53C2×C22⋊C411yescharacteristic
126C23×C53#12011yescharacteristic
127C23×C53#12311yescharacteristic
128D4×C53—81no—
129D4×C53—81no—
130D4×C53—82no—
131D4×C53—82no—
132C2×C4×C53—24no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—42no—
142C2×C4×C53—41no—
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53#12111yescharacteristic
146C2×C4×C53#12111yescharacteristic
147C2×Dic53—24no—
148C2×Dic53—22no—
149C2×Dic53—22no—
150C2×Dic53—22no—
151C2×Dic53—22no—
152C2×Dic53—24no—
153C2×Dic53—22no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—42no—
158C2×Dic53—42no—
159C2×Dic53—42no—
160C2×Dic53—42no—
161C2×Dic53—81no—
162C2×Dic53#10414yes—
163C8×C53—81no—
164#88322—81no—
165#9410—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—21no—
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—44no—
174C22×C53—84no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53#38611yescharacteristic
179C22×C53C2×#10412yes—
180C22×C53#45412yes—
181C4×C53—21no—
182C4×C53—21no—
183C4×C53—21no—
184C4×C53—21no—
185C4×C53—44no—
186C4×C53—81no—
187Dic53—24no—
188Dic53—44no—
189Dic53—42no—
190Dic53—82no—
191#2591—531no—
192#2274—531no—
193#1507—531no—
194#1507D5311yescharacteristic
195#1463—531no—
196#1297—531no—
197#1297—531no—
198C2×C53—22no—
199C2×C53—22no—
200C2×C53—41no—
201C2×C53—161no—
202C2×C53#188811yescharacteristic
203C2×C53#189611yescharacteristic
204C2×C53#191311yescharacteristic
205C2×#121C2×D5311yescharacteristic
206C2×#117—1061no—
207C22×C4⋊C4—531no—
208#407—531no—
209#407C2×D5311yescharacteristic
210#400—532no—
211#394—532no—
212#388—1062no—
213#386—1061no—
214#386—1061no—
215C4×C4⋊C4—1062no—
216C4×C4⋊C4—1062no—
217C4×C22⋊C4—531no—
218C4×C22⋊C4—531no—
219C4×C22⋊C4—1061no—
220C4×C22⋊C4C2×D5311yescharacteristic
221#327—22no—
222#327—1062no—
223#324—1062no—
224C53#941011yescharacteristic, complemented, Sylow
225C22×D4—21no—
226C22×D4C4×D5311yescharacteristic
227C23×C4—531no—
228C23×C4—531no—
229C23×C4C22×D5311yescharacteristic
230C2×M16—21no—
231C2×M16—1061no—
232#121—21no—
233#121—21no—
234#121—21no—
235#121—21no—
236#117—1068no—
237C2×C4⋊C4—532no—
238C2×C4⋊C4—532no—
239C2×C4⋊C4—1062no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4—1061no—
242C2×C4⋊C4—1061no—
243C2×C4⋊C4—1061no—
244C2×C4⋊C4C4×D5311yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1061no—
248C2×C22⋊C4—1062no—
249C2×C22⋊C4—1062no—
250C2×C22⋊C4—1061no—
251C2×C42—21no—
252C2×C42—1062no—
253C2×C42—1062no—
254C2×C42—1061no—
255C2×C42—1061no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258C22⋊C8—1061no—
259C22⋊C8—1061no—
260#95—532no—
261#95—532no—
262#95—1062no—
263#95—1062no—
264#95—1062no—
265#95C22⋊D5312yes—
266C24—21no—
267C24D4×D5311yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—81no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—81no—
277C22×C4—532no—
278C22×C4—532no—
279C22×C4—1064no—
280C22×C4—1061no—
281C22×C4—1061no—
282C22×C4—1061no—
283C22×C4—1062no—
284C22×C4—1061no—
285C22×C4—1062no—
286C22×C4—4241no—
287C22×C4C2×C4×D5311yescharacteristic
288C22×C4#1171994511yescharacteristic
289C22×C4D4×D5311yescharacteristic
290C22×C4D4×D5311yescharacteristic
291C22×C4#1171994711yescharacteristic
292C22×C4C2×C22⋊D5311yescharacteristic
293M16—81no—
294M16—4241no—
295C2×C8—41no—
296C2×C8—2121no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—1068no—
300C4⋊C4—1062no—
301C4⋊C4—1062no—
302C4⋊C4—1062no—
303C4⋊C4—1062no—
304C22⋊C4—24no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—1064no—
308C22⋊C4—4242no—
309C22⋊C4—2124no—
310C22⋊C4—2122no—
311C42—81no—
312C42—1068no—
313C42—4241no—
314C42—4241no—
315C42—4242no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—42no—
320C23—42no—
321C23—41no—
322C23—42no—
323C23C22⋊C4×D5311yescharacteristic
324C23#1313318511yescharacteristic
325C23#1313319911yescharacteristic
326D4—81no—
327D4—81no—
328D4—82no—
329D4—82no—
330C2×C4—24no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—534no—
342C2×C4—1062no—
343C2×C4—1062no—
344C2×C4—1064no—
345C2×C4—1062no—
346C2×C4—1062no—
347C2×C4—1064no—
348C2×C4—1062no—
349C2×C4—4241no—
350C2×C4—2122no—
351C2×C4—2122no—
352C2×C4—2122no—
353C2×C4—2122no—
354C2×C4—2122no—
355C2×C4—2122no—
356C2×C4—2122no—
357C2×C4#1313317711yescharacteristic
358C2×C4C4×C22⋊D5311yescharacteristic
359C2×C4#1313318811yescharacteristic
360C2×C4#1313319011yescharacteristic
361C8—81no—
362C8—4241no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—84no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22#1681049911yescharacteristic
376C22#104×D5312yes—
377C22#1681079412yes—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—44no—
383C4—81no—
384C4—1064no—
385C4—4242no—
386C4—2124no—
387C4—2122no—
388C2—22no—
389C2—22no—
390C2—41no—
391C2—161no—
392C2#2027511511yescharacteristic
393C2#2027515511yescharacteristic
394C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1507C2
socleC22×C53#386
prime radicalC2×#121×C53C22
cosocleC23×C4×C53C23
hypercenter#1507D53
hypocenterC53#9410
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1507

Complements

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

NH ≅ G/Ntype
2. #20277070C2semidirect
4. #20273044C2semidirect
5. #20272601C2semidirect
6. #20272600C2semidirect
11. #16810561C4semidirect
12. #16810537C4semidirect
224. C53#9410semidirect

Maximal subgroups

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

subgroupindex
#202801962
#202770702
#202732312
#202730442
#202726012
#202726002
C53×#15072
#941053