Finite Group Databasebeta

← #27281529

Subgroups — #27281529

Order 13568.

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

normal subgroups only

This group has 618 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#20280096C211yesmaximal, characteristic
2#20277070C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic
4#20273044C211yesmaximal, characteristic
5#20272601C211yesmaximal, characteristic
6#20272600C211yesmaximal, characteristic
7C53×#1508C211yesmaximal, characteristic
8C2×#13133173—21no—
9C22×#11719919C2211yescharacteristic
10#16810587C2211yescharacteristic
11#16810554C412yescomplemented
12#16810542C412yescomplemented
13#16810518—22no—
14#16810495—21no—
15#16810495—21no—
16C4×#11719919—22no—
17#16810474—22no—
18C22⋊C4×Dic53—21no—
19C4×C22⋊Dic53C2211yescharacteristic
20C4×C22⋊Dic53C2211yescharacteristic
21#16810360—22no—
22#16810348—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C53×#407C2211yescharacteristic
25C4×C22⋊C4×C53C2211yescharacteristic
26C53×#328—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×#11719919—22no—
37C2×#11719919—22no—
38C2×#11719919C2×C412yes—
39C2×#11719919D412yes—
40C2×C22⋊Dic53—21no—
41C2×C22⋊Dic53—21no—
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×Q8×C53C2×C411yescharacteristic
54C23×C4×C53C2311yescharacteristic
55C2×M16×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C2×C4⋊C4×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×C53C2×D411yescharacteristic
69C2×Q8×C53—21no—
70C2×Q8×C53—21no—
71C2×Q8×C53—81no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—81no—
76C22×C4×C53C22×C411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—24no—
85C22×Dic53—22no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—21no—
89C22×Dic53—81no—
90C22×Dic53C22⋊C412yes—
91C22×Dic53C22⋊C412yes—
92M16×C53—81no—
93C2×C8×C53—41no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—41no—
97C4⋊C4×C53—41no—
98C22⋊C4×C53—24no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C42×C53—81no—
102C22⋊Dic53—24no—
103C22⋊Dic53—24no—
104#11719919—28no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—44no—
110#11719919—82no—
111C4×Dic53—28no—
112C4×Dic53—82no—
113C4×Dic53—81no—
114C4×Dic53—81no—
115#11719917—81no—
116C2×#88322—41no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53C2×C22⋊C411yescharacteristic
121C23×C53#12011yescharacteristic
122C23×C53#12311yescharacteristic
123Q8×C53—81no—
124Q8×C53—81no—
125C2×C4×C53—24no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—42no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12111yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×Dic53—24no—
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—24no—
148C2×Dic53—24no—
149C2×Dic53—22no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—81no—
157C2×Dic53#10414yes—
158C8×C53—81no—
159#88322—81no—
160#9408—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—44no—
169C22×C53#38611yescharacteristic
170C22×C53C2×#10412yes—
171C22×C53#45612yes—
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—44no—
177C4×C53—81no—
178C4×C53—81no—
179Dic53—24no—
180Dic53—44no—
181Dic53—44no—
182Dic53—42no—
183#2583—531no—
184#2274—531no—
185#1508—531no—
186#1508D5311yescharacteristic
187#1463—531no—
188#1297—531no—
189#1297—531no—
190C2×C53—22no—
191C2×C53—22no—
192C2×C53—41no—
193C2×C53#188611yescharacteristic
194C2×C53#189611yescharacteristic
195C2×C53#191311yescharacteristic
196C2×C22⋊Q8C2×D5311yescharacteristic
197C2×#117—1061no—
198C22×C4⋊C4—531no—
199#407—531no—
200#407C2×D5311yescharacteristic
201#398—532no—
202#395—532no—
203#389—1062no—
204#385—1061no—
205#385—1061no—
206C4×C4⋊C4—1062no—
207C4×C4⋊C4—1062no—
208C4×C22⋊C4—531no—
209C4×C22⋊C4—531no—
210C4×C22⋊C4—1061no—
211C4×C22⋊C4C2×D5311yescharacteristic
212#328—22no—
213#328—1062no—
214#324—1062no—
215C53#940811yescharacteristic, complemented, Sylow
216C22×Q8C4×D5311yescharacteristic
217C23×C4—531no—
218C23×C4—531no—
219C23×C4C22×D5311yescharacteristic
220C2×M16—21no—
221C2×M16—1061no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226#117—1068no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—532no—
229C2×C4⋊C4—532no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1062no—
236C2×C4⋊C4—1062no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4—1061no—
242C2×C42—21no—
243C2×C42—1062no—
244C2×C42—1062no—
245C2×C42—1061no—
246C2×C42—1061no—
247C22⋊C8—21no—
248C22⋊C8—21no—
249C22⋊C8—1061no—
250C22⋊C8—1061no—
251#95—532no—
252#95—532no—
253#95—1062no—
254#95—1062no—
255#95—1062no—
256#95C22⋊D5312yes—
257C24D4×D5311yescharacteristic
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—81no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—81no—
265C22×C4—532no—
266C22×C4—532no—
267C22×C4—1064no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4—1061no—
271C22×C4—1062no—
272C22×C4—1062no—
273C22×C4—1061no—
274C22×C4—4241no—
275C22×C4C2×C4×D5311yescharacteristic
276C22×C4#1171994511yescharacteristic
277C22×C4D4×D5311yescharacteristic
278C22×C4D4×D5311yescharacteristic
279C22×C4#1171994711yescharacteristic
280C22×C4C2×C22⋊D5311yescharacteristic
281M16—81no—
282M16—4241no—
283C2×C8—41no—
284C2×C8—2121no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—1068no—
290C4⋊C4—1062no—
291C4⋊C4—1062no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—4242no—
295C4⋊C4—2124no—
296C22⋊C4—24no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—1064no—
300C22⋊C4—1064no—
301C42—81no—
302C42—1068no—
303C42—4241no—
304C42—4241no—
305C42—4242no—
306C23—22no—
307C23—22no—
308C23—41no—
309C23C22⋊C4×D5311yescharacteristic
310C23#1313318511yescharacteristic
311C23#1313319911yescharacteristic
312Q8—81no—
313Q8—81no—
314C2×C4—24no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—534no—
328C2×C4—1062no—
329C2×C4—1062no—
330C2×C4—1064no—
331C2×C4—1062no—
332C2×C4—1062no—
333C2×C4—1064no—
334C2×C4—1064no—
335C2×C4—1062no—
336C2×C4—4241no—
337C2×C4—2122no—
338C2×C4—2122no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2122no—
342C2×C4—2122no—
343C2×C4#1313317711yescharacteristic
344C2×C4C4×C22⋊D5311yescharacteristic
345C2×C4#1313318811yescharacteristic
346C2×C4#1313319011yescharacteristic
347C8—81no—
348C8—4241no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—44no—
357C22#1681049911yescharacteristic
358C22#104×D5312yes—
359C22#1681080412yes—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—44no—
365C4—81no—
366C4—81no—
367C4—1064no—
368C4—2124no—
369C4—2124no—
370C4—2122no—
371C2—22no—
372C2—22no—
373C2—41no—
374C2#2027510511yescharacteristic
375C2#2027515511yescharacteristic
376C2#2027525411yescharacteristic
subgroupG/H
centerC22#16810499
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1508C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1508D53
hypocenterC53#9408
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1508

Complements

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

NH ≅ G/Ntype
11. #16810554C4semidirect
12. #16810542C4semidirect
215. C53#9408semidirect

Maximal subgroups

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

subgroupindex
#202800962
#202770702
#202732352
#202730442
#202726012
#202726002
C53×#15082
#940853