Finite Group Databasebeta

← #27281559

Subgroups — #27281559

Order 13568.

552 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 93 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#20277315C211yesmaximal, characteristic, complemented
3#20273231C211yesmaximal, characteristic
4#20273044C211yesmaximal, characteristic, complemented
5#20272601C211yesmaximal, characteristic, complemented
6#20272600C211yesmaximal, characteristic, complemented
7C53×#1507C211yesmaximal, characteristic
8C22×#11719919C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810561C412yescomplemented
11#16810537C412yescomplemented
12#16810517—22no—
13#16810507—22no—
14#16810505—22no—
15#16810503—21no—
16#16810503—21no—
17#16810484—21no—
18#16810481—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—
30C2×#11719919—22no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919C2×C412yes—
36C2×#11719919D412yes—
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—22no—
40C2×C22⋊Dic53—22no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—21no—
43C22⋊#88322—21no—
44C22⋊#88322—21no—
45#13133124—22no—
46#13133124—24no—
47#13133124—22no—
48#13133124—22no—
49#13133124C2×C412yes—
50#13133124D412yes—
51C22×D4×C53—21no—
52C22×D4×C53C2×C411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×M16×C53—21no—
55#121×C53—21no—
56#121×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C42×C53—21no—
63C22⋊C8×C53—21no—
64C22⋊C8×C53—21no—
65#95×C53D412yes—
66C24×C53—21no—
67C24×C53C2×D411yescharacteristic
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53—44no—
71C2×D4×C53—41no—
72C2×D4×C53—41no—
73C2×D4×C53—81no—
74C22×C4×C53—21no—
75C22×C4×C53—22no—
76C22×C4×C53—81no—
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C22×C4×C53#4111yescharacteristic
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53—22no—
88C22×Dic53—22no—
89C22×Dic53—42no—
90C22×Dic53—81no—
91C22×Dic53C22⋊C412yes—
92C22×Dic53C22⋊C412yes—
93M16×C53—81no—
94C2×C8×C53—41no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—21no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53—81no—
101C22⋊Dic53—44no—
102C22⋊Dic53—42no—
103C22⋊Dic53—42no—
104C22⋊Dic53—82no—
105#11719919—22no—
106#11719919—22no—
107#11719919—22no—
108#11719919—22no—
109#11719919—82no—
110C4×Dic53—81no—
111C4×Dic53—81no—
112#11719917—81no—
113C2×#88322—41no—
114C23×C53—22no—
115C23×C53—22no—
116C23×C53—41no—
117C23×C53—42no—
118C23×C53—42no—
119C23×C53—41no—
120C23×C53—42no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53#12011yescharacteristic
123C23×C53#12311yescharacteristic
124D4×C53—81no—
125D4×C53—81no—
126D4×C53—82no—
127D4×C53—82no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—42no—
137C2×C4×C53—41no—
138C2×C4×C53—42no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×Dic53—22no—
144C2×Dic53—22no—
145C2×Dic53—22no—
146C2×Dic53—22no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—42no—
153C2×Dic53—42no—
154C2×Dic53—42no—
155C2×Dic53—42no—
156C2×Dic53—42no—
157C2×Dic53—81no—
158C8×C53—81no—
159#88322—81no—
160#9414—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—84no—
170C22×C53—81no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53#38611yescharacteristic
174C22×C53#42112yes—
175C22×C53#45412yes—
176C4×C53—21no—
177C4×C53—21no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—82no—
181C4×C53—81no—
182Dic53—42no—
183Dic53—42no—
184Dic53—82no—
185Dic53—82no—
186#2591—531no—
187#2313—531no—
188#1507—531no—
189#1507D5311yescharacteristic
190#1463—531no—
191#1297—531no—
192#1297—531no—
193C2×C53—22no—
194C2×C53—22no—
195C2×C53—41no—
196C2×C53—161no—
197C2×C53#188811yescharacteristic
198C2×C53#189711yescharacteristic
199C2×C53#191311yescharacteristic
200C2×#121C2×D5311yescharacteristic
201C22×C4⋊C4—531no—
202#407—531no—
203#407C2×D5311yescharacteristic
204#400—532no—
205#394—532no—
206#388—1062no—
207#387—1062no—
208#387—1062no—
209#386—1061no—
210#386—1061no—
211#381—1061no—
212#380—1061no—
213C4×C22⋊C4—531no—
214C4×C22⋊C4—531no—
215C4×C22⋊C4C2×D5311yescharacteristic
216#327—22no—
217#327—1062no—
218#324—1062no—
219C53#941411yescharacteristic, complemented, Sylow
220C22×D4—21no—
221C22×D4C4×D5311yescharacteristic
222C23×C4—531no—
223C23×C4—531no—
224C23×C4C22×D5311yescharacteristic
225C2×M16—21no—
226C2×M16—1061no—
227#121—21no—
228#121—21no—
229#121—21no—
230#121—21no—
231C2×C4⋊C4—532no—
232C2×C4⋊C4—532no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—1061no—
235C2×C4⋊C4—1061no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1062no—
238C2×C4⋊C4C4×D5311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—1062no—
242C2×C22⋊C4—1062no—
243C2×C22⋊C4—1061no—
244C2×C22⋊C4—1061no—
245C2×C42—21no—
246C2×C42—1061no—
247C2×C42—1061no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—1061no—
251C22⋊C8—1061no—
252#95—532no—
253#95—532no—
254#95—1062no—
255#95—1062no—
256#95—1062no—
257#95—1064no—
258#95C22⋊D5312yes—
259C24—21no—
260C24D4×D5311yescharacteristic
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—44no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—81no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—81no—
270C22×C4—532no—
271C22×C4—532no—
272C22×C4—1061no—
273C22×C4—1061no—
274C22×C4—1061no—
275C22×C4—1061no—
276C22×C4—1062no—
277C22×C4—1062no—
278C22×C4—4241no—
279C22×C4—2122no—
280C22×C4C2×C4×D5311yescharacteristic
281C22×C4#1171994511yescharacteristic
282C22×C4D4×D5311yescharacteristic
283C22×C4D4×D5311yescharacteristic
284C22×C4#1171994711yescharacteristic
285C22×C4C2×C22⋊D5311yescharacteristic
286M16—81no—
287M16—4241no—
288C2×C8—41no—
289C2×C8—2121no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—1062no—
293C4⋊C4—1062no—
294C4⋊C4—1062no—
295C4⋊C4—1062no—
296C4⋊C4—4242no—
297C22⋊C4—42no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—4242no—
301C22⋊C4—2124no—
302C22⋊C4—2122no—
303C22⋊C4—2122no—
304C42—81no—
305C42—4241no—
306C42—4241no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23—42no—
311C23—42no—
312C23—41no—
313C23—42no—
314C23C22⋊C4×D5311yescharacteristic
315C23#1313318511yescharacteristic
316C23#1313319911yescharacteristic
317D4—81no—
318D4—81no—
319D4—82no—
320D4—82no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—1062no—
333C2×C4—1062no—
334C2×C4—1062no—
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—2122no—
344C2×C4—2122no—
345C2×C4—2122no—
346C2×C4—2122no—
347C2×C4#1313317711yescharacteristic
348C2×C4C4×C22⋊D5311yescharacteristic
349C2×C4#1313318811yescharacteristic
350C2×C4#1313319011yescharacteristic
351C8—81no—
352C8—4241no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—84no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#1681049911yescharacteristic
366C22#1681063812yes—
367C22#1681079412yes—
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—81no—
374C4—4242no—
375C4—4242no—
376C4—2122no—
377C4—2122no—
378C2—22no—
379C2—22no—
380C2—41no—
381C2—161no—
382C2#2027511511yescharacteristic
383C2#2027516011yescharacteristic
384C2#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#9414
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. #20277315C2semidirect
4. #20273044C2semidirect
5. #20272601C2semidirect
6. #20272600C2semidirect
10. #16810561C4semidirect
11. #16810537C4semidirect
219. C53#9414semidirect

Maximal subgroups

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

subgroupindex
#202801962
#202773152
#202732312
#202730442
#202726012
#202726002
C53×#15072
#941453