Finite Group Databasebeta

← #27281549

Subgroups — #27281549

Order 13568.

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

normal subgroups only

This group has 526 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#20280096C211yesmaximal, characteristic
2#20277315C211yesmaximal, characteristic
3#20273235C211yesmaximal, characteristic
4#20273044C211yesmaximal, characteristic
5#20272601C211yesmaximal, characteristic
6#20272600C211yesmaximal, characteristic
7C53×#1508C211yesmaximal, characteristic
8C22×#11719919C2211yescharacteristic
9#16810587C2211yescharacteristic
10#16810554C412yescomplemented
11#16810542C412yescomplemented
12#16810518—22no—
13#16810507—22no—
14#16810505—22no—
15#16810495—21no—
16#16810495—21no—
17#16810484—21no—
18#16810481—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—
30C2×#11719919—22no—
31C2×#11719919—21no—
32C2×#11719919—21no—
33C2×#11719919—21no—
34C2×#11719919—21no—
35C2×#11719919—22no—
36C2×#11719919—22no—
37C2×#11719919C2×C412yes—
38C2×#11719919D412yes—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
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×Q8×C53C2×C411yescharacteristic
52C23×C4×C53C2311yescharacteristic
53C2×M16×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C2×C4⋊C4×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×C53C2×D411yescharacteristic
67C2×Q8×C53—21no—
68C2×Q8×C53—21no—
69C2×Q8×C53—81no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—81no—
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C22×C4×C53#4111yescharacteristic
80C22×Dic53—22no—
81C22×Dic53—21no—
82C22×Dic53—21no—
83C22×Dic53—21no—
84C22×Dic53—21no—
85C22×Dic53—22no—
86C22×Dic53—42no—
87C22×Dic53—81no—
88C22×Dic53C22⋊C412yes—
89C22×Dic53C22⋊C412yes—
90M16×C53—81no—
91C2×C8×C53—41no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—41no—
95C4⋊C4×C53—41no—
96C22⋊C4×C53—42no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C42×C53—81no—
100C22⋊Dic53—24no—
101C22⋊Dic53—42no—
102#11719919—22no—
103#11719919—22no—
104#11719919—22no—
105#11719919—22no—
106#11719919—44no—
107#11719919—82no—
108#11719919—82no—
109C4×Dic53—81no—
110C4×Dic53—81no—
111#11719917—81no—
112C2×#88322—41no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—41no—
116C23×C53C2×C22⋊C411yescharacteristic
117C23×C53#12011yescharacteristic
118C23×C53#12311yescharacteristic
119Q8×C53—81no—
120Q8×C53—81no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53#12111yescharacteristic
137C2×C4×C53#12111yescharacteristic
138C2×Dic53—24no—
139C2×Dic53—22no—
140C2×Dic53—22no—
141C2×Dic53—22no—
142C2×Dic53—22no—
143C2×Dic53—42no—
144C2×Dic53—42no—
145C2×Dic53—42no—
146C2×Dic53—42no—
147C2×Dic53—42no—
148C2×Dic53—42no—
149C2×Dic53—42no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—81no—
153C8×C53—81no—
154#88322—81no—
155#9412—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—44no—
164C22×C53#38611yescharacteristic
165C22×C53#42112yes—
166C22×C53#45612yes—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—82no—
172C4×C53—81no—
173C4×C53—81no—
174Dic53—44no—
175Dic53—42no—
176Dic53—42no—
177Dic53—82no—
178#2583—531no—
179#2313—531no—
180#1508—531no—
181#1508D5311yescharacteristic
182#1463—531no—
183#1297—531no—
184#1297—531no—
185C2×C53—22no—
186C2×C53—22no—
187C2×C53—41no—
188C2×C53#188611yescharacteristic
189C2×C53#189711yescharacteristic
190C2×C53#191311yescharacteristic
191C2×C22⋊Q8C2×D5311yescharacteristic
192C22×C4⋊C4—531no—
193#407—531no—
194#407C2×D5311yescharacteristic
195#398—532no—
196#395—532no—
197#389—1062no—
198#387—1062no—
199#387—1062no—
200#385—1061no—
201#385—1061no—
202#381—1061no—
203#380—1061no—
204C4×C22⋊C4—531no—
205C4×C22⋊C4—531no—
206C4×C22⋊C4C2×D5311yescharacteristic
207#328—22no—
208#328—1062no—
209#324—1062no—
210C53#941211yescharacteristic, complemented, Sylow
211C22×Q8C4×D5311yescharacteristic
212C23×C4—531no—
213C23×C4—531no—
214C23×C4C22×D5311yescharacteristic
215C2×M16—21no—
216C2×M16—1061no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—532no—
223C2×C4⋊C4—532no—
224C2×C4⋊C4—1061no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1062no—
230C2×C4⋊C4—1062no—
231C2×C4⋊C4C4×D5311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1061no—
235C2×C22⋊C4—1061no—
236C2×C42—21no—
237C2×C42—1061no—
238C2×C42—1061no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—1061no—
242C22⋊C8—1061no—
243#95—532no—
244#95—532no—
245#95—1062no—
246#95—1062no—
247#95—1062no—
248#95—1064no—
249#95C22⋊D5312yes—
250C24D4×D5311yescharacteristic
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—81no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—81no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—1061no—
261C22×C4—1062no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1062no—
266C22×C4—4241no—
267C22×C4—2122no—
268C22×C4C2×C4×D5311yescharacteristic
269C22×C4#1171994511yescharacteristic
270C22×C4D4×D5311yescharacteristic
271C22×C4D4×D5311yescharacteristic
272C22×C4#1171994711yescharacteristic
273C22×C4C2×C22⋊D5311yescharacteristic
274M16—81no—
275M16—4241no—
276C2×C8—41no—
277C2×C8—2121no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—1062no—
283C4⋊C4—1062no—
284C4⋊C4—1062no—
285C4⋊C4—1062no—
286C4⋊C4—4242no—
287C4⋊C4—4242no—
288C4⋊C4—2124no—
289C22⋊C4—42no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—1064no—
293C22⋊C4—2122no—
294C42—81no—
295C42—4241no—
296C42—4241no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23C22⋊C4×D5311yescharacteristic
301C23#1313318511yescharacteristic
302C23#1313319911yescharacteristic
303Q8—81no—
304Q8—81no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—1064no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1062no—
322C2×C4—1062no—
323C2×C4—4241no—
324C2×C4—2122no—
325C2×C4—2122no—
326C2×C4—2122no—
327C2×C4—2122no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2122no—
331C2×C4—2122no—
332C2×C4—2122no—
333C2×C4#1313317711yescharacteristic
334C2×C4C4×C22⋊D5311yescharacteristic
335C2×C4#1313318811yescharacteristic
336C2×C4#1313319011yescharacteristic
337C8—81no—
338C8—4241no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22#1681049911yescharacteristic
348C22#1681063812yes—
349C22#1681080412yes—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—82no—
355C4—81no—
356C4—81no—
357C4—4242no—
358C4—2124no—
359C4—2122no—
360C4—2122no—
361C2—22no—
362C2—22no—
363C2—41no—
364C2#2027510511yescharacteristic
365C2#2027516011yescharacteristic
366C2#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#9412
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
10. #16810554C4semidirect
11. #16810542C4semidirect
210. C53#9412semidirect

Maximal subgroups

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

subgroupindex
#202800962
#202773152
#202732352
#202730442
#202726012
#202726002
C53×#15082
#941253