Finite Group Databasebeta

← #27281882

Subgroups — #27281882

Order 13568.

556 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 123 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#20283744C211yesmaximal, characteristic
2#20274805C212yesmaximal, complemented
3#20274703C211yesmaximal, characteristic
4#20272582C212yesmaximal, complemented
5C53×#2487C211yesmaximal, characteristic
6#16810695—24no—
7#16810680—21no—
8C2×#13133132—21no—
9#16810587—21no—
10C2×C22⋊#88322C2212yes—
11#16810483C2212yes—
12#16810390—21no—
13#16810385—24no—
14C2×C4×D4×C53C2211yescharacteristic
15C22×C4⋊C4×C53C2211yescharacteristic
16C53×#400C412yescomplemented
17C53×#399C412yescomplemented
18C53×#389—21no—
19C53×#386—21no—
20C53×#385—21no—
21C53×#380—21no—
22C53×#380C2211yescharacteristic
23C2×#11719917—22no—
24C22×#88322D414yes—
25C2×C4⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27C4⋊#88322—22no—
28#13133132—42no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32#13133128—22no—
33#13133124—22no—
34C22×D4×C53D411yescharacteristic
35C23×C4×C53—21no—
36C23×C4×C53D411yescharacteristic
37C23×C4×C53C2311yescharacteristic
38C4×D4×C53—22no—
39C4×D4×C53—22no—
40C2×C4⋊C4×C53—22no—
41C2×C4⋊C4×C53—21no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53C2×C412yes—
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C22⋊C4×C53—22no—
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52#95×C53—22no—
53#95×C53—22no—
54#95×C53C2×C412yes—
55C24×C53C22⋊C411yescharacteristic
56C24×C53C2×D411yescharacteristic
57C2×D4×C53—22no—
58C2×D4×C53—21no—
59C2×D4×C53—21no—
60C2×D4×C53—22no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—22no—
64C22×C4×C53—22no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—82no—
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×C4×C53C2×D412yes—
74C22×C4×C53C2×D411yescharacteristic
75C22×C4×C53#4112yes—
76C22×Dic53—42no—
77C4⋊C4×C53—22no—
78C4⋊C4×C53—24no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—41no—
81C4⋊C4×C53—41no—
82C4⋊C4×C53—82no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—41no—
86C22⋊C4×C53—41no—
87C22⋊C4×C53—82no—
88C42×C53—21no—
89C42×C53—21no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—44no—
93#11719917—82no—
94C2×#88322—24no—
95C2×#88322—44no—
96C23×C53—21no—
97C23×C53—21no—
98C23×C53—21no—
99C23×C53—41no—
100C23×C53—41no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53C2×C22⋊C411yescharacteristic
104C23×C53#12112yes—
105C23×C53AGL(1,ℤ/8)11yescharacteristic
106C23×C53#13711yescharacteristic
107C23×C53#13712yes—
108D4×C53—42no—
109D4×C53—42no—
110C2×C4×C53—22no—
111C2×C4×C53—24no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—81no—
129C2×C4×C53—84no—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12311yescharacteristic
133C2×Dic53—82no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136#88322—44no—
137#9473—531nomaximal, Sylow
138C22×C53—22no—
139C22×C53—21no—
140C22×C53—22no—
141C22×C53—21no—
142C22×C53—41no—
143C22×C53—41no—
144C22×C53—42no—
145C22×C53—42no—
146C22×C53—41no—
147C22×C53—41no—
148C22×C53—41no—
149C22×C53—81no—
150C22×C53#38611yescharacteristic
151C22×C53#41711yescharacteristic
152C22×C53#41911yescharacteristic
153C22×C53#46812yes—
154C22×C53#47012yes—
155C4×C53—21no—
156C4×C53—22no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160C4×C53—82no—
161C4×C53—82no—
162C4×C53#45812yes—
163Dic53—162no—
164#2973—531no—
165#2487D5311yescharacteristic
166#1824—532no—
167#1799—531no—
168#1289—532no—
169C2×C53—22no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—22no—
173C2×C53—81no—
174C2×C53#189011yescharacteristic
175C2×C53#189411yescharacteristic
176C2×C53#191011yescharacteristic
177C2×C4×D4C2×D5311yescharacteristic
178C22×C4⋊C4C2×D5311yescharacteristic
179#435—1064no—
180#432—1061no—
181C2×#99—1061no—
182#407—1061no—
183C2×C22⋊C8—532no—
184#400Dic5312yescomplemented
185#399Dic5312yescomplemented
186#389—21no—
187#386—21no—
188#385—21no—
189#380—21no—
190#380—532no—
191#380C2×D5311yescharacteristic
192#342—1061no—
193#340—1064no—
194C53#947311yescharacteristic, complemented, Sylow
195C22×D4C22⋊D5311yescharacteristic
196C23×C4—21no—
197C23×C4C22⋊D5311yescharacteristic
198C23×C4C22×D5311yescharacteristic
199C2×M16—1062no—
200C22×C8—534no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4C2×Dic5312yes—
210C2×C4⋊C4C2×Dic5311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4C22⋊D5311yescharacteristic
216C2×C42C22⋊D5311yescharacteristic
217C4⋊C8—1062no—
218#99—2122no—
219C22⋊C8—1062no—
220C22⋊C8—1062no—
221C22⋊C8—1062no—
222#97—1062no—
223#95—22no—
224#95—22no—
225#95—1062no—
226#95C2×Dic5312yes—
227C24C22⋊Dic5311yescharacteristic
228C24C2×C22⋊D5311yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—22no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—82no—
241C22×C4—2122no—
242C22×C4C22⋊Dic5311yescharacteristic
243C22×C4C22⋊Dic5311yescharacteristic
244C22×C4C22⋊Dic5311yescharacteristic
245C22×C4D4×D5312yes—
246C22×C4#1171994712yes—
247C22×C4C22×Dic5311yescharacteristic
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—4242no—
250C2×C8—1064no—
251C2×C8—2124no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—82no—
258C4⋊C4—2124no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C22⋊C4—82no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×C22⋊Dic5311yescharacteristic
276C23#1313318912yes—
277C23#1313324211yescharacteristic
278C23#1313324711yescharacteristic
279C23#1313324812yes—
280D4—42no—
281D4—42no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—84no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4D4×Dic5312yes—
306C2×C4#1313318511yescharacteristic
307C2×C4#1313319811yescharacteristic
308C8—2124no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1681050311yescharacteristic
322C22#1681062411yescharacteristic
323C22#1681063211yescharacteristic
324C22#1681087112yes—
325C22#1681087912yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4—8482no—
334C4#1681081212yes—
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2027512911yescharacteristic
341C2#2027514911yescharacteristic
342C2#2027524311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2487C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2487D53
hypocenterC53#9473
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #20274805C2semidirect
4. #20272582C2semidirect
16. C53×#400C4semidirect
17. C53×#399C4semidirect
184. #400Dic53semidirect
185. #399Dic53semidirect
194. C53#9473semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202748052
#202747032
#202725822
C53×#24872
#947353