Finite Group Databasebeta

← #27281887

Subgroups — #27281887

Order 13568.

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

normal subgroups only

This group has 576 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#20274801C212yesmaximal, complemented
3#20274703C211yesmaximal, characteristic
4#20272582C212yesmaximal, complemented
5C53×#2486C211yesmaximal, 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×#123×C53—21no—
15C2×C4×D4×C53C2211yescharacteristic
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#398C412yescomplemented
18C53×#397C412yescomplemented
19C53×#389—21no—
20C53×#381—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
38#123×C53—44no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C2×C4⋊C4×C53—22no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53C2×C412yes—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53D411yescharacteristic
52C2×C42×C53D411yescharacteristic
53#95×C53—22no—
54#95×C53—22no—
55#95×C53C2×C412yes—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C2×D411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—21no—
61C2×D4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—22no—
65C22×C4×C53—22no—
66C22×C4×C53—42no—
67C22×C4×C53—42no—
68C22×C4×C53—42no—
69C22×C4×C53—42no—
70C22×C4×C53—41no—
71C22×C4×C53—41no—
72C22×C4×C53C22⋊C411yescharacteristic
73C22×C4×C53C22⋊C411yescharacteristic
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×C4×C53C2×D412yes—
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4112yes—
79C22×Dic53—42no—
80C4⋊C4×C53—22no—
81C4⋊C4×C53—24no—
82C4⋊C4×C53—42no—
83C4⋊C4×C53—44no—
84C4⋊C4×C53—41no—
85C4⋊C4×C53—41no—
86C22⋊C4×C53—22no—
87C22⋊C4×C53—22no—
88C22⋊C4×C53—44no—
89C22⋊C4×C53—42no—
90C42×C53—21no—
91C42×C53—21no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—44no—
95#11719917—82no—
96C2×#88322—24no—
97C2×#88322—44no—
98C23×C53—21no—
99C23×C53—21no—
100C23×C53—21no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53C2×C22⋊C411yescharacteristic
106C23×C53#12112yes—
107C23×C53AGL(1,ℤ/8)11yescharacteristic
108C23×C53AGL(1,ℤ/8)12yes—
109C23×C53#13711yescharacteristic
110D4×C53—42no—
111D4×C53—42no—
112C2×C4×C53—22no—
113C2×C4×C53—24no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—42no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—42no—
129C2×C4×C53—42no—
130C2×C4×C53—81no—
131C2×C4×C53—82no—
132C2×C4×C53—82no—
133C2×C4×C53C4×D412yes—
134C2×C4×C53#12011yescharacteristic
135C2×C4×C53#12311yescharacteristic
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139#88322—44no—
140#9474—531nomaximal, Sylow
141C22×C53—22no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—21no—
145C22×C53—41no—
146C22×C53—42no—
147C22×C53—41no—
148C22×C53—42no—
149C22×C53—41no—
150C22×C53—41no—
151C22×C53—41no—
152C22×C53—81no—
153C22×C53#38611yescharacteristic
154C22×C53#41711yescharacteristic
155C22×C53#41811yescharacteristic
156C22×C53#46812yes—
157C22×C53#46912yes—
158C4×C53—21no—
159C4×C53—22no—
160C4×C53—41no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—82no—
164C4×C53—82no—
165C4×C53#45812yes—
166Dic53—162no—
167#2973—531no—
168#2486D5311yescharacteristic
169#1823—532no—
170#1799—531no—
171#1289—532no—
172C2×C53—22no—
173C2×C53—21no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—81no—
177C2×C53#189011yescharacteristic
178C2×C53#189511yescharacteristic
179C2×C53#190911yescharacteristic
180C2×#123—21no—
181C2×C4×D4C2×D5311yescharacteristic
182C22×C4⋊C4C2×D5311yescharacteristic
183#435—1064no—
184#432—1061no—
185C2×#99—1061no—
186#407—1061no—
187C2×C22⋊C8—532no—
188#398Dic5312yescomplemented
189#397Dic5312yescomplemented
190#389—21no—
191#381—21no—
192#380—21no—
193#380—532no—
194#380C2×D5311yescharacteristic
195#342—1061no—
196#340—1064no—
197C53#947411yescharacteristic, complemented, Sylow
198C22×D4C22⋊D5311yescharacteristic
199C23×C4—21no—
200C23×C4C22⋊D5311yescharacteristic
201C23×C4C22×D5311yescharacteristic
202C2×M16—1062no—
203C22×C8—534no—
204#123—44no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4C2×Dic5312yes—
214C2×C4⋊C4C2×Dic5311yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C2×Dic5311yescharacteristic
219C2×C22⋊C4C22⋊D5311yescharacteristic
220C2×C42C22⋊D5311yescharacteristic
221C4⋊C8—1062no—
222#99—2122no—
223C22⋊C8—1062no—
224C22⋊C8—1062no—
225C22⋊C8—1062no—
226#97—1062no—
227#95—22no—
228#95—22no—
229#95—1062no—
230#95C2×Dic5312yes—
231C24C22⋊Dic5311yescharacteristic
232C24C2×C22⋊D5311yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—2122no—
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4D4×D5312yes—
252C22×C4#1171994712yes—
253C22×C4C22×Dic5311yescharacteristic
254C22×C4C2×C22⋊D5311yescharacteristic
255M16—4242no—
256C2×C8—1064no—
257C2×C8—2124no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—44no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2124no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic5311yescharacteristic
281C23#1313318912yes—
282C23#1313324211yescharacteristic
283C23#1313324312yes—
284C23#1313324711yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—4242no—
309C2×C4—4242no—
310C2×C4—4242no—
311C2×C4D4×Dic5312yes—
312C2×C4#1313318511yescharacteristic
313C2×C4#1313319811yescharacteristic
314C8—2124no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1681050311yescharacteristic
328C22#1681062411yescharacteristic
329C22#1681062811yescharacteristic
330C22#1681086712yes—
331C22#1681087412yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—8482no—
340C4#1681081212yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2027512911yescharacteristic
347C2#2027515411yescharacteristic
348C2#2027523811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2486C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2486D53
hypocenterC53#9474
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #20274801C2semidirect
4. #20272582C2semidirect
17. C53×#398C4semidirect
18. C53×#397C4semidirect
188. #398Dic53semidirect
189. #397Dic53semidirect
197. C53#9474semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202748012
#202747032
#202725822
C53×#24862
#947453