Finite Group Databasebeta

← #27281886

Subgroups — #27281886

Order 13568.

576 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 119 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#20279023C211yesmaximal, characteristic
2#20274802C212yesmaximal, complemented
3#20274703C211yesmaximal, characteristic
4#20272583C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133198—21no—
7C22×C4⋊Dic53C2211yescharacteristic
8C2×#13133132—21no—
9#16810553C412yescomplemented
10#16810548C412yescomplemented
11#16810521—21no—
12#16810484—21no—
13#16810483—21no—
14#16810483C2212yes—
15#16810483C2211yescharacteristic
16#16810390—21no—
17#16810386—24no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432—21no—
21C53×#407—21no—
22C2×C22⋊C8×C53C2212yes—
23C23×Dic53—21no—
24#13133198—44no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53C2×C412yes—
31C2×#11719919—22no—
32C2×C22⋊Dic53—22no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35#13133132—42no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124C2×C412yes—
40C22×D4×C53D411yescharacteristic
41C23×C4×C53D411yescharacteristic
42C23×C4×C53C2311yescharacteristic
43C2×M16×C53—22no—
44C22×C8×C53D414yes—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C2×C4⋊C4×C53C2×C411yescharacteristic
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C42×C53D411yescharacteristic
51C4⋊C8×C53—22no—
52C22⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55#97×C53—22no—
56C24×C53C22⋊C411yescharacteristic
57C24×C53C2×D411yescharacteristic
58C2×D4×C53—22no—
59C2×D4×C53—21no—
60C2×D4×C53—22no—
61C2×D4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—21no—
64C22×C4×C53—42no—
65C22×C4×C53—42no—
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22×C411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53C2×D411yescharacteristic
72C22×C4×C53#4112yes—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—42no—
76C22×Dic53—42no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—42no—
80M16×C53—82no—
81C2×C8×C53—24no—
82C2×C8×C53—44no—
83C4⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C42×C53—21no—
87C42×C53—21no—
88C22⋊Dic53—44no—
89C22⋊Dic53—42no—
90C22⋊Dic53—42no—
91C22⋊Dic53—42no—
92C4⋊Dic53—24no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96C4⋊Dic53—44no—
97#11719919—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—81no—
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12311yescharacteristic
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—82no—
135C2×Dic53—82no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C8×C53—44no—
140#9474—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—22no—
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—41no—
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#45812yes—
164Dic53—82no—
165Dic53—82no—
166Dic53—162no—
167#2973D5311yescharacteristic
168#2486—531no—
169#1823—532no—
170#1799—531no—
171#1289—532no—
172C2×C53—21no—
173C2×C53—22no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—81no—
177C2×C53#189011yescharacteristic
178C2×C53#189511yescharacteristic
179C2×C53#190911yescharacteristic
180C2×#123—1061no—
181C2×C4×D4C2×D5311yescharacteristic
182C22×C4⋊C4—531no—
183#435—24no—
184#432—21no—
185C2×#99—1061no—
186#407—21no—
187C2×C22⋊C8C2×D5312yes—
188#398—532no—
189#397—532no—
190#389—1061no—
191#381—1061no—
192#380—531no—
193#380—532no—
194#380—1061no—
195#342—1061no—
196#340—1064no—
197C53#947411yescharacteristic, complemented, Sylow
198C22×D4D21211yescharacteristic
199C23×C4—1061no—
200C23×C4C22⋊D5311yescharacteristic
201C23×C4C22×D5311yescharacteristic
202C2×M16—22no—
203C22×C8C22⋊D5314yes—
204#123—2124no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—532no—
208C2×C4⋊C4—1062no—
209C2×C4⋊C4—1062no—
210C2×C4⋊C4—1061no—
211C2×C4⋊C4—1061no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1062no—
214C2×C4⋊C4C4×D5311yescharacteristic
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1062no—
217C2×C22⋊C4—1062no—
218C2×C22⋊C4C4×D5311yescharacteristic
219C2×C22⋊C4D21211yescharacteristic
220C2×C42C22⋊D5311yescharacteristic
221C4⋊C8—22no—
222#99—2122no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#97—22no—
227#95—532no—
228#95—1062no—
229#95—1062no—
230#95—1062no—
231C24#1171992111yescharacteristic
232C24C2×D21211yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—1062no—
242C22×C4—1062no—
243C22×C4—2122no—
244C22×C4—2122no—
245C22×C4—2122no—
246C22×C4—2121no—
247C22×C4—2121no—
248C22×C4#1171992111yescharacteristic
249C22×C4#1171992111yescharacteristic
250C22×C4#1171992111yescharacteristic
251C22×C4C2×C4×D5311yescharacteristic
252C22×C4#1171994512yes—
253C22×C4C2×C22⋊D5312yes—
254C22×C4C2×C22⋊D5311yescharacteristic
255M16—82no—
256C2×C8—24no—
257C2×C8—44no—
258C4⋊C4—22no—
259C4⋊C4—1064no—
260C4⋊C4—2124no—
261C4⋊C4—2124no—
262C4⋊C4—2122no—
263C4⋊C4—2121no—
264C4⋊C4—2121no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—2122no—
268C22⋊C4—2122no—
269C22⋊C4—2124no—
270C22⋊C4—2122no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×#1171992111yescharacteristic
281C23#1313319112yes—
282C23#1313324512yes—
283C23#1313324511yescharacteristic
284C23#1313325011yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—4242no—
301C2×C4—4242no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4—2122no—
306C2×C4—2122no—
307C2×C4—2122no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4C4×C22⋊D5312yes—
312C2×C4#1313318611yescharacteristic
313C2×C4#1313320011yescharacteristic
314C8—44no—
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#1681050211yescharacteristic
328C22#1681062511yescharacteristic
329C22#1681062911yescharacteristic
330C22#1681086912yes—
331C22#1681087612yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—8482no—
338C4—4242no—
339C4—4242no—
340C4#1681081312yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2027512811yescharacteristic
347C2#2027515311yescharacteristic
348C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2973D53
hypocenterC53#9474
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20274802C2semidirect
4. #20272583C2semidirect
9. #16810553C4semidirect
10. #16810548C4semidirect
197. C53#9474semidirect

Maximal subgroups

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

subgroupindex
#202790232
#202748022
#202747032
#202725832
C53×#29732
#947453