Finite Group Databasebeta

← #27281866

Subgroups — #27281866

Order 13568.

598 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2972C211yesmaximal, 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
19C8×D4×C53—24no—
20C53×#431—21no—
21C2×C22⋊C8×C53—21no—
22C2×C22⋊C8×C53C2212yes—
23C23×Dic53—21no—
24#13133198—42no—
25#13133198—42no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53—22no—
29C2×C4⋊Dic53—22no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53C2×C412yes—
32C2×#11719919—22no—
33C2×C22⋊Dic53—22no—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36#13133132—42no—
37#13133124—22no—
38#13133124—22no—
39#13133124—22no—
40#13133124C2×C412yes—
41C22×D4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C22×C8×C53—22no—
45C22×C8×C53D414yes—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C2×C4⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53C2×C411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C42×C53D411yescharacteristic
52C4⋊C8×C53—22no—
53C22⋊C8×C53—22no—
54C22⋊C8×C53—22no—
55C22⋊C8×C53—22no—
56C4×C8×C53—22no—
57C24×C53C22⋊C411yescharacteristic
58C24×C53C2×D411yescharacteristic
59C2×D4×C53—22no—
60C2×D4×C53—22no—
61C2×D4×C53D812yes—
62C2×D4×C53SD1612yes—
63C22×C4×C53—21no—
64C22×C4×C53—21no—
65C22×C4×C53—42no—
66C22×C4×C53—42no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22×C411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53#4112yes—
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—42no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—42no—
80C22×Dic53—42no—
81C2×C8×C53—24no—
82C2×C8×C53—44no—
83C2×C8×C53—82no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—44no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C4⋊Dic53—24no—
95C4⋊Dic53—42no—
96C4⋊Dic53—41no—
97C4⋊Dic53—41no—
98C4⋊Dic53—44no—
99#11719919—44no—
100C23×C53—21no—
101C23×C53—21no—
102C23×C53—41no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53#10212yes—
107C23×C53C2×C22⋊C411yescharacteristic
108C23×C53#12112yes—
109C23×C53C2×D811yescharacteristic
110C23×C53C2×SD1611yescharacteristic
111C23×C53#13512yes—
112D4×C53—24no—
113D4×C53—24no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—24no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—81no—
126C2×C4×C53#10212yes—
127C2×C4×C53C4×D412yes—
128C2×C4×C53#12011yescharacteristic
129C2×C4×C53#12311yescharacteristic
130C2×Dic53—42no—
131C2×Dic53—42no—
132C2×Dic53—42no—
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141C8×C53—44no—
142#9470—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—22no—
145C22×C53—21no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—41no—
149C22×C53—41no—
150C22×C53—42no—
151C22×C53—42no—
152C22×C53—41no—
153C22×C53—41no—
154C22×C53—81no—
155C22×C53#38611yescharacteristic
156C22×C53C2×#10211yescharacteristic
157C22×C53#41611yescharacteristic
158C22×C53#46512yes—
159C22×C53#46612yes—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—22no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53#45812yes—
166Dic53—82no—
167Dic53—82no—
168Dic53—162no—
169#2972D5311yescharacteristic
170#2486—531no—
171#1823—532no—
172#1799—531no—
173#1289—532no—
174C2×C53—21no—
175C2×C53—22no—
176C2×C53—21no—
177C2×C53—22no—
178C2×C53—42no—
179C2×C53#189011yescharacteristic
180C2×C53#189311yescharacteristic
181C2×C53#190911yescharacteristic
182C2×#123—1061no—
183C2×C4×D4C2×D5311yescharacteristic
184C22×C4⋊C4—531no—
185C8×D4—24no—
186#431—21no—
187C2×#99—1061no—
188C2×C22⋊C8—21no—
189C2×C22⋊C8C2×D5312yes—
190#398—532no—
191#397—532no—
192#389—1061no—
193#381—1061no—
194#380—531no—
195#380—532no—
196#380—1061no—
197#342—1061no—
198#340—1064no—
199C53#947011yescharacteristic, complemented, Sylow
200C22×D4D21211yescharacteristic
201C23×C4—1061no—
202C23×C4C22⋊D5311yescharacteristic
203C23×C4C22×D5311yescharacteristic
204C22×C8—22no—
205C22×C8C22⋊D5314yes—
206#123—2122no—
207#123—2122no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—532no—
211C2×C4⋊C4—1062no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1061no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1062no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4C4×D5311yescharacteristic
218C2×C22⋊C4—1061no—
219C2×C22⋊C4—1062no—
220C2×C22⋊C4—1062no—
221C2×C22⋊C4C4×D5311yescharacteristic
222C2×C22⋊C4D21211yescharacteristic
223C2×C42C22⋊D5311yescharacteristic
224C4⋊C8—22no—
225#99—2122no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C4×C8—22no—
230#95—532no—
231#95—1062no—
232#95—1062no—
233#95—1062no—
234C24#1171992111yescharacteristic
235C24C2×D21211yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1171991312yes—
239C2×D4D42412yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—1062no—
245C22×C4—1062no—
246C22×C4—2122no—
247C22×C4—2122no—
248C22×C4—2121no—
249C22×C4—2121no—
250C22×C4—2122no—
251C22×C4#1171992111yescharacteristic
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4C2×C4×D5311yescharacteristic
255C22×C4#1171994512yes—
256C22×C4C2×C22⋊D5312yes—
257C22×C4C2×C22⋊D5311yescharacteristic
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—1064no—
263C4⋊C4—2124no—
264C4⋊C4—2122no—
265C4⋊C4—2124no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—2122no—
271C22⋊C4—2122no—
272C22⋊C4—2124no—
273C22⋊C4—2121no—
274C22⋊C4—2121no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1313313912yes—
284C23C2×#1171992111yescharacteristic
285C23#1313319112yes—
286C23C2×D42411yescharacteristic
287C23C2×#1171991311yescharacteristic
288C23#1313323912yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—4242no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—2122no—
309C2×C4—2122no—
310C2×C4—2122no—
311C2×C4—2122no—
312C2×C4—2122no—
313C2×C4—2122no—
314C2×C4#1313313912yes—
315C2×C4C4×C22⋊D5312yes—
316C2×C4#1313318611yescharacteristic
317C2×C4#1313320011yescharacteristic
318C8—44no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1681050211yescharacteristic
332C22C2×#1313313911yescharacteristic
333C22#1681062011yescharacteristic
334C22#1681085212yes—
335C22#1681085912yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—8482no—
342C4—4242no—
343C4—4242no—
344C4#1681081312yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2027512811yescharacteristic
351C2#2027514311yescharacteristic
352C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2972C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#2972D53
hypocenterC53#9470
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #20274802C2semidirect
4. #20272583C2semidirect
9. #16810553C4semidirect
10. #16810548C4semidirect
199. C53#9470semidirect

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×#29722
#947053