Finite Group Databasebeta

← #27281864

Subgroups — #27281864

Order 13568.

598 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 131 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#20283739C211yesmaximal, characteristic
2#20279023C211yesmaximal, characteristic
3#20274799C212yesmaximal, complemented
4#20272582C212yesmaximal, complemented
5C53×#1799C211yesmaximal, characteristic
6C2×#13133198—21no—
7C22×C4⋊Dic53C2211yescharacteristic
8D4×#88322—24no—
9#16810676—21no—
10C2×C22⋊#88322—21no—
11C2×C22⋊#88322C2212yes—
12#16810553C412yescomplemented
13#16810548C412yescomplemented
14#16810521—21no—
15#16810484—21no—
16#16810483—21no—
17#16810483C2211yescharacteristic
18#16810385—24no—
19C2×C4×D4×C53C2211yescharacteristic
20C2×#99×C53—21no—
21C53×#380C2212yes—
22C53×#342—21no—
23C23×Dic53—21no—
24C22×#88322—22no—
25C22×#88322D414yes—
26#13133198—42no—
27#13133198—42no—
28C2×C4⋊Dic53—21no—
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53—22no—
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719919—22no—
34C2×C22⋊Dic53—22no—
35C2×C22⋊Dic53—21no—
36C4⋊#88322—22no—
37C22⋊#88322—22no—
38C22⋊#88322—22no—
39C22⋊#88322—22no—
40C4×#88322—22no—
41#13133124—22no—
42#13133124—22no—
43#13133124C2×C412yes—
44C22×D4×C53D411yescharacteristic
45C23×C4×C53D411yescharacteristic
46C23×C4×C53C2311yescharacteristic
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C22⋊C4×C53—22no—
52C2×C22⋊C4×C53C2×C411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55#99×C53—42no—
56#95×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×C53—42no—
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D412yes—
73C22×C4×C53C2×D411yescharacteristic
74C22×C4×C53#4112yes—
75C22×Dic53—22no—
76C22×Dic53—22no—
77C22×Dic53—42no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—42no—
81C4⋊C4×C53—22no—
82C4⋊C4×C53—44no—
83C22⋊C4×C53—22no—
84C22⋊C4×C53—22no—
85C22⋊C4×C53—42no—
86C22⋊C4×C53—42no—
87C42×C53—21no—
88C42×C53—21no—
89C22⋊Dic53—44no—
90C22⋊Dic53—41no—
91C22⋊Dic53—41no—
92C4⋊Dic53—24no—
93C4⋊Dic53—42no—
94C4⋊Dic53—41no—
95C4⋊Dic53—41no—
96#11719919—44no—
97C2×#88322—24no—
98C2×#88322—44no—
99C2×#88322—82no—
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—82no—
127C2×C4×C53—82no—
128C2×C4×C53—82no—
129C2×C4×C53#10212yes—
130C2×C4×C53C4×D412yes—
131C2×C4×C53#12011yescharacteristic
132C2×C4×C53#12311yescharacteristic
133C2×Dic53—42no—
134C2×Dic53—42no—
135C2×Dic53—42no—
136C2×Dic53—42no—
137C2×Dic53—42no—
138C2×Dic53—42no—
139C2×Dic53—82no—
140C2×Dic53—82no—
141#88322—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—162no—
166C4×C53#45812yes—
167Dic53—82no—
168Dic53—82no—
169#2972—531no—
170#2486—531no—
171#1823—532no—
172#1799D5311yescharacteristic
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—1064no—
186#431—1061no—
187C2×#99—21no—
188C2×C22⋊C8—532no—
189C2×C22⋊C8—1061no—
190#398—532no—
191#397—532no—
192#389—1061no—
193#381—1061no—
194#380—531no—
195#380—1061no—
196#380C2×D5312yes—
197#342—21no—
198#340—1064no—
199C53#947011yescharacteristic, complemented, Sylow
200C22×D4C22⋊D5311yescharacteristic
201C23×C4—1061no—
202C23×C4D21211yescharacteristic
203C23×C4C22×D5311yescharacteristic
204C22×C8—534no—
205C22×C8—1062no—
206#123—2122no—
207#123—2122no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—532no—
212C2×C4⋊C4—1062no—
213C2×C4⋊C4—1062no—
214C2×C4⋊C4—1061no—
215C2×C4⋊C4—1061no—
216C2×C4⋊C4—1062no—
217C2×C4⋊C4C4×D5311yescharacteristic
218C2×C22⋊C4—22no—
219C2×C22⋊C4—1061no—
220C2×C22⋊C4—1062no—
221C2×C22⋊C4C4×D5311yescharacteristic
222C2×C22⋊C4C22⋊D5311yescharacteristic
223C2×C42D21211yescharacteristic
224C4⋊C8—1062no—
225#99—42no—
226C22⋊C8—1062no—
227C22⋊C8—1062no—
228C22⋊C8—1062no—
229C4×C8—1062no—
230#95—22no—
231#95—532no—
232#95—1062no—
233#95—1062no—
234C24#1171992111yescharacteristic
235C24C2×C22⋊D5311yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4#1171992212yes—
239C2×D4#1171992412yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—1062no—
246C22×C4—1062no—
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×C4C2×D21211yescharacteristic
256C22×C4D4×D5312yes—
257C22×C4#1171994712yes—
258C2×C8—1064no—
259C2×C8—4242no—
260C2×C8—2124no—
261C4⋊C4—22no—
262C4⋊C4—44no—
263C4⋊C4—1064no—
264C4⋊C4—2122no—
265C4⋊C4—2124no—
266C4⋊C4—2121no—
267C4⋊C4—2121no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—42no—
271C22⋊C4—42no—
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#1313313712yes—
284C23C2×#1171992111yescharacteristic
285C23#1313318912yes—
286C23C2×#1171992211yescharacteristic
287C23C2×#1171992411yescharacteristic
288C23#1313323712yes—
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—82no—
304C2×C4—82no—
305C2×C4—82no—
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#1313313712yes—
315C2×C4#1313317712yes—
316C2×C4#1313318711yescharacteristic
317C2×C4#1313320111yescharacteristic
318C8—2124no—
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#1681050011yescharacteristic
332C22C2×#1313313711yescharacteristic
333C22#1681061711yescharacteristic
334C22#1681085412yes—
335C22#1681085712yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—162no—
342C4—4242no—
343C4—4242no—
344C4#1681081412yes—
345C2—22no—
346C2—21no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2027512611yescharacteristic
351C2#2027514111yescharacteristic
352C2#2027523511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1799C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC23×C4×C53C23
hypercenter#1799D53
hypocenterC53#9470
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #20274799C2semidirect
4. #20272582C2semidirect
12. #16810553C4semidirect
13. #16810548C4semidirect
199. C53#9470semidirect

Maximal subgroups

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

subgroupindex
#202837392
#202790232
#202747992
#202725822
C53×#17992
#947053