Finite Group Databasebeta

← #27281781

Subgroups — #27281781

Order 13568.

498 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 107 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#20279888C211yesmaximal, characteristic
2#20273431C212yesmaximal, complemented
3#20273056C211yesmaximal, characteristic
4#20272572C212yesmaximal, complemented
5C53×#2973C211yesmaximal, characteristic
6C2×#13133217C2211yescharacteristic
7#16810680C2212yes—
8#16810535—21no—
9#16810535—21no—
10#16810526C412yes—
11#16810488C412yes—
12#16810483—21no—
13#16810483—21no—
14C4×C4⋊Dic53C2211yescharacteristic
15#16810368—24no—
16#16810347—21no—
17#16810347—21no—
18C2×C4×D4×C53C2211yescharacteristic
19C53×#435—24no—
20C53×#432C2212yes—
21C53×#407—21no—
22C2×C22⋊C8×C53—21no—
23C22×Dic106—21no—
24#13133217—22no—
25#13133217—22no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—21no—
28C2×C4⋊Dic53C2×C412yes—
29C2×C22⋊Dic53—22no—
30C2×C22⋊Dic53—22no—
31C2×C4×Dic53C2×C412yes—
32C4⋊#88322—22no—
33C22⋊#88322—22no—
34#13133128—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—21no—
38C22×D4×C53C2×C411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53D411yescharacteristic
41C2×M16×C53—22no—
42C22×C8×C53—22no—
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C22⋊C4×C53D411yescharacteristic
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53D414yes—
50C22⋊C8×C53—22no—
51C22⋊C8×C53—22no—
52C22⋊C8×C53—22no—
53#97×C53—22no—
54C24×C53C22⋊C411yescharacteristic
55C24×C53C22⋊C411yescharacteristic
56C2×D4×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—41no—
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—42no—
64C22×C4×C53—42no—
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22×C411yescharacteristic
68C22×C4×C53C2×D411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×Dic53—22no—
71C22×Dic53—22no—
72C2×Dic106—42no—
73C2×Dic106—82no—
74M16×C53—82no—
75C2×C8×C53—24no—
76C2×C8×C53—82no—
77C4⋊C4×C53—22no—
78C22⋊C4×C53—22no—
79C22⋊C4×C53—22no—
80C42×C53C2×D412yes—
81C42×C53#4112yes—
82C22⋊Dic53—82no—
83C22⋊Dic53—82no—
84C4⋊Dic53—24no—
85C4⋊Dic53—42no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88C4×Dic53—42no—
89C2×#88322—42no—
90C23×C53—21no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—41no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53—41no—
99C23×C53#10112yes—
100C23×C53C2×C22⋊C411yescharacteristic
101D4×C53—44no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—22no—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—41no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53—81no—
113C2×C4×C53—81no—
114C2×C4×C53C4×D412yes—
115C2×C4×C53#12011yescharacteristic
116C2×C4×C53#12112yes—
117C2×C4×C53#12311yescharacteristic
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124Dic106—84no—
125C8×C53—44no—
126#88322—82no—
127#9453—531nomaximal, Sylow
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—22no—
131C22×C53—42no—
132C22×C53—42no—
133C22×C53—41no—
134C22×C53—41no—
135C22×C53—41no—
136C22×C53—41no—
137C22×C53—41no—
138C22×C53—81no—
139C22×C53—81no—
140C22×C53#38611yescharacteristic
141C22×C53C2×#10111yescharacteristic
142C22×C53#41311yescharacteristic
143C4×C53—21no—
144C4×C53—22no—
145C4×C53—22no—
146C4×C53—41no—
147C4×C53—41no—
148C4×C53#45512yes—
149Dic53—82no—
150Dic53—82no—
151#2973D5311yescharacteristic
152#2563—531no—
153#1556—532no—
154#1466—531no—
155#1285—532no—
156C2×C53—21no—
157C2×C53—21no—
158C2×C53—42no—
159C2×C53—81no—
160C2×C53#188511yescharacteristic
161C2×C53#190111yescharacteristic
162C2×C53#191011yescharacteristic
163C2×C4⋊Q8—531no—
164C2×C4×D4C2×D5311yescharacteristic
165#435—24no—
166#432—532no—
167#432C2×D5312yes—
168#407—21no—
169C2×C22⋊C8—21no—
170#393—1061no—
171#393—1061no—
172#390—532no—
173#382—532no—
174#380—1061no—
175#380—1061no—
176C4×C4⋊C4—531no—
177#332—1064no—
178#323—1061no—
179#323—1061no—
180C53#945311yescharacteristic, complemented, Sylow
181C22×Q8—1061no—
182C22×D4C4×D5311yescharacteristic
183C23×C4C22⋊D5311yescharacteristic
184C23×C4C22⋊D5311yescharacteristic
185C2×M16—22no—
186C22×C8—22no—
187C4⋊Q8—1062no—
188C4⋊Q8—1062no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—532no—
192C2×C4⋊C4—1061no—
193C2×C4⋊C4—1061no—
194C2×C4⋊C4C4×D5311yescharacteristic
195C2×C22⋊C4—1062no—
196C2×C22⋊C4—1062no—
197C2×C22⋊C4D21211yescharacteristic
198C2×C22⋊C4D21211yescharacteristic
199C2×C42—532no—
200C2×C42C22×D5311yescharacteristic
201C4⋊C8—1062no—
202C4⋊C8C22⋊D5314yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—1062no—
207#97—22no—
208#97—1062no—
209#95—1061no—
210#95—1062no—
211#95—1062no—
212C24#1171992111yescharacteristic
213C24#1171992111yescharacteristic
214C2×Q8—4242no—
215C2×Q8—2122no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—1062no—
226C22×C4—1062no—
227C22×C4#1171992111yescharacteristic
228C22×C4#1171992111yescharacteristic
229C22×C4C2×C4×D5311yescharacteristic
230C22×C4C2×D21211yescharacteristic
231C22×C4C2×C22⋊D5311yescharacteristic
232M16—82no—
233C2×C8—24no—
234C2×C8—82no—
235C2×C8—2122no—
236C4⋊C4—22no—
237C4⋊C4—1064no—
238C4⋊C4—2122no—
239C4⋊C4—2121no—
240C4⋊C4—2121no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—4242no—
244C22⋊C4—4242no—
245C42—2122no—
246C42#1171994512yes—
247C42C2×C22⋊D5312yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#1313313512yes—
258C23C2×#1171992111yescharacteristic
259Q8—4244no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—2122no—
274C2×C4—2122no—
275C2×C4—2122no—
276C2×C4—2122no—
277C2×C4—2122no—
278C2×C4—2122no—
279C2×C4C4×C22⋊D5312yes—
280C2×C4#1313318611yescharacteristic
281C2×C4#1313319112yes—
282C2×C4#1313320011yescharacteristic
283C8—44no—
284C8—4242no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#1681050211yescharacteristic
298C22C2×#1313313511yescharacteristic
299C22#1681060511yescharacteristic
300C4—22no—
301C4—21no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—4242no—
306C4—4242no—
307C4#1681080112yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#2027510311yescharacteristic
313C2#2027518311yescharacteristic
314C2#2027524211yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2973C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2973D53
hypocenterC53#9453
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2973

Complements

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

NH ≅ G/Ntype
2. #20273431C2semidirect
4. #20272572C2semidirect
180. C53#9453semidirect

Maximal subgroups

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

subgroupindex
#202798882
#202734312
#202730562
#202725722
C53×#29732
#945353