Finite Group Databasebeta

← #27293488

Subgroups — #27293488

Order 13568.

490 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×#16810488C211yesmaximal, characteristic
2#20273045C211yesmaximal, characteristic
3#20272658C214yesmaximal, complemented
4C53×#1463C211yesmaximal, characteristic
5C22×C4⋊Dic53C411yescharacteristic
6C22×C4×Dic53C411yescharacteristic
7#16810587—21no—
8#16810488—21no—
9#16810488—21no—
10#16810488C412yes—
11#16810488C2212yes—
12C2×#13133124—21no—
13C2×#13133124—21no—
14#16810349—21no—
15#16810349C2212yes—
16C22×C4⋊C4×C53C2211yescharacteristic
17C53×#407—21no—
18C53×#324—21no—
19C53×#324C2212yes—
20C23×Dic53—21no—
21C23×Dic53—21no—
22C2×#11719917—41no—
23C2×C4⋊Dic53—21no—
24C2×C4⋊Dic53—41no—
25C2×C4⋊Dic53—41no—
26C2×C4⋊Dic53C2×C412yes—
27C2×C4×Dic53—24no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C412yes—
30C22⋊#88322—22no—
31C22⋊#88322—41no—
32#13133124—22no—
33#13133124—22no—
34#13133124—22no—
35#13133124—22no—
36C23×C4×C53C2×C411yescharacteristic
37C23×C4×C53D411yescharacteristic
38C23×C4×C53D411yescharacteristic
39C2×M16×C53—41no—
40C2×C4⋊C4×C53—41no—
41C2×C4⋊C4×C53—41no—
42C2×C4⋊C4×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53D411yescharacteristic
44C2×C4⋊C4×C53D411yescharacteristic
45C2×C4⋊C4×C53C2311yescharacteristic
46C22⋊C8×C53—22no—
47C22⋊C8×C53—41no—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—21no—
50C22×C4×C53—21no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—42no—
58C22×C4×C53—42no—
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53C2×D411yescharacteristic
65C22×Dic53—24no—
66C22×Dic53—22no—
67C22×Dic53—22no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53—44no—
71C22×Dic53—41no—
72C22×Dic53—41no—
73M16×C53—82no—
74C2×C8×C53—42no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊Dic53—42no—
80C4⋊Dic53—42no—
81C4×Dic53—24no—
82C4×Dic53—22no—
83C4×Dic53M1614yescomplemented
84#11719917—82no—
85C2×#88322—42no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53#9911yescharacteristic
91C23×C53#9911yescharacteristic
92C23×C53C2×C22⋊C411yescharacteristic
93C2×C4×C53—24no—
94C2×C4×C53—24no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—81no—
110C2×C4×C53—81no—
111C2×C4×C53#10112yes—
112C2×C4×C53C2×M1612yes—
113C2×Dic53—24no—
114C2×Dic53—48no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117C2×Dic53—44no—
118C2×Dic53—42no—
119C2×Dic53—82no—
120C2×Dic53—82no—
121C8×C53—82no—
122#88322—82no—
123#11511—531nomaximal, Sylow
124C22×C53—24no—
125C22×C53—24no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—21no—
130C22×C53—21no—
131C22×C53—21no—
132C22×C53#35412yes—
133C22×C53#35512yes—
134C22×C53#40711yescharacteristic
135C22×C53C2×#9911yescharacteristic
136C22×C53C2×#10111yescharacteristic
137C4×C53—22no—
138C4×C53—22no—
139C4×C53—41no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53—41no—
143Dic53—44no—
144Dic53—82no—
145C2×#382—531no—
146#1463—531no—
147#1463D5311yescharacteristic
148#1324—534no—
149C2×C53—21no—
150C2×C53—22no—
151C2×C53—22no—
152C2×C53—21no—
153C2×C53#146311yescharacteristic
154C2×C53C2×#35411yescharacteristic
155C2×C53C2×#35511yescharacteristic
156C22×C4⋊C4—531no—
157C22×C4⋊C4C2×D5311yescharacteristic
158C22×C42—531no—
159#407—21no—
160#407—1061no—
161#382—532no—
162#382—532no—
163#382—1061no—
164#382—1061no—
165C2×#95—1061no—
166C2×#95—1061no—
167#324—21no—
168#324—532no—
169#324—1061no—
170#324C2×D5312yes—
171C53#1151111yescharacteristic, complemented, Sylow
172C23×C4—1061no—
173C23×C4—1061no—
174C23×C4C4×D5311yescharacteristic
175C23×C4D21211yescharacteristic
176C23×C4C22⋊D5311yescharacteristic
177C2×M16—41no—
178C2×M16—2121no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—1061no—
183C2×C4⋊C4—2121no—
184C2×C4⋊C4—2121no—
185C2×C4⋊C4C4×D5311yescharacteristic
186C2×C4⋊C4D21211yescharacteristic
187C2×C4⋊C4C22⋊D5311yescharacteristic
188C2×C4⋊C4C22×D5311yescharacteristic
189C2×C42—532no—
190C2×C42—1064no—
191C2×C42—1061no—
192C22⋊C8—22no—
193C22⋊C8—41no—
194C22⋊C8—1062no—
195C22⋊C8—2121no—
196#95—1062no—
197#95—1062no—
198#95—1062no—
199#95—1062no—
200C24#1171992111yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—1064no—
212C22×C4—1062no—
213C22×C4—1062no—
214C22×C4—2124no—
215C22×C4—2121no—
216C22×C4—2121no—
217C22×C4—2122no—
218C22×C4—2122no—
219C22×C4#1171992111yescharacteristic
220C22×C4#1171992111yescharacteristic
221C22×C4#1171992111yescharacteristic
222C22×C4C2×C4×D5311yescharacteristic
223C22×C4C2×D21211yescharacteristic
224C22×C4C2×C22⋊D5311yescharacteristic
225M16—82no—
226M16—4242no—
227C2×C8—42no—
228C2×C8—2122no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—2122no—
234C4⋊C4—2122no—
235C42—534no—
236C42—1064no—
237C42—1062no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—22no—
242C23#1313313111yescharacteristic
243C23#1313313111yescharacteristic
244C23C2×#1171992111yescharacteristic
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—1064no—
264C2×C4—4242no—
265C2×C4—4242no—
266C2×C4—2124no—
267C2×C4—2122no—
268C2×C4—2128no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4#1313313512yes—
272C2×C4M16×D5312yes—
273C8—82no—
274C8—4242no—
275C22—24no—
276C22—24no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22#1681041712yes—
284C22#1681041912yes—
285C22#1681058611yescharacteristic
286C22C2×#1313313111yescharacteristic
287C22C2×#1313313511yescharacteristic
288C4—22no—
289C4—22no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—4242no—
295C4—2124no—
296C2—21no—
297C2—21no—
298C2—22no—
299C2—22no—
300C2#2027304411yescharacteristic
301C2C2×#1681041711yescharacteristic
302C2C2×#1681041911yescharacteristic
subgroupG/H
centerC22C2×#13133131
commutatorC22×C4×C53C22×C4
FrattiniC2×C4⋊C4C22×D53
FittingC53×#1463C2
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊C4×C53C23
hypercenter#1463D53
hypocenterC53#11511
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #20272658C2semidirect
83. C4×Dic53M16semidirect
171. C53#11511semidirect

Maximal subgroups

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

subgroupindex
C2×#168104882
#202730452
#202726582
C53×#14632
#1151153