Finite Group Databasebeta

← #27281819

Subgroups — #27281819

Order 13568.

528 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 99 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#20279189C211yesmaximal, characteristic
2#20277318C211yesmaximal, characteristic
3#20273249C212yesmaximal
4#20272569C212yesmaximal
5C53×#1821C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810554—22no—
8#16810528C412yescomplemented
9#16810509—24no—
10#16810496—21no—
11#16810496—21no—
12#16810492C412yescomplemented
13#16810486C2212yes—
14#16810483—21no—
15C4×C22⋊Dic53—21no—
16C4×C22⋊Dic53C2211yescharacteristic
17#16810361—24no—
18C22×C4⋊C4×C53C2211yescharacteristic
19C53×#407C2212yes—
20C53×#343—22no—
21C22×Dic106C2×C411yescharacteristic
22#13133196—22no—
23#13133196—22no—
24C2×C4⋊Dic53—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C2×C22⋊Dic53—22no—
32C2×C4×Dic53—22no—
33C2×C4×Dic53C2×C412yes—
34#13133124—22no—
35#13133124—22no—
36#13133124—22no—
37#13133124—22no—
38#13133124—22no—
39#13133124—21no—
40#13133124D414yes—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C2×M16×C53—22no—
45C2×C4⋊C4×C53—22no—
46C2×C4⋊C4×C53—22no—
47C2×C4⋊C4×C53C2×C412yes—
48C2×C4⋊C4×C53D412yes—
49C22⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C24×C53C2×D411yescharacteristic
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—42no—
57C22×C4×C53—81no—
58C22×C4×C53—81no—
59C22×C4×C53C22⋊C412yes—
60C22×C4×C53C22⋊C412yes—
61C22×C4×C53C22×C411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D412yes—
64C22×C4×C53#4112yes—
65C22×Dic53—22no—
66C22×Dic53—21no—
67C22×Dic53—21no—
68C22×Dic53—24no—
69C2×Dic106—22no—
70C2×Dic106—42no—
71M16×C53—44no—
72C2×C8×C53—42no—
73C4⋊C4×C53—24no—
74C4⋊C4×C53—24no—
75C22⋊Dic53—24no—
76C22⋊Dic53—44no—
77C22⋊Dic53—41no—
78C22⋊Dic53—41no—
79C4⋊Dic53—22no—
80C4⋊Dic53—82no—
81#11719919—42no—
82#11719919—82no—
83#11719919—82no—
84C4×Dic53—42no—
85C4×Dic53—82no—
86C23×C53—22no—
87C23×C53—22no—
88C23×C53—41no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—22no—
95C2×C4×C53—22no—
96C2×C4×C53—42no—
97C2×C4×C53—42no—
98C2×C4×C53—42no—
99C2×C4×C53—42no—
100C2×C4×C53—42no—
101C2×C4×C53—42no—
102C2×C4×C53—81no—
103C2×C4×C53—81no—
104C2×C4×C53C4×D412yes—
105C2×C4×C53#12112yes—
106C2×Dic53—24no—
107C2×Dic53—42no—
108C2×Dic53—42no—
109C2×Dic53—44no—
110C2×Dic53—42no—
111C2×Dic53—44no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115C2×Dic53—44no—
116Dic106—82no—
117C8×C53—82no—
118#9461—531nomaximal, Sylow
119C22×C53—21no—
120C22×C53—21no—
121C22×C53—21no—
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53#38611yescharacteristic
128C22×C53#41012yes—
129C22×C53#45612yes—
130C4×C53—22no—
131C4×C53—22no—
132C4×C53—42no—
133C4×C53—42no—
134Dic53—44no—
135Dic53—84no—
136Dic53—81no—
137Dic53—81no—
138#2502—531no—
139#2313—531no—
140#1821D5311yescharacteristic
141#1511—532no—
142#1284—532no—
143C2×C53—22no—
144C2×C53—22no—
145C2×C53—41no—
146C2×C53#188611yescharacteristic
147C2×C53#189911yescharacteristic
148C2×C53#191111yescharacteristic
149C2×C22⋊Q8—531no—
150C22×C4⋊C4C2×D5311yescharacteristic
151#407C2×D5312yes—
152#398—1062no—
153#391—532no—
154#387—1064no—
155#385—1061no—
156#385—1061no—
157#384—532no—
158#381—532no—
159#380—1061no—
160C4×C22⋊C4—531no—
161C4×C22⋊C4—1061no—
162#343—22no—
163#328—1064no—
164C53#946111yescharacteristic, complemented, Sylow
165C22×Q8—531no—
166C23×C4D21211yescharacteristic
167C23×C4D21211yescharacteristic
168C23×C4C22×D5311yescharacteristic
169C2×M16—22no—
170C22⋊Q8—1062no—
171C22⋊Q8—1062no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—531no—
175C2×C4⋊C4—1062no—
176C2×C4⋊C4—1062no—
177C2×C4⋊C4—1062no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4C4×D5312yes—
180C2×C4⋊C4C22⋊D5312yes—
181C2×C22⋊C4—1061no—
182C2×C22⋊C4—1061no—
183C2×C22⋊C4—1062no—
184C2×C42—532no—
185C2×C42—1062no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188#95—534no—
189#95—1062no—
190#95—1062no—
191#95—1062no—
192#95—1061no—
193#95—1062no—
194#95—1062no—
195C24C2×D21211yescharacteristic
196C2×Q8—1062no—
197C2×Q8—2122no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—1062no—
206C22×C4—1061no—
207C22×C4—1061no—
208C22×C4—1064no—
209C22×C4#1171992112yes—
210C22×C4#1171992112yes—
211C22×C4C2×C4×D5311yescharacteristic
212C22×C4D4×D5312yes—
213C22×C4#1171994712yes—
214C22×C4C2×C22⋊D5311yescharacteristic
215M16—44no—
216C2×C8—42no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—1062no—
220C4⋊C4—4242no—
221C4⋊C4—4242no—
222C4⋊C4—4242no—
223C4⋊C4—2122no—
224C22⋊C4—1064no—
225C22⋊C4—2124no—
226C22⋊C4—2121no—
227C22⋊C4—2121no—
228C42—4242no—
229C42—2122no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23C2×#1171992111yescharacteristic
234C23#1313318711yescharacteristic
235C23#1313320111yescharacteristic
236Q8—4242no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—1064no—
250C2×C4—2124no—
251C2×C4—2124no—
252C2×C4—2122no—
253C2×C4—2121no—
254C2×C4—2121no—
255C2×C4—2124no—
256C2×C4—2122no—
257C2×C4—2122no—
258C2×C4—2122no—
259C2×C4#1313317712yes—
260C2×C4#1313318912yes—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—44no—
270C22#1681050011yescharacteristic
271C22#1681059612yes—
272C22#1681080812yes—
273C4—22no—
274C4—22no—
275C4—42no—
276C4—42no—
277C4—4244no—
278C4—4241no—
279C4—4241no—
280C4—2124no—
281C2—22no—
282C2—22no—
283C2—41no—
284C2#2027510611yescharacteristic
285C2#2027517111yescharacteristic
286C2#2027524511yescharacteristic
subgroupG/H
centerC22#16810500
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1821C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#1821D53
hypocenterC53#9461
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
8. #16810528C4semidirect
12. #16810492C4semidirect
164. C53#9461semidirect

Maximal subgroups

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

subgroupindex
#202791892
#202773182
#202732492
#202725692
C53×#18212
#946153