Finite Group Databasebeta

← #28311200

Subgroups — #28311200

Order 13568.

468 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 86 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×#16811533C212yesmaximal
2#20290582C212yesmaximal
3#20289812C212yesmaximal
4#20277284C211yesmaximal, characteristic
5#16811626—22no—
6C2×#13133278C2214yes—
7#16811533—24no—
8#16811503—22no—
9C2×#13133215C2211yescharacteristic
10C2×Q8×Dic53C411yescharacteristic
11C2×#13133182C2211yescharacteristic
12#16810529C2211yescharacteristic
13#16810520—22no—
14#16810492C411yescharacteristic
15#16810474—22no—
16C42×Dic53C411yescharacteristic
17C53×#391C411yescharacteristic
18C2×#11719940—24no—
19#13133284—42no—
20C4⋊#88324—42no—
21#13133278—24no—
22#13133278—24no—
23C22×Dic106C2311yescharacteristic
24#13133215—22no—
25#13133215D412yes—
26#13133215Q812yes—
27Q8×Dic53—41no—
28Q8×Dic53—41no—
29#13133182—41no—
30#13133182—41no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919D412yes—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41C2×C4×Dic53D412yes—
42#13133124—22no—
43C22×Q8×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46#95×C53—22no—
47C2×Q8×C53—21no—
48C2×Q8×C53—41no—
49C2×Q8×C53—41no—
50C2×Q8×C53C4⋊C412yes—
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53C22⋊C411yescharacteristic
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C2×D412yes—
60C2×Dic106—41no—
61C2×Dic106—41no—
62C2×Dic106C2×D411yescharacteristic
63C2×Dic106C2×Q811yescharacteristic
64C2×Dic106#4112yes—
65#11719940—44no—
66#11719940—44no—
67C2×#88324—44no—
68C4⋊C4×C53—41no—
69C42×C53—41no—
70C4⋊Dic53—42no—
71C4⋊Dic53—41no—
72#11719919—24no—
73#11719919—24no—
74#11719919—42no—
75#11719919—44no—
76C4×Dic53—24no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—22no—
81C4×Dic53—22no—
82C4×Dic53—42no—
83C4×Dic53—41no—
84C4×Dic53—41no—
85C4×Dic53C4⋊C412yes—
86C23×C53C2×C22⋊C411yescharacteristic
87Q8×C53—42no—
88Q8×C53—42no—
89C2×C4×C53—21no—
90C2×C4×C53—21no—
91C2×C4×C53—22no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—41no—
95C2×C4×C53—41no—
96C2×C4×C53—41no—
97C2×C4×C53—41no—
98C2×C4×C53—42no—
99C2×C4×C53C2×C4⋊C411yescharacteristic
100C2×C4×C53#11711yescharacteristic
101C2×Dic53—22no—
102C2×Dic53—22no—
103C2×Dic53—22no—
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—21no—
108C2×Dic53—41no—
109C2×Dic53—42no—
110C2×Dic53—41no—
111C2×Dic53#9911yescharacteristic
112C2×Dic53#9911yescharacteristic
113C2×Dic53#12112yes—
114C2×Dic53C22⋊Q812yes—
115Dic106—41no—
116Dic106—41no—
117Dic106—42no—
118#88324—84no—
119#12802—531nomaximal, Sylow
120C22×C53—21no—
121C22×C53—21no—
122C22×C53#38011yescharacteristic
123C22×C53C2×#9911yescharacteristic
124C22×C53#41011yescharacteristic
125C4×C53—21no—
126C4×C53—21no—
127C4×C53—42no—
128C4×C53—42no—
129C4×C53—42no—
130Dic53—24no—
131Dic53—22no—
132Dic53—22no—
133Dic53—21no—
134Dic53—41no—
135Dic53—81no—
136Dic53#35612yescomplemented
137C4×C4⋊Q8—531no—
138C2×#356—532no—
139#1801—532no—
140#1394—532no—
141C2×C53—21no—
142C2×C53—21no—
143C2×C53#179911yescharacteristic
144C2×C53C2×#35611yescharacteristic
145C2×C53#213211yescharacteristic
146C2×C4⋊Q8—531no—
147C2×C4×Q8—531no—
148C2×C4×Q8—531no—
149#423—1062no—
150C2×#101—534no—
151#391—531no—
152#391#615711yescharacteristic
153#389—1062no—
154#384—531no—
155C4×C4⋊C4—1062no—
156C43—531no—
157#356—1064no—
158#343—1062no—
159C53#1280211yescharacteristic, complemented, Sylow
160C22×Q8—531no—
161C22×Q8C2×#615711yescharacteristic
162C2×M16—1064no—
163C4⋊Q8—532no—
164C4⋊Q8—532no—
165C4⋊Q8—1062no—
166C4×Q8—2121no—
167C4×Q8—2121no—
168C4×Q8—2121no—
169C4×Q8—2121no—
170C2×C4⋊C4—532no—
171C2×C4⋊C4—531no—
172C2×C4⋊C4—1062no—
173C2×C4⋊C4—1061no—
174C2×C4⋊C4—1062no—
175C2×C4⋊C4C2×#615711yescharacteristic
176C2×C42—531no—
177C2×C42—532no—
178C2×C42—531no—
179C2×C42—1061no—
180C2×C42—1061no—
181C2×C42—1061no—
182C2×C42C2×#615711yescharacteristic
183C4⋊C8—2122no—
184C4⋊C8—2122no—
185#101—1064no—
186#101—1064no—
187#95—22no—
188#95—1062no—
189C2×Q8—21no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—531no—
193C2×Q8—531no—
194C2×Q8—532no—
195C2×Q8—2121no—
196C2×Q8—2121no—
197C2×Q8C4⋊#615712yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—531no—
201C22×C4—532no—
202C22×C4—531no—
203C22×C4—1061no—
204C22×C4—1061no—
205C22×C4C22⋊#615712yes—
206C22×C4C22×#615711yescharacteristic
207M16—2124no—
208M16—2124no—
209C2×C8—2124no—
210C4⋊C4—41no—
211C4⋊C4—1064no—
212C4⋊C4—1064no—
213C4⋊C4—2122no—
214C4⋊C4—2121no—
215C4⋊C4—2122no—
216C4⋊C4—2124no—
217C42—41no—
218C42—532no—
219C42—1062no—
220C42—1062no—
221C42—1061no—
222C42—1061no—
223C42—1064no—
224C42—1061no—
225C42—2121no—
226C42—2121no—
227C42—2122no—
228C23C2×C22⋊#615711yescharacteristic
229Q8—42no—
230Q8—42no—
231Q8—2121no—
232Q8—2121no—
233Q8—2122no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—531no—
245C2×C4—532no—
246C2×C4—532no—
247C2×C4—531no—
248C2×C4—1062no—
249C2×C4—1062no—
250C2×C4—1062no—
251C2×C4—1062no—
252C2×C4—1062no—
253C2×C4—1062no—
254C2×C4—1061no—
255C2×C4—2122no—
256C2×C4—2121no—
257C2×C4—2121no—
258C2×C4C2×C4⋊#615711yescharacteristic
259C2×C4#1313329611yescharacteristic
260C8—4244no—
261C22—21no—
262C22—21no—
263C22#1681156511yescharacteristic
264C22D53⋊#9911yescharacteristic
265C22#1681160311yescharacteristic
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—532no—
272C4—1064no—
273C4—1062no—
274C4—1062no—
275C4—1061no—
276C4—4241no—
277C4—2121no—
278C2—21no—
279C2—21no—
280C2#2029057711yescharacteristic
281C2D53⋊#35611yescharacteristic
282C2#2029100911yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radicalC23×C53C2×C22⋊C4
cosocleC22×Dic106C23
hypercenter#391#6157
hypocenterC53#12802
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #391

Complements

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

NH ≅ G/Ntype
136. Dic53#356semidirect
159. C53#12802semidirect

Maximal subgroups

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

subgroupindex
C2×#168115332
#202905822
#202898122
#202772842
#1280253