Finite Group Databasebeta

← #28311196

Subgroups — #28311196

Order 13568.

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

normal subgroups only

This group has 500 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×#16811531C212yesmaximal
2#20290582C212yesmaximal, complemented
3#20289812C212yesmaximal, complemented
4#20277262C211yesmaximal, characteristic
5#16811626—22no—
6C2×#13133278C2214yes—
7#16811531—24no—
8#16811503—22no—
9C2×#13133204C2211yescharacteristic
10C2×#13133182C2211yescharacteristic
11C2×D4×Dic53C411yescharacteristic
12#16810529C2211yescharacteristic
13#16810511—22no—
14#16810488C411yescharacteristic
15C22⋊C4×Dic53—22no—
16C42×Dic53C411yescharacteristic
17C53×#390C411yescharacteristic
18C2×#11719940—24no—
19#13133284—42no—
20C4⋊#88324—42no—
21#13133278—24no—
22#13133278—24no—
23C22×Dic106C2311yescharacteristic
24C23×Dic53—21no—
25#13133204—22no—
26#13133204D412yes—
27#13133204Q812yes—
28#13133182—41no—
29#13133182—41no—
30D4×Dic53—41no—
31D4×Dic53—41no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919D412yes—
34C2×C22⋊Dic53—21no—
35C2×C22⋊Dic53—22no—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53D412yes—
41#13133124—22no—
42#13133124—22no—
43C22×D4×C53C2×C411yescharacteristic
44C2×C4⋊C4×C53C2×C411yescharacteristic
45C2×C22⋊C4×C53—22no—
46C2×C42×C53C2×C411yescharacteristic
47C24×C53—21no—
48C2×D4×C53—21no—
49C2×D4×C53—41no—
50C2×D4×C53—41no—
51C2×D4×C53C4⋊C412yes—
52C22×C4×C53—21no—
53C22×C4×C53C22⋊C412yes—
54C22×C4×C53C22×C411yescharacteristic
55C22×Dic53—21no—
56C22×Dic53—21no—
57C22×Dic53—44no—
58C22×Dic53—41no—
59C22×Dic53—41no—
60C22×Dic53C22⋊C411yescharacteristic
61C22×Dic53C22⋊C411yescharacteristic
62C22×Dic53C2×D412yes—
63C2×Dic106—41no—
64C2×Dic106—41no—
65C2×Dic106C2×D411yescharacteristic
66C2×Dic106C2×Q811yescharacteristic
67C2×Dic106#4112yes—
68#11719940—44no—
69#11719940—44no—
70C2×#88324—44no—
71C4⋊C4×C53—41no—
72C22⋊C4×C53—44no—
73C42×C53—41no—
74C22⋊Dic53—24no—
75C22⋊Dic53—24no—
76C22⋊Dic53—42no—
77C4⋊Dic53—41no—
78#11719919—42no—
79C4×Dic53—21no—
80C4×Dic53—21no—
81C4×Dic53—21no—
82C4×Dic53—22no—
83C4×Dic53—22no—
84C4×Dic53—42no—
85C4×Dic53—42no—
86C4×Dic53C4⋊C412yes—
87C23×C53—22no—
88C23×C53—42no—
89C23×C53—42no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53C2×C22⋊C411yescharacteristic
93D4×C53—42no—
94D4×C53—42no—
95C2×C4×C53—21no—
96C2×C4×C53—21no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53C2×C4⋊C411yescharacteristic
103C2×C4×C53#11711yescharacteristic
104C2×Dic53—22no—
105C2×Dic53—22no—
106C2×Dic53—22no—
107C2×Dic53—22no—
108C2×Dic53—21no—
109C2×Dic53—41no—
110C2×Dic53—41no—
111C2×Dic53—41no—
112C2×Dic53—42no—
113C2×Dic53—41no—
114C2×Dic53—82no—
115C2×Dic53—82no—
116C2×Dic53#9911yescharacteristic
117C2×Dic53#9911yescharacteristic
118C2×Dic53#12112yes—
119C2×Dic53C22⋊Q812yes—
120Dic106—41no—
121Dic106—41no—
122Dic106—42no—
123#88324—84no—
124#12801—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—44no—
128C22×C53—42no—
129C22×C53—82no—
130C22×C53—81no—
131C22×C53—81no—
132C22×C53#38011yescharacteristic
133C22×C53C2×#9911yescharacteristic
134C22×C53#41011yescharacteristic
135C4×C53—21no—
136C4×C53—21no—
137C4×C53—42no—
138C4×C53—42no—
139Dic53—22no—
140Dic53—22no—
141Dic53—21no—
142Dic53—42no—
143Dic53—41no—
144Dic53—81no—
145Dic53#35512yescomplemented
146C4×#124—531no—
147C2×#355—532no—
148#1801—532no—
149#1394—532no—
150C2×C53—21no—
151C2×C53—21no—
152C2×C53—82no—
153C2×C53#179911yescharacteristic
154C2×C53C2×#35511yescharacteristic
155C2×C53#213211yescharacteristic
156C2×#124—531no—
157C2×C4×Q8—531no—
158C2×C4×D4—531no—
159#423—1062no—
160C2×#101—534no—
161#391—531no—
162#390#615711yescharacteristic
163#388—1062no—
164#382—531no—
165C4×C22⋊C4—1062no—
166C43—531no—
167#355—1064no—
168#343—1062no—
169C53#1280111yescharacteristic, complemented, Sylow
170C22×Q8—531no—
171C22×D4C2×#615711yescharacteristic
172C23×C4—1061no—
173C2×M16—1064no—
174#124—532no—
175#124—532no—
176#124—1062no—
177C4×Q8—2121no—
178C4×Q8—2121no—
179C4×D4—2121no—
180C4×D4—2121no—
181C2×C4⋊C4—532no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4C2×#615711yescharacteristic
184C2×C22⋊C4—22no—
185C2×C22⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C42—531no—
188C2×C42—532no—
189C2×C42—531no—
190C2×C42—1061no—
191C2×C42—1061no—
192C2×C42C2×#615711yescharacteristic
193C4⋊C8—2122no—
194C4⋊C8—2122no—
195#101—1064no—
196#101—1064no—
197#95—1062no—
198#95—1062no—
199C24—21no—
200C2×Q8—531no—
201C2×Q8—531no—
202C2×Q8—532no—
203C2×Q8—2121no—
204C2×Q8—2121no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4C4⋊#615712yes—
209C22×C4—21no—
210C22×C4—531no—
211C22×C4—532no—
212C22×C4—531no—
213C22×C4—1061no—
214C22×C4—1061no—
215C22×C4—2121no—
216C22×C4—2121no—
217C22×C4—2124no—
218C22×C4C22⋊#615712yes—
219C22×C4C22×#615711yescharacteristic
220M16—2124no—
221M16—2124no—
222C2×C8—2124no—
223C4⋊C4—41no—
224C4⋊C4—2122no—
225C4⋊C4—2121no—
226C22⋊C4—44no—
227C22⋊C4—1064no—
228C22⋊C4—1064no—
229C22⋊C4—2122no—
230C42—41no—
231C42—532no—
232C42—1062no—
233C42—1062no—
234C42—1061no—
235C42—1061no—
236C42—1061no—
237C42—2122no—
238C42—2122no—
239C23—22no—
240C23—42no—
241C23—42no—
242C23—41no—
243C23—41no—
244C23C2×C22⋊#615711yescharacteristic
245Q8—2121no—
246Q8—2121no—
247Q8—2122no—
248D4—42no—
249D4—42no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—531no—
258C2×C4—532no—
259C2×C4—532no—
260C2×C4—531no—
261C2×C4—1062no—
262C2×C4—1062no—
263C2×C4—1062no—
264C2×C4—1062no—
265C2×C4—1061no—
266C2×C4—4242no—
267C2×C4—4242no—
268C2×C4—2122no—
269C2×C4—2121no—
270C2×C4—2121no—
271C2×C4—2121no—
272C2×C4—2121no—
273C2×C4C2×C4⋊#615711yescharacteristic
274C2×C4#1313329611yescharacteristic
275C8—4244no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—82no—
281C22—81no—
282C22—81no—
283C22#1681156511yescharacteristic
284C22D53⋊#9911yescharacteristic
285C22#1681160311yescharacteristic
286C4—21no—
287C4—21no—
288C4—42no—
289C4—42no—
290C4—532no—
291C4—1062no—
292C4—1062no—
293C4—1061no—
294C4—4241no—
295C4—2122no—
296C4—2121no—
297C2—21no—
298C2—21no—
299C2—82no—
300C2#2029057711yescharacteristic
301C2D53⋊#35511yescharacteristic
302C2#2029100711yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC53×#390C4
socleC22×C53C2×#99
prime radicalC22×D4×C53C2×C4
cosocleC22×Dic106C23
hypercenter#390#6157
hypocenterC53#12801
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. #20290582C2semidirect
3. #20289812C2semidirect
145. Dic53#355semidirect
169. C53#12801semidirect

Maximal subgroups

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

subgroupindex
C2×#168115312
#202905822
#202898122
#202772622
#1280153