Finite Group Databasebeta

← #28310616

Subgroups — #28310616

Order 13568.

584 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 102 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×#16811529C212yesmaximal
2#20290460C212yesmaximal
3#20289809C212yesmaximal
4#20277258C211yesmaximal, characteristic
5C2×#13133276C2214yes—
6C4×C22⋊#6157—22no—
7#16811529C418yescomplemented
8#16811501—22no—
9C2×#13133206C2211yescharacteristic
10C2×Q8×Dic53C411yescharacteristic, complemented
11C2×#13133179C2211yescharacteristic
12#16810525C2211yescharacteristic
13#16810515—22no—
14#16810488C411yescharacteristic, complemented
15#16810470—22no—
16C42×Dic53C411yescharacteristic, complemented
17C53×#391C411yescharacteristic, complemented
18C2×C22⋊#6157—24no—
19C2×C4×#6157—42no—
20#13133276—24no—
21#13133276C2×C418yes—
22(C2×#6157)⋊C4—42no—
23C22×D212C2311yescharacteristic
24C22×C4×D53D412yes—
25#13133206—22no—
26#13133206C2×C414yes—
27Q8×Dic53—41no—
28Q8×Dic53—41no—
29#13133179—41no—
30#13133179—41no—
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719921—22no—
34C2×#11719921—22no—
35C2×#11719921D412yes—
36C2×C4×Dic53—21no—
37C2×C4×Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53C2×C411yescharacteristic
40C2×C4×Dic53C2×C411yescharacteristic
41#13133124—22no—
42C22×Q8×C53C2×C411yescharacteristic
43C2×C4⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45#95×C53—22no—
46C23×D53C2×D412yes—
47C22×#6157—44no—
48C2×Q8×C53—21no—
49C2×Q8×C53—41no—
50C2×Q8×C53—41no—
51C2×Q8×C53C4212yescomplemented
52C22×C4×C53—21no—
53C22×C4×C53—21no—
54C22×C4×C53C22⋊C412yes—
55C22×C4×C53C22×C411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53C22⋊C411yescharacteristic
59C22×Dic53C22⋊C411yescharacteristic
60C2×D212—41no—
61C2×D212—41no—
62C2×D212C22×C412yes—
63C2×D212#4112yes—
64C2×C4×D53—22no—
65C2×C4×D53—22no—
66C2×C4×D53—44no—
67C22⋊#6157—28no—
68C22⋊#6157—44no—
69C4×#6157—84no—
70C4⋊C4×C53—41no—
71C42×C53—41no—
72#11719921—28no—
73#11719921—42no—
74#11719921—44no—
75C4⋊Dic53—42no—
76C4⋊Dic53—41no—
77C4×Dic53—21no—
78C4×Dic53—21no—
79C4×Dic53—21no—
80C4×Dic53—42no—
81C4×Dic53—42no—
82C4×Dic53—41no—
83C4×Dic53—41no—
84C4×Dic53C4212yescomplemented
85C23×C53C2×C22⋊C411yescharacteristic
86C22×D53—22no—
87C22×D53—22no—
88C22×D53—42no—
89C22×D53—42no—
90C22×D53C4×D414yes—
91C2×#6157—48no—
92C2×#6157—84no—
93C2×#6157—84no—
94Q8×C53—42no—
95Q8×C53—42no—
96C2×C4×C53—21no—
97C2×C4×C53—21no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—41no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—42no—
106C2×C4×C53C2×C4211yescharacteristic
107C2×C4×C53#11711yescharacteristic
108C2×Dic53—22no—
109C2×Dic53—22no—
110C2×Dic53—21no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—42no—
115C2×Dic53—41no—
116C2×Dic53#9911yescharacteristic
117C2×Dic53#9911yescharacteristic
118D212—42no—
119D212—42no—
120C4×D53—44no—
121C4×D53—82no—
122C4×#354—531nomaximal, Sylow
123C22×C53—21no—
124C22×C53—21no—
125C22×C53C4×C22⋊C411yescharacteristic
126C22×C53C2×#9911yescharacteristic
127C22×C53#41011yescharacteristic
128C2×D53—24no—
129C2×D53—42no—
130C2×D53—42no—
131C2×D53—44no—
132C2×D53—82no—
133#6157—88no—
134C4×C53—21no—
135C4×C53—21no—
136C4×C53—42no—
137C4×C53—42no—
138C4×C53—42no—
139Dic53—21no—
140Dic53—42no—
141Dic53—41no—
142Dic53—81no—
143Dic53#35412yescomplemented
144C4×#124—531no—
145C2×#354—532no—
146C4×#99—532no—
147#1393—532no—
148C2×C53—21no—
149C2×C53—21no—
150C2×C53C4×#9911yescharacteristic
151C2×C53C2×#35411yescharacteristic
152C2×C53#212611yescharacteristic
153D53—44no—
154C2×#124—531no—
155C2×C4×Q8—531no—
156C2×C4×D4—531no—
157C2×#99—534no—
158#391#615711yescharacteristic, complemented
159#390—531no—
160#388—1062no—
161#382—531no—
162C4×C22⋊C4—1062no—
163C4×C22⋊C4—1062no—
164C43—531no—
165#354—538no—
166#342—1062no—
167C53C4×#35411yescharacteristic, complemented, Sylow
168C22×Q8C2×#615711yescharacteristic
169C22×D4—531no—
170C23×C4—532no—
171#124—534no—
172#124—1062no—
173C4×Q8—2121no—
174C4×Q8—2121no—
175C4×D4—2121no—
176C4×D4—2121no—
177C2×C4⋊C4—531no—
178C2×C4⋊C4—1061no—
179C2×C4⋊C4C2×#615711yescharacteristic
180C2×C22⋊C4—532no—
181C2×C22⋊C4—1062no—
182C2×C22⋊C4—1064no—
183C2×C22⋊C4—1062no—
184C2×C42—531no—
185C2×C42—531no—
186C2×C42—1061no—
187C2×C42—1061no—
188C2×C42—1061no—
189C2×C42—2122no—
190C2×C42C2×#615711yescharacteristic
191#99—538no—
192#99—1064no—
193#95—22no—
194#95—1062no—
195#95—2122no—
196C24—532no—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C4×#615712yescomplemented
201C2×D4—532no—
202C2×D4—532no—
203C2×D4—2121no—
204C2×D4—2121no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—531no—
208C22×C4—531no—
209C22×C4—1062no—
210C22×C4—1062no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4—2124no—
214C22×C4—2124no—
215C22×C4C22⋊#615712yes—
216C22×C4C22×#615711yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—2121no—
219C4⋊C4—2122no—
220C22⋊C4—1068no—
221C22⋊C4—1068no—
222C22⋊C4—2122no—
223C22⋊C4—2124no—
224C22⋊C4—2124no—
225C42—41no—
226C42—532no—
227C42—1061no—
228C42—1061no—
229C42—1061no—
230C42—4244no—
231C42—2122no—
232C42—2121no—
233C42—2121no—
234C42—2122no—
235C23—534no—
236C23—1062no—
237C23—1062no—
238C23—2122no—
239C23—2122no—
240C23C2×C22⋊#615711yescharacteristic
241Q8—42no—
242Q8—42no—
243D4—2122no—
244D4—2122no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—531no—
256C2×C4—531no—
257C2×C4—1062no—
258C2×C4—1062no—
259C2×C4—1061no—
260C2×C4—4242no—
261C2×C4—4244no—
262C2×C4—4244no—
263C2×C4—2124no—
264C2×C4—2122no—
265C2×C4—2128no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2121no—
269C2×C4—2121no—
270C2×C4C2×C4×#615711yescharacteristic
271C2×C4#1313329611yescharacteristic
272C22—21no—
273C22—21no—
274C22—1064no—
275C22—4242no—
276C22—2124no—
277C22—2122no—
278C22—2122no—
279C22C4×C22⋊#615711yescharacteristic
280C22D53⋊#9911yescharacteristic
281C22#1681160311yescharacteristic
282C4—21no—
283C4—21no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—532no—
288C4—1061no—
289C4—4248no—
290C4—4241no—
291C4—2122no—
292C4—2121no—
293C2—21no—
294C2—21no—
295C2—2124no—
296C2#2029046211yescharacteristic
297C2D53⋊#35411yescharacteristic
298C2#2029098911yescharacteristic
subgroupG/H
centerC22D53⋊#99
commutatorC2×C4×C53C2×C42
FrattiniC22×C4C22×#6157
FittingC53×#391C4
socleC22×C53C2×#99
prime radical, cosocleC22×D212C23
hypercenter#391#6157
hypocenterC53C4×#354
LCSG ▷ C2×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
7. #16811529C4semidirect
10. C2×Q8×Dic53C4semidirect
14. #16810488C4semidirect
16. C42×Dic53C4semidirect
17. C53×#391C4semidirect
51. C2×Q8×C53C42semidirect
84. C4×Dic53C42semidirect
143. Dic53#354semidirect
158. #391#6157semidirect
167. C53C4×#354semidirect
200. C2×Q8C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×#168115292
#202904602
#202898092
#202772582
C4×#35453