Finite Group Databasebeta

← #13125668

Subgroups — #13125668

Order 1664.

414 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 121 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#11719496C211yesmaximal, characteristic
2C2×#88178C211yesmaximal, characteristic, complemented
3C13×#407C211yesmaximal, characteristic, complemented
4#11718798C212yesmaximal, complemented
5#11718795C212yesmaximal, complemented
6#11718400C211yesmaximal, characteristic, complemented
7#11718399C211yesmaximal, characteristic, complemented
8#11718396C211yesmaximal, characteristic, complemented
9#11718394C212yesmaximal, complemented
10#11718384C212yesmaximal, complemented
11#11718383C211yesmaximal, characteristic, complemented
12C23×C4×C13C2211yescharacteristic
13C2×#6127C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×M16C2212yes—
16#88207—21no—
17C2×C22⋊Dic13—21no—
18#88184C2211yescharacteristic
19#88184C2211yescharacteristic
20#88183—21no—
21C4×C22⋊D13—21no—
22C4×C22⋊D13—21no—
23#88180C2211yescharacteristic
24C2×C4⋊Dic13C2211yescharacteristic
25#88178—21no—
26#88178—21no—
27#88178C2211yescharacteristic
28#88178C2212yes—
29#88178C2212yes—
30#88178C2211yescharacteristic
31C2×#6101—21no—
32C2×Dic52C2212yescomplemented
33C2×#6095C2212yescomplemented
34C13×C22⋊C8C2211yescharacteristic
35C13×C22⋊C8C2211yescharacteristic
36C13×C22⋊C8C2212yes—
37#88061C2212yescomplemented
38#88058C2212yescomplemented
39#88057C2212yescomplemented
40#88056C2212yescomplemented
41#88056C2212yescomplemented
42#88056C2212yescomplemented
43C24×C13D411yescharacteristic
44C22×C4×C13—42no—
45C22×C4×C13D411yescharacteristic
46C22×C4×C13D411yescharacteristic
47C22×C4×C13D411yescharacteristic
48C22×C4×C13C2311yescharacteristic
49C22×C4×C13C2311yescharacteristic
50C22×C4×C13C2311yescharacteristic
51C2×C22⋊D13—21no—
52C2×C22⋊D13—21no—
53C2×C22⋊D13—21no—
54C22×Dic13—21no—
55C22×Dic13—21no—
56#6127—42no—
57C2×D52C2311yescharacteristic
58C2×C4×D13—21no—
59C2×Dic26—21no—
60C2×Dic26—42no—
61C2×Dic26C2312yes—
62C2×Dic26C2311yescharacteristic
63C13×M16—42no—
64C2×C8×C13C2312yes—
65C2×C8×C13C2312yes—
66C22⋊Dic13—22no—
67C22⋊Dic13—22no—
68C22⋊Dic13—21no—
69C22⋊Dic13—21no—
70C22⋊Dic13—21no—
71#6103—21no—
72#6103—21no—
73C4⋊Dic13—21no—
74C4⋊Dic13C2312yes—
75C4⋊Dic13C2311yescharacteristic
76C4⋊Dic13C2311yescharacteristic
77#6101—22no—
78#6101—22no—
79#6101—21no—
80#6101—21no—
81C4×Dic13—21no—
82Dic52—42no—
83#6095—42no—
84#3196—131nomaximal, Sylow
85C23×C13—22no—
86C23×C13—41no—
87C23×C13—41no—
88C23×C13C2×D411yescharacteristic
89C23×C13C2×D411yescharacteristic
90C23×C13C2×D411yescharacteristic
91C22×D13—21no—
92C2×C4×C13—22no—
93C2×C4×C13—22no—
94C2×C4×C13—22no—
95C2×C4×C13—41no—
96C2×C4×C13—41no—
97C2×C4×C13—41no—
98C2×C4×C13C2×D411yescharacteristic
99C2×C4×C13C2×D411yescharacteristic
100C2×C4×C13C2×D411yescharacteristic
101C2×C4×C13C2411yescharacteristic
102C22⋊D13—42no—
103C22⋊D13—42no—
104C22⋊D13—42no—
105C2×Dic13—21no—
106C2×Dic13—21no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—21no—
110C2×Dic13—42no—
111C2×Dic13—42no—
112D52—22no—
113C4×D13—42no—
114Dic26—22no—
115Dic26—22no—
116Dic26—22no—
117Dic26—42no—
118C8×C13—22no—
119C8×C13—22no—
120#525—131no—
121C2×C22⋊Q8—131no—
122#485—132no—
123#484—131no—
124#481—131no—
125#470—132no—
126#468—132no—
127#452—132no—
128#451—131no—
129#448—131no—
130#407D1311yescharacteristic, complemented
131C22×C13—21no—
132C22×C13—21no—
133C22×C13—21no—
134C22×C13—21no—
135C22×C13—21no—
136C22×C13—21no—
137C22×C13—21no—
138C22×C13—21no—
139C22×C13—44no—
140C22×C13#13712yes—
141C22×C13C22×D411yescharacteristic
142C2×D13—41no—
143C2×D13—41no—
144C2×D13—41no—
145C4×C13—21no—
146C4×C13—21no—
147C4×C13—21no—
148C4×C13H(5,𝔽2)12yes—
149Dic13—42no—
150Dic13—41no—
151Dic13—41no—
152Dic13—42no—
153Dic13—41no—
154C2×#41—131no—
155C22×Q8—131no—
156C23×C4C2×D1311yescharacteristic
157C2×Q16—132no—
158C2×SD16—132no—
159C2×M16C2×D1312yes—
160#123—261no—
161C22⋊Q8—132no—
162C22⋊Q8—132no—
163C22⋊Q8—131no—
164C22⋊Q8—131no—
165C22⋊Q8—261no—
166C22⋊Q8—261no—
167#121—131no—
168#121—131no—
169#120—261no—
170C4×D4—261no—
171C4×D4—261no—
172#117—131no—
173C2×C4⋊C4—131no—
174C2×C4⋊C4—261no—
175C2×C22⋊C4—261no—
176#107—132no—
177#106—132no—
178#103—132no—
179#103—132no—
180#103—132no—
181#102—132no—
182C22⋊C8C2×D1311yescharacteristic
183C22⋊C8C2×D1311yescharacteristic
184C22⋊C8C2×D1312yes—
185C2×C13—22no—
186C2×C13—41no—
187C2×C13—41no—
188C2×C13#53411yescharacteristic
189C2×C13C2×#13711yescharacteristic
190C2×C13#57511yescharacteristic
191D13—81no—
192C24D5211yescharacteristic
193#41—522no—
194C2×Q8—132no—
195C2×Q8—131no—
196C2×Q8—261no—
197C2×Q8—522no—
198C2×D4—131no—
199C2×D4—261no—
200C2×D4—261no—
201C2×D4—261no—
202C22×C4—42no—
203C22×C4—261no—
204C22×C4—261no—
205C22×C4—261no—
206C22×C4D5211yescharacteristic
207C22×C4D5211yescharacteristic
208C22×C4D5211yescharacteristic
209C22×C4C22×D1311yescharacteristic
210C22×C4C22×D1311yescharacteristic
211C22×C4C22×D1311yescharacteristic
212Q16—522no—
213SD16—522no—
214M16—42no—
215C2×C8C22×D1312yes—
216C2×C8C22×D1312yes—
217C4⋊C4—132no—
218C4⋊C4—131no—
219C4⋊C4—131no—
220C4⋊C4—261no—
221C4⋊C4—262no—
222C4⋊C4—262no—
223C4⋊C4—261no—
224C4⋊C4—261no—
225C22⋊C4—262no—
226C22⋊C4—262no—
227C22⋊C4—261no—
228C22⋊C4—261no—
229C22⋊C4—261no—
230C22⋊C4—261no—
231C22⋊C4—261no—
232C42—261no—
233C13#319611yescharacteristic, complemented, Sylow
234C23—22no—
235C23—41no—
236C23—41no—
237C23—261no—
238C23C2×D5211yescharacteristic
239C23C2×D5211yescharacteristic
240C23C2×D5211yescharacteristic
241Q8—262no—
242Q8—262no—
243Q8—262no—
244Q8—522no—
245D4—262no—
246D4—522no—
247D4—522no—
248D4—522no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—261no—
256C2×C4—261no—
257C2×C4—262no—
258C2×C4—261no—
259C2×C4—262no—
260C2×C4—522no—
261C2×C4—522no—
262C2×C4—522no—
263C2×C4C2×D5211yescharacteristic
264C2×C4C2×D5211yescharacteristic
265C2×C4C2×D5211yescharacteristic
266C2×C4C23×D1311yescharacteristic
267C8—22no—
268C8—22no—
269C22—21no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—44no—
278C22—521no—
279C22—521no—
280C22—521no—
281C22#8816312yes—
282C22C22×D5211yescharacteristic
283C4—21no—
284C4—21no—
285C4—21no—
286C4—521no—
287C4—522no—
288C4—521no—
289C4—521no—
290C4—522no—
291C4#8825112yes—
292C2—22no—
293C2—41no—
294C2—41no—
295C2—1041no—
296C2#1171919811yescharacteristic
297C2C2×#8816311yescharacteristic
298C2#1171945311yescharacteristic
subgroupG/H
centerC22C22×D52
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#407C2
socleC22×C13C22×D4
prime radical#11719496C2
hypercenter#407D13
hypocenterC13#3196
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C2×#88178C2semidirect
3. C13×#407C2semidirect
4. #11718798C2semidirect
5. #11718795C2semidirect
6. #11718400C2semidirect
7. #11718399C2semidirect
8. #11718396C2semidirect
9. #11718394C2semidirect
10. #11718384C2semidirect
11. #11718383C2semidirect
32. C2×Dic52C22semidirect
33. C2×#6095C22semidirect
37. #88061C22semidirect
38. #88058C22semidirect
39. #88057C22semidirect
40. #88056C22semidirect
41. #88056C22semidirect
42. #88056C22semidirect
130. #407D13semidirect
233. C13#3196semidirect

Maximal subgroups

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

subgroupindex
#117194962
C2×#881782
C13×#4072
#117187982
#117187952
#117184002
#117183992
#117183962
#117183942
#117183842
#117183832
#319613