Finite Group Databasebeta

← #13125838

Subgroups — #13125838

Order 1664.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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×#88201C211yesmaximal, characteristic
2#11719327C211yesmaximal, characteristic, complemented
3C13×#451C211yesmaximal, characteristic, complemented
4#11718889C211yesmaximal, characteristic, complemented
5#11718851C212yesmaximal, complemented
6#11718846C212yesmaximal, complemented
7#11718707C211yesmaximal, characteristic
8#11718459C212yesmaximal, complemented
9#11718456C212yesmaximal, complemented
10#11718392C211yesmaximal, characteristic
11#11718380C211yesmaximal, characteristic, complemented
12C22×Q8×C13C2211yescharacteristic
13C22×D52C2211yescharacteristic
14C2×C13×Q16C2212yescomplemented
15C13×C22⋊Q8C2211yescharacteristic
16#88201—21no—
17#88201—21no—
18#88201C2212yes—
19#88201C2212yes—
20Q8×Dic13C2211yescharacteristic, complemented
21#88198C2211yescharacteristic, complemented
22C2×#6107C2212yescomplemented
23C2×#6106C2212yes—
24C2×#6103—21no—
25C2×#6103—21no—
26#88178C2211yescharacteristic
27C2×C4×Dic13C2211yescharacteristic
28C2×#6095C2212yes—
29#88143—21no—
30#88142C2211yescharacteristic, complemented
31#88141—21no—
32#88141C2211yescharacteristic, complemented
33#88132—21no—
34#88131—21no—
35C13×#103C2212yescomplemented
36C13×C22⋊C8C2211yescharacteristic
37#88075C2212yescomplemented
38C22⋊#1055C2211yescharacteristic
39#88061C2212yes—
40#88055C2212yescomplemented
41#88054C2212yescomplemented
42#88049C2212yes—
43C23×D13—21no—
44C2×Q8×C13—21no—
45C2×Q8×C13—42no—
46C2×Q8×C13C2312yes—
47C2×Q8×C13C2311yescharacteristic
48C22×C4×C13—21no—
49C22×C4×C13C2311yescharacteristic
50C22×Dic13D411yescharacteristic
51C22×Dic13D411yescharacteristic
52C2×D52—21no—
53C2×D52—42no—
54C2×D52C2312yes—
55C2×Dic26—21no—
56C2×Dic26C2311yescharacteristic
57C13×Q16—42no—
58C2×C8×C13C2312yes—
59C13×C4⋊C4—21no—
60C13×C4⋊C4C2311yescharacteristic
61C13×C22⋊C4—21no—
62C22⋊Dic13—21no—
63#6107—42no—
64#6106—42no—
65#6103—22no—
66#6103—22no—
67#6103—22no—
68#6103—22no—
69C4⋊Dic13—21no—
70C4⋊Dic13C2311yescharacteristic
71#6101—21no—
72#6101—21no—
73#6101—21no—
74#6101—21no—
75#6101—21no—
76C4×Dic13—21no—
77C4×Dic13—21no—
78C4×Dic13D411yescharacteristic
79C4×Dic13D411yescharacteristic
80C4×Dic13C2311yescharacteristic
81C4×Dic13C2311yescharacteristic
82C2×#1055C2312yes—
83#6095—42no—
84#3215—131nomaximal, Sylow
85C23×C13C2×D411yescharacteristic
86C22×D13—22no—
87C22×D13—42no—
88C22×D13—42no—
89C22×D13—42no—
90Q8×C13—22no—
91Q8×C13—22no—
92Q8×C13—22no—
93Q8×C13—42no—
94C2×C4×C13—22no—
95C2×C4×C13—21no—
96C2×C4×C13—21no—
97C2×C4×C13—22no—
98C2×C4×C13—42no—
99C2×C4×C13C2×D411yescharacteristic
100C2×C4×C13C2411yescharacteristic
101C2×Dic13—22no—
102C2×Dic13—21no—
103C2×Dic13—21no—
104C2×Dic13—41no—
105C2×Dic13C2×D411yescharacteristic
106C2×Dic13C2×D411yescharacteristic
107C2×Dic13C2×D411yescharacteristic
108C2×Dic13C2×D411yescharacteristic
109D52—22no—
110D52—22no—
111D52—42no—
112Dic26—22no—
113Dic26—42no—
114C8×C13—22no—
115#1055—22no—
116#533—131no—
117C2×#124—131no—
118#497—132no—
119#489—132no—
120#464—132no—
121#464—132no—
122#451—131no—
123#451D1311yescharacteristic, complemented
124#450—131no—
125#450—131no—
126#432—131no—
127C22×C13—21no—
128C22×C13—21no—
129C22×C13#13712yes—
130C22×C13C22×D411yescharacteristic
131C2×D13—42no—
132C2×D13—42no—
133C2×D13—42no—
134C2×D13—84no—
135C4×C13—21no—
136C4×C13—42no—
137C4×C13—41no—
138C4×C13—41no—
139C4×C13H(5,𝔽2)12yes—
140Dic13—21no—
141Dic13—21no—
142Dic13—21no—
143Dic13—21no—
144Dic13—41no—
145Dic13—41no—
146C22×Q8C2×D1311yescharacteristic
147C22×D4—131no—
148C2×Q16—132no—
149C2×Q16C2×D1312yescomplemented
150C2×SD16—132no—
151C2×SD16—132no—
152C4⋊Q8—131no—
153C4⋊Q8—131no—
154#125—261no—
155#124—132no—
156#124—132no—
157#124—261no—
158#124—261no—
159C22⋊Q8—131no—
160C22⋊Q8—261no—
161C22⋊Q8C2×D1311yescharacteristic
162C4×Q8—131no—
163C4×Q8—131no—
164C4×Q8—261no—
165#117—261no—
166C2×C22⋊C4—261no—
167C2×C22⋊C4—261no—
168C2×C42—131no—
169C4⋊C8—132no—
170#103—132no—
171#103C2×D1312yescomplemented
172#102—132no—
173#102—132no—
174C22⋊C8—131no—
175C22⋊C8C2×D1311yescharacteristic
176#97—132no—
177C2×C13—22no—
178C2×C13#53511yescharacteristic
179C2×C13C2×#13711yescharacteristic
180C2×C13#57511yescharacteristic
181D13—82no—
182C24—261no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—131no—
186C2×Q8—261no—
187C2×Q8C22×D1312yes—
188C2×Q8C22×D1311yescharacteristic
189C2×D4—132no—
190C2×D4—261no—
191C2×D4—522no—
192C22×C4—21no—
193C22×C4—131no—
194C22×C4—131no—
195C22×C4C22×D1311yescharacteristic
196Q16—42no—
197Q16—522no—
198SD16—522no—
199SD16—522no—
200C2×C8—132no—
201C2×C8C22×D1312yes—
202C4⋊C4—21no—
203C4⋊C4—131no—
204C4⋊C4—261no—
205C4⋊C4—261no—
206C4⋊C4—261no—
207C4⋊C4—261no—
208C4⋊C4—261no—
209C4⋊C4—261no—
210C4⋊C4C22×D1311yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—262no—
213C22⋊C4—262no—
214C22⋊C4—262no—
215C22⋊C4—262no—
216C22⋊C4—261no—
217C42—131no—
218C42—131no—
219C42—131no—
220C42—131no—
221C42—261no—
222C42—261no—
223C13#321511yescharacteristic, complemented, Sylow
224C23—262no—
225C23—522no—
226C23—522no—
227C23—522no—
228C23D4×D1311yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—22no—
232Q8—42no—
233Q8—262no—
234Q8—522no—
235D4—262no—
236D4—262no—
237D4—522no—
238C2×C4—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—131no—
244C2×C4—131no—
245C2×C4—131no—
246C2×C4—131no—
247C2×C4—262no—
248C2×C4—261no—
249C2×C4—261no—
250C2×C4—521no—
251C2×C4D4×D1311yescharacteristic
252C2×C4C23×D1311yescharacteristic
253C8—22no—
254C8—262no—
255C22—21no—
256C22—21no—
257C22—522no—
258C22—522no—
259C22—522no—
260C22—1044no—
261C22#8817212yes—
262C22C2×D4×D1311yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—261no—
268C4—261no—
269C4—261no—
270C4—261no—
271C4—521no—
272C4—521no—
273C4#8825112yes—
274C2—22no—
275C2—1042no—
276C2#1171929711yescharacteristic
277C2C2×#8817211yescharacteristic
278C2#1171948211yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#451C2
socleC22×C13C22×D4
prime radicalC22×D52C22
hypercenter#451D13
hypocenterC13#3215
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
2. #11719327C2semidirect
3. C13×#451C2semidirect
4. #11718889C2semidirect
5. #11718851C2semidirect
6. #11718846C2semidirect
8. #11718459C2semidirect
9. #11718456C2semidirect
11. #11718380C2semidirect
14. C2×C13×Q16C22semidirect
20. Q8×Dic13C22semidirect
21. #88198C22semidirect
22. C2×#6107C22semidirect
30. #88142C22semidirect
32. #88141C22semidirect
35. C13×#103C22semidirect
37. #88075C22semidirect
40. #88055C22semidirect
41. #88054C22semidirect
123. #451D13semidirect
149. C2×Q16C2×D13semidirect
171. #103C2×D13semidirect
223. C13#3215semidirect

Maximal subgroups

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

subgroupindex
C2×#882012
#117193272
C13×#4512
#117188892
#117188512
#117188462
#117187072
#117184592
#117184562
#117183922
#117183802
#321513