Finite Group Databasebeta

← #13120645

Subgroups — #13120645

Order 1664.

668 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×C22⋊Dic13C211yesmaximal, characteristic
2C2×#88132C211yesmaximal, characteristic, complemented
3C13×#379C211yesmaximal, characteristic
4#11718908C211yesmaximal, characteristic
5#11718611C211yesmaximal, characteristic, complemented
6#11718610C212yesmaximal, complemented
7#11718609C212yesmaximal, complemented
8#11718609C212yesmaximal, complemented
9#11718608C212yesmaximal, complemented
10#11718607C211yesmaximal, characteristic, complemented
11#11718603C211yesmaximal, characteristic, complemented
12C25×C13C2211yescharacteristic
13C23×Dic13C2211yescharacteristic
14C23×Dic13C2211yescharacteristic
15C22×Dic26C2211yescharacteristic, complemented
16C2×C13×C22⋊C4C2211yescharacteristic
17C2×C13×C22⋊C4C2211yescharacteristic
18C2×C13×C22⋊C4C2212yes—
19C2×C13×C22⋊C4C2212yes—
20C2×C22⋊Dic13—24no—
21C2×C22⋊Dic13C2211yescharacteristic
22C2×C22⋊Dic13C2212yes—
23C2×C22⋊Dic13C2212yes—
24C2×C22⋊Dic13C2211yescharacteristic
25C2×C22⋊Dic13C2212yes—
26C2×C22⋊Dic13C2212yes—
27C2×C4⋊Dic13C2211yescharacteristic, complemented
28C2×C4⋊Dic13C2212yescomplemented
29C2×#6101C2211yescharacteristic, complemented
30C2×#6101C2211yescharacteristic, complemented
31C2×#6101C2212yescomplemented
32C2×#6101C2212yescomplemented
33#88132—24no—
34#88071C2212yescomplemented
35#88071C2212yescomplemented
36#88071C2212yescomplemented
37C24×C13—22no—
38C24×C13—22no—
39C24×C13—42no—
40C24×C13—42no—
41C24×C13Q814yescomplemented
42C24×C13C2312yes—
43C24×C13C2311yescharacteristic
44C22×C4×C13C2312yes—
45C22×C4×C13C2311yescharacteristic
46C22×C4×C13C2311yescharacteristic
47C22×Dic13—22no—
48C22×Dic13—42no—
49C22×Dic13—42no—
50C22×Dic13D414yes—
51C22×Dic13C2312yes—
52C22×Dic13C2312yes—
53C22×Dic13C2311yescharacteristic
54C22×Dic13C2311yescharacteristic
55C22×Dic13C2311yescharacteristic
56C22×Dic13C2311yescharacteristic
57C2×Dic26—22no—
58C2×Dic26—42no—
59C13×C22⋊C4—22no—
60C13×C22⋊C4—22no—
61C13×C22⋊C4—42no—
62C13×C22⋊C4—42no—
63C22⋊Dic13—22no—
64C22⋊Dic13—22no—
65C22⋊Dic13—24no—
66C22⋊Dic13—24no—
67C22⋊Dic13—42no—
68C22⋊Dic13—42no—
69C4⋊Dic13—22no—
70C4⋊Dic13—42no—
71#6101—22no—
72#6101—22no—
73#6101—42no—
74#6101—42no—
75#2626—131nomaximal, Sylow
76C23×C13—22no—
77C23×C13—22no—
78C23×C13—28no—
79C23×C13—22no—
80C23×C13—22no—
81C23×C13—22no—
82C23×C13—22no—
83C23×C13—22no—
84C23×C13—22no—
85C23×C13—48no—
86C23×C13—44no—
87C23×C13—44no—
88C23×C13—48no—
89C23×C13C2×D412yes—
90C23×C13C2×Q814yes—
91C23×C13C2×Q812yes—
92C23×C13#4112yes—
93C23×C13C2411yescharacteristic
94C2×C4×C13—21no—
95C2×C4×C13—21no—
96C2×C4×C13—22no—
97C2×C4×C13—22no—
98C2×C4×C13—22no—
99C2×C4×C13—22no—
100C2×C4×C13—22no—
101C2×Dic13—22no—
102C2×Dic13—22no—
103C2×Dic13—21no—
104C2×Dic13—21no—
105C2×Dic13—22no—
106C2×Dic13—22no—
107C2×Dic13—22no—
108C2×Dic13—22no—
109C2×Dic13—22no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—44no—
114C2×Dic13—44no—
115C2×Dic13C2×D412yes—
116C2×Dic13C2×D412yes—
117Dic26—44no—
118C2×C22⋊Q8—131no—
119C22×C22⋊C4—131no—
120#399—132no—
121#396—132no—
122#396—132no—
123#393—132no—
124#385—131no—
125#385—131no—
126#380—131no—
127#379—131no—
128#379D1311yescharacteristic
129C22×C13—22no—
130C22×C13—22no—
131C22×C13—24no—
132C22×C13—24no—
133C22×C13—24no—
134C22×C13—24no—
135C22×C13—24no—
136C22×C13—416no—
137C22×C13—42no—
138C22×C13—42no—
139C22×C13—42no—
140C22×C13—42no—
141C22×C13C22⋊Q814yes—
142C22×C13C22×D411yescharacteristic
143C22×C13C22×Q811yescharacteristic
144C22×C13C2×#4111yescharacteristic
145C22×C13H(5,𝔽2)12yes—
146C22×C13H(5,𝔽2)12yes—
147C4×C13—41no—
148C4×C13—41no—
149C4×C13—42no—
150Dic13—22no—
151Dic13—22no—
152Dic13—41no—
153Dic13—41no—
154Dic13—42no—
155Dic13—42no—
156C25C2×D1311yescharacteristic
157C22×Q8—131no—
158C23×C4—131no—
159C23×C4—131no—
160C22⋊Q8—264no—
161C2×C4⋊C4—131no—
162C2×C4⋊C4—131no—
163C2×C4⋊C4—131no—
164C2×C4⋊C4—132no—
165C2×C4⋊C4—132no—
166C2×C4⋊C4—132no—
167C2×C22⋊C4—132no—
168C2×C22⋊C4—132no—
169C2×C22⋊C4—131no—
170C2×C22⋊C4—131no—
171C2×C22⋊C4—132no—
172C2×C22⋊C4—132no—
173C2×C22⋊C4—264no—
174C2×C22⋊C4C2×D1311yescharacteristic
175C2×C22⋊C4C2×D1311yescharacteristic
176C2×C22⋊C4C2×D1312yes—
177C2×C22⋊C4C2×D1312yes—
178#95—132no—
179#95—132no—
180#95—132no—
181C2×C13—24no—
182C2×C13—44no—
183C2×C13C2×C22⋊Q811yescharacteristic
184C2×C13#53411yescharacteristic
185C2×C13#53511yescharacteristic
186C2×C13#53812yes—
187C2×C13#54312yes—
188C24—22no—
189C24—22no—
190C24—42no—
191C24—42no—
192C24Dic2614yescomplemented
193C24C22×D1312yes—
194C24C22×D1311yescharacteristic
195C2×Q8—262no—
196C2×Q8—522no—
197C22×C4—131no—
198C22×C4—131no—
199C22×C4—134no—
200C22×C4—132no—
201C22×C4—131no—
202C22×C4—132no—
203C22×C4—131no—
204C22×C4—262no—
205C22×C4—522no—
206C22×C4—522no—
207C22×C4C22×D1312yes—
208C22×C4C22×D1311yescharacteristic
209C22×C4C22×D1311yescharacteristic
210C4⋊C4—262no—
211C4⋊C4—262no—
212C4⋊C4—262no—
213C4⋊C4—522no—
214C4⋊C4—522no—
215C4⋊C4—522no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—264no—
221C22⋊C4—264no—
222C22⋊C4—262no—
223C22⋊C4—262no—
224C22⋊C4—522no—
225C22⋊C4—522no—
226C13#262611yescharacteristic, complemented, Sylow
227C23—22no—
228C23—22no—
229C23—28no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—48no—
237C23—44no—
238C23—44no—
239C23—48no—
240C23C2×Dic2614yes—
241C23C2×Dic2612yes—
242C23D4×D1312yes—
243C23#612912yes—
244C23C23×D1311yescharacteristic
245Q8—524no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—132no—
254C2×C4—132no—
255C2×C4—262no—
256C2×C4—262no—
257C2×C4—262no—
258C2×C4—261no—
259C2×C4—261no—
260C2×C4—262no—
261C2×C4—262no—
262C2×C4—262no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—262no—
266C2×C4—262no—
267C2×C4—524no—
268C2×C4—524no—
269C22—24no—
270C22—22no—
271C22—22no—
272C22—24no—
273C22—24no—
274C22—24no—
275C22—24no—
276C22—416no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22#8813214yes—
282C22C22×Dic2611yescharacteristic
283C22C2×D4×D1311yescharacteristic
284C22C2×#612911yescharacteristic
285C22#8825112yes—
286C22#8825112yes—
287C4—41no—
288C4—41no—
289C4—42no—
290C4—262no—
291C4—262no—
292C4—521no—
293C4—521no—
294C4—522no—
295C4—522no—
296C2—24no—
297C2—44no—
298C2C2×#8813211yescharacteristic
299C2#1171918812yes—
300C2#1171929411yescharacteristic
301C2#1171929711yescharacteristic
302C2#1171929812yes—
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#379C2
prime radicalC25×C13C22
hypercenter#379D13
hypocenterC13#2626
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #379

Complements

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

NH ≅ G/Ntype
2. C2×#88132C2semidirect
5. #11718611C2semidirect
6. #11718610C2semidirect
7. #11718609C2semidirect
8. #11718609C2semidirect
9. #11718608C2semidirect
10. #11718607C2semidirect
11. #11718603C2semidirect
15. C22×Dic26C22semidirect
27. C2×C4⋊Dic13C22semidirect
28. C2×C4⋊Dic13C22semidirect
29. C2×#6101C22semidirect
30. C2×#6101C22semidirect
31. C2×#6101C22semidirect
32. C2×#6101C22semidirect
34. #88071C22semidirect
35. #88071C22semidirect
36. #88071C22semidirect
41. C24×C13Q8semidirect
192. C24Dic26semidirect
226. C13#2626semidirect

Maximal subgroups

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

subgroupindex
C22×C22⋊Dic132
C2×#881322
C13×#3792
#117189082
#117186112
#117186102
#117186092
#117186082
#117186072
#117186032
#262613