Finite Group Databasebeta

← #13124467

Subgroups — #13124467

Order 1664.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#88163C211yesmaximal, characteristic, complemented
2C2×M16×D13C211yesmaximal, characteristic, complemented
3#11719229C211yesmaximal, characteristic
4C2×#88151C211yesmaximal, characteristic, complemented
5C13×#428C211yesmaximal, characteristic, complemented
6#11718665C211yesmaximal, characteristic, complemented
7#11718627C211yesmaximal, characteristic, complemented
8#11718548C212yesmaximal, complemented
9#11718547C212yesmaximal, complemented
10#11718529C212yesmaximal, complemented
11#11718525C212yesmaximal, complemented
12C2×#6127C2211yescharacteristic
13C22×C4×D13C2211yescharacteristic
14C22×Dic26C2211yescharacteristic
15C2×C13×M16C2211yescharacteristic
16C13×#117C2211yescharacteristic
17C2×C13×C4⋊C4C2211yescharacteristic
18C2×#6103—21no—
19C2×C4⋊Dic13—21no—
20C2×#6099C2211yescharacteristic
21#88163—24no—
22M16×D13—24no—
23C2×Dic52C2212yescomplemented
24C2×#6095C2212yescomplemented
25C2×#6094C2212yes—
26C2×C8×D13C2212yes—
27#88151—21no—
28#88151—21no—
29#88151C2212yes—
30#88151C2212yes—
31#88151C2211yescharacteristic
32#88151C2211yescharacteristic
33#88149C2211yescharacteristic
34#88149C2211yescharacteristic
35#88140—21no—
36#88136—21no—
37C4×D52—21no—
38C4×D52—21no—
39C13×#107C2212yescomplemented
40C13×#106C2212yescomplemented
41#88050C2212yescomplemented
42#88050C2212yescomplemented
43#88050C2212yescomplemented
44#88049C2212yescomplemented
45C23×D13D411yescharacteristic
46C22×C4×C13—21no—
47C22×C4×C13C2311yescharacteristic
48C2×C22⋊D13—21no—
49C22×Dic13—21no—
50C22×Dic13D411yescharacteristic
51#6127—22no—
52#6127—22no—
53C2×D52—21no—
54C2×D52—21no—
55C2×D52C2311yescharacteristic
56C2×C4×D13—22no—
57C2×C4×D13—22no—
58C2×C4×D13—21no—
59C2×C4×D13D411yescharacteristic
60C2×C4×D13D411yescharacteristic
61C2×C4×D13C2311yescharacteristic
62C2×C4×D13C2311yescharacteristic
63C2×Dic26—22no—
64C2×Dic26—22no—
65C2×Dic26—21no—
66C2×Dic26C2312yes—
67C2×Dic26C2311yescharacteristic
68C13×M16D414yescomplemented
69C2×C8×C13C2312yes—
70C13×C4⋊C4—21no—
71C13×C4⋊C4C2311yescharacteristic
72C13×C4⋊C4C2312yes—
73C13×C4⋊C4C2311yescharacteristic
74C13×C22⋊C4—21no—
75C42×C13—21no—
76#6103—21no—
77#6103—21no—
78#6103—22no—
79#6103—22no—
80#6103—21no—
81#6103—21no—
82C4⋊Dic13—21no—
83C4⋊Dic13—21no—
84C4⋊Dic13—22no—
85C4⋊Dic13—22no—
86#6099—22no—
87C2×#1055C2312yes—
88Dic52—24no—
89#6095—24no—
90#6094—24no—
91C8×D13—24no—
92#3056—131nomaximal, Sylow
93C23×C13C2×D411yescharacteristic
94C22×D13—22no—
95C22×D13—22no—
96C22×D13—21no—
97C22×D13—41no—
98C22×D13C2×D411yescharacteristic
99C22×D13C2×D411yescharacteristic
100C2×C4×C13—22no—
101C2×C4×C13—21no—
102C2×C4×C13—21no—
103C2×C4×C13—42no—
104C2×C4×C13C2×D411yescharacteristic
105C2×C4×C13C2×D412yes—
106C2×C4×C13#4112yes—
107C2×C4×C13C2411yescharacteristic
108C22⋊D13—42no—
109C2×Dic13—22no—
110C2×Dic13—22no—
111C2×Dic13—21no—
112C2×Dic13—42no—
113C2×Dic13C2×D411yescharacteristic
114C2×Dic13C2×D411yescharacteristic
115D52—21no—
116D52—21no—
117D52—42no—
118D52—42no—
119C4×D13—22no—
120C4×D13—22no—
121C4×D13—21no—
122C4×D13—21no—
123C4×D13—21no—
124C4×D13—21no—
125C4×D13—42no—
126Dic26—22no—
127Dic26—21no—
128Dic26—22no—
129Dic26—21no—
130Dic26—42no—
131C8×C13C2×D414yes—
132#1055—22no—
133C2×#137—131no—
134C22×M16—131no—
135#525—131no—
136C2×C22⋊Q8—131no—
137#470—132no—
138#468—132no—
139#467—132no—
140#465—132no—
141#428D1311yescharacteristic, complemented
142#419—131no—
143#417—131no—
144C22×C13—21no—
145C22×C13—21no—
146C22×C13#12112yes—
147C22×C13C22×D411yescharacteristic
148C2×D13—21no—
149C2×D13—21no—
150C2×D13—21no—
151C2×D13—21no—
152C2×D13—21no—
153C2×D13—44no—
154C2×D13—41no—
155C2×D13—41no—
156C2×D13—41no—
157C2×D13#13712yescomplemented
158C4×C13—41no—
159C4×C13—41no—
160C4×C13—42no—
161C4×C13#12112yes—
162C4×C13C22×D411yescharacteristic
163C4×C13C2×#4111yescharacteristic
164Dic13—21no—
165Dic13—21no—
166Dic13—42no—
167Dic13—41no—
168C2×#41—131no—
169C22×Q8—131no—
170C23×C4—131no—
171#137—264no—
172C2×Q16—132no—
173C2×SD16—132no—
174C2×M16—131no—
175C2×M16—132no—
176C2×M16—264no—
177C2×M16C2×D1311yescharacteristic
178C22×C8—132no—
179#123—261no—
180C22⋊Q8—132no—
181C22⋊Q8—132no—
182C22⋊Q8—131no—
183C22⋊Q8—131no—
184C22⋊Q8—261no—
185C22⋊Q8—261no—
186#121—131no—
187#121—131no—
188#120—261no—
189C4×D4—261no—
190C4×D4—261no—
191#117C2×D1311yescharacteristic
192C2×C4⋊C4—261no—
193C2×C4⋊C4C2×D1311yescharacteristic
194C2×C22⋊C4—261no—
195#107C2×D1312yescomplemented
196#106C2×D1312yescomplemented
197#103—132no—
198#103—132no—
199#103—132no—
200#102—132no—
201C2×C13—22no—
202C2×C13C2×#12111yescharacteristic
203C2×C13C2×#13711yescharacteristic
204C2×C13#57511yescharacteristic
205D13—22no—
206D13—41no—
207D13—81no—
208C24—131no—
209#41—262no—
210#41—262no—
211C2×Q8—132no—
212C2×Q8—131no—
213C2×Q8—261no—
214C2×Q8—262no—
215C2×Q8—262no—
216C2×D4—131no—
217C2×D4—261no—
218C2×D4—261no—
219C2×D4—261no—
220C22×C4—21no—
221C22×C4—131no—
222C22×C4—131no—
223C22×C4—131no—
224C22×C4—131no—
225C22×C4—131no—
226C22×C4—262no—
227C22×C4—262no—
228C22×C4—261no—
229C22×C4—261no—
230C22×C4C22×D1311yescharacteristic
231Q16—264no—
232SD16—264no—
233M16—262no—
234M16—264no—
235M16D5214yescomplemented
236C2×C8—132no—
237C2×C8—264no—
238C2×C8C22×D1312yes—
239C4⋊C4—21no—
240C4⋊C4—262no—
241C4⋊C4—262no—
242C4⋊C4—261no—
243C4⋊C4—261no—
244C4⋊C4C22×D1312yes—
245C4⋊C4C22×D1311yescharacteristic
246C4⋊C4C22×D1311yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—261no—
249C22⋊C4—261no—
250C22⋊C4—262no—
251C22⋊C4—262no—
252C22⋊C4—261no—
253C22⋊C4—261no—
254C42—21no—
255C13#305611yescharacteristic, complemented, Sylow
256C23—131no—
257C23—131no—
258C23—262no—
259C23—262no—
260C23—261no—
261C23—521no—
262C23D4×D1311yescharacteristic
263Q8—262no—
264Q8—261no—
265Q8—262no—
266Q8—261no—
267Q8—522no—
268D4—261no—
269D4—261no—
270D4—522no—
271D4—522no—
272D4—522no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—131no—
278C2×C4—131no—
279C2×C4—262no—
280C2×C4—262no—
281C2×C4—262no—
282C2×C4—261no—
283C2×C4—261no—
284C2×C4—261no—
285C2×C4—261no—
286C2×C4—261no—
287C2×C4—262no—
288C2×C4—522no—
289C2×C4—522no—
290C2×C4C2×D5212yes—
291C2×C4D4×D1311yescharacteristic
292C2×C4#613112yes—
293C2×C4C23×D1311yescharacteristic
294C8—262no—
295C8C2×D5214yes—
296C22—21no—
297C22—21no—
298C22—132no—
299C22—261no—
300C22—261no—
301C22—261no—
302C22—261no—
303C22—261no—
304C22—524no—
305C22—521no—
306C22—521no—
307C22—521no—
308C22#8814912yes—
309C22C2×D4×D1311yescharacteristic
310C4—42no—
311C4—41no—
312C4—41no—
313C4—261no—
314C4—261no—
315C4—521no—
316C4—522no—
317C4#8814912yes—
318C4C22×D5211yescharacteristic
319C4C2×#613111yescharacteristic
320C2—22no—
321C2—262no—
322C2—521no—
323C2—1041no—
324C2C2×#8814911yescharacteristic
325C2#1171947811yescharacteristic
326C2D13×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D13
commutator, cosocleC2×C4×C13C24
FrattiniC2×C4C23×D13
FittingC13×#428C2
socleC22×C13C22×D4
prime radical#11719229C2
hypercenter#428D13
hypocenterC13#3056
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#88163C2semidirect
2. C2×M16×D13C2semidirect
4. C2×#88151C2semidirect
5. C13×#428C2semidirect
6. #11718665C2semidirect
7. #11718627C2semidirect
8. #11718548C2semidirect
9. #11718547C2semidirect
10. #11718529C2semidirect
11. #11718525C2semidirect
23. C2×Dic52C22semidirect
24. C2×#6095C22semidirect
39. C13×#107C22semidirect
40. C13×#106C22semidirect
41. #88050C22semidirect
42. #88050C22semidirect
43. #88050C22semidirect
44. #88049C22semidirect
68. C13×M16D4semidirect
141. #428D13semidirect
157. C2×D13#137semidirect
195. #107C2×D13semidirect
196. #106C2×D13semidirect
235. M16D52semidirect
255. C13#3056semidirect

Maximal subgroups

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

subgroupindex
C2×#881632
C2×M16×D132
#117192292
C2×#881512
C13×#4282
#117186652
#117186272
#117185482
#117185472
#117185292
#117185252
#305613