Finite Group Databasebeta

← #13118219

Subgroups — #13118219

Order 1664.

786 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 161 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×#6103C211yesmaximal, characteristic, complemented
2C2×#88140C211yesmaximal, characteristic, complemented
3C2×#88136C211yesmaximal, characteristic
4C2×#88131C211yesmaximal, characteristic, complemented
5C13×#385C211yesmaximal, characteristic, complemented
6#11718874C211yesmaximal, characteristic, complemented
7#11718655C212yesmaximal, complemented
8#11718649C212yesmaximal, complemented
9#11718616C211yesmaximal, characteristic, complemented
10#11718333C212yesmaximal, complemented
11#11718331C212yesmaximal, complemented
12C24×D13C2211yescharacteristic
13C23×C4×C13C2211yescharacteristic
14C22×C22⋊D13C2211yescharacteristic
15C23×Dic13C2211yescharacteristic
16C22×D52C2212yes—
17C2×C13×C4⋊C4C2212yescomplemented
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic
21C2×#6103—24no—
22C2×#6103C2212yes—
23C2×#6103C2212yes—
24C2×#6103C2212yes—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×#6103C2211yescharacteristic, complemented
28C2×#6103C2211yescharacteristic, complemented
29C2×#6103C2212yescomplemented
30C2×C4⋊Dic13C2211yescharacteristic, complemented
31C2×C4⋊Dic13C2211yescharacteristic, complemented
32C2×#6101C2212yescomplemented
33C2×C4×Dic13C2212yescomplemented
34#88140—24no—
35#88136—24no—
36#88131—24no—
37C13×#95C2212yescomplemented
38#88071C2212yescomplemented
39C24×C13C2311yescharacteristic
40C23×D13—22no—
41C23×D13—24no—
42C23×D13—22no—
43C23×D13—22no—
44C23×D13—42no—
45C23×D13C2311yescharacteristic
46C23×D13C2312yes—
47C22×C4×C13—22no—
48C22×C4×C13—42no—
49C22×C4×C13C2312yes—
50C22×C4×C13C2311yescharacteristic
51C22×C4×C13C2311yescharacteristic
52C22×C4×C13C2312yes—
53C2×C22⋊D13—22no—
54C2×C22⋊D13—22no—
55C2×C22⋊D13—22no—
56C22×Dic13—22no—
57C22×Dic13—22no—
58C22×Dic13D414yescomplemented
59C22×Dic13C2311yescharacteristic
60C22×Dic13C2312yes—
61C22×Dic13C2311yescharacteristic
62C22×Dic13C2311yescharacteristic
63C2×D52—24no—
64C2×D52—24no—
65C2×D52—42no—
66C13×C4⋊C4—42no—
67C13×C22⋊C4—22no—
68C13×C22⋊C4D414yes—
69C22⋊Dic13—22no—
70#6103—24no—
71#6103—24no—
72#6103—22no—
73#6103—22no—
74#6103—24no—
75#6103—24no—
76#6103—42no—
77#6103—42no—
78C4⋊Dic13—22no—
79C4⋊Dic13—22no—
80#6101—24no—
81C4×Dic13—24no—
82#2411—131nomaximal, Sylow
83C23×C13—22no—
84C23×C13—22no—
85C23×C13C2×D412yes—
86C23×C13C2×D412yes—
87C23×C13#4112yes—
88C23×C13C2411yescharacteristic
89C22×D13—28no—
90C22×D13—22no—
91C22×D13—22no—
92C22×D13—22no—
93C22×D13—22no—
94C22×D13—22no—
95C22×D13—22no—
96C22×D13—21no—
97C22×D13—21no—
98C22×D13—22no—
99C22×D13—21no—
100C22×D13—22no—
101C22×D13—48no—
102C22×D13—48no—
103C22×D13—44no—
104C22×D13—44no—
105C22×D13#4114yes—
106C2×C4×C13—22no—
107C2×C4×C13—22no—
108C2×C4×C13—21no—
109C2×C4×C13—21no—
110C2×C4×C13—22no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—44no—
116C2×C4×C13C2×D414yes—
117C22⋊D13—44no—
118C2×Dic13—28no—
119C2×Dic13—21no—
120C2×Dic13—21no—
121C2×Dic13—21no—
122C2×Dic13—21no—
123C2×Dic13—22no—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—22no—
127C2×Dic13—21no—
128C2×Dic13—22no—
129C2×Dic13C2×D412yes—
130C2×Dic13C2×D412yes—
131D52—48no—
132C2×#123—131no—
133C2×#120—131no—
134C2×#117—131no—
135C22×C22⋊C4—131no—
136#396—132no—
137#394—132no—
138#390—132no—
139#388—132no—
140#386—131no—
141#385D1311yescharacteristic, complemented
142#379—131no—
143C22×C13—22no—
144C22×C13—22no—
145C22×C13—24no—
146C22×C13—24no—
147C22×C13#12114yes—
148C22×C13C22×D411yescharacteristic
149C22×C13C22×D411yescharacteristic
150C22×C13C2×#4111yescharacteristic
151C22×C13C2×#4112yes—
152C22×C13H(5,𝔽2)12yes—
153C2×D13—24no—
154C2×D13—24no—
155C2×D13—24no—
156C2×D13—416no—
157C2×D13—42no—
158C2×D13—42no—
159C2×D13—42no—
160C2×D13—42no—
161C2×D13—41no—
162C2×D13—41no—
163C2×D13—41no—
164C2×D13—42no—
165C2×D13—42no—
166C4×C13—24no—
167C4×C13—41no—
168C4×C13—41no—
169C4×C13—42no—
170Dic13—22no—
171Dic13—22no—
172Dic13—41no—
173Dic13—42no—
174C25—131no—
175C22×D4—131no—
176C22×D4—132no—
177C23×C4—131no—
178C23×C4C2×D1311yescharacteristic
179#123—264no—
180#120—264no—
181#117—264no—
182C2×C4⋊C4—132no—
183C2×C4⋊C4—131no—
184C2×C4⋊C4—131no—
185C2×C4⋊C4C2×D1312yescomplemented
186C2×C22⋊C4—131no—
187C2×C22⋊C4—132no—
188C2×C22⋊C4—132no—
189C2×C22⋊C4—132no—
190C2×C22⋊C4—131no—
191C2×C22⋊C4—131no—
192C2×C22⋊C4—132no—
193C2×C22⋊C4—132no—
194C2×C22⋊C4—132no—
195C2×C22⋊C4—264no—
196C2×C22⋊C4C2×D1311yescharacteristic
197C2×C22⋊C4C2×D1311yescharacteristic
198C2×C42—132no—
199#95—132no—
200#95C2×D1312yescomplemented
201C2×C13—24no—
202C2×C13C2×#12111yescharacteristic
203C2×C13#52511yescharacteristic
204C2×C13#53511yescharacteristic
205C2×C13#54612yes—
206C2×C13#55112yes—
207D13—44no—
208D13—81no—
209C24—132no—
210C24—131no—
211C24—264no—
212C24—262no—
213C24—262no—
214C24—262no—
215C24—522no—
216C24C22×D1311yescharacteristic
217C2×D4—264no—
218C2×D4—262no—
219C2×D4—262no—
220C2×D4—262no—
221C2×D4—264no—
222C2×D4—522no—
223C22×C4—22no—
224C22×C4—42no—
225C22×C4—134no—
226C22×C4—131no—
227C22×C4—131no—
228C22×C4—131no—
229C22×C4—132no—
230C22×C4—262no—
231C22×C4—262no—
232C22×C4C22×D1312yes—
233C22×C4C22×D1311yescharacteristic
234C22×C4C22×D1311yescharacteristic
235C22×C4C22×D1312yes—
236C4⋊C4—42no—
237C4⋊C4—262no—
238C4⋊C4—262no—
239C4⋊C4—264no—
240C22⋊C4—22no—
241C22⋊C4—264no—
242C22⋊C4—264no—
243C22⋊C4—262no—
244C22⋊C4—262no—
245C22⋊C4—264no—
246C22⋊C4—262no—
247C22⋊C4—264no—
248C22⋊C4—522no—
249C22⋊C4—522no—
250C22⋊C4D5214yes—
251C42—264no—
252C13#241111yescharacteristic, complemented, Sylow
253C23—22no—
254C23—22no—
255C23—134no—
256C23—268no—
257C23—262no—
258C23—262no—
259C23—262no—
260C23—262no—
261C23—262no—
262C23—262no—
263C23—261no—
264C23—261no—
265C23—262no—
266C23—261no—
267C23—262no—
268C23—528no—
269C23—528no—
270C23—524no—
271C23—524no—
272C23C2×D5212yes—
273C23D4×D1312yes—
274C23#613112yes—
275C23C23×D1311yescharacteristic
276D4—528no—
277D4—524no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—44no—
288C2×C4—132no—
289C2×C4—132no—
290C2×C4—268no—
291C2×C4—261no—
292C2×C4—261no—
293C2×C4—261no—
294C2×C4—261no—
295C2×C4—262no—
296C2×C4—261no—
297C2×C4—261no—
298C2×C4—262no—
299C2×C4—261no—
300C2×C4—262no—
301C2×C4C2×D5214yes—
302C22—24no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—264no—
307C22—264no—
308C22—264no—
309C22—5216no—
310C22—522no—
311C22—522no—
312C22—522no—
313C22—522no—
314C22—521no—
315C22—522no—
316C22—521no—
317C22—522no—
318C22—521no—
319C22#8814914yes—
320C22C22×D5211yescharacteristic
321C22C2×D4×D1311yescharacteristic
322C22#8825112yes—
323C22C2×#613111yescharacteristic
324C22#41×D1312yes—
325C4—24no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—262no—
330C4—262no—
331C4—521no—
332C4—522no—
333C2—24no—
334C2—524no—
335C2—1041no—
336C2C2×#8814911yescharacteristic
337C2#1171925912yes—
338C2#1171929711yescharacteristic
339C2#1171934212yes—
340C2#1171936111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#385C2
prime radicalC2×#88136C2
hypercenter#385D13
hypocenterC13#2411
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C22×#6103C2semidirect
2. C2×#88140C2semidirect
4. C2×#88131C2semidirect
5. C13×#385C2semidirect
6. #11718874C2semidirect
7. #11718655C2semidirect
8. #11718649C2semidirect
9. #11718616C2semidirect
10. #11718333C2semidirect
11. #11718331C2semidirect
17. C2×C13×C4⋊C4C22semidirect
27. C2×#6103C22semidirect
28. C2×#6103C22semidirect
29. C2×#6103C22semidirect
30. C2×C4⋊Dic13C22semidirect
31. C2×C4⋊Dic13C22semidirect
32. C2×#6101C22semidirect
33. C2×C4×Dic13C22semidirect
37. C13×#95C22semidirect
38. #88071C22semidirect
58. C22×Dic13D4semidirect
141. #385D13semidirect
185. C2×C4⋊C4C2×D13semidirect
200. #95C2×D13semidirect
252. C13#2411semidirect

Maximal subgroups

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

subgroupindex
C22×#61032
C2×#881402
C2×#881362
C2×#881312
C13×#3852
#117188742
#117186552
#117186492
#117186162
#117183332
#117183312
#241113