Finite Group Databasebeta

← #13120613

Subgroups — #13120613

Order 1664.

702 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 141 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×#88191C211yesmaximal, characteristic
2C22×#6103C211yesmaximal, characteristic, complemented
3C13×#386C211yesmaximal, characteristic, complemented
4#11718881C211yesmaximal, characteristic, complemented
5#11718871C211yesmaximal, characteristic, complemented
6#11718622C212yesmaximal, complemented
7#11718620C212yesmaximal, complemented
8#11718616C211yesmaximal, characteristic, complemented
9#11718611C211yesmaximal, characteristic, complemented
10#11718336C212yesmaximal, complemented
11#11718335C212yesmaximal, complemented
12C24×D13C2211yescharacteristic
13C22×D4×C13C2211yescharacteristic
14C23×C4×C13C2211yescharacteristic
15C22×C22⋊D13C2211yescharacteristic
16C22×C22⋊D13C2211yescharacteristic
17C23×Dic13C2211yescharacteristic
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2212yescomplemented
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C22⋊Dic13C2211yescharacteristic
22C2×C22⋊Dic13C2212yescomplemented
23#88191—24no—
24C2×#6103—24no—
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×#6103C2211yescharacteristic
28C2×#6103C2211yescharacteristic
29C2×#6103C2212yes—
30C2×#6103C2212yes—
31C2×C4⋊Dic13C2212yescomplemented
32C2×#6101C2212yescomplemented
33C13×#95C2212yescomplemented
34#88071C2212yescomplemented
35#88071C2212yescomplemented
36#88071C2212yescomplemented
37C24×C13C2311yescharacteristic
38C24×C13C2311yescharacteristic
39C23×D13—22no—
40C23×D13—22no—
41C23×D13—42no—
42C23×D13—42no—
43C23×D13D414yes—
44C23×D13C2311yescharacteristic
45C23×D13C2311yescharacteristic
46C2×D4×C13—22no—
47C2×D4×C13—22no—
48C2×D4×C13—22no—
49C22×C4×C13—22no—
50C22×C4×C13—42no—
51C22×C4×C13C2311yescharacteristic
52C22×C4×C13C2312yes—
53C22×C4×C13C2312yes—
54C2×C22⋊D13—22no—
55C2×C22⋊D13—22no—
56C2×C22⋊D13—42no—
57C2×C22⋊D13—42no—
58C22×Dic13—22no—
59C22×Dic13—42no—
60C22×Dic13C2311yescharacteristic
61C22×Dic13C2312yes—
62C22×Dic13C2311yescharacteristic
63C22×Dic13C2312yes—
64C13×C22⋊C4—22no—
65C13×C22⋊C4—42no—
66C22⋊Dic13—22no—
67C22⋊Dic13—22no—
68C22⋊Dic13—42no—
69#6103—24no—
70#6103—24no—
71#6103—22no—
72#6103—22no—
73#6103—42no—
74#6103—42no—
75C4⋊Dic13—42no—
76#6101—42no—
77#2622—131nomaximal, Sylow
78C23×C13—22no—
79C23×C13—22no—
80C23×C13—21no—
81C23×C13—21no—
82C23×C13—21no—
83C23×C13—22no—
84C23×C13—22no—
85C23×C13C2×D412yes—
86C23×C13#4112yes—
87C23×C13#4112yes—
88C23×C13C2411yescharacteristic
89C22×D13—28no—
90C22×D13—21no—
91C22×D13—21no—
92C22×D13—21no—
93C22×D13—21no—
94C22×D13—22no—
95C22×D13—22no—
96C22×D13—22no—
97C22×D13—22no—
98C22×D13—44no—
99C22×D13—44no—
100C22×D13—48no—
101C22×D13—48no—
102C22×D13C2×D412yes—
103C22×D13C2×D412yes—
104D4×C13—44no—
105C2×C4×C13—22no—
106C2×C4×C13—21no—
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—44no—
115C22⋊D13—44no—
116C22⋊D13—44no—
117C2×Dic13—22no—
118C2×Dic13—21no—
119C2×Dic13—21no—
120C2×Dic13—22no—
121C2×Dic13—22no—
122C2×Dic13—22no—
123C2×Dic13—22no—
124C2×Dic13—22no—
125C2×Dic13—22no—
126C2×Dic13—22no—
127C2×Dic13—44no—
128C2×#120—131no—
129C22×C22⋊C4—131no—
130#397—132no—
131#397—132no—
132#394—132no—
133#394—132no—
134#386—131no—
135#386D1311yescharacteristic, complemented
136#385—131no—
137#385—131no—
138#379—131no—
139C22×C13—22no—
140C22×C13—22no—
141C22×C13—24no—
142C22×C13—24no—
143C22×C13—41no—
144C22×C13—41no—
145C22×C13—41no—
146C22×C13—42no—
147C22×C13—42no—
148C22×C13#12314yes—
149C22×C13C22×D411yescharacteristic
150C22×C13C2×#4111yescharacteristic
151C22×C13C2×#4111yescharacteristic
152C22×C13H(5,𝔽2)12yes—
153C22×C13H(5,𝔽2)12yes—
154C2×D13—22no—
155C2×D13—22no—
156C2×D13—22no—
157C2×D13—22no—
158C2×D13—22no—
159C2×D13—22no—
160C2×D13—416no—
161C2×D13—42no—
162C2×D13—42no—
163C2×D13—42no—
164C2×D13—42no—
165C4×C13—41no—
166C4×C13—42no—
167C4×C13—42no—
168Dic13—42no—
169Dic13—41no—
170Dic13—41no—
171Dic13—42no—
172C25—131no—
173C22×D4—131no—
174C22×D4—131no—
175C22×D4C2×D1311yescharacteristic
176C23×C4—131no—
177C23×C4C2×D1311yescharacteristic
178#120—264no—
179C2×C4⋊C4—132no—
180C2×C4⋊C4—132no—
181C2×C22⋊C4—131no—
182C2×C22⋊C4—131no—
183C2×C22⋊C4—132no—
184C2×C22⋊C4—132no—
185C2×C22⋊C4—131no—
186C2×C22⋊C4—131no—
187C2×C22⋊C4—132no—
188C2×C22⋊C4—132no—
189C2×C22⋊C4—132no—
190C2×C22⋊C4—264no—
191C2×C22⋊C4C2×D1311yescharacteristic
192C2×C22⋊C4C2×D1312yescomplemented
193#95—132no—
194#95—132no—
195#95—132no—
196#95C2×D1312yescomplemented
197C2×C13—24no—
198C2×C13—81no—
199C2×C13C2×#12311yescharacteristic
200C2×C13#53411yescharacteristic
201C2×C13#53411yescharacteristic
202C2×C13#53812yes—
203C2×C13#53812yes—
204D13—42no—
205D13—42no—
206C24—131no—
207C24—131no—
208C24—134no—
209C24—262no—
210C24—262no—
211C24—522no—
212C24—522no—
213C24C22×D1311yescharacteristic
214C24C22×D1311yescharacteristic
215C2×D4—22no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—262no—
219C2×D4—262no—
220C2×D4—522no—
221C2×D4—522no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—132no—
225C22×C4—132no—
226C22×C4—131no—
227C22×C4—131no—
228C22×C4—262no—
229C22×C4—522no—
230C22×C4C22×D1311yescharacteristic
231C22×C4C22×D1312yes—
232C22×C4C22×D1312yes—
233C4⋊C4—522no—
234C4⋊C4—522no—
235C22⋊C4—22no—
236C22⋊C4—42no—
237C22⋊C4—264no—
238C22⋊C4—264no—
239C22⋊C4—262no—
240C22⋊C4—262no—
241C22⋊C4—262no—
242C22⋊C4—262no—
243C22⋊C4—522no—
244C22⋊C4—522no—
245C22⋊C4—522no—
246C13#262211yescharacteristic, complemented, Sylow
247C23—22no—
248C23—22no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—132no—
255C23—132no—
256C23—268no—
257C23—261no—
258C23—261no—
259C23—261no—
260C23—261no—
261C23—262no—
262C23—262no—
263C23—262no—
264C23—262no—
265C23—524no—
266C23—524no—
267C23—528no—
268C23—528no—
269C23#612712yes—
270C23D4×D1312yes—
271C23#612912yes—
272C23C23×D1311yescharacteristic
273D4—44no—
274D4—524no—
275D4—524no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—262no—
287C2×C4—262no—
288C2×C4—261no—
289C2×C4—261no—
290C2×C4—262no—
291C2×C4—262no—
292C2×C4—262no—
293C2×C4—262no—
294C2×C4—262no—
295C2×C4—262no—
296C2×C4—524no—
297C22—24no—
298C22—22no—
299C22—24no—
300C22—22no—
301C22—41no—
302C22—42no—
303C22—41no—
304C22—42no—
305C22—41no—
306C22—262no—
307C22—262no—
308C22—262no—
309C22—262no—
310C22—262no—
311C22—262no—
312C22—5216no—
313C22—522no—
314C22—522no—
315C22—522no—
316C22—522no—
317C22#8813714yes—
318C22C2×#612711yescharacteristic
319C22C2×D4×D1311yescharacteristic
320C22C2×#612911yescharacteristic
321C22#8825112yes—
322C22#8825612yes—
323C4—41no—
324C4—42no—
325C4—42no—
326C4—522no—
327C4—521no—
328C4—522no—
329C4—521no—
330C2—24no—
331C2—81no—
332C2—522no—
333C2—522no—
334C2C2×#8813711yescharacteristic
335C2#1171926512yes—
336C2#1171929411yescharacteristic
337C2#1171932212yes—
338C2#1171935811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#386C2
prime radicalC2×#88191C2
hypercenter#386D13
hypocenterC13#2622
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C22×#6103C2semidirect
3. C13×#386C2semidirect
4. #11718881C2semidirect
5. #11718871C2semidirect
6. #11718622C2semidirect
7. #11718620C2semidirect
8. #11718616C2semidirect
9. #11718611C2semidirect
10. #11718336C2semidirect
11. #11718335C2semidirect
19. C2×C13×C22⋊C4C22semidirect
22. C2×C22⋊Dic13C22semidirect
31. C2×C4⋊Dic13C22semidirect
32. C2×#6101C22semidirect
33. C13×#95C22semidirect
34. #88071C22semidirect
35. #88071C22semidirect
36. #88071C22semidirect
135. #386D13semidirect
192. C2×C22⋊C4C2×D13semidirect
196. #95C2×D13semidirect
246. C13#2622semidirect

Maximal subgroups

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

subgroupindex
C2×#881912
C22×#61032
C13×#3862
#117188812
#117188712
#117186222
#117186202
#117186162
#117186112
#117183362
#117183352
#262213