Finite Group Databasebeta

← #13119116

Subgroups — #13119116

Order 1664.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C2×C22⋊C4×D13C211yesmaximal, characteristic
3C13×#385C211yesmaximal, characteristic, complemented
4#11718871C211yesmaximal, characteristic, complemented
5#11718654C212yesmaximal, complemented
6#11718650C212yesmaximal, complemented
7#11718616C211yesmaximal, characteristic
8#11718604C211yesmaximal, characteristic, complemented
9#11718604C211yesmaximal, characteristic, complemented
10#11718336C212yesmaximal, complemented
11#11718325C212yesmaximal, complemented
12C24×D13C2211yescharacteristic
13C23×C4×C13C2211yescharacteristic
14C23×Dic13C2211yescharacteristic
15C22×C4×D13C2211yescharacteristic
16C22×C4×D13C2211yescharacteristic
17C2×C13×C4⋊C4C2212yescomplemented
18C2×C13×C22⋊C4C2211yescharacteristic
19C2×C13×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic13C2211yescharacteristic
21C2×C22⋊Dic13C2211yescharacteristic
22C2×#6103—24no—
23C2×#6103C2211yescharacteristic
24C2×#6103C2211yescharacteristic
25C2×#6103C2212yes—
26C2×#6103C2212yes—
27C2×#6103C2212yes—
28C2×#6103C2212yes—
29C2×#6101C2212yescomplemented
30C22⋊C4×D13—24no—
31C13×#95C2212yescomplemented
32#88071C2212yescomplemented
33#88071C2212yescomplemented
34#88071C2212yescomplemented
35#88071C2212yescomplemented
36#88071C2212yescomplemented
37C24×C13C2311yescharacteristic
38C23×D13—22no—
39C23×D13—22no—
40C23×D13—24no—
41C23×D13—42no—
42C23×D13D414yes—
43C23×D13C2311yescharacteristic
44C23×D13C2311yescharacteristic
45C22×C4×C13—22no—
46C22×C4×C13—42no—
47C22×C4×C13C2311yescharacteristic
48C22×C4×C13C2312yes—
49C22×C4×C13C2311yescharacteristic
50C22×C4×C13C2312yes—
51C22×Dic13—22no—
52C22×Dic13—22no—
53C22×Dic13—22no—
54C22×Dic13C2312yes—
55C22×Dic13C2311yescharacteristic
56C22×Dic13C2312yes—
57C22×Dic13C2311yescharacteristic
58C2×C4×D13—22no—
59C2×C4×D13—22no—
60C2×C4×D13—24no—
61C2×C4×D13—42no—
62C13×C4⋊C4—42no—
63C13×C22⋊C4—22no—
64C13×C22⋊C4—22no—
65C22⋊Dic13—22no—
66C22⋊Dic13—22no—
67#6103—22no—
68#6103—22no—
69#6103—24no—
70#6103—24no—
71#6103—24no—
72#6103—42no—
73#6101—42no—
74#2483—131nomaximal, Sylow
75C23×C13—22no—
76C23×C13—22no—
77C23×C13C2×D412yes—
78C23×C13#4112yes—
79C23×C13#4112yes—
80C23×C13C2411yescharacteristic
81C22×D13—21no—
82C22×D13—21no—
83C22×D13—22no—
84C22×D13—28no—
85C22×D13—21no—
86C22×D13—21no—
87C22×D13—22no—
88C22×D13—22no—
89C22×D13—28no—
90C22×D13—44no—
91C22×D13—48no—
92C22×D13—48no—
93C22×D13C2×D412yes—
94C22×D13C2×D412yes—
95C22×D13#4114yes—
96C2×C4×C13—22no—
97C2×C4×C13—21no—
98C2×C4×C13—21no—
99C2×C4×C13—22no—
100C2×C4×C13—22no—
101C2×C4×C13—22no—
102C2×C4×C13—22no—
103C2×C4×C13—22no—
104C2×C4×C13—22no—
105C2×C4×C13—22no—
106C2×C4×C13—44no—
107C2×Dic13—22no—
108C2×Dic13—21no—
109C2×Dic13—21no—
110C2×Dic13—22no—
111C2×Dic13—22no—
112C2×Dic13—22no—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—22no—
116C2×Dic13—22no—
117C2×Dic13—44no—
118C4×D13—44no—
119C4×D13—44no—
120C22×C22⋊C4—131no—
121C22×C22⋊C4—131no—
122#397—132no—
123#397—132no—
124#385—131no—
125#385—132no—
126#385D1311yescharacteristic, complemented
127#381—131no—
128#381—131no—
129#381—132no—
130#379—131no—
131C22×C13—22no—
132C22×C13—22no—
133C22×C13—24no—
134C22×C13—24no—
135C22×C13#12314yes—
136C22×C13C22×D411yescharacteristic
137C22×C13C2×#4111yescharacteristic
138C22×C13C2×#4111yescharacteristic
139C22×C13C2×#4112yes—
140C22×C13H(5,𝔽2)12yes—
141C2×D13—22no—
142C2×D13—22no—
143C2×D13—24no—
144C2×D13—24no—
145C2×D13—22no—
146C2×D13—22no—
147C2×D13—22no—
148C2×D13—416no—
149C2×D13—42no—
150C2×D13—42no—
151C2×D13#12314yes—
152C4×C13—41no—
153C4×C13—41no—
154C4×C13—42no—
155C4×C13—42no—
156Dic13—41no—
157Dic13—42no—
158Dic13—41no—
159Dic13—42no—
160C25—131no—
161C23×C4—131no—
162C23×C4—131no—
163C23×C4—131no—
164C23×C4C2×D1311yescharacteristic
165C2×C4⋊C4—132no—
166C2×C4⋊C4C2×D1312yescomplemented
167C2×C22⋊C4—131no—
168C2×C22⋊C4—131no—
169C2×C22⋊C4—132no—
170C2×C22⋊C4—131no—
171C2×C22⋊C4—132no—
172C2×C22⋊C4—131no—
173C2×C22⋊C4—132no—
174C2×C22⋊C4—132no—
175C2×C22⋊C4—264no—
176C2×C22⋊C4—264no—
177C2×C22⋊C4C2×D1311yescharacteristic
178C2×C22⋊C4C2×D1311yescharacteristic
179#95—132no—
180#95—132no—
181#95—132no—
182#95—132no—
183#95—132no—
184#95C2×D1312yescomplemented
185C2×C13—24no—
186C2×C13C2×#12311yescharacteristic
187C2×C13C2×#12311yescharacteristic
188C2×C13#53411yescharacteristic
189C2×C13#55112yes—
190C2×C13#55112yes—
191D13—24no—
192D13—42no—
193C24—131no—
194C24—131no—
195C24—134no—
196C24—264no—
197C24—262no—
198C24—262no—
199C24—522no—
200C24C22×D1311yescharacteristic
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—132no—
204C22×C4—131no—
205C22×C4—132no—
206C22×C4—131no—
207C22×C4—262no—
208C22×C4—262no—
209C22×C4—264no—
210C22×C4—262no—
211C22×C4—262no—
212C22×C4—262no—
213C22×C4—522no—
214C22×C4C22×D1311yescharacteristic
215C22×C4C22×D1312yes—
216C22×C4C22×D1311yescharacteristic
217C22×C4C22×D1312yes—
218C4⋊C4—42no—
219C4⋊C4—522no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—262no—
223C22⋊C4—262no—
224C22⋊C4—264no—
225C22⋊C4—264no—
226C22⋊C4—264no—
227C22⋊C4—262no—
228C22⋊C4—262no—
229C22⋊C4—522no—
230C13#248311yescharacteristic, complemented, Sylow
231C23—22no—
232C23—22no—
233C23—132no—
234C23—134no—
235C23—132no—
236C23—268no—
237C23—261no—
238C23—262no—
239C23—261no—
240C23—268no—
241C23—261no—
242C23—262no—
243C23—261no—
244C23—262no—
245C23—524no—
246C23—528no—
247C23—528no—
248C23#612712yes—
249C23D4×D1312yes—
250C23#613112yes—
251C23C23×D1311yescharacteristic
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—44no—
263C2×C4—262no—
264C2×C4—262no—
265C2×C4—261no—
266C2×C4—261no—
267C2×C4—262no—
268C2×C4—262no—
269C2×C4—262no—
270C2×C4—262no—
271C2×C4—262no—
272C2×C4—262no—
273C2×C4—524no—
274C2×C4—524no—
275C2×C4—524no—
276C22—24no—
277C22—22no—
278C22—24no—
279C22—22no—
280C22—134no—
281C22—262no—
282C22—264no—
283C22—262no—
284C22—264no—
285C22—262no—
286C22—262no—
287C22—262no—
288C22—5216no—
289C22—522no—
290C22—522no—
291C22#8814814yes—
292C22C2×#612711yescharacteristic
293C22C2×D4×D1311yescharacteristic
294C22#8825112yes—
295C22C2×#613111yescharacteristic
296C22#41×D1312yes—
297C4—42no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—522no—
302C4—521no—
303C4—522no—
304C4—521no—
305C2—24no—
306C2—264no—
307C2—522no—
308C2C2×#8814811yescharacteristic
309C2#1171926112yes—
310C2#1171929411yescharacteristic
311C2#1171934212yes—
312C2D13×#12311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D13
commutator, socle, cosocleC23×C13C24
FittingC13×#385C2
prime radicalC24×D13C22
hypercenter#385D13
hypocenterC13#2483
LCSG ▷ C23×C13 ▷ C13
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C13×#385C2semidirect
4. #11718871C2semidirect
5. #11718654C2semidirect
6. #11718650C2semidirect
8. #11718604C2semidirect
9. #11718604C2semidirect
10. #11718336C2semidirect
11. #11718325C2semidirect
17. C2×C13×C4⋊C4C22semidirect
29. C2×#6101C22semidirect
31. C13×#95C22semidirect
32. #88071C22semidirect
33. #88071C22semidirect
34. #88071C22semidirect
35. #88071C22semidirect
36. #88071C22semidirect
126. #385D13semidirect
166. C2×C4⋊C4C2×D13semidirect
184. #95C2×D13semidirect
230. C13#2483semidirect

Maximal subgroups

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

subgroupindex
C22×#61032
C2×C22⋊C4×D132
C13×#3852
#117188712
#117186542
#117186502
#117186162
#117186042
#117183362
#117183252
#248313