Finite Group Databasebeta

← #15758208

Subgroups — #15758208

Order 3328.

654 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 155 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×#11718746C211yesmaximal, characteristic
2#13115763C212yesmaximal
3C2×#11718138C211yesmaximal, characteristic, complemented
4#13112677C212yesmaximal, complemented
5C13×#2369C211yesmaximal, characteristic, complemented
6C2×C13×C4⋊Q8C2211yescharacteristic
7C22×#6106—22no—
8C2×#88128C2211yescharacteristic
9C13×#391—22no—
10C13×#389—22no—
11C4×C13×C4⋊C4—22no—
12C4×C13×C4⋊C4C2212yes—
13#11718746—22no—
14#11718746C414yescomplemented
15C2×#88049—22no—
16C2×#88049—22no—
17C2×C4⋊#1055C2212yes—
18C2×C4×#1055C2211yescharacteristic
19#11718138—22no—
20C22×Q8×C13D412yes—
21C22×D52—21no—
22C22×D52—22no—
23C13×C4⋊Q8—22no—
24C13×C4⋊Q8C2×C412yes—
25C13×C4⋊Q8D412yes—
26C2×C13×C4⋊C4—22no—
27C2×C13×C4⋊C4—22no—
28C2×C13×C4⋊C4D412yes—
29C2×C13×C4⋊C4D412yes—
30C2×C42×C13—22no—
31C2×C42×C13—22no—
32C2×C42×C13C2311yescharacteristic
33C2×#6106—28no—
34C2×#6106—44no—
35C2×#6106—44no—
36C22×#1055—21no—
37C22×#1055—22no—
38#88128—21no—
39#88128—22no—
40#88128C2×C412yes—
41C13×#95—24no—
42C13×#95—22no—
43#88049—44no—
44#88049—44no—
45C4⋊#1055—22no—
46C4⋊#1055—22no—
47C4×#1055—21no—
48C4×#1055C2×C412yes—
49#9063—131nomaximal, Sylow
50C23×D13—41no—
51C2×Q8×C13—22no—
52C2×Q8×C13—42no—
53C2×Q8×C13—42no—
54C2×Q8×C13C22⋊C414yes—
55C22×C4×C13—22no—
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13C2×D411yescharacteristic
59C22×C4×C13C2×D412yes—
60C2×D52—22no—
61C2×D52—24no—
62C2×D52—42no—
63C2×D52—42no—
64C2×D52—42no—
65C2×D52—42no—
66C2×D52—42no—
67C2×D52—42no—
68C2×D52—42no—
69C2×D52—41no—
70C13×C4⋊C4—24no—
71C13×C4⋊C4—24no—
72C13×C4⋊C4—22no—
73C13×C4⋊C4—22no—
74C13×C4⋊C4—22no—
75C13×C4⋊C4—82no—
76C13×C4⋊C4SD1614yescomplemented
77C42×C13—42no—
78C42×C13—42no—
79C42×C13—82no—
80C42×C13C22×C411yescharacteristic
81C42×C13C2×D411yescharacteristic
82C42×C13#4112yes—
83#6106—416no—
84C2×#1055—22no—
85C2×#1055—22no—
86C2×#1055—22no—
87C2×#1055—42no—
88C2×#1055—42no—
89C2×#1055—42no—
90C2×#492—131no—
91#2369D1311yescharacteristic, complemented
92#1982—132no—
93C2×#333—131no—
94#1279—132no—
95C23×C13#12011yescharacteristic
96C22×D13—42no—
97C22×D13—82no—
98C22×D13—82no—
99C22×D13—81no—
100C22×D13—81no—
101Q8×C13—44no—
102Q8×C13—82no—
103C2×C4×C13—21no—
104C2×C4×C13—21no—
105C2×C4×C13—24no—
106C2×C4×C13—24no—
107C2×C4×C13—42no—
108C2×C4×C13—42no—
109C2×C4×C13—42no—
110C2×C4×C13—42no—
111C2×C4×C13—42no—
112C2×C4×C13—42no—
113C2×C4×C13—42no—
114C2×C4×C13C2×C22⋊C411yescharacteristic
115C2×C4×C13C4×D412yes—
116C2×C4×C13#12112yes—
117C2×C4×C13#12311yescharacteristic
118C2×C4×C13C2×SD1612yes—
119C2×C4×C13#13512yes—
120C2×C4×C13AGL(1,ℤ/8)12yes—
121C2×C4×C13#13712yes—
122D52—44no—
123D52—44no—
124D52—44no—
125D52—82no—
126D52—82no—
127D52—82no—
128#1055—24no—
129#1055—82no—
130C22×SD16—262no—
131C2×C4⋊Q8C2×D1311yescharacteristic
132C2×#127—131no—
133#492—134no—
134#492—262no—
135C2×C4⋊C8—132no—
136C2×#102—262no—
137C2×#102—262no—
138C2×C4×C8—131no—
139#391—22no—
140#389—22no—
141C4×C4⋊C4—22no—
142C4×C4⋊C4C2×D1312yes—
143#333—262no—
144C22×C13#38611yescharacteristic
145C22×C13#44812yes—
146C22×C13#44912yes—
147C22×C13#45311yescharacteristic
148C22×C13#45311yescharacteristic
149C2×D13—84no—
150C2×D13—82no—
151C2×D13—162no—
152C2×D13—161no—
153C2×D13—161no—
154C4×C13—21no—
155C4×C13—21no—
156C4×C13—44no—
157C4×C13—44no—
158C4×C13—82no—
159C4×C13C4×SD1612yes—
160C4×C13#44012yes—
161C4×C13#46012yes—
162C4×C13#46412yes—
163C22×Q8C22⋊D1312yes—
164C22×D4—261no—
165C22×D4—262no—
166C2×SD16—268no—
167C2×SD16—524no—
168C2×SD16—524no—
169C22×C8—261no—
170C22×C8—262no—
171C4⋊Q8—22no—
172C4⋊Q8C4×D1312yes—
173C4⋊Q8D5212yes—
174#127—132no—
175#127—261no—
176#127—262no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊D1312yes—
180C2×C4⋊C4C22⋊D1312yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42C22×D1311yescharacteristic
184C4⋊C8—262no—
185C4⋊C8—262no—
186#102—524no—
187#102—524no—
188C4×C8—132no—
189C4×C8—261no—
190#95—24no—
191#95—22no—
192C2×C13#162212yes—
193C2×C13#162311yescharacteristic
194C2×C13#162411yescharacteristic
195C2×C13#188012yes—
196C2×C13#188711yescharacteristic
197D13—162no—
198C24—521no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8#610314yes—
203C2×D4—262no—
204C2×D4—264no—
205C2×D4—522no—
206C2×D4—522no—
207C2×D4—521no—
208C2×D4—522no—
209C2×D4—522no—
210C2×D4—522no—
211C2×D4—522no—
212C2×D4—522no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4C2×C22⋊D1311yescharacteristic
217C22×C4C2×C22⋊D1312yes—
218SD16—5216no—
219C2×C8—262no—
220C2×C8—262no—
221C2×C8—262no—
222C2×C8—522no—
223C2×C8—522no—
224C2×C8—522no—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—82no—
231C4⋊C4#610614yescomplemented
232C42—42no—
233C42—42no—
234C42—82no—
235C42C2×C4×D1311yescharacteristic
236C42C2×D5211yescharacteristic
237C42#612712yes—
238C13#906311yescharacteristic, complemented, Sylow
239C23—522no—
240C23—1042no—
241C23—1042no—
242C23—1041no—
243C23—1041no—
244C23#8820711yescharacteristic
245Q8—44no—
246Q8—82no—
247D4—524no—
248D4—524no—
249D4—524no—
250D4—1042no—
251D4—1042no—
252D4—1042no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4C2×#610311yescharacteristic
265C2×C4C4×C22⋊D1312yes—
266C2×C4#8818311yescharacteristic
267C2×C4#8818412yes—
268C2×C4C2×#610612yes—
269C2×C4#8819612yes—
270C2×C4#8820312yes—
271C2×C4#8820412yes—
272C8—264no—
273C8—1042no—
274C22—1044no—
275C22—1042no—
276C22—2082no—
277C22—2081no—
278C22—2081no—
279C22#1171887411yescharacteristic
280C22#1171888812yes—
281C22#1171889912yes—
282C22#1171890211yescharacteristic
283C22#1171890211yescharacteristic
284C4—21no—
285C4—21no—
286C4—44no—
287C4—44no—
288C4—82no—
289C4C4×#610612yes—
290C4#1171868712yes—
291C4#1171868812yes—
292C4#1171868912yes—
293C2—2082no—
294C2#1311390812yes—
295C2#1311391411yescharacteristic
296C2#1311391911yescharacteristic
297C2#1311519112yes—
298C2#1311523411yescharacteristic
subgroupG/H
centerC23#88207
commutatorC42×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#2369C2
socleC23×C13#120
prime radicalC2×#88128C22
cosocleC2×C42×C13C23
hypercenter#2369D13
hypocenterC13#9063
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. C2×#11718138C2semidirect
4. #13112677C2semidirect
5. C13×#2369C2semidirect
14. #11718746C4semidirect
76. C13×C4⋊C4SD16semidirect
91. #2369D13semidirect
231. C4⋊C4#6106semidirect
238. C13#9063semidirect

Maximal subgroups

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

subgroupindex
C2×#117187462
#131157632
C2×#117181382
#131126772
C13×#23692
#906313