Finite Group Databasebeta

← #15758645

Subgroups — #15758645

Order 3328.

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

normal subgroups only

This group has 694 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×#11718593C211yesmaximal, characteristic
2#13114904C211yesmaximal, characteristic
3C2×#11718130C212yesmaximal
4#13112714C212yesmaximal
5C13×#2855C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C2211yescharacteristic
7C22×C42×C13C2211yescharacteristic
8C22×C4×Dic13—21no—
9C13×#391C412yescomplemented
10C13×#389C412yescomplemented
11C13×#387—22no—
12C13×#385—21no—
13C13×#381—21no—
14C13×#380C2211yescharacteristic
15C2×#88071—21no—
16C2×#88071C2212yes—
17C2×C22⋊#1055C2212yes—
18#11718593—24no—
19#11718130—24no—
20#11718116—22no—
21C22×Q8×C13C2×C411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13D411yescharacteristic
24C23×C4×C13C2311yescharacteristic
25C23×Dic13—22no—
26C13×C22⋊Q8—22no—
27C13×C22⋊Q8—22no—
28C2×C13×C4⋊C4—22no—
29C2×C13×C4⋊C4—21no—
30C2×C13×C4⋊C4C2×C411yescharacteristic
31C2×C13×C4⋊C4C2×C412yes—
32C2×C13×C22⋊C4—21no—
33C2×C13×C22⋊C4—21no—
34C2×C42×C13—22no—
35C2×C42×C13—22no—
36C2×C42×C13C2×C412yes—
37C2×C42×C13D412yes—
38C2×C4×Dic13—28no—
39C2×C4×Dic13—24no—
40C22×#1055—22no—
41C13×#95—22no—
42C13×#95—22no—
43C13×#95—22no—
44C13×#95—21no—
45#88071—24no—
46#88071—22no—
47#88071D414yes—
48C22⋊#1055—22no—
49C22⋊#1055—22no—
50#9145—131nomaximal, Sylow
51C24×C13C2×D411yescharacteristic
52C2×Q8×C13—22no—
53C2×Q8×C13—42no—
54C22×C4×C13—21no—
55C22×C4×C13—21no—
56C22×C4×C13—24no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—22no—
60C22×C4×C13—22no—
61C22×C4×C13—22no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—42no—
65C22×C4×C13C22⋊C412yes—
66C22×C4×C13C22⋊C412yes—
67C22×C4×C13C22×C411yescharacteristic
68C22×C4×C13C2×D412yes—
69C22×C4×C13C2×D411yescharacteristic
70C22×C4×C13#4112yes—
71C22×Dic13—28no—
72C22×Dic13—24no—
73C22×Dic13—44no—
74C22×Dic13—42no—
75C22×Dic13—42no—
76C13×C4⋊C4—22no—
77C13×C4⋊C4—42no—
78C13×C4⋊C4—44no—
79C13×C4⋊C4—42no—
80C13×C22⋊C4—41no—
81C13×C22⋊C4—41no—
82C13×C22⋊C4—42no—
83C42×C13—24no—
84C42×C13—24no—
85C4×Dic13—216no—
86C2×#1055—44no—
87C2×#1055—42no—
88#2855D1311yescharacteristic
89C2×#376—131no—
90#1817—131no—
91C2×#328—132no—
92#1294—132no—
93C23×C13—22no—
94C23×C13—22no—
95C23×C13C2×C22⋊C411yescharacteristic
96C23×C13#12011yescharacteristic
97C23×C13#12311yescharacteristic
98C23×C13#13512yes—
99C23×C13#13712yes—
100Q8×C13—82no—
101C2×C4×C13—24no—
102C2×C4×C13—24no—
103C2×C4×C13—22no—
104C2×C4×C13—22no—
105C2×C4×C13—22no—
106C2×C4×C13—22no—
107C2×C4×C13—22no—
108C2×C4×C13—22no—
109C2×C4×C13—22no—
110C2×C4×C13—42no—
111C2×C4×C13—42no—
112C2×C4×C13—42no—
113C2×C4×C13—42no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—42no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13#10414yes—
122C2×C4×C13C4×D412yes—
123C2×C4×C13#12112yes—
124C2×Dic13—28no—
125C2×Dic13—44no—
126C2×Dic13—48no—
127C2×Dic13—48no—
128C2×Dic13—44no—
129#1055—82no—
130C2×C22⋊Q8C2×D1311yescharacteristic
131C22×C42—261no—
132C22×C42C2×D1311yescharacteristic
133C2×C22⋊C8—132no—
134#391Dic1312yescomplemented
135#389Dic1312yescomplemented
136#387—22no—
137#385—21no—
138#381—21no—
139#380C2×D1311yescharacteristic
140#376—264no—
141C2×#95—132no—
142C2×#95—261no—
143#336—262no—
144#328—264no—
145C22×C13—21no—
146C22×C13—21no—
147C22×C13—21no—
148C22×C13—21no—
149C22×C13—24no—
150C22×C13—24no—
151C22×C13#38611yescharacteristic
152C22×C13#41012yes—
153C22×C13#41611yescharacteristic
154C22×C13#41911yescharacteristic
155C22×C13C2×#10412yes—
156C22×C13#45212yes—
157C22×C13#48512yes—
158C4×C13—24no—
159C4×C13—22no—
160C4×C13—22no—
161C4×C13—42no—
162C4×C13—81no—
163C4×C13—81no—
164C4×C13—82no—
165Dic13—48no—
166C22×Q8C2×Dic1311yescharacteristic
167C23×C4—262no—
168C23×C4C22⋊D1311yescharacteristic
169C23×C4C22⋊D1311yescharacteristic
170C23×C4C22×D1311yescharacteristic
171C22×C8—262no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×Dic1311yescharacteristic
177C2×C4⋊C4C2×Dic1312yes—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—264no—
183C2×C42—268no—
184C2×C42C2×Dic1312yes—
185C2×C42C22⋊D1312yes—
186C22⋊C8—262no—
187C22⋊C8—262no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—21no—
192#95—134no—
193#95—264no—
194#95—262no—
195C2×C13—22no—
196C2×C13—22no—
197C2×C13#150012yes—
198C2×C13#150612yes—
199C2×C13#187311yescharacteristic
200C2×C13#189611yescharacteristic
201C2×C13#189911yescharacteristic
202C24C2×C22⋊D1311yescharacteristic
203C2×Q8—22no—
204C2×Q8—42no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—42no—
216C22×C4—268no—
217C22×C4—264no—
218C22×C4—524no—
219C22×C4—522no—
220C22×C4—522no—
221C22×C4C22⋊Dic1312yes—
222C22×C4C22⋊Dic1312yes—
223C22×C4D4×D1312yes—
224C22×C4#612912yes—
225C22×C4C22×Dic1311yescharacteristic
226C22×C4C2×C22⋊D1311yescharacteristic
227C2×C8—524no—
228C2×C8—522no—
229C4⋊C4—22no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C22⋊C4—41no—
234C22⋊C4—41no—
235C22⋊C4—42no—
236C42—24no—
237C42—24no—
238C42—2616no—
239C13#914511yescharacteristic, complemented, Sylow
240C23—22no—
241C23—22no—
242C23#8818911yescharacteristic
243C23#8819111yescharacteristic
244C23#8819612yes—
245C23#8820412yes—
246C23C2×C22⋊Dic1311yescharacteristic
247Q8—82no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—268no—
269C2×C4—528no—
270C2×C4—524no—
271C2×C4—528no—
272C2×C4—524no—
273C2×C4#8807714yes—
274C2×C4D4×Dic1312yes—
275C2×C4#8819312yes—
276C8—1042no—
277C22—24no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—24no—
283C22#1171870612yes—
284C22#1171870812yes—
285C22#1171888111yescharacteristic
286C22#1171888611yescharacteristic
287C22#1171889411yescharacteristic
288C22C2×#8807712yes—
289C22#1171890112yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—42no—
294C4—81no—
295C4—81no—
296C4—82no—
297C4—528no—
298C2—22no—
299C2—22no—
300C2#1311332212yes—
301C2#1311334812yes—
302C2#1311514911yescharacteristic
303C2#1311528011yescharacteristic
304C2#1311529511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic13
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2855C2
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#2855D13
hypocenterC13#9145
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
9. C13×#391C4semidirect
10. C13×#389C4semidirect
134. #391Dic13semidirect
135. #389Dic13semidirect
239. C13#9145semidirect

Maximal subgroups

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

subgroupindex
C2×#117185932
#131149042
C2×#117181302
#131127142
C13×#28552
#914513