Finite Group Databasebeta

← #15758968

Subgroups — #15758968

Order 3328.

560 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 135 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
1#13117454C211yesmaximal, characteristic
2#13113408C212yesmaximal
3C2×#11718120C211yesmaximal, characteristic
4#13112703C212yesmaximal
5C13×#2627C211yesmaximal, characteristic
6C2×C13×C22⋊Q8C2211yescharacteristic
7C22×C13×C4⋊C4C2211yescharacteristic
8C2×#88180—21no—
9C13×#400C412yescomplemented
10C13×#398—22no—
11C13×#395C412yescomplemented
12C13×#385—21no—
13C13×#381—21no—
14C13×#380C2211yescharacteristic
15#11718873—21no—
16#11718873C2212yes—
17C2×C22⋊#1055C2212yes—
18#11718646—24no—
19#11718130—24no—
20#11718120—22no—
21C22×Q8×C13C2×C411yescharacteristic
22C23×C4×C13D411yescharacteristic
23C23×C4×C13D411yescharacteristic
24C23×C4×C13C2311yescharacteristic
25C13×C22⋊Q8—22no—
26C13×C22⋊Q8—22no—
27C2×C13×C4⋊C4—22no—
28C2×C13×C4⋊C4—22no—
29C2×C13×C4⋊C4—21no—
30C2×C13×C4⋊C4—22no—
31C2×C13×C4⋊C4—21no—
32C2×C13×C4⋊C4C2×C411yescharacteristic
33C2×C13×C4⋊C4C2×C412yes—
34C2×C13×C4⋊C4D412yes—
35C2×C13×C22⋊C4—21no—
36C2×C13×C22⋊C4—21no—
37C2×C22⋊Dic13—22no—
38#88180—42no—
39#88180—42no—
40C2×C4⋊Dic13—22no—
41C2×C4⋊Dic13—22no—
42C2×C4⋊Dic13—22no—
43C2×C4×Dic13—22no—
44C22×#1055—22no—
45C13×#95—22no—
46C13×#95—22no—
47C13×#95—22no—
48C13×#95C2×C412yes—
49#88071—22no—
50#88071D414yes—
51C22⋊#1055—22no—
52C22⋊#1055—22no—
53#9200—131nomaximal, Sylow
54C24×C13C2×D411yescharacteristic
55C2×Q8×C13—22no—
56C2×Q8×C13—81no—
57C22×C4×C13—21no—
58C22×C4×C13—21no—
59C22×C4×C13—21no—
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—21no—
64C22×C4×C13—22no—
65C22×C4×C13—21no—
66C22×C4×C13—81no—
67C22×C4×C13—81no—
68C22×C4×C13C22⋊C412yes—
69C22×C4×C13C22⋊C412yes—
70C22×C4×C13SD1612yes—
71C22×C4×C13Q1612yes—
72C22×C4×C13C22×C411yescharacteristic
73C22×C4×C13C2×D412yes—
74C22×C4×C13C2×D411yescharacteristic
75C22×C4×C13#4112yes—
76C22×Dic13—24no—
77C13×C4⋊C4—24no—
78C13×C4⋊C4—24no—
79C13×C4⋊C4—22no—
80C13×C4⋊C4—42no—
81C13×C4⋊C4—81no—
82C13×C4⋊C4—82no—
83C13×C22⋊C4—41no—
84C13×C22⋊C4—41no—
85C13×C22⋊C4—41no—
86C13×C22⋊C4—41no—
87C22⋊Dic13—42no—
88C22⋊Dic13—42no—
89C4⋊Dic13—44no—
90C4⋊Dic13—82no—
91C4⋊Dic13—82no—
92C4×Dic13—44no—
93C2×#1055—42no—
94C2×#1055—82no—
95#2627D1311yescharacteristic
96#2316—131no—
97#1520—132no—
98C2×#324—131no—
99#1289—132no—
100C23×C13—41no—
101C23×C13—41no—
102C23×C13C2×C22⋊C411yescharacteristic
103C23×C13#12011yescharacteristic
104C23×C13#12311yescharacteristic
105C23×C13C2×SD1611yescharacteristic
106C23×C13C2×Q1611yescharacteristic
107C23×C13AGL(1,ℤ/8)11yescharacteristic
108C23×C13#13711yescharacteristic
109Q8×C13—82no—
110C2×C4×C13—22no—
111C2×C4×C13—22no—
112C2×C4×C13—22no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—41no—
120C2×C4×C13—41no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—41no—
128C2×C4×C13—41no—
129C2×C4×C13—42no—
130C2×C4×C13—81no—
131C2×C4×C13#10314yes—
132C2×C4×C13C4×D412yes—
133C2×C4×C13#12112yes—
134C2×Dic13—44no—
135C2×Dic13—44no—
136C2×Dic13—44no—
137#1055—82no—
138C2×C22⋊Q8C2×D1311yescharacteristic
139C2×#117—261no—
140C22×C4⋊C4C2×D1311yescharacteristic
141C2×C22⋊C8—132no—
142#400Dic1312yescomplemented
143#398—22no—
144#395Dic1312yescomplemented
145#389—264no—
146#385—21no—
147#381—21no—
148#380—132no—
149#380—261no—
150#380C2×D1311yescharacteristic
151#328—264no—
152#324—262no—
153C22×C13—21no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—41no—
160C22×C13—41no—
161C22×C13—41no—
162C22×C13—41no—
163C22×C13#38611yescharacteristic
164C22×C13C2×#10311yescharacteristic
165C22×C13#41711yescharacteristic
166C22×C13#45011yescharacteristic
167C22×C13#45111yescharacteristic
168C22×C13#48311yescharacteristic
169C22×C13#48411yescharacteristic
170C4×C13—24no—
171C4×C13—22no—
172C4×C13—22no—
173C4×C13—42no—
174C4×C13—82no—
175C4×C13—81no—
176C4×C13—81no—
177Dic13—84no—
178C22×Q8C2×Dic1311yescharacteristic
179C23×C4C22⋊D1311yescharacteristic
180C23×C4C22⋊D1311yescharacteristic
181C23×C4C22×D1311yescharacteristic
182C22×C8—262no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185#117—522no—
186#117—522no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—262no—
193C2×C4⋊C4—262no—
194C2×C4⋊C4—262no—
195C2×C4⋊C4C2×Dic1311yescharacteristic
196C2×C4⋊C4C2×Dic1312yes—
197C2×C4⋊C4C22⋊D1312yes—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—262no—
201C2×C42—262no—
202C22⋊C8—262no—
203C22⋊C8—262no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—134no—
208#95—262no—
209#95C2×Dic1312yes—
210C2×C13—21no—
211C2×C13—21no—
212C2×C13—21no—
213C2×C13—21no—
214C2×C13—21no—
215C2×C13—21no—
216C2×C13#187211yescharacteristic
217C2×C13#191211yescharacteristic
218C2×C13#191311yescharacteristic
219C24C2×C22⋊D1311yescharacteristic
220C2×Q8—22no—
221C2×Q8—81no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—264no—
234C22×C4#610612yes—
235C22×C4#610712yes—
236C22×C4C22⋊Dic1312yes—
237C22×C4C22⋊Dic1312yes—
238C22×C4D4×D1312yes—
239C22×C4#612912yes—
240C22×C4C22×Dic1311yescharacteristic
241C22×C4C2×C22⋊D1311yescharacteristic
242C2×C8—522no—
243C2×C8—1042no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—42no—
248C4⋊C4—81no—
249C4⋊C4—82no—
250C4⋊C4—524no—
251C4⋊C4—1042no—
252C4⋊C4—1042no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—522no—
258C22⋊C4—522no—
259C42—524no—
260C13#920011yescharacteristic, complemented, Sylow
261C23—41no—
262C23—41no—
263C23#8818911yescharacteristic
264C23#8819111yescharacteristic
265C23C2×#610611yescharacteristic
266C23C2×#610711yescharacteristic
267C23#8820311yescharacteristic
268C23#8820511yescharacteristic
269C23C2×C22⋊Dic1311yescharacteristic
270Q8—82no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—524no—
293C2×C4—524no—
294C2×C4—524no—
295C2×C4#8807514yes—
296C2×C4D4×Dic1312yes—
297C2×C4#8819312yes—
298C8—1042no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22#1171870411yescharacteristic
310C22#1171870511yescharacteristic
311C22#1171870711yescharacteristic
312C22#1171871111yescharacteristic
313C22#1171888111yescharacteristic
314C22C2×#8807511yescharacteristic
315C22#1171889311yescharacteristic
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4—81no—
323C4—1044no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#1311514411yescharacteristic
331C2#1311537411yescharacteristic
332C2#1311537911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2627C2
socleC22×C13#386
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#2627D13
hypocenterC13#9200
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
9. C13×#400C4semidirect
11. C13×#395C4semidirect
142. #400Dic13semidirect
144. #395Dic13semidirect
260. C13#9200semidirect

Maximal subgroups

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

subgroupindex
#131174542
#131134082
C2×#117181202
#131127032
C13×#26272
#920013