Finite Group Databasebeta

← #15760651

Subgroups — #15760651

Order 3328.

700 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 107 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#13118476C211yesmaximal, characteristic
2#13117422C211yesmaximal, characteristic
3#13113370C212yesmaximal
4#13112690C212yesmaximal
5C13×#1784C211yesmaximal, characteristic
6C23×Q8×C13C2211yescharacteristic
7C2×#88198C412yescomplemented
8C2×#88180—21no—
9C2×#88178C2211yescharacteristic
10C2×C13×#101—22no—
11C13×#407C2212yes—
12#11718891—24no—
13#11718891C412yescomplemented
14#11718890—22no—
15#11718872—21no—
16#11718872C2212yes—
17#11718871—21no—
18#11718871—21no—
19C4×C22⋊Dic13C2211yescharacteristic
20#11718059—24no—
21C22×Q8×C13—22no—
22C22×Q8×C13—22no—
23C22×Q8×C13—44no—
24C22×Q8×C13C2×C412yes—
25C22×Q8×C13D412yes—
26C23×C4×C13D411yescharacteristic
27C23×C4×C13D411yescharacteristic
28C23×C4×C13C2311yescharacteristic
29C22×Dic26C2×C411yescharacteristic
30C2×C13×M16—22no—
31C2×C22⋊Dic13—21no—
32C2×C22⋊Dic13—21no—
33C2×C22⋊Dic13—22no—
34#88198—44no—
35#88180—44no—
36C2×C4⋊Dic13—22no—
37C2×C4⋊Dic13C2×C411yescharacteristic
38#88178—22no—
39#88178—22no—
40C2×#6101—22no—
41C2×#6101—22no—
42C2×#6101—21no—
43C2×C4×Dic13—22no—
44C2×C4×Dic13C2×C412yes—
45C13×#101—44no—
46C13×C22⋊C8—22no—
47C13×C22⋊C8—22no—
48#88071—24no—
49#88071—22no—
50#88071—22no—
51#88071—21no—
52#88071—22no—
53#88071D414yes—
54#9462—131nomaximal, Sylow
55C24×C13C2×D411yescharacteristic
56C2×Q8×C13—24no—
57C2×Q8×C13—24no—
58C2×Q8×C13—24no—
59C2×Q8×C13—24no—
60C2×Q8×C13—42no—
61C2×Q8×C13—42no—
62C2×Q8×C13—44no—
63C2×Q8×C13—44no—
64C2×Q8×C13—48no—
65C22×C4×C13—21no—
66C22×C4×C13—21no—
67C22×C4×C13—24no—
68C22×C4×C13—21no—
69C22×C4×C13—21no—
70C22×C4×C13—42no—
71C22×C4×C13—42no—
72C22×C4×C13C22⋊C412yes—
73C22×C4×C13C22⋊C412yes—
74C22×C4×C13C22×C411yescharacteristic
75C22×C4×C13C2×D412yes—
76C22×C4×C13C2×D411yescharacteristic
77C22×C4×C13#4112yes—
78C22×Dic13—22no—
79C22×Dic13—21no—
80C22×Dic13—21no—
81C22×Dic13—24no—
82C2×Dic26—22no—
83C2×Dic26—42no—
84C13×M16—44no—
85C2×C8×C13—42no—
86C22⋊Dic13—24no—
87C22⋊Dic13—44no—
88C22⋊Dic13—41no—
89C22⋊Dic13—41no—
90C4⋊Dic13—22no—
91C4⋊Dic13—44no—
92#6101—44no—
93#6101—44no—
94#6101—42no—
95C4×Dic13—44no—
96C4×Dic13—42no—
97#2434—131no—
98#2310—131no—
99#1784D1311yescharacteristic
100#1511—132no—
101#1284—132no—
102C23×C13—22no—
103C23×C13—22no—
104C23×C13—41no—
105C23×C13C2×C22⋊C411yescharacteristic
106C23×C13#12011yescharacteristic
107C23×C13#12311yescharacteristic
108Q8×C13—48no—
109Q8×C13—48no—
110C2×C4×C13—22no—
111C2×C4×C13—22no—
112C2×C4×C13—24no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—22no—
118C2×C4×C13—22no—
119C2×C4×C13—22no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13—42no—
124C2×C4×C13#9914yes—
125C2×C4×C13C4×D412yes—
126C2×C4×C13#12112yes—
127C2×Dic13—24no—
128C2×Dic13—42no—
129C2×Dic13—42no—
130C2×Dic13—44no—
131C2×Dic13—44no—
132C2×Dic13—42no—
133C2×Dic13—41no—
134C2×Dic13—41no—
135C2×Dic13—42no—
136C2×Dic13—44no—
137Dic26—82no—
138C8×C13—82no—
139C23×Q8C2×D1311yescharacteristic
140C2×C4⋊Q8—132no—
141C2×C22⋊Q8—131no—
142C2×#117—261no—
143C2×#101—22no—
144#407C2×D1312yes—
145#395—262no—
146#391—132no—
147#391—264no—
148#385—261no—
149#385—261no—
150#381—132no—
151#381—261no—
152C4×C22⋊C4—131no—
153#328—264no—
154C22×C13—21no—
155C22×C13—21no—
156C22×C13—21no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13—44no—
162C22×C13#38611yescharacteristic
163C22×C13C2×#9912yes—
164C22×C13#45612yes—
165C4×C13—22no—
166C4×C13—22no—
167C4×C13—24no—
168C4×C13—42no—
169Dic13—44no—
170Dic13—84no—
171Dic13—81no—
172Dic13—81no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—44no—
176C22×Q8—131no—
177C22×Q8C4×D1312yes—
178C22×Q8C22⋊D1312yes—
179C23×C4D5211yescharacteristic
180C23×C4D5211yescharacteristic
181C23×C4C22×D1311yescharacteristic
182C2×M16—22no—
183C4⋊Q8—524no—
184C22⋊Q8—262no—
185C22⋊Q8—262no—
186#117—524no—
187C2×C4⋊C4—131no—
188C2×C4⋊C4—262no—
189C2×C4⋊C4—262no—
190C2×C4⋊C4—262no—
191C2×C4⋊C4—261no—
192C2×C22⋊C4—261no—
193C2×C22⋊C4—262no—
194C2×C22⋊C4—261no—
195C2×C42—132no—
196C2×C42—262no—
197#101—44no—
198C22⋊C8—22no—
199C22⋊C8—22no—
200#95—134no—
201#95—264no—
202#95—262no—
203#95—262no—
204#95—262no—
205#95—261no—
206C2×C13—22no—
207C2×C13—22no—
208C2×C13—41no—
209C2×C13#188611yescharacteristic
210C2×C13#189811yescharacteristic
211C2×C13#191211yescharacteristic
212C24C2×D5211yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—48no—
222C2×Q8—262no—
223C2×Q8—522no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—24no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—262no—
232C22×C4—264no—
233C22×C4—261no—
234C22×C4—261no—
235C22×C4#610312yes—
236C22×C4#610312yes—
237C22×C4C2×C4×D1311yescharacteristic
238C22×C4D4×D1312yes—
239C22×C4#612912yes—
240C22×C4C2×C22⋊D1311yescharacteristic
241M16—44no—
242C2×C8—42no—
243C4⋊C4—262no—
244C4⋊C4—524no—
245C4⋊C4—524no—
246C4⋊C4—522no—
247C4⋊C4—524no—
248C22⋊C4—264no—
249C22⋊C4—524no—
250C22⋊C4—521no—
251C22⋊C4—521no—
252C42—522no—
253C42—524no—
254C13#946211yescharacteristic, complemented, Sylow
255C23—22no—
256C23—22no—
257C23—41no—
258C23#8813611yescharacteristic
259C23#8814011yescharacteristic
260C23C2×#610311yescharacteristic
261Q8—48no—
262Q8—48no—
263Q8—1042no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—264no—
279C2×C4—522no—
280C2×C4—522no—
281C2×C4—524no—
282C2×C4—524no—
283C2×C4—522no—
284C2×C4—524no—
285C2×C4—522no—
286C2×C4—521no—
287C2×C4—521no—
288C2×C4#8804614yes—
289C2×C4#8813512yes—
290C2×C4#8819312yes—
291C8—82no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—44no—
300C22#1171841312yes—
301C22C2×#8804612yes—
302C22#1171861911yescharacteristic
303C4—22no—
304C4—24no—
305C4—22no—
306C4—42no—
307C4—524no—
308C4—1044no—
309C4—1041no—
310C4—1041no—
311C2—22no—
312C2—22no—
313C2—41no—
314C2#1311522711yescharacteristic
315C2#1311528711yescharacteristic
316C2#1311537111yescharacteristic
subgroupG/H
centerC22#11718619
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#1784C2
socleC22×C13#386
prime radicalC24×C13C2×D4
cosocleC23×C4×C13C23
hypercenter#1784D13
hypocenterC13#9462
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
7. C2×#88198C4semidirect
13. #11718891C4semidirect
254. C13#9462semidirect

Maximal subgroups

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

subgroupindex
#131184762
#131174222
#131133702
#131126902
C13×#17842
#946213