Finite Group Databasebeta

← #15754697

Subgroups — #15754697

Order 3328.

782 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 159 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#13114832C211yesmaximal, characteristic, complemented
2#13113521C212yesmaximal, complemented
3#13113261C212yesmaximal
4#13113160C211yesmaximal, characteristic, complemented
5C2×C13×#554C211yesmaximal, characteristic
6C13×#554—24no—
7C13×#554—24no—
8C2×C13×C4⋊Q8C2211yescharacteristic
9C2×C13×#125C412yes—
10C2×C13×C22⋊Q8—21no—
11C2×C13×C22⋊Q8—22no—
12C2×C4×Q8×C13C2211yescharacteristic
13C2×C4×D4×C13C412yes—
14C2×C4×D4×C13C2211yescharacteristic
15C22×C13×C4⋊C4—21no—
16C2×#88076—21no—
17#11718586C2212yes—
18#11718585C2212yes—
19#11718139—24no—
20D4⋊#1055—24no—
21#11718122—21no—
22#11718120—22no—
23C22×Q8×C13—21no—
24C22×Q8×C13D411yescharacteristic
25C22×D4×C13C2×C411yescharacteristic
26C23×C4×C13—21no—
27C23×C4×C13—22no—
28C13×C4⋊Q8—22no—
29C13×C4⋊Q8D414yes—
30C13×#125—24no—
31C13×#125—24no—
32C13×C22⋊Q8—28no—
33C13×C22⋊Q8—216no—
34C4×Q8×C13—22no—
35C4×Q8×C13—22no—
36C4×D4×C13—24no—
37C4×D4×C13—24no—
38C4×D4×C13—22no—
39C4×D4×C13D414yes—
40C2×C13×C4⋊C4—28no—
41C2×C13×C4⋊C4—22no—
42C2×C13×C4⋊C4—22no—
43C2×C13×C4⋊C4—21no—
44C2×C13×C4⋊C4C2×C412yes—
45C2×C13×C4⋊C4C2×C411yescharacteristic
46C2×C13×C4⋊C4D411yescharacteristic
47C2×C13×C4⋊C4D412yes—
48C2×C13×C22⋊C4—22no—
49C2×C13×C22⋊C4—21no—
50C2×C42×C13C2×C412yes—
51C2×C42×C13C2311yescharacteristic
52C2×#6099—22no—
53#88076—42no—
54C22⋊#1055—22no—
55C4⋊#1055—22no—
56C4⋊#1055—22no—
57C4⋊#1055—22no—
58C4×#1055—22no—
59#8398—131nomaximal, Sylow
60C24×C13—21no—
61C2×Q8×C13—22no—
62C2×Q8×C13—24no—
63C2×Q8×C13—22no—
64C2×Q8×C13—21no—
65C2×Q8×C13—21no—
66C2×Q8×C13—82no—
67C2×D4×C13—24no—
68C2×D4×C13—21no—
69C2×D4×C13C22⋊C412yes—
70C22×C4×C13—21no—
71C22×C4×C13—22no—
72C22×C4×C13—28no—
73C22×C4×C13—24no—
74C22×C4×C13—42no—
75C22×C4×C13—41no—
76C22×C4×C13—41no—
77C22×C4×C13—84no—
78C22×C4×C13C22⋊C412yes—
79C22×C4×C13C22⋊C412yes—
80C22×C4×C13C22×C411yescharacteristic
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13C2×D411yescharacteristic
83C13×C4⋊C4—28no—
84C13×C4⋊C4—28no—
85C13×C4⋊C4—21no—
86C13×C4⋊C4—22no—
87C13×C4⋊C4—22no—
88C13×C4⋊C4—24no—
89C13×C4⋊C4—24no—
90C13×C4⋊C4C22⋊C414yes—
91C13×C4⋊C4C22⋊C412yes—
92C13×C22⋊C4—28no—
93C13×C22⋊C4—24no—
94C42×C13—22no—
95C42×C13—22no—
96C42×C13C2×D412yes—
97C42×C13C2×D412yes—
98#6099—42no—
99#6099—42no—
100C2×#1055—42no—
101C2×#1055—42no—
102C2×#554D1311yescharacteristic
103#1801—131no—
104#1549—132no—
105#1486—132no—
106#1461—131no—
107C23×C13—24no—
108C23×C13—22no—
109C23×C13—42no—
110C23×C13—41no—
111C23×C13—41no—
112C23×C13C2×C22⋊C411yescharacteristic
113Q8×C13—42no—
114Q8×C13—42no—
115Q8×C13—84no—
116D4×C13—24no—
117D4×C13—22no—
118D4×C13#10114yescomplemented
119C2×C4×C13—22no—
120C2×C4×C13—22no—
121C2×C4×C13—22no—
122C2×C4×C13—22no—
123C2×C4×C13—24no—
124C2×C4×C13—22no—
125C2×C4×C13—22no—
126C2×C4×C13—41no—
127C2×C4×C13—44no—
128C2×C4×C13—41no—
129C2×C4×C13—41no—
130C2×C4×C13—44no—
131C2×C4×C13—44no—
132C2×C4×C13—42no—
133C2×C4×C13—88no—
134C2×C4×C13C2×C22⋊C411yescharacteristic
135C2×C4×C13C2×C22⋊C411yescharacteristic
136C2×C4×C13#12012yes—
137C2×C4×C13#12012yes—
138C2×C4×C13#13512yes—
139C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C13#13711yescharacteristic
141#1055—82no—
142#1055—82no—
143#554—24no—
144#554—24no—
145C2×C4⋊Q8C2×D1311yescharacteristic
146C2×#125Dic1312yes—
147C2×C22⋊Q8—21no—
148C2×C22⋊Q8—22no—
149C2×C4×Q8C2×D1311yescharacteristic
150C2×C4×D4Dic1312yes—
151C2×C4×D4C2×D1311yescharacteristic
152C22×C4⋊C4—21no—
153#431—132no—
154#423—132no—
155C2×#101—261no—
156#343—261no—
157#332—264no—
158D4⋊C8—264no—
159#324—262no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—24no—
163C22×C13—44no—
164C22×C13—44no—
165C22×C13—42no—
166C22×C13—42no—
167C22×C13#37911yescharacteristic
168C22×C13#41611yescharacteristic
169C22×C13#41711yescharacteristic
170C4×C13—22no—
171C4×C13—21no—
172C4×C13—44no—
173C4×C13—42no—
174C4×C13—42no—
175C4×C13—42no—
176C4×C13C2×#10112yes—
177C4×C13#44912yes—
178C4×C13#45212yes—
179C22×Q8—21no—
180C22×Q8C22⋊D1311yescharacteristic
181C22×D4C2×Dic1311yescharacteristic
182C23×C4—21no—
183C23×C4—22no—
184C2×M16—262no—
185C4⋊Q8—22no—
186C4⋊Q8C22⋊D1314yes—
187#125—24no—
188#125—24no—
189C22⋊Q8—28no—
190C22⋊Q8—216no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4C22⋊D1314yes—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic1312yes—
202C2×C4⋊C4C2×Dic1311yescharacteristic
203C2×C4⋊C4C22⋊D1311yescharacteristic
204C2×C4⋊C4C22⋊D1312yes—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C42C2×Dic1312yes—
208C2×C42C22×D1311yescharacteristic
209C4⋊C8—262no—
210C4⋊C8—262no—
211C4⋊C8—262no—
212#101—522no—
213C22⋊C8—262no—
214C4×C8—262no—
215C2×C13—21no—
216C2×C13—21no—
217C2×C13—44no—
218C2×C13#178411yescharacteristic
219C2×C13#178811yescharacteristic
220C2×C13#179511yescharacteristic
221C24—21no—
222C2×Q8—21no—
223C2×Q8—22no—
224C2×Q8—21no—
225C2×Q8—22no—
226C2×Q8—24no—
227C2×Q8—82no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4C22⋊Dic1312yes—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—24no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—84no—
239C22×C4C22⋊Dic1312yes—
240C22×C4C22⋊Dic1312yes—
241C22×C4C22×Dic1311yescharacteristic
242C22×C4C2×C22⋊D1311yescharacteristic
243C22×C4C2×C22⋊D1311yescharacteristic
244M16—522no—
245M16—522no—
246C2×C8—522no—
247C2×C8—522no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4C22⋊Dic1314yes—
256C4⋊C4C22⋊Dic1312yes—
257C22⋊C4—28no—
258C22⋊C4—24no—
259C42—22no—
260C42—22no—
261C42C2×C22⋊D1312yes—
262C42C2×C22⋊D1312yes—
263C13#839811yescharacteristic, complemented, Sylow
264C23—24no—
265C23—22no—
266C23—42no—
267C23—41no—
268C23—41no—
269C23C2×C22⋊Dic1311yescharacteristic
270Q8—42no—
271Q8—42no—
272Q8—84no—
273D4—24no—
274D4—22no—
275D4#8807614yescomplemented
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—41no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—42no—
290C2×C4—88no—
291C2×C4#8820311yescharacteristic
292C2×C4#8820412yes—
293C2×C4#8820511yescharacteristic
294C2×C4C2×C22⋊Dic1311yescharacteristic
295C2×C4C2×C22⋊Dic1311yescharacteristic
296C2×C4#8820712yes—
297C2×C4#8820712yes—
298C8—1042no—
299C8—1042no—
300C22—21no—
301C22—21no—
302C22—24no—
303C22—44no—
304C22—44no—
305C22—42no—
306C22—42no—
307C22#1171889311yescharacteristic
308C22#1171889411yescharacteristic
309C22#1171890811yescharacteristic
310C4—21no—
311C4—22no—
312C4—42no—
313C4—42no—
314C4—44no—
315C4—42no—
316C4C2×#8807612yes—
317C4#1171889912yes—
318C4#1171890012yes—
319C2—21no—
320C2—21no—
321C2—44no—
322C2#1311476011yescharacteristic
323C2#1311477611yescharacteristic
324C2#1311480611yescharacteristic
subgroupG/H
centerC22#11718908
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC2×C13×#554C2
socleC22×C13#379
prime radicalC22×D4×C13C2×C4
cosocleC2×C42×C13C23
hypercenterC2×#554D13
hypocenterC13#8398
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #13114832C2semidirect
2. #13113521C2semidirect
4. #13113160C2semidirect
118. D4×C13#101semidirect
263. C13#8398semidirect
275. D4#88076semidirect

Maximal subgroups

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

subgroupindex
#131148322
#131135212
#131132612
#131131602
C2×C13×#5542
#839813