Finite Group Databasebeta

← #15754657

Subgroups — #15754657

Order 3328.

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

normal subgroups only

This group has 706 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
2#13113508C212yesmaximal
3#13113265C212yesmaximal
4#13113160C211yesmaximal, characteristic
5C2×C13×#557C211yesmaximal, characteristic
6C13×#557—24no—
7C13×#557—24no—
8C2×C13×C4⋊Q8—21no—
9C2×C13×C4⋊Q8C2211yescharacteristic
10C2×C13×#125—22no—
11C2×C13×#125C412yes—
12C2×C4×Q8×C13—21no—
13C2×C4×Q8×C13C412yes—
14C2×C4×Q8×C13C2211yescharacteristic
15C2×C4×Q8×C13C2211yescharacteristic
16C2×#88076—21no—
17#11718586C2212yes—
18#11718585C2212yes—
19#11718139—24no—
20Q8⋊#1055—24no—
21#11718122—21no—
22#11718120—22no—
23C22×Q8×C13—21no—
24C22×Q8×C13C2×C411yescharacteristic
25C22×Q8×C13D411yescharacteristic
26C13×C4⋊Q8—28no—
27C13×C4⋊Q8—22no—
28C13×C4⋊Q8D414yes—
29C13×#125—216no—
30C13×#125—24no—
31C13×#125—24no—
32C4×Q8×C13—28no—
33C4×Q8×C13—24no—
34C4×Q8×C13—24no—
35C4×Q8×C13—22no—
36C4×Q8×C13—22no—
37C4×Q8×C13—22no—
38C4×Q8×C13D414yes—
39C2×C13×C4⋊C4—22no—
40C2×C13×C4⋊C4—22no—
41C2×C13×C4⋊C4—22no—
42C2×C13×C4⋊C4—21no—
43C2×C13×C4⋊C4—21no—
44C2×C13×C4⋊C4C2×C412yes—
45C2×C13×C4⋊C4C2×C411yescharacteristic
46C2×C13×C4⋊C4D411yescharacteristic
47C2×C13×C4⋊C4D412yes—
48C2×C42×C13—22no—
49C2×C42×C13—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#8390—131nomaximal, Sylow
60C2×Q8×C13—24no—
61C2×Q8×C13—21no—
62C2×Q8×C13—22no—
63C2×Q8×C13—24no—
64C2×Q8×C13—22no—
65C2×Q8×C13—21no—
66C2×Q8×C13—21no—
67C2×Q8×C13—42no—
68C2×Q8×C13—41no—
69C2×Q8×C13—41no—
70C2×Q8×C13C22⋊C412yes—
71C22×C4×C13—21no—
72C22×C4×C13—21no—
73C22×C4×C13—22no—
74C22×C4×C13C22⋊C412yes—
75C22×C4×C13C22⋊C412yes—
76C22×C4×C13C22×C411yescharacteristic
77C22×C4×C13C2×D411yescharacteristic
78C22×C4×C13C2×D411yescharacteristic
79C13×C4⋊C4—28no—
80C13×C4⋊C4—28no—
81C13×C4⋊C4—28no—
82C13×C4⋊C4—21no—
83C13×C4⋊C4—22no—
84C13×C4⋊C4—24no—
85C13×C4⋊C4—24no—
86C13×C4⋊C4—22no—
87C13×C4⋊C4—24no—
88C13×C4⋊C4C22⋊C414yes—
89C13×C4⋊C4C22⋊C412yes—
90C42×C13—28no—
91C42×C13—22no—
92C42×C13—22no—
93C42×C13—24no—
94C42×C13C2×D412yes—
95C42×C13C2×D412yes—
96#6099—42no—
97#6099—42no—
98C2×#1055—42no—
99C2×#1055—42no—
100C2×#557D1311yescharacteristic
101#1801—131no—
102#1546—132no—
103#1487—132no—
104#1461—131no—
105C23×C13C2×C22⋊C411yescharacteristic
106Q8×C13—24no—
107Q8×C13—22no—
108Q8×C13—42no—
109Q8×C13—44no—
110Q8×C13—42no—
111Q8×C13—44no—
112Q8×C13#10114yescomplemented
113C2×C4×C13—24no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—22no—
118C2×C4×C13—22no—
119C2×C4×C13—24no—
120C2×C4×C13—22no—
121C2×C4×C13—22no—
122C2×C4×C13—41no—
123C2×C4×C13—42no—
124C2×C4×C13—42no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13C2×C22⋊C411yescharacteristic
128C2×C4×C13C2×C22⋊C411yescharacteristic
129C2×C4×C13#12012yes—
130C2×C4×C13#12012yes—
131C2×C4×C13#13512yes—
132C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C13#13711yescharacteristic
134#1055—82no—
135#1055—82no—
136#557—24no—
137#557—24no—
138C2×C4⋊Q8—21no—
139C2×C4⋊Q8C2×D1311yescharacteristic
140C2×#125—22no—
141C2×#125Dic1312yes—
142C2×C4×Q8—21no—
143C2×C4×Q8Dic1312yes—
144C2×C4×Q8C2×D1311yescharacteristic
145C2×C4×Q8C2×D1311yescharacteristic
146#431—132no—
147#423—132no—
148C2×#101—261no—
149#343—261no—
150#332—264no—
151Q8⋊C8—264no—
152#324—262no—
153C22×C13—21no—
154C22×C13—21no—
155C22×C13#37911yescharacteristic
156C22×C13#41611yescharacteristic
157C22×C13#41711yescharacteristic
158C4×C13—22no—
159C4×C13—24no—
160C4×C13—21no—
161C4×C13—42no—
162C4×C13—42no—
163C4×C13—42no—
164C4×C13—44no—
165C4×C13C2×#10112yes—
166C4×C13#44912yes—
167C4×C13#45212yes—
168C22×Q8—21no—
169C22×Q8C2×Dic1311yescharacteristic
170C22×Q8C22⋊D1311yescharacteristic
171C2×M16—262no—
172C4⋊Q8—28no—
173C4⋊Q8—22no—
174C4⋊Q8C22⋊D1314yes—
175#125—216no—
176#125—24no—
177#125—24no—
178C4×Q8—28no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8C22⋊D1314yes—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C2×Dic1312yes—
191C2×C4⋊C4C2×Dic1311yescharacteristic
192C2×C4⋊C4C22⋊D1311yescharacteristic
193C2×C4⋊C4C22⋊D1312yes—
194C2×C42—22no—
195C2×C42—21no—
196C2×C42C2×Dic1312yes—
197C2×C42C22×D1311yescharacteristic
198C4⋊C8—262no—
199C4⋊C8—262no—
200C4⋊C8—262no—
201#101—522no—
202C22⋊C8—262no—
203C4×C8—262no—
204C2×C13—21no—
205C2×C13—21no—
206C2×C13#178411yescharacteristic
207C2×C13#178811yescharacteristic
208C2×C13#179611yescharacteristic
209C2×Q8—24no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—24no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8C22⋊Dic1312yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4C22⋊Dic1312yes—
224C22×C4C22⋊Dic1312yes—
225C22×C4C22×Dic1311yescharacteristic
226C22×C4C2×C22⋊D1311yescharacteristic
227C22×C4C2×C22⋊D1311yescharacteristic
228M16—522no—
229M16—522no—
230C2×C8—522no—
231C2×C8—522no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—21no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4C22⋊Dic1314yes—
242C4⋊C4C22⋊Dic1312yes—
243C42—28no—
244C42—22no—
245C42—22no—
246C42—24no—
247C42C2×C22⋊D1312yes—
248C42C2×C22⋊D1312yes—
249C13#839011yescharacteristic, complemented, Sylow
250C23C2×C22⋊Dic1311yescharacteristic
251Q8—24no—
252Q8—22no—
253Q8—42no—
254Q8—44no—
255Q8—42no—
256Q8—44no—
257Q8#8807614yescomplemented
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4#8820311yescharacteristic
273C2×C4#8820412yes—
274C2×C4#8820511yescharacteristic
275C2×C4C2×C22⋊Dic1311yescharacteristic
276C2×C4C2×C22⋊Dic1311yescharacteristic
277C2×C4#8820712yes—
278C2×C4#8820712yes—
279C8—1042no—
280C8—1042no—
281C22—21no—
282C22—21no—
283C22#1171889311yescharacteristic
284C22#1171889411yescharacteristic
285C22#1171890811yescharacteristic
286C4—24no—
287C4—21no—
288C4—22no—
289C4—42no—
290C4—44no—
291C4—42no—
292C4—42no—
293C4C2×#8807612yes—
294C4#1171889912yes—
295C4#1171890012yes—
296C2—21no—
297C2—21no—
298C2#1311476011yescharacteristic
299C2#1311477611yescharacteristic
300C2#1311481111yescharacteristic
subgroupG/H
centerC22#11718908
commutatorC22×C4×C13C22×C4
FrattiniC2×C42C22×D13
FittingC2×C13×#557C2
socleC22×C13#379
prime radicalC23×C13C2×C22⋊C4
cosocleC2×C42×C13C23
hypercenterC2×#557D13
hypocenterC13#8390
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
112. Q8×C13#101semidirect
249. C13#8390semidirect
257. Q8#88076semidirect

Maximal subgroups

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

subgroupindex
#131148322
#131135082
#131132652
#131131602
C2×C13×#5572
#839013