Finite Group Databasebeta

← #15763478

Subgroups — #15763478

Order 3328.

804 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 121 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×#11718711C212yesmaximal
2C2×#11718130C212yesmaximal
3#13113106C211yesmaximal, characteristic
4#13112679C211yesmaximal, characteristic
5C13×#2659C211yesmaximal, characteristic
6C23×Dic26C2211yescharacteristic
7C2×C13×C22⋊Q8C2212yes—
8C22×#6107—22no—
9C13×#398—22no—
10C13×#395—22no—
11C13×#380—22no—
12C2×C13×#95C2211yescharacteristic
13C2×#88071—22no—
14C2×#88071C2211yescharacteristic
15C2×C22⋊#1055C2212yes—
16#11718711—24no—
17#11718711—24no—
18C2×#88050—22no—
19#11718130—24no—
20C22×Q8×C13D412yes—
21C23×C4×C13D412yes—
22C23×C4×C13C2311yescharacteristic
23C23×Dic13—21no—
24C23×Dic13—21no—
25C22×Dic26—22no—
26C22×Dic26—22no—
27C22×Dic26—21no—
28C22×Dic26—44no—
29C22×Dic26D412yes—
30C13×C22⋊Q8—22no—
31C13×C22⋊Q8—22no—
32C13×C22⋊Q8—22no—
33C13×C22⋊Q8—22no—
34C2×C13×C4⋊C4—22no—
35C2×C13×C4⋊C4—22no—
36C2×C13×C4⋊C4—22no—
37C2×C13×C4⋊C4D412yes—
38C2×C13×C22⋊C4—22no—
39C2×#6107—28no—
40C2×#6107—48no—
41C22×#1055—22no—
42C13×#95—24no—
43C13×#95—22no—
44#88071—22no—
45#88071—44no—
46C22⋊#1055—22no—
47C22⋊#1055D414yes—
48#88050—28no—
49#9967—131nomaximal, Sylow
50C24×C13C2×D411yescharacteristic
51C2×Q8×C13—22no—
52C2×Q8×C13—22no—
53C2×Q8×C13—44no—
54C22×C4×C13—22no—
55C22×C4×C13—22no—
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13—42no—
59C22×C4×C13—42no—
60C22×C4×C13—42no—
61C22×C4×C13Q1614yescomplemented
62C22×C4×C13C2×D411yescharacteristic
63C22×C4×C13C2×D412yes—
64C22×C4×C13C2×D412yes—
65C22×C4×C13#4111yescharacteristic
66C22×Dic13—24no—
67C22×Dic13—22no—
68C22×Dic13—22no—
69C22×Dic13—42no—
70C22×Dic13—41no—
71C22×Dic13—41no—
72C22×Dic13—42no—
73C22×Dic13—41no—
74C22×Dic13—41no—
75C22×Dic13—42no—
76C2×Dic26—28no—
77C2×Dic26—24no—
78C2×Dic26—44no—
79C2×Dic26—48no—
80C2×Dic26—44no—
81C2×Dic26—44no—
82C2×Dic26—42no—
83C13×C4⋊C4—22no—
84C13×C4⋊C4—22no—
85C13×C4⋊C4—42no—
86C13×C4⋊C4—42no—
87C13×C4⋊C4—82no—
88C13×C4⋊C4—82no—
89C13×C22⋊C4—42no—
90C13×C22⋊C4—42no—
91#6107—416no—
92C2×#1055—24no—
93C2×#1055—44no—
94C2×#451—132no—
95#2659D1311yescharacteristic
96C2×#328—132no—
97#1446—131no—
98#1280—131no—
99C23×C13—21no—
100C23×C13—21no—
101C23×C13—22no—
102C23×C13#12011yescharacteristic
103C23×C13#12111yescharacteristic
104C23×C13#12711yescharacteristic
105C23×C13C2×Q1612yes—
106C23×C13#13712yes—
107Q8×C13—44no—
108Q8×C13—82no—
109C2×C4×C13—24no—
110C2×C4×C13—24no—
111C2×C4×C13—41no—
112C2×C4×C13—42no—
113C2×C4×C13—41no—
114C2×C4×C13—42no—
115C2×C4×C13—42no—
116C2×C4×C13—44no—
117C2×C4×C13—42no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—44no—
122C2×C4×C13—82no—
123C2×C4×C13#12012yes—
124C2×C4×C13#12112yes—
125C2×Dic13—24no—
126C2×Dic13—42no—
127C2×Dic13—44no—
128C2×Dic13—42no—
129C2×Dic13—44no—
130C2×Dic13—44no—
131C2×Dic13—42no—
132C2×Dic13—82no—
133C2×Dic13—81no—
134C2×Dic13—81no—
135Dic26—28no—
136Dic26—44no—
137Dic26—48no—
138#1055—44no—
139C23×Q8—131no—
140C22×Q16—262no—
141C2×C22⋊Q8C2×D1312yes—
142#451—264no—
143#451—264no—
144C2×#103—262no—
145C2×C22⋊C8—132no—
146#398—22no—
147#395—22no—
148#380—22no—
149C2×#95—131no—
150C2×#95—262no—
151C2×#95C2×D1311yescharacteristic
152#328—264no—
153C22×C13—22no—
154C22×C13—22no—
155C22×C13—22no—
156C22×C13—22no—
157C22×C13—21no—
158C22×C13—21no—
159C22×C13—21no—
160C22×C13—21no—
161C22×C13#39211yescharacteristic
162C22×C13#45112yes—
163C22×C13#45511yescharacteristic
164C22×C13#45611yescharacteristic
165C22×C13B(4,𝔽2)12yes—
166C22×C13#46212yes—
167C22×C13#49411yescharacteristic
168C22×C13#49711yescharacteristic
169C4×C13—22no—
170C4×C13—22no—
171C4×C13—44no—
172C4×C13—82no—
173C4×C13—82no—
174Dic13—44no—
175Dic13—82no—
176C22×Q8—132no—
177C22×Q8—261no—
178C22×Q8—262no—
179C22×Q8—262no—
180C22×Q8—524no—
181C22×Q8C22⋊D1312yes—
182C23×C4—261no—
183C23×C4—261no—
184C23×C4C22⋊D1312yes—
185C23×C4C22×D1311yescharacteristic
186C2×Q16—268no—
187C2×Q16—528no—
188C22×C8—262no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4C22⋊D1312yes—
197C2×C22⋊C4—22no—
198#103—268no—
199C22⋊C8—134no—
200C22⋊C8—262no—
201#95—24no—
202#95—22no—
203#95—262no—
204#95—524no—
205C2×C13—21no—
206C2×C13—22no—
207C2×C13—21no—
208C2×C13#160212yes—
209C2×C13#160512yes—
210C2×C13#200711yescharacteristic
211C2×C13#201611yescharacteristic
212C2×C13#202111yescharacteristic
213C24C2×C22⋊D1311yescharacteristic
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—44no—
217C2×Q8—268no—
218C2×Q8—264no—
219C2×Q8—524no—
220C2×Q8—524no—
221C2×Q8—524no—
222C2×Q8—528no—
223C2×Q8—522no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—264no—
232C22×C4—262no—
233C22×C4—262no—
234C22×C4—522no—
235C22×C4—521no—
236C22×C4—521no—
237C22×C4—522no—
238C22×C4—521no—
239C22×C4—521no—
240C22×C4—522no—
241C22×C4#610714yescomplemented
242C22×C4D4×D1311yescharacteristic
243C22×C4D4×D1312yes—
244C22×C4#612911yescharacteristic
245C22×C4C2×C22⋊D1312yes—
246Q16—5216no—
247C2×C8—264no—
248C2×C8—524no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C13#996711yescharacteristic, complemented, Sylow
258C23—21no—
259C23—22no—
260C23—21no—
261C23#8819211yescharacteristic
262C23#8819411yescharacteristic
263C23#8819612yes—
264C23C2×#610712yes—
265C23#8820711yescharacteristic
266Q8—44no—
267Q8—82no—
268Q8—268no—
269Q8—524no—
270Q8—528no—
271C2×C4—24no—
272C2×C4—24no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—44no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—42no—
284C2×C4—82no—
285C2×C4—264no—
286C2×C4—522no—
287C2×C4—524no—
288C2×C4—522no—
289C2×C4—524no—
290C2×C4—524no—
291C2×C4—522no—
292C2×C4—1041no—
293C2×C4—1042no—
294C2×C4—1041no—
295C2×C4#8819112yes—
296C2×C4#8819312yes—
297C8—524no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22#1171874711yescharacteristic
307C22#1171875112yes—
308C22#1171875311yescharacteristic
309C22#1171888411yescharacteristic
310C22#1171888912yes—
311C22#1171890311yescharacteristic
312C22#1171890512yes—
313C22#1171890711yescharacteristic
314C4—22no—
315C4—22no—
316C4—44no—
317C4—82no—
318C4—82no—
319C4—524no—
320C4—1042no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2#1311379312yes—
325C2#1311381012yes—
326C2#1311588311yescharacteristic
327C2#1311593611yescharacteristic
328C2#1311596011yescharacteristic
subgroupG/H
centerC23#88207
commutator, cosocleC23×C4×C13C23
FrattiniC23×C4C22×D13
FittingC13×#2659C2
socleC23×C13#120
prime radicalC24×C13C2×D4
hypercenter#2659D13
hypocenterC13#9967
LCSG ▷ C23×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
61. C22×C4×C13Q16semidirect
241. C22×C4#6107semidirect
257. C13#9967semidirect

Maximal subgroups

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

subgroupindex
C2×#117187112
C2×#117181302
#131131062
#131126792
C13×#26592
#996713