Finite Group Databasebeta

← #15757923

Subgroups — #15757923

Order 3328.

836 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 131 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×#11718608C211yesmaximal, characteristic
2#13114770C212yesmaximal, complemented
3C2×#11718117C211yesmaximal, characteristic
4#13112714C212yesmaximal, complemented
5C13×#2312C211yesmaximal, characteristic
6C23×Dic26C2211yescharacteristic
7C22×C13×C22⋊C4C2211yescharacteristic
8C22×C22⋊Dic13—21no—
9C13×#388—24no—
10C13×#380C2212yes—
11C13×#379—21no—
12C4×C13×C22⋊C4—21no—
13C2×#88071—21no—
14C2×#88071C2211yescharacteristic
15C2×C22⋊#1055C2212yes—
16C2×#88050—24no—
17#11718608—22no—
18#11718608C414yescomplemented
19#11718117—22no—
20C25×C13D411yescharacteristic
21C23×C4×C13D411yescharacteristic
22C23×C4×C13C2311yescharacteristic
23C23×Dic13—21no—
24C23×Dic13—21no—
25C22×Dic26—22no—
26C22×Dic26—22no—
27C22×Dic26—21no—
28C22×Dic26—44no—
29C22×Dic26C2×C412yes—
30C2×C13×C4⋊C4D414yes—
31C2×C13×C22⋊C4—22no—
32C2×C13×C22⋊C4—22no—
33C2×C13×C22⋊C4—22no—
34C2×C13×C22⋊C4—22no—
35C2×C13×C22⋊C4—22no—
36C2×C13×C22⋊C4C2×C412yes—
37C2×C13×C22⋊C4D412yes—
38C2×C42×C13—22no—
39C2×C22⋊Dic13—24no—
40C2×C22⋊Dic13—44no—
41C22×#1055—22no—
42C13×#95—22no—
43C13×#95—22no—
44#88071—24no—
45#88071—21no—
46#88071C2×C412yes—
47C22⋊#1055—22no—
48C22⋊#1055—22no—
49#88050—48no—
50#9016—131nomaximal, Sylow
51C24×C13—21no—
52C24×C13—22no—
53C24×C13—21no—
54C24×C13—22no—
55C24×C13—42no—
56C24×C13—42no—
57C24×C13C22⋊C412yes—
58C24×C13C2×D411yescharacteristic
59C22×C4×C13—21no—
60C22×C4×C13—21no—
61C22×C4×C13—24no—
62C22×C4×C13—42no—
63C22×C4×C13—42no—
64C22×C4×C13C22⋊C412yes—
65C22×C4×C13C22×C411yescharacteristic
66C22×C4×C13C2×D411yescharacteristic
67C22×C4×C13C2×D412yes—
68C22×C4×C13#4112yes—
69C22×Dic13—24no—
70C22×Dic13—22no—
71C22×Dic13—22no—
72C22×Dic13—42no—
73C22×Dic13—41no—
74C22×Dic13—41no—
75C22×Dic13—42no—
76C22×Dic13—42no—
77C22×Dic13—42no—
78C2×Dic26—28no—
79C2×Dic26—24no—
80C2×Dic26—44no—
81C2×Dic26—48no—
82C2×Dic26—44no—
83C2×Dic26—44no—
84C2×Dic26—42no—
85C13×C4⋊C4—44no—
86C13×C22⋊C4—24no—
87C13×C22⋊C4—24no—
88C13×C22⋊C4—42no—
89C13×C22⋊C4—42no—
90C13×C22⋊C4—82no—
91C13×C22⋊C4—82no—
92C42×C13—82no—
93C22⋊Dic13—48no—
94C2×#1055—44no—
95C2×#1055—42no—
96C2×#393—131no—
97#2312D1311yescharacteristic
98#1787—132no—
99C2×#323—131no—
100#1294—132no—
101C23×C13—22no—
102C23×C13—22no—
103C23×C13—22no—
104C23×C13—22no—
105C23×C13—22no—
106C23×C13—42no—
107C23×C13—42no—
108C23×C13—48no—
109C23×C13—42no—
110C23×C13—42no—
111C23×C13—42no—
112C23×C13—42no—
113C23×C13—81no—
114C23×C13—81no—
115C23×C13—84no—
116C23×C13C2×C22⋊C411yescharacteristic
117C23×C13#12011yescharacteristic
118C23×C13#12311yescharacteristic
119C23×C13#13512yes—
120C23×C13#13712yes—
121C2×C4×C13—24no—
122C2×C4×C13—22no—
123C2×C4×C13—41no—
124C2×C4×C13—41no—
125C2×C4×C13—42no—
126C2×C4×C13—42no—
127C2×C4×C13—42no—
128C2×C4×C13—44no—
129C2×C4×C13—44no—
130C2×C4×C13—44no—
131C2×C4×C13—81no—
132C2×C4×C13C4×D412yes—
133C2×C4×C13#12112yes—
134C2×Dic13—24no—
135C2×Dic13—42no—
136C2×Dic13—44no—
137C2×Dic13—42no—
138C2×Dic13—44no—
139C2×Dic13—42no—
140C2×Dic13—42no—
141C2×Dic13—42no—
142C2×Dic13—84no—
143Dic26—28no—
144Dic26—48no—
145Dic26—44no—
146#1055—82no—
147C23×Q8—131no—
148C22×C22⋊C4—261no—
149C22×C22⋊C4C2×D1311yescharacteristic
150C2×#103—264no—
151C2×C22⋊C8—132no—
152#393—134no—
153#393—262no—
154#388—24no—
155#380C2×D1312yes—
156#379—21no—
157C4×C22⋊C4—21no—
158C2×#95—131no—
159C2×#95—261no—
160#323—262no—
161C22×C13—24no—
162C22×C13—24no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—21no—
167C22×C13—44no—
168C22×C13—42no—
169C22×C13—42no—
170C22×C13—42no—
171C22×C13—82no—
172C22×C13—84no—
173C22×C13—84no—
174C22×C13#38611yescharacteristic
175C22×C13#41611yescharacteristic
176C22×C13#41911yescharacteristic
177C22×C13#45212yes—
178C22×C13#45512yes—
179C22×C13B(4,𝔽2)12yes—
180C22×C13#48512yes—
181C4×C13—22no—
182C4×C13—22no—
183C4×C13—42no—
184C4×C13—84no—
185Dic13—44no—
186Dic13—82no—
187C25C22⋊D1311yescharacteristic
188C22×Q8—132no—
189C22×Q8—261no—
190C22×Q8—262no—
191C22×Q8—262no—
192C22×Q8—524no—
193C23×C4—261no—
194C23×C4—261no—
195C23×C4C22⋊D1311yescharacteristic
196C23×C4C22×D1311yescharacteristic
197C22×C8—262no—
198C2×C4⋊C4C22⋊D1314yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—264no—
205C2×C22⋊C4—524no—
206C2×C22⋊C4C4×D1312yes—
207C2×C22⋊C4D5212yes—
208C2×C42—22no—
209#103—528no—
210C22⋊C8—262no—
211C22⋊C8—262no—
212#95—22no—
213#95—22no—
214#95—132no—
215#95—264no—
216#95—261no—
217C2×C13—21no—
218C2×C13—21no—
219C2×C13—22no—
220C2×C13—82no—
221C2×C13#160614yes—
222C2×C13#187311yescharacteristic
223C2×C13#188511yescharacteristic
224C2×C13#188911yescharacteristic
225C24—22no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24#610312yes—
232C24C2×C22⋊D1311yescharacteristic
233C2×Q8—268no—
234C2×Q8—264no—
235C2×Q8—524no—
236C2×Q8—524no—
237C2×Q8—524no—
238C2×Q8—528no—
239C2×Q8—522no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—264no—
246C22×C4—262no—
247C22×C4—262no—
248C22×C4—522no—
249C22×C4—521no—
250C22×C4—521no—
251C22×C4—522no—
252C22×C4—522no—
253C22×C4—522no—
254C22×C4#610312yes—
255C22×C4C2×C4×D1311yescharacteristic
256C22×C4C2×D5211yescharacteristic
257C22×C4#612712yes—
258C22×C4C2×C22⋊D1312yes—
259C2×C8—524no—
260C2×C8—522no—
261C4⋊C4—44no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—82no—
267C22⋊C4—82no—
268C22⋊C4—528no—
269C42—82no—
270C13#901611yescharacteristic, complemented, Sylow
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—42no—
277C23—42no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—81no—
284C23—81no—
285C23—84no—
286C23C2×#610311yescharacteristic
287C23#8818311yescharacteristic
288C23#8820412yes—
289C23#8820512yes—
290C23#8820711yescharacteristic
291Q8—268no—
292Q8—528no—
293Q8—524no—
294C2×C4—24no—
295C2×C4—22no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—81no—
305C2×C4—264no—
306C2×C4—522no—
307C2×C4—524no—
308C2×C4—522no—
309C2×C4—524no—
310C2×C4—522no—
311C2×C4—522no—
312C2×C4—522no—
313C2×C4—1044no—
314C2×C4C4×C22⋊D1312yes—
315C2×C4#8818412yes—
316C8—1042no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—24no—
322C22—24no—
323C22—44no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—82no—
328C22—84no—
329C22—84no—
330C22#1171865712yes—
331C22#1171866311yescharacteristic
332C22#1171866511yescharacteristic
333C22#1171887411yescharacteristic
334C22#1171890012yes—
335C22#1171890312yes—
336C22#1171890512yes—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—84no—
341C4—524no—
342C4—1042no—
343C2—21no—
344C2—21no—
345C2—22no—
346C2—82no—
347C2#1311381614yes—
348C2#1311514811yescharacteristic
349C2#1311522411yescharacteristic
350C2#1311524411yescharacteristic
subgroupG/H
centerC23#88207
commutatorC22×C4×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2312C2
socleC23×C13#120
prime radicalC25×C13D4
cosocleC23×C4×C13C23
hypercenter#2312D13
hypocenterC13#9016
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #13114770C2semidirect
4. #13112714C2semidirect
18. #11718608C4semidirect
270. C13#9016semidirect

Maximal subgroups

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

subgroupindex
C2×#117186082
#131147702
C2×#117181172
#131127142
C13×#23122
#901613