Finite Group Databasebeta

← #15837768

Subgroups — #15837768

Order 3328.

998 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 207 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#13125774C212yesmaximal
2#13119835C211yesmaximal, characteristic
3#13113776C218yesmaximal, complemented
4#13113335C212yesmaximal, complemented
5#13113160C211yesmaximal, characteristic, complemented
6C13×#3502C211yesmaximal, characteristic, complemented
7Q82×C13—22no—
8C13×#557—22no—
9C13×#557—24no—
10C13×#544—21no—
11C13×#544C2214yes—
12C13×#533—22no—
13C13×#533C2212yes—
14C13×#519—21no—
15C2×C4×Q8×C13C2211yescharacteristic
16C2×#88201—21no—
17C2×#88128C2211yescharacteristic
18#11718749—22no—
19#11718746—22no—
20#11718707C2218yes—
21#11718688—24no—
22#11718652C2214yes—
23#11718649—21no—
24#11718600—22no—
25C4×C4⋊Dic13C2211yescharacteristic
26#11718586C2212yes—
27#11718130C2218yes—
28#11718120C2214yes—
29C22×Q8×C13D411yescharacteristic
30C22×D52—21no—
31C22×D52—21no—
32C22×D52C2312yes—
33C13×C4⋊Q8—22no—
34C13×C4⋊Q8—22no—
35C13×C4⋊Q8—24no—
36C13×C4⋊Q8—28no—
37C13×C4⋊Q8D412yes—
38C13×C4⋊Q8D412yes—
39C13×#125—24no—
40C13×#125—24no—
41C13×#125—28no—
42C13×#125—22no—
43C13×C22⋊Q8—22no—
44C13×C22⋊Q8—24no—
45C13×C22⋊Q8C2314yes—
46C4×Q8×C13—24no—
47C4×Q8×C13—24no—
48C4×Q8×C13—24no—
49C4×Q8×C13—24no—
50C4×Q8×C13—22no—
51C4×Q8×C13—22no—
52C4×Q8×C13—22no—
53C4×Q8×C13D414yes—
54C13×#117—24no—
55C13×#117—22no—
56C2×C13×C4⋊C4D411yescharacteristic
57C2×C13×C4⋊C4C2312yes—
58C2×C42×C13D412yes—
59C2×C42×C13C2311yescharacteristic
60#88201—44no—
61C2×#6106—28no—
62C2×#6103—24no—
63C2×#6103—24no—
64C2×C4⋊Dic13—21no—
65C2×C4×Dic13—21no—
66#88128—22no—
67#88128—22no—
68#88071C2312yes—
69C22⋊#1055C2314yes—
70#88049—28no—
71C4⋊#1055—22no—
72C4×#1055—22no—
73#18403—131nomaximal, Sylow
74C23×D13—22no—
75C2×Q8×C13—24no—
76C2×Q8×C13—22no—
77C2×Q8×C13—22no—
78C2×Q8×C13—22no—
79C2×Q8×C13SD1614yescomplemented
80C2×Q8×C13C2×D414yes—
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13C2×D412yes—
83C22×C4×C13C2×D412yes—
84C22×C4×C13C2×D411yescharacteristic
85C22×C4×C13C2411yescharacteristic
86C22×Dic13—22no—
87C2×D52—24no—
88C2×D52—44no—
89C2×D52—41no—
90C2×D52—42no—
91C2×D52—41no—
92C2×D52—42no—
93C2×D52—42no—
94C2×D52—82no—
95C13×C4⋊C4—24no—
96C13×C4⋊C4—24no—
97C13×C4⋊C4—24no—
98C13×C4⋊C4—28no—
99C13×C4⋊C4—22no—
100C13×C4⋊C4—22no—
101C13×C4⋊C4—22no—
102C13×C4⋊C4—22no—
103C13×C4⋊C4—24no—
104C13×C4⋊C4—24no—
105C13×C4⋊C4—24no—
106C13×C4⋊C4C2×D414yes—
107C13×C22⋊C4—24no—
108C42×C13—28no—
109C42×C13—22no—
110C42×C13—22no—
111C42×C13—24no—
112C42×C13C2×D412yes—
113C42×C13C2×D412yes—
114#6106—88no—
115#6103—48no—
116#6103—84no—
117C4⋊Dic13—42no—
118C4×Dic13—42no—
119C2×#1055—24no—
120#3502D1311yescharacteristic, complemented
121#3208—132no—
122#2548—131no—
123#1599—138no—
124#1503—132no—
125#1461—131no—
126C23×C13C22×D411yescharacteristic
127C22×D13—44no—
128C22×D13—44no—
129C22×D13—42no—
130C22×D13—42no—
131C22×D13—42no—
132Q8×C13—28no—
133Q8×C13—44no—
134Q8×C13—44no—
135Q8×C13—84no—
136C2×C4×C13—22no—
137C2×C4×C13—22no—
138C2×C4×C13—22no—
139C2×C4×C13—24no—
140C2×C4×C13—24no—
141C2×C4×C13—22no—
142C2×C4×C13—41no—
143C2×C4×C13—42no—
144C2×C4×C13#12011yescharacteristic
145C2×C4×C13#12012yes—
146C2×C4×C13#12011yescharacteristic
147C2×C4×C13C2×SD1614yes—
148C2×C4×C13C2×SD1612yes—
149C2×C4×C13#13712yes—
150C2×C4×C13C22×D412yes—
151C2×Dic13—42no—
152C2×Dic13—42no—
153C2×Dic13—42no—
154D52—48no—
155D52—82no—
156D52—84no—
157D52—82no—
158#1055—44no—
159Q82—22no—
160#557—22no—
161#557—24no—
162#544—21no—
163#544C2×D1314yes—
164#533—22no—
165#533C2×D1312yes—
166C2×#127—131no—
167C2×#124—261no—
168#519—21no—
169C2×C4×Q8C2×D1311yescharacteristic
170#492—262no—
171#486—262no—
172#460—264no—
173#450—138no—
174#431—132no—
175#394—134no—
176#390—261no—
177#390—262no—
178C4×C4⋊C4—131no—
179#328—138no—
180#324—134no—
181C22×C13—21no—
182C22×C13—21no—
183C22×C13C2×#12011yescharacteristic
184C22×C13C22×SD1611yescharacteristic
185C22×C13C2×#13711yescharacteristic
186C2×D13—84no—
187C2×D13—84no—
188C2×D13—82no—
189C2×D13—82no—
190C2×D13—82no—
191C4×C13—24no—
192C4×C13—21no—
193C4×C13—22no—
194C4×C13—44no—
195C4×C13—42no—
196C4×C13—44no—
197C4×C13#44814yes—
198C4×C13#45412yes—
199Dic13—82no—
200C22×Q8C22⋊D1311yescharacteristic
201C22×D4—132no—
202C22×D4—261no—
203C22×D4—261no—
204C2×SD16—268no—
205C4⋊Q8—22no—
206C4⋊Q8—22no—
207C4⋊Q8—24no—
208C4⋊Q8—28no—
209C4⋊Q8C22⋊D1312yes—
210C4⋊Q8C22⋊D1312yes—
211#127—262no—
212#127—262no—
213#125—24no—
214#125—24no—
215#125—28no—
216#125—22no—
217#124—524no—
218C22⋊Q8—22no—
219C22⋊Q8—24no—
220C22⋊Q8C22×D1314yes—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8C22⋊D1314yes—
229#117—24no—
230#117—22no—
231C2×C4⋊C4—261no—
232C2×C4⋊C4C22⋊D1311yescharacteristic
233C2×C4⋊C4C22×D1312yes—
234C2×C22⋊C4—264no—
235C2×C22⋊C4—264no—
236C2×C42—261no—
237C2×C42C22⋊D1312yes—
238C2×C42C22×D1311yescharacteristic
239C4⋊C8—262no—
240#102—268no—
241C22⋊C8—134no—
242C4×C8—262no—
243#95—132no—
244C2×C13—21no—
245C2×C13—21no—
246C2×C13C2×#44811yescharacteristic
247C2×C13C2×#45411yescharacteristic
248C2×C13#302711yescharacteristic
249D13—162no—
250C24—262no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8#610614yescomplemented
256C2×Q8C2×C22⋊D1314yes—
257C2×D4—264no—
258C2×D4—524no—
259C2×D4—522no—
260C2×D4—522no—
261C2×D4—521no—
262C2×D4—522no—
263C2×D4—521no—
264C2×D4—1042no—
265C22×C4—262no—
266C22×C4C2×C22⋊D1312yes—
267C22×C4C2×C22⋊D1311yescharacteristic
268C22×C4C2×C22⋊D1312yes—
269C22×C4C2×C22⋊D1311yescharacteristic
270C22×C4C23×D1311yescharacteristic
271SD16—1048no—
272C2×C8—264no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—28no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—522no—
285C4⋊C4C2×C22⋊D1314yes—
286C22⋊C4—24no—
287C22⋊C4—528no—
288C22⋊C4—1044no—
289C42—28no—
290C42—22no—
291C42—22no—
292C42—24no—
293C42—522no—
294C42C2×C22⋊D1312yes—
295C42C2×C22⋊D1312yes—
296C13#1840311yescharacteristic, complemented, Sylow
297C23—524no—
298C23—524no—
299C23—522no—
300C23—522no—
301C23—522no—
302C23C22×C22⋊D1311yescharacteristic
303Q8—28no—
304Q8—44no—
305Q8—44no—
306Q8—84no—
307D4—528no—
308D4—1042no—
309D4—1044no—
310D4—1042no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—522no—
320C2×C4—522no—
321C2×C4—522no—
322C2×C4C2×#610614yes—
323C2×C4C2×#610612yes—
324C2×C4#8819612yes—
325C2×C4#8820711yescharacteristic
326C2×C4#8820712yes—
327C2×C4#8820711yescharacteristic
328C2×C4C22×C22⋊D1312yes—
329C8—524no—
330C22—21no—
331C22—21no—
332C22—1044no—
333C22—1044no—
334C22—1042no—
335C22—1042no—
336C22—1042no—
337C22C22×#610611yescharacteristic
338C22C2×#8819611yescharacteristic
339C22C2×#8820711yescharacteristic
340C4—21no—
341C4—22no—
342C4—24no—
343C4—44no—
344C4—42no—
345C4—44no—
346C4—1042no—
347C4#1171888814yes—
348C4#1171890612yes—
349C2—21no—
350C2—21no—
351C2—2082no—
352C2C2×#1171888811yescharacteristic
353C2C2×#1171890611yescharacteristic
354C2#1312423811yescharacteristic
subgroupG/H
centerC22C2×#88207
commutator, cosocleC22×C4×C13C24
FrattiniC22×C4C23×D13
FittingC13×#3502C2
socleC22×C13C2×#120
prime radicalC2×#88128C22
hypercenter#3502D13
hypocenterC13#18403
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
3. #13113776C2semidirect
4. #13113335C2semidirect
5. #13113160C2semidirect
6. C13×#3502C2semidirect
79. C2×Q8×C13SD16semidirect
120. #3502D13semidirect
255. C2×Q8#6106semidirect
296. C13#18403semidirect

Maximal subgroups

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

subgroupindex
#131257742
#131198352
#131137762
#131133352
#131131602
C13×#35022
#1840313