Finite Group Databasebeta

← #16778706

Subgroups — #16778706

Order 3328.

444 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 90 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#13130553C211yesmaximal, characteristic, complemented
2#13130550C211yesmaximal, characteristic, complemented
3#13129654C212yesmaximal, complemented
4#13129653C212yesmaximal
5#13123204C211yesmaximal, characteristic
6C2×#88152—21no—
7C2×#88110—21no—
8C2×C4×#1057—21no—
9C13×#389C411yescharacteristic
10#11718651—21no—
11#11718649C411yescharacteristic
12#11718649C2211yescharacteristic
13#11718646C2211yescharacteristic
14#11718634C411yescharacteristic
15#11718600C411yescharacteristic
16C42×Dic13C2211yescharacteristic
17#11718331—21no—
18#11718330—21no—
19#11718290—21no—
20#11718290—21no—
21#11718290C2212yes—
22#11718290C2212yes—
23C22×D52C2×C411yescharacteristic
24C22×#1057—22no—
25C22×#1057—22no—
26C2×C13×C4⋊C4—21no—
27C2×C13×C4⋊C4C2×C411yescharacteristic
28C2×C13×C4⋊C4C2×C411yescharacteristic
29C2×C42×C13C2×C411yescharacteristic
30C2×#6103—21no—
31C2×#6103—21no—
32C2×#6103—21no—
33C2×#6103—21no—
34C2×#6103—21no—
35C2×#6103—21no—
36C2×C4⋊Dic13—21no—
37C2×C4⋊Dic13C2×C411yescharacteristic
38C2×#6101—21no—
39C2×C4×Dic13C2×C411yescharacteristic
40C2×C4×Dic13D411yescharacteristic
41C2×C4×Dic13D411yescharacteristic
42C2×C4×Dic13D411yescharacteristic
43C2×C4×Dic13D411yescharacteristic
44C2×C4×Dic13C2311yescharacteristic
45#88152—44no—
46#88110—24no—
47C4×#1057—24no—
48C13×#95—21no—
49#88071—21no—
50#88071—21no—
51#7692—131nomaximal, Sylow
52C23×D13—21no—
53C22×C4×C13—21no—
54C22×C4×C13—21no—
55C22×C4×C13C22⋊C411yescharacteristic
56C22×C4×C13C22⋊C411yescharacteristic
57C22×C4×C13C22×C411yescharacteristic
58C22×Dic13—21no—
59C22×Dic13C22⋊C411yescharacteristic
60C22×Dic13C22⋊C411yescharacteristic
61C22×Dic13C2×D411yescharacteristic
62C22×Dic13C2×D411yescharacteristic
63C2×D52—22no—
64C2×D52—41no—
65C2×D52—41no—
66C2×#1057—24no—
67C2×#1057—24no—
68C2×#1057—42no—
69C2×#1057—42no—
70C2×#1057—42no—
71C2×#1057—42no—
72C13×C4⋊C4—22no—
73C13×C4⋊C4—41no—
74C13×C4⋊C4—42no—
75C42×C13—22no—
76#6103—42no—
77#6103—42no—
78#6103—42no—
79#6103—42no—
80#6103—42no—
81#6103—42no—
82C4⋊Dic13—42no—
83C4⋊Dic13—41no—
84#6101—42no—
85C4×Dic13—22no—
86C4×Dic13—22no—
87C4×Dic13—22no—
88C4×Dic13—22no—
89C4×Dic13—41no—
90C4×Dic13—41no—
91#2860—131no—
92#1745—131no—
93#1744—131no—
94#1271—132no—
95#1270—132no—
96C23×C13C2×C22⋊C411yescharacteristic
97C22×D13—41no—
98C22×D13—41no—
99C22×D13—41no—
100C22×D13—41no—
101C22×D13—41no—
102C22×D13—41no—
103C22×D13—41no—
104C2×C4×C13—21no—
105C2×C4×C13—21no—
106C2×C4×C13—21no—
107C2×C4×C13—21no—
108C2×C4×C13—21no—
109C2×C4×C13—21no—
110C2×C4×C13—21no—
111C2×C4×C13—41no—
112C2×C4×C13—41no—
113C2×C4×C13—41no—
114C2×C4×C13—41no—
115C2×C4×C13—41no—
116C2×C4×C13—41no—
117C2×C4×C13#10412yes—
118C2×C4×C13#13112yes—
119C2×Dic13—21no—
120C2×Dic13—21no—
121C2×Dic13—21no—
122C2×Dic13—21no—
123C2×Dic13—21no—
124C2×Dic13—21no—
125C2×Dic13—21no—
126C2×Dic13—21no—
127C2×Dic13—21no—
128C2×Dic13—41no—
129C2×Dic13—41no—
130C2×Dic13—41no—
131C2×Dic13#10412yes—
132C2×Dic13#13512yes—
133C2×Dic13AGL(1,ℤ/8)12yes—
134D52—82no—
135#1057—44no—
136#1057—44no—
137C2×#126—261no—
138C2×#97—261no—
139C2×C4×C8—261no—
140#390—131no—
141#390—131no—
142#390—131no—
143#389—131no—
144#389—131no—
145#389#26111yescharacteristic
146#388—261no—
147#388—261no—
148#388—261no—
149C43—131no—
150#336—132no—
151#336—132no—
152#336—261no—
153#336—261no—
154C22×C13#40811yescharacteristic
155C22×C13#41011yescharacteristic
156C22×C13#41311yescharacteristic
157C22×C13#41611yescharacteristic
158C22×C13#41811yescharacteristic
159C22×C13C2×#10411yescharacteristic
160C22×C13C2×#10411yescharacteristic
161C2×D13—81no—
162C2×D13—81no—
163C2×D13—81no—
164C2×D13—81no—
165C2×D13—81no—
166C2×D13—81no—
167C2×D13—81no—
168C4×C13—22no—
169C4×C13—22no—
170C4×C13—41no—
171C4×C13—81no—
172C4×C13—81no—
173Dic13—22no—
174Dic13—22no—
175Dic13—22no—
176Dic13—41no—
177Dic13—81no—
178C22×D4—131no—
179C22×C8—262no—
180C22×C8—262no—
181#126—524no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—131no—
184C2×C4⋊C4—261no—
185C2×C4⋊C4—261no—
186C2×C4⋊C4C2×#26111yescharacteristic
187C2×C4⋊C4C2×#26111yescharacteristic
188C2×C22⋊C4—261no—
189C2×C22⋊C4—261no—
190C2×C22⋊C4—261no—
191C2×C22⋊C4—261no—
192C2×C22⋊C4—261no—
193C2×C22⋊C4—261no—
194C2×C42—131no—
195C2×C42—131no—
196C2×C42—131no—
197C2×C42—131no—
198C2×C42—131no—
199C2×C42—131no—
200C2×C42C2×#26111yescharacteristic
201#97—264no—
202C4×C8—264no—
203#95—21no—
204#95—261no—
205#95—261no—
206C2×C13#147311yescharacteristic
207C2×C13#148411yescharacteristic
208C2×C13#149411yescharacteristic
209C2×C13#150011yescharacteristic
210C2×C13#150511yescharacteristic
211C2×C13#152411yescharacteristic
212C2×C13#155311yescharacteristic
213D13—161no—
214C24—261no—
215C2×D4—262no—
216C2×D4—521no—
217C2×D4—521no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—131no—
221C22×C4—131no—
222C22×C4—131no—
223C22×C4—131no—
224C22×C4—261no—
225C22×C4C22⋊#26111yescharacteristic
226C22×C4C22⋊#26111yescharacteristic
227C22×C4C22×#26111yescharacteristic
228C2×C8—264no—
229C2×C8—264no—
230C2×C8—522no—
231C2×C8—522no—
232C2×C8—522no—
233C2×C8—522no—
234C4⋊C4—22no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—521no—
238C4⋊C4—522no—
239C4⋊C4—522no—
240C22⋊C4—522no—
241C22⋊C4—522no—
242C22⋊C4—522no—
243C22⋊C4—522no—
244C22⋊C4—522no—
245C22⋊C4—522no—
246C42—22no—
247C42—262no—
248C42—262no—
249C42—262no—
250C42—262no—
251C42—521no—
252C42—521no—
253C13#769211yescharacteristic, complemented, Sylow
254C23—521no—
255C23—521no—
256C23—521no—
257C23—521no—
258C23—521no—
259C23—521no—
260C23—521no—
261C23C2×C22⋊#26111yescharacteristic
262D4—1042no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—132no—
277C2×C4—132no—
278C2×C4—132no—
279C2×C4—261no—
280C2×C4—261no—
281C2×C4—261no—
282C2×C4—261no—
283C2×C4—261no—
284C2×C4—261no—
285C2×C4—261no—
286C2×C4—261no—
287C2×C4—261no—
288C2×C4—521no—
289C2×C4—521no—
290C2×C4—521no—
291C2×C4#8811812yes—
292C2×C4#8823812yes—
293C8—524no—
294C8—524no—
295C22—1041no—
296C22—1041no—
297C22—1041no—
298C22—1041no—
299C22—1041no—
300C22—1041no—
301C22—1041no—
302C22#1171913211yescharacteristic
303C22#1171913811yescharacteristic
304C22#1171914611yescharacteristic
305C22C2×#8811811yescharacteristic
306C22#1171915611yescharacteristic
307C22D13⋊#10411yescharacteristic
308C22#1171916511yescharacteristic
309C4—22no—
310C4—22no—
311C4—41no—
312C4—81no—
313C4—81no—
314C4—262no—
315C4—262no—
316C4—262no—
317C4—521no—
318C4—1041no—
319C2—2081no—
320C2#1313012011yescharacteristic
321C2#1313015511yescharacteristic
322C2#1313018911yescharacteristic
323C2#1313021011yescharacteristic
324C2#1313022711yescharacteristic
325C2#1313028811yescharacteristic
326C2#1313035911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC13×#389C4
socleC23×C13C2×C22⋊C4
prime radicalC22×D52C2×C4
cosocleC2×C4×Dic13C23
hypercenter#389#261
hypocenterC13#7692
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
1. #13130553C2semidirect
2. #13130550C2semidirect
3. #13129654C2semidirect
253. C13#7692semidirect

Maximal subgroups

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

subgroupindex
#131305532
#131305502
#131296542
#131296532
#131232042
#769213