Finite Group Databasebeta

← #15757396

Subgroups — #15757396

Order 3328.

426 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123582C211yesmaximal, characteristic
2#13115611C211yesmaximal, characteristic, complemented
3#13113611C211yesmaximal, characteristic, complemented
4#13113483C211yesmaximal, characteristic
5#13113288C211yesmaximal, characteristic
6#13112671C211yesmaximal, characteristic, complemented
7C13×#1278C211yesmaximal, characteristic, complemented
8C2×#88161—21no—
9#11719182C2211yescharacteristic
10C13×#431C2211yescharacteristic
11C4×C13×M16C2211yescharacteristic
12C2×C4×C8×C13C2211yescharacteristic
13#11718790—21no—
14C2×#88062—21no—
15#11718585C2211yescharacteristic
16C2×C4⋊#1055C2211yescharacteristic
17C2×C4⋊#1055C2211yescharacteristic
18#11718367—22no—
19#11718349—21no—
20C8×D52—21no—
21#11718150—21no—
22#11718078C411yescharacteristic
23#11718078C411yescharacteristic
24#11718077C411yescharacteristic
25#11718077C411yescharacteristic
26#11718051—22no—
27#11718050—21no—
28#11718050—21no—
29C2×#6127—21no—
30C2×C13×M16D411yescharacteristic
31C2×C13×M16D411yescharacteristic
32C22×C8×C13—21no—
33C2×C42×C13C2311yescharacteristic
34#88184—21no—
35#88184Q812yes—
36C2×#6099—21no—
37C22×#1055—21no—
38C22×#1055—21no—
39#88161—42no—
40#88161—42no—
41C2×#6094—22no—
42C2×C8×D13—22no—
43#88129—21no—
44#88128C2×C411yescharacteristic
45C4×D52—21no—
46C4×D52D412yes—
47#88123C2×C411yescharacteristic
48C13×C4⋊C8C2×C411yescharacteristic
49C13×C4⋊C8C2×C411yescharacteristic
50C13×C22⋊C8—21no—
51C13×#97—21no—
52C4×C8×C13—21no—
53C4×C8×C13—21no—
54C4×C8×C13—21no—
55C4×C8×C13—21no—
56#88062—42no—
57#88060—22no—
58#88060—22no—
59C4⋊#1055—21no—
60C4⋊#1055—21no—
61C4⋊#1055—21no—
62C4⋊#1055—21no—
63C4⋊#1055C2×C411yescharacteristic
64C4⋊#1055C2×C411yescharacteristic
65C4⋊#1055D411yescharacteristic
66C4⋊#1055D411yescharacteristic
67#8930—131nomaximal, Sylow
68C22×C4×C13C2×D411yescharacteristic
69C22×C4×C13C2×Q811yescharacteristic
70C22×C4×C13#4111yescharacteristic
71C2×C22⋊D13—22no—
72#6127—41no—
73#6127—41no—
74#6127—41no—
75#6127—41no—
76C2×D52—21no—
77C2×D52—21no—
78C2×D52C4⋊C412yes—
79C2×C4×D13—22no—
80C2×Dic26—21no—
81C13×M16—21no—
82C13×M16—21no—
83C13×M16—21no—
84C13×M16D812yes—
85C2×C8×C13—21no—
86C2×C8×C13—22no—
87C2×C8×C13—21no—
88C2×C8×C13—21no—
89C2×C8×C13—21no—
90C2×C8×C13—41no—
91C2×C8×C13—41no—
92C42×C13C22×C411yescharacteristic
93C42×C13C2×D411yescharacteristic
94C42×C13C2×D411yescharacteristic
95C42×C13#4111yescharacteristic
96#6103—22no—
97#6103—22no—
98C4⋊Dic13—21no—
99C4⋊Dic13C4⋊C412yes—
100#6099—41no—
101#6099—41no—
102C2×#1055—21no—
103C2×#1055—21no—
104C2×#1055—21no—
105C2×#1055—21no—
106C2×#1055—41no—
107C2×#1055—41no—
108C2×#1055—41no—
109C2×#1055—41no—
110C2×#1055—41no—
111#6094—44no—
112C8×D13—44no—
113#2935—131no—
114#1950—131no—
115#1571—131no—
116#1540—131no—
117#1492—131no—
118#1278D1311yescharacteristic, complemented
119#1277—131no—
120C23×C13C22⋊Q811yescharacteristic
121C22×D13—22no—
122C2×C4×C13—21no—
123C2×C4×C13—21no—
124C2×C4×C13—41no—
125C2×C4×C13C2×C4⋊C411yescharacteristic
126C2×C4×C13C4×D411yescharacteristic
127C2×C4×C13C4×D411yescharacteristic
128C2×C4×C13#12011yescharacteristic
129C2×C4×C13C22⋊Q811yescharacteristic
130C2×C4×C13#12311yescharacteristic
131C2×C4×C13C2×D811yescharacteristic
132C2×C4×C13#13511yescharacteristic
133C2×C4×C13AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C13#13711yescharacteristic
135C22⋊D13—42no—
136C22⋊D13—42no—
137C2×Dic13—22no—
138D52—42no—
139D52—41no—
140D52—41no—
141D52—41no—
142D52—41no—
143C4×D13—42no—
144C4×D13—42no—
145Dic26—41no—
146Dic26—41no—
147C8×C13—22no—
148C8×C13—42no—
149C8×C13—41no—
150C8×C13—41no—
151C8×C13—41no—
152#1055—41no—
153#1055—41no—
154#1055—41no—
155#1055—41no—
156#1055—81no—
157C2×#131—261no—
158#532—131no—
159#436—261no—
160#435—262no—
161C8×D4—261no—
162#431C2×D1311yescharacteristic
163C2×#108—261no—
164#423—131no—
165C2×C4⋊C8—131no—
166C2×C4⋊C8—131no—
167#408—261no—
168C4×M16C2×D1311yescharacteristic
169C2×C4×C8C2×D1311yescharacteristic
170#341—261no—
171#335—261no—
172#335—261no—
173#332—131no—
174#332—131no—
175#331—131no—
176#331—131no—
177D4⋊C8—262no—
178C22×C13—21no—
179C22×C13—21no—
180C22×C13#38511yescharacteristic
181C22×C13#47411yescharacteristic
182C22×C13#47811yescharacteristic
183C2×D13—42no—
184C2×D13—42no—
185C2×D13—42no—
186C4×C13—21no—
187C4×C13—21no—
188C4×C13—21no—
189C4×C13C4×D811yescharacteristic
190C4×C13#44011yescharacteristic
191C4×C13#44311yescharacteristic
192C4×C13#44411yescharacteristic
193C4×C13#44711yescharacteristic
194C4×C13#45211yescharacteristic
195Dic13—42no—
196C2×#41—261no—
197#131—522no—
198#131—522no—
199C2×M16—262no—
200C2×M16—261no—
201C2×M16D5211yescharacteristic
202C2×M16D5211yescharacteristic
203C22×C8—21no—
204C22×C8—262no—
205C22×C8—261no—
206C22×C8—261no—
207C4⋊Q8—131no—
208#127—131no—
209#124—261no—
210#121—132no—
211#121—261no—
212C4×D4—132no—
213C4×D4—261no—
214C2×C42C22×D1311yescharacteristic
215#108—522no—
216C4⋊C8—131no—
217C4⋊C8—131no—
218C4⋊C8—131no—
219C4⋊C8—131no—
220C4⋊C8—261no—
221C4⋊C8—261no—
222C4⋊C8—261no—
223C4⋊C8—261no—
224C4⋊C8C4×D1311yescharacteristic
225C4⋊C8C4×D1311yescharacteristic
226C22⋊C8—21no—
227C22⋊C8—262no—
228C22⋊C8—262no—
229#97—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C2×C13—21no—
235C2×C13—21no—
236C2×C13#185811yescharacteristic
237C2×C13#186211yescharacteristic
238C2×C13#186611yescharacteristic
239D13—82no—
240#41—521no—
241#41—521no—
242#41—521no—
243#41—521no—
244C2×Q8—261no—
245C2×D4—132no—
246C2×D4—262no—
247C2×D4—261no—
248C2×D4—261no—
249C22×C4—262no—
250C22×C4C2×D5211yescharacteristic
251C22×C4#612911yescharacteristic
252C22×C4Q8×D1311yescharacteristic
253M16—21no—
254M16—21no—
255M16—21no—
256M16—524no—
257M16—521no—
258M16—521no—
259M16D10412yes—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—261no—
268C2×C8—261no—
269C2×C8—261no—
270C2×C8—261no—
271C2×C8—524no—
272C2×C8—521no—
273C2×C8—521no—
274C2×C8—521no—
275C2×C8—521no—
276C2×C8—521no—
277C4⋊C4—132no—
278C4⋊C4—261no—
279C22⋊C4—262no—
280C22⋊C4—262no—
281C42C2×C4×D1311yescharacteristic
282C42#612711yescharacteristic
283C42D4×D1311yescharacteristic
284C42D4×D1311yescharacteristic
285C13#893011yescharacteristic, complemented, Sylow
286C23—262no—
287C23#8815111yescharacteristic
288Q8—521no—
289Q8—521no—
290D4—522no—
291D4—522no—
292D4—522no—
293D4—521no—
294D4—521no—
295D4—521no—
296D4—521no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—262no—
301C2×C4—522no—
302C2×C4—522no—
303C2×C4C4×D5211yescharacteristic
304C2×C4#8813511yescharacteristic
305C2×C4#8813611yescharacteristic
306C2×C4#8813711yescharacteristic
307C2×C4C4⋊C4×D1311yescharacteristic
308C2×C4#8815011yescharacteristic
309C2×C4C2×D10411yescharacteristic
310C2×C4#8815811yescharacteristic
311C2×C4#8816211yescharacteristic
312C2×C4#8816311yescharacteristic
313C8—22no—
314C8—42no—
315C8—41no—
316C8—41no—
317C8—41no—
318C8—521no—
319C8—521no—
320C8—521no—
321C8—521no—
322C8—1041no—
323C22—21no—
324C22—21no—
325C22—522no—
326C22—522no—
327C22—522no—
328C22#1171832911yescharacteristic
329C22#1171850711yescharacteristic
330C22#1171850811yescharacteristic
331C4—21no—
332C4—21no—
333C4—21no—
334C4—522no—
335C4C4×D10411yescharacteristic
336C4#1171837111yescharacteristic
337C4#1171839311yescharacteristic
338C4#1171839411yescharacteristic
339C4#1171848411yescharacteristic
340C4#1171848511yescharacteristic
341C2—21no—
342C2—21no—
343C2—1042no—
344C2#1311505711yescharacteristic
345C2#1311508611yescharacteristic
346C2#1311511211yescharacteristic
subgroupG/H
centerC2×C4#88136
commutatorC42×C13C22×C4
FrattiniC2×C42C22×D13
FittingC13×#1278C2
socleC22×C13#385
prime radical#11719182C22
cosocleC2×C42×C13C23
hypercenter#1278D13
hypocenterC13#8930
LCSG ▷ C42×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1278

Complements

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

NH ≅ G/Ntype
2. #13115611C2semidirect
3. #13113611C2semidirect
6. #13112671C2semidirect
7. C13×#1278C2semidirect
118. #1278D13semidirect
285. C13#8930semidirect

Maximal subgroups

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

subgroupindex
#131235822
#131156112
#131136112
#131134832
#131132882
#131126712
C13×#12782
#893013