Finite Group Databasebeta

← #16022980

Subgroups — #16022980

Order 3328.

2806 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13117713C218yesmaximal, complemented
2#13117705C218yesmaximal, complemented
3#13117454C212yesmaximal, complemented
4#13117439C214yesmaximal, complemented
5#13117219C212yesmaximal
6#13117219C214yesmaximal
7#13117193C212yesmaximal, complemented
8C23×C13×C4⋊C4C211yesmaximal, characteristic
9C24×C4×C13C2211yescharacteristic
10C24×C4×C13C2212yes—
11C22×C13×C4⋊C4C4116yescomplemented
12C22×C13×C4⋊C4C2211yescharacteristic
13C22×C13×C4⋊C4C2214yes—
14C22×C13×C4⋊C4C2211yescharacteristic
15C22×C13×C4⋊C4C2214yes—
16C22×C13×C4⋊C4C2212yes—
17C2×#88180C2214yes—
18C2×#88180C2218yes—
19#11718908C2214yes—
20#11718873C2214yes—
21#11718873C2218yes—
22#11718873C2218yes—
23#11718873C2214yes—
24#11718872C2218yes—
25#11718872C2214yes—
26C4×C22⋊Dic13C2218yes—
27C4×C22⋊Dic13C2214yes—
28C4×C22⋊Dic13C2214yes—
29C4×C22⋊Dic13C2218yes—
30#11718646C22116yescomplemented
31#11718642C22132yescomplemented
32C4⋊C4×Dic13C22116yescomplemented
33C25×C13C2311yescharacteristic
34C23×C4×C13—28no—
35C23×C4×C13—48no—
36C23×C4×C13C2×C418yes—
37C23×C4×C13C2×C4116yes—
38C23×C4×C13C2314yes—
39C23×C4×C13C2312yes—
40C23×C4×C13C2312yes—
41C23×C4×C13C2311yescharacteristic
42C23×C4×C13C2312yes—
43C23×C4×C13C2314yes—
44C23×C4×C13C2312yes—
45C23×C4×C13C2311yescharacteristic
46C2×C13×C4⋊C4C2×C4132yes—
47C2×C13×C4⋊C4C2×C4132yes—
48C2×C13×C4⋊C4C2×C4116yes—
49C2×C13×C4⋊C4C2×C418yes—
50C2×C13×C4⋊C4C2×C418yes—
51C2×C13×C4⋊C4C2314yes—
52C2×C13×C4⋊C4C2318yes—
53C2×C13×C4⋊C4C2314yes—
54C2×C22⋊Dic13C2318yes—
55C2×C22⋊Dic13C2318yes—
56C2×C22⋊Dic13C2318yes—
57#88180—216no—
58#88180—416no—
59C2×C4⋊Dic13C2318yes—
60C2×C4⋊Dic13C23116yes—
61C2×C4×Dic13C2318yes—
62C2×C4×Dic13C23116yes—
63#88071C23116yes—
64#88071C2318yes—
65#88071C2318yes—
66#88071C23116yes—
67#34178—131nomaximal, Sylow
68C24×C13—24no—
69C24×C13—24no—
70C24×C13—24no—
71C24×C13C22×C414yes—
72C24×C13C2411yescharacteristic
73C24×C13C2411yescharacteristic
74C24×C13C2411yescharacteristic
75C22×C4×C13—24no—
76C22×C4×C13—216no—
77C22×C4×C13—232no—
78C22×C4×C13—28no—
79C22×C4×C13—28no—
80C22×C4×C13—432no—
81C22×C4×C13—44no—
82C22×C4×C13—48no—
83C22×C4×C13—44no—
84C22×C4×C13C22×C414yes—
85C22×C4×C13C22×C4116yes—
86C22×C4×C13C22×C418yes—
87C22×C4×C13C22×C418yes—
88C22×C4×C13C22×C418yes—
89C22×C4×C13C22×C418yes—
90C22×C4×C13C22×C4116yes—
91C22×C4×C13C22×C414yes—
92C22×C4×C13#4118yes—
93C22×C4×C13C2412yes—
94C22×C4×C13C2414yes—
95C22×C4×C13C2414yes—
96C22×C4×C13C2412yes—
97C22×Dic13C24116yes—
98C13×C4⋊C4C22×C4132yes—
99C13×C4⋊C4C22×C4116yes—
100C13×C4⋊C4C22×C4116yes—
101C22⋊Dic13—216no—
102C22⋊Dic13—216no—
103C22⋊Dic13—216no—
104C4⋊Dic13—216no—
105C4⋊Dic13—416no—
106C4×Dic13—216no—
107C4×Dic13—416no—
108C23×C4⋊C4D1311yescharacteristic
109#2345—138no—
110#2344—138no—
111#2316—132no—
112#2313—134no—
113#2280—132no—
114#2280—134no—
115#2274—132no—
116C23×C13—232no—
117C23×C13—28no—
118C23×C13—28no—
119C23×C13—28no—
120C23×C13—22no—
121C23×C13—22no—
122C23×C13—22no—
123C23×C13—22no—
124C23×C13—22no—
125C23×C13—22no—
126C23×C13C23×C412yes—
127C23×C13C23×C412yes—
128C23×C13C23×C412yes—
129C23×C13C2×#4114yes—
130C23×C13H(5,𝔽2)14yes—
131C23×C132−1+414yes—
132C23×C13C2511yescharacteristic
133C2×C4×C13—28no—
134C2×C4×C13—232no—
135C2×C4×C13—24no—
136C2×C4×C13—28no—
137C2×C4×C13—28no—
138C2×C4×C13—216no—
139C2×C4×C13—216no—
140C2×C4×C13—48no—
141C2×C4×C13—416no—
142C2×C4×C13—48no—
143C2×C4×C13#117116yescomplemented
144C2×C4×C13C23×C414yes—
145C2×C4×C13C23×C418yes—
146C2×C4×C13C23×C418yes—
147C2×C4×C13C23×C414yes—
148C2×C4×C13C2×#4118yes—
149C2×Dic13—216no—
150C2×Dic13—216no—
151C2×Dic13—216no—
152C24×C4C2×D1311yescharacteristic
153C24×C4C2×D1312yes—
154C2×#117—134no—
155C2×#117—138no—
156C22×C4⋊C4Dic13116yescomplemented
157C22×C4⋊C4C2×D1312yes—
158C22×C4⋊C4C2×D1311yescharacteristic
159C22×C4⋊C4C2×D1314yes—
160C22×C4⋊C4C2×D1314yes—
161C22×C4⋊C4C2×D1311yescharacteristic
162#389—1316no—
163#387—1332no—
164#381—134no—
165#381—138no—
166#380—138no—
167#380—134no—
168#380—138no—
169#380—134no—
170#379—134no—
171C4×C4⋊C4—1316no—
172C4×C22⋊C4—138no—
173C4×C22⋊C4—134no—
174C4×C22⋊C4—134no—
175C4×C22⋊C4—138no—
176C22×C13—232no—
177C22×C13—22no—
178C22×C13—22no—
179C22×C13—22no—
180C22×C13—22no—
181C22×C13—22no—
182C22×C13—22no—
183C22×C13—28no—
184C22×C13—28no—
185C22×C13—28no—
186C22×C13C2×#11714yes—
187C22×C13#51814yes—
188C22×C13#51914yes—
189C22×C13C24×C411yescharacteristic
190C22×C13C22×#4112yes—
191C22×C13C2×H(5,𝔽2)12yes—
192C22×C13C2×2−1+412yes—
193C4×C13—24no—
194C4×C13—28no—
195C4×C13—28no—
196C4×C13C2×#11718yes—
197Dic13—416no—
198C25C22×D1311yescharacteristic
199C23×C4—28no—
200C23×C4—48no—
201C23×C4C2×Dic1318yes—
202C23×C4C2×Dic13116yes—
203C23×C4C22×D1314yes—
204C23×C4C22×D1312yes—
205C23×C4C22×D1312yes—
206C23×C4C22×D1311yescharacteristic
207C23×C4C22×D1312yes—
208C23×C4C22×D1314yes—
209C23×C4C22×D1312yes—
210C23×C4C22×D1311yescharacteristic
211#117—2616no—
212#117—5216no—
213C2×C4⋊C4—138no—
214C2×C4⋊C4—1316no—
215C2×C4⋊C4C2×Dic13132yes—
216C2×C4⋊C4C2×Dic13116yes—
217C2×C4⋊C4C2×Dic13132yes—
218C2×C4⋊C4C2×Dic1318yes—
219C2×C4⋊C4C2×Dic1318yes—
220C2×C4⋊C4C22×D1314yes—
221C2×C4⋊C4C22×D1318yes—
222C2×C4⋊C4C22×D1314yes—
223C2×C22⋊C4—138no—
224C2×C22⋊C4—138no—
225C2×C22⋊C4—138no—
226C2×C42—138no—
227C2×C42—1316no—
228#95—1316no—
229#95—138no—
230#95—138no—
231#95—1316no—
232C2×C13—24no—
233C2×C13—24no—
234C2×C13—24no—
235C2×C13C22×#11711yescharacteristic
236C2×C13C2×#51811yescharacteristic
237C2×C13C2×#51911yescharacteristic
238C2×C13#349214yes—
239C24—24no—
240C24—24no—
241C24—24no—
242C24C22×Dic1314yes—
243C24C23×D1311yescharacteristic
244C24C23×D1311yescharacteristic
245C24C23×D1311yescharacteristic
246C22×C4—24no—
247C22×C4—216no—
248C22×C4—232no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—432no—
252C22×C4—44no—
253C22×C4—48no—
254C22×C4—44no—
255C22×C4—1316no—
256C22×C4#612718yes—
257C22×C4C22×Dic1314yes—
258C22×C4C22×Dic13116yes—
259C22×C4C22×Dic1318yes—
260C22×C4C22×Dic1318yes—
261C22×C4C22×Dic1318yes—
262C22×C4C22×Dic1318yes—
263C22×C4C22×Dic13116yes—
264C22×C4C22×Dic1314yes—
265C22×C4C23×D1312yes—
266C22×C4C23×D1314yes—
267C22×C4C23×D1314yes—
268C22×C4C23×D1312yes—
269C4⋊C4—2616no—
270C4⋊C4—5216no—
271C4⋊C4C22×Dic13132yes—
272C4⋊C4C22×Dic13116yes—
273C4⋊C4C22×Dic13116yes—
274C22⋊C4—2616no—
275C22⋊C4—2616no—
276C22⋊C4—2616no—
277C42—2616no—
278C42—5216no—
279C13#3417811yescharacteristic, complemented, Sylow
280C23—232no—
281C23—28no—
282C23—28no—
283C23—28no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23C2×#612714yes—
291C23#8825114yes—
292C23#8825414yes—
293C23C23×Dic1312yes—
294C23C23×Dic1312yes—
295C23C23×Dic1312yes—
296C23C24×D1311yescharacteristic
297C2×C4—28no—
298C2×C4—232no—
299C2×C4—24no—
300C2×C4—28no—
301C2×C4—28no—
302C2×C4—216no—
303C2×C4—216no—
304C2×C4—48no—
305C2×C4—416no—
306C2×C4—48no—
307C2×C4—2616no—
308C2×C4—2616no—
309C2×C4—2616no—
310C2×C4#88180116yescomplemented
311C2×C4C2×#612718yes—
312C2×C4C23×Dic1314yes—
313C2×C4C23×Dic1318yes—
314C2×C4C23×Dic1318yes—
315C2×C4C23×Dic1314yes—
316C22—232no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—28no—
324C22—28no—
325C22—28no—
326C22C2×#8818014yes—
327C22#1171950314yes—
328C22#1171951714yes—
329C22C22×#612712yes—
330C22C2×#8825112yes—
331C22C2×#8825412yes—
332C22C24×Dic1311yescharacteristic
333C4—24no—
334C4—28no—
335C4—28no—
336C4—5216no—
337C4C2×#8818018yes—
338C2—24no—
339C2—24no—
340C2—24no—
341C2C22×#8818011yescharacteristic
342C2C2×#1171950311yescharacteristic
343C2C2×#1171951711yescharacteristic
344C2#1312854514yes—
subgroupG/H
center, FrattiniC23C24×D13
commutatorC22×C13C24×C4
FittingC23×C13×C4⋊C4C2
socle, cosocleC23×C13C25
prime radicalC25×C13C23
hypercenterC23×C4⋊C4D13
hypocenterC13#34178
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #13117713C2semidirect
2. #13117705C2semidirect
3. #13117454C2semidirect
4. #13117439C2semidirect
7. #13117193C2semidirect
11. C22×C13×C4⋊C4C4semidirect
30. #11718646C22semidirect
31. #11718642C22semidirect
32. C4⋊C4×Dic13C22semidirect
143. C2×C4×C13#117semidirect
156. C22×C4⋊C4Dic13semidirect
279. C13#34178semidirect
310. C2×C4#88180semidirect

Maximal subgroups

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

subgroupindex
#131177132
#131177052
#131174542
#131174392
#131172192
#131171932
C23×C13×C4⋊C42
#3417813