Finite Group Databasebeta

← #15863510

Subgroups — #15863510

Order 3328.

898 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#13117243C211yesmaximal, characteristic
2#13117234C211yesmaximal, characteristic
3#13115004C212yesmaximal, complemented
4#13114970C212yesmaximal, complemented
5#13114865C212yesmaximal
6#13114861C212yesmaximal
7#13114733C214yesmaximal, complemented
8C22×C13×#97C211yesmaximal, characteristic
9C23×C8×C13C2212yes—
10C22×C42×C13C2211yescharacteristic
11C2×#88180—21no—
12C2×C13×#97C418yescomplemented
13C2×C13×#97C2211yescharacteristic
14C2×C13×#97C2212yes—
15C2×C13×#97C2211yescharacteristic
16#11718873—21no—
17#11718873C2211yescharacteristic
18#11718873C2211yescharacteristic
19#11718873C2212yes—
20#11718872—21no—
21C4×C22⋊Dic13—21no—
22C2×#88058C2214yes—
23C2×#88057C2214yes—
24#11718598—21no—
25#11718597—21no—
26#11718597C2212yes—
27#11718596C2212yes—
28C4×C4⋊Dic13—21no—
29C4×C4⋊Dic13C2212yes—
30C4×C4⋊Dic13C2212yes—
31#11718593—21no—
32#11718143C2214yes—
33#11718143C2214yes—
34C23×C4×C13D411yescharacteristic
35C23×C4×C13D411yescharacteristic
36C23×C4×C13C2311yescharacteristic
37C22×C8×C13—28no—
38C22×C8×C13C2×C418yes—
39C22×C8×C13C2312yes—
40C22×C8×C13C2312yes—
41C2×C42×C13C2×C414yes—
42C2×C42×C13D411yescharacteristic
43C2×C42×C13D411yescharacteristic
44C2×C42×C13Q814yes—
45C2×C42×C13C2311yescharacteristic
46C2×C42×C13C2311yescharacteristic
47C2×C22⋊Dic13—22no—
48C2×C22⋊Dic13—21no—
49C2×C22⋊Dic13—21no—
50#88180—44no—
51C2×C4⋊Dic13—22no—
52C2×C4⋊Dic13—22no—
53C2×C4⋊Dic13C2312yes—
54C2×C4⋊Dic13C2314yes—
55C2×C4⋊Dic13C2312yes—
56C2×C4×Dic13—22no—
57C2×C4×Dic13—22no—
58C13×#97C2×C414yes—
59C13×#97C2×C418yes—
60C13×#97C2×C414yes—
61#88071—22no—
62#88071—21no—
63#88071—21no—
64#88071—22no—
65#88071—22no—
66#88071—22no—
67#88071—22no—
68#88058—28no—
69#88057—28no—
70#20823—131nomaximal, Sylow
71C24×C13C2×D411yescharacteristic
72C22×C4×C13—22no—
73C22×C4×C13—22no—
74C22×C4×C13—24no—
75C22×C4×C13—42no—
76C22×C4×C13C4⋊C414yes—
77C22×C4×C13C4⋊C414yes—
78C22×C4×C13C22×C412yes—
79C22×C4×C13C2×D411yescharacteristic
80C22×C4×C13C2×D411yescharacteristic
81C22×C4×C13C2×D411yescharacteristic
82C22×C4×C13C2×D411yescharacteristic
83C22×C4×C13C2×D411yescharacteristic
84C22×C4×C13C2×Q812yes—
85C22×C4×C13C2411yescharacteristic
86C22×Dic13—22no—
87C22×Dic13—22no—
88C22×Dic13—24no—
89C2×C8×C13—24no—
90C2×C8×C13—216no—
91C2×C8×C13C22×C414yes—
92C2×C8×C13C22×C414yes—
93C2×C8×C13#4118yes—
94C42×C13C4⋊C414yes—
95C42×C13C4⋊C414yes—
96C42×C13C22×C412yes—
97C42×C13C22×C412yes—
98C42×C13C2×Q812yes—
99C42×C13C2×Q812yes—
100C22⋊Dic13—44no—
101C22⋊Dic13—42no—
102C22⋊Dic13—42no—
103C4⋊Dic13—24no—
104C4⋊Dic13—24no—
105C4⋊Dic13—24no—
106C4⋊Dic13—24no—
107C4⋊Dic13—44no—
108C4⋊Dic13—44no—
109C4×Dic13—44no—
110C4×Dic13—44no—
111C22×#97D1311yescharacteristic
112#2285—131no—
113#2283—131no—
114#1847—132no—
115#1836—132no—
116#1808—132no—
117#1807—132no—
118#1776—134no—
119C23×C13—22no—
120C23×C13—22no—
121C23×C13C2×C4⋊C412yes—
122C23×C13AGL(1,ℤ/8)12yes—
123C23×C13#13712yes—
124C23×C13C22×D411yescharacteristic
125C2×C4×C13—22no—
126C2×C4×C13—21no—
127C2×C4×C13—21no—
128C2×C4×C13—21no—
129C2×C4×C13—21no—
130C2×C4×C13—22no—
131C2×C4×C13—22no—
132C2×C4×C13—28no—
133C2×C4×C13—24no—
134C2×C4×C13—24no—
135C2×C4×C13—44no—
136C2×C4×C13C2×C4⋊C412yes—
137C2×C4×C13C2×C4⋊C412yes—
138C2×C4×C13C2×C4⋊C412yes—
139C2×C4×C13C2×C4⋊C412yes—
140C2×C4×C13C2×C4⋊C412yes—
141C2×C4×C13C23×C411yescharacteristic
142C2×C4×C13C22×Q811yescharacteristic
143C2×C4×C13C2×#4114yes—
144C2×Dic13—44no—
145C2×Dic13—42no—
146C2×Dic13—42no—
147C2×Dic13—44no—
148C2×Dic13—42no—
149C2×Dic13—42no—
150C2×Dic13—42no—
151C2×Dic13—42no—
152C2×Dic13—44no—
153C8×C13—24no—
154C8×C13#11718yescomplemented
155C23×C8C2×D1312yes—
156C2×#117—261no—
157C22×C42C2×D1311yescharacteristic
158C2×#107—134no—
159C2×#106—134no—
160C2×#97Dic1318yescomplemented
161C2×#97C2×D1311yescharacteristic
162C2×#97C2×D1311yescharacteristic
163C2×#97C2×D1312yes—
164#384—132no—
165#383—261no—
166#382—132no—
167#382—261no—
168#381—261no—
169#380—131no—
170#380—132no—
171#380—131no—
172#380—261no—
173C4×C4⋊C4—132no—
174C4×C4⋊C4—132no—
175C4×C4⋊C4—261no—
176C4×C22⋊C4—261no—
177#376—261no—
178#340—134no—
179#340—134no—
180C22×C13—22no—
181C22×C13—24no—
182C22×C13—24no—
183C22×C13—22no—
184C22×C13#42814yes—
185C22×C13C22×C4⋊C411yescharacteristic
186C22×C13#53212yes—
187C22×C13C2×AGL(1,ℤ/8)11yescharacteristic
188C22×C13C2×#13711yescharacteristic
189C22×C13#57512yes—
190C4×C13—22no—
191C4×C13—22no—
192C4×C13—22no—
193C4×C13—22no—
194C4×C13—22no—
195C4×C13C2×#11712yes—
196C4×C13#53312yes—
197Dic13—84no—
198Dic13—82no—
199Dic13—82no—
200C23×C4D5211yescharacteristic
201C23×C4D5211yescharacteristic
202C23×C4C22×D1311yescharacteristic
203C22×C8—28no—
204C22×C8C2×Dic1318yes—
205C22×C8C22×D1312yes—
206C22×C8C22×D1312yes—
207#117—524no—
208C2×C4⋊C4—132no—
209C2×C4⋊C4—132no—
210C2×C4⋊C4—134no—
211C2×C4⋊C4—262no—
212C2×C4⋊C4—262no—
213C2×C22⋊C4—262no—
214C2×C22⋊C4—261no—
215C2×C22⋊C4—261no—
216C2×C42—262no—
217C2×C42—262no—
218C2×C42Dic2614yes—
219C2×C42D5211yescharacteristic
220C2×C42D5211yescharacteristic
221C2×C42C2×Dic1314yes—
222C2×C42C22×D1311yescharacteristic
223C2×C42C22×D1311yescharacteristic
224#107—268no—
225#106—268no—
226#97C2×Dic1314yes—
227#97C2×Dic1318yes—
228#97C2×Dic1314yes—
229#95—261no—
230#95—261no—
231#95—262no—
232#95—262no—
233#95—262no—
234#95—262no—
235#95—262no—
236C2×C13—24no—
237C2×C13#228511yescharacteristic
238C2×C13C2×#42811yescharacteristic
239C2×C13#291111yescharacteristic
240C2×C13#318012yes—
241C2×C13#318112yes—
242C24C2×D5211yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—42no—
247C22×C4—262no—
248C22×C4—262no—
249C22×C4—264no—
250C22×C4C4⋊Dic1314yes—
251C22×C4C4⋊Dic1314yes—
252C22×C4C2×Dic2612yes—
253C22×C4C2×D5211yescharacteristic
254C22×C4C2×D5211yescharacteristic
255C22×C4C2×D5211yescharacteristic
256C22×C4C2×D5211yescharacteristic
257C22×C4C2×D5211yescharacteristic
258C22×C4C22×Dic1312yes—
259C22×C4C23×D1311yescharacteristic
260C2×C8—24no—
261C2×C8—216no—
262C2×C8#612718yes—
263C2×C8C22×Dic1314yes—
264C2×C8C22×Dic1314yes—
265C4⋊C4—264no—
266C4⋊C4—264no—
267C4⋊C4—264no—
268C4⋊C4—264no—
269C4⋊C4—524no—
270C4⋊C4—524no—
271C22⋊C4—524no—
272C22⋊C4—522no—
273C22⋊C4—522no—
274C42—524no—
275C42—524no—
276C42C4⋊Dic1314yes—
277C42C4⋊Dic1314yes—
278C42C2×Dic2612yes—
279C42C2×Dic2612yes—
280C42C22×Dic1312yes—
281C42C22×Dic1312yes—
282C13#2082311yescharacteristic, complemented, Sylow
283C23—22no—
284C23—22no—
285C23#8816212yes—
286C23#8816312yes—
287C23C2×C4⋊Dic1312yes—
288C23C22×D5211yescharacteristic
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—28no—
297C2×C4—24no—
298C2×C4—24no—
299C2×C4—44no—
300C2×C4—524no—
301C2×C4—522no—
302C2×C4—522no—
303C2×C4—524no—
304C2×C4—522no—
305C2×C4—522no—
306C2×C4—522no—
307C2×C4—522no—
308C2×C4—524no—
309C2×C4C2×C4⋊Dic1312yes—
310C2×C4C2×C4⋊Dic1312yes—
311C2×C4C2×C4⋊Dic1312yes—
312C2×C4C2×C4⋊Dic1312yes—
313C2×C4C2×C4⋊Dic1312yes—
314C2×C4C22×Dic2611yescharacteristic
315C2×C4C2×#612714yes—
316C2×C4C23×Dic1311yescharacteristic
317C8—24no—
318C8#8818018yescomplemented
319C22—22no—
320C22—22no—
321C22—24no—
322C22—24no—
323C22#1171878214yes—
324C22#1171918212yes—
325C22C2×#8816211yescharacteristic
326C22C2×#8816311yescharacteristic
327C22#1171945312yes—
328C22C22×C4⋊Dic1311yescharacteristic
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—1044no—
335C4—1042no—
336C4—1042no—
337C4#1171917512yes—
338C4C2×#8818012yes—
339C2—24no—
340C2#1311724311yescharacteristic
341C2C2×#1171878211yescharacteristic
342C2#1312342611yescharacteristic
343C2#1312553412yes—
344C2#1312554512yes—
subgroupG/H
centerC23C22×D52
commutatorC2×C4×C13C23×C4
FrattiniC22×C4C23×D13
FittingC22×C13×#97C2
socleC23×C13C22×D4
prime radicalC24×C13C2×D4
cosocleC22×C4×C13C24
hypercenterC22×#97D13
hypocenterC13#20823
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. #13115004C2semidirect
4. #13114970C2semidirect
7. #13114733C2semidirect
12. C2×C13×#97C4semidirect
154. C8×C13#117semidirect
160. C2×#97Dic13semidirect
282. C13#20823semidirect
318. C8#88180semidirect

Maximal subgroups

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

subgroupindex
#131172432
#131172342
#131150042
#131149702
#131148652
#131148612
#131147332
C22×C13×#972
#2082313