Finite Group Databasebeta

← #27381863

Subgroups — #27381863

Order 13568.

1630 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#11719945C211yesmaximal, characteristic
2C2×#16810620C212yesmaximal
3C4×#13133176C211yesmaximal, characteristic, complemented
4#20274546C214yesmaximal, complemented
5#20274541C214yesmaximal, complemented
6#20274432C212yesmaximal, complemented
7C22×#97×C53C211yesmaximal, characteristic, complemented
8C22×#11719945C2211yescharacteristic
9C4×#11719945—24no—
10C4×#11719945C418yescomplemented
11C2×C4×Dic106C2212yes—
12C2×C4×D212C2212yes—
13C2×C4×C22⋊D53—22no—
14C2×#13133176C2211yescharacteristic
15C2×#13133176C2212yes—
16C2×#13133174—21no—
17C2×C42×D53—21no—
18#16810620C4116yescomplemented
19C2×#13133142C2214yes—
20C2×#13133139C2214yes—
21C4×C4⋊Dic53C2214yes—
22C4×C22⋊Dic53—22no—
23#16810469—21no—
24C42×Dic53—21no—
25#16810386C2218yes—
26C23×C8×C53C2212yes—
27C22×C42×C53C2211yescharacteristic
28C2×#97×C53—24no—
29C2×#97×C53C2214yes—
30C2×#11719945—22no—
31C2×#11719945—24no—
32C2×#11719945C2×C414yes—
33C2×#11719945C2×C418yes—
34C22×Dic106C2311yescharacteristic
35C22×D212C2311yescharacteristic
36C22×C22⋊D53—21no—
37C22×C4×D53—21no—
38C22×C4×D53—22no—
39C4×Dic106—24no—
40C4×Dic106C2×C418yes—
41C4×D212—24no—
42C4×D212C2×C418yes—
43C4×C22⋊D53—216no—
44#13133176—22no—
45#13133176—24no—
46#13133176C2×C414yes—
47#13133176C2×C418yescomplemented
48#13133174—28no—
49C2×C4⋊Dic53C2312yes—
50C2×C4⋊Dic53C2314yes—
51C2×#11719919—22no—
52C2×C22⋊Dic53—21no—
53C2×C22⋊Dic53—22no—
54C2×#11719921—22no—
55C2×C4×Dic53—21no—
56C2×C4×Dic53—22no—
57C2×C4×Dic53—22no—
58C2×C4×Dic53—24no—
59C42×D53—28no—
60#13133142C2×C4116yescomplemented
61#13133139C2×C4116yescomplemented
62#13133124—24no—
63C23×C4×C53D412yes—
64C23×C4×C53C2311yescharacteristic
65C22×C8×C53—24no—
66C22×C8×C53—24no—
67C22×C8×C53C2×C418yescomplemented
68C22×C8×C53C2314yes—
69C2×C42×C53C2×C414yes—
70C2×C42×C53D414yes—
71C2×C42×C53D411yescharacteristic
72C2×C42×C53D411yescharacteristic
73C2×C42×C53C2312yes—
74#97×C53—28no—
75C24×C53C2×D411yescharacteristic
76C23×D53—21no—
77C22×C4×C53—44no—
78C22×C4×C53C22⋊C418yes—
79C22×C4×C53C22×C412yes—
80C22×C4×C53C22×C414yes—
81C22×C4×C53C2×D412yes—
82C22×C4×C53C2×D412yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D412yes—
85C22×C4×C53#4114yes—
86C22×C4×C53C2411yescharacteristic
87C2×C22⋊D53—24no—
88C2×C22⋊D53—28no—
89C22×Dic53—21no—
90C22×Dic53—24no—
91C22×Dic53—22no—
92#11719945—28no—
93#11719945—216no—
94#11719945C42116yescomplemented
95C2×D212—21no—
96C2×D212—22no—
97C2×D212C22×C414yes—
98C2×D212C22×C412yes—
99C2×C4×D53—24no—
100C2×C4×D53—28no—
101C2×C4×D53—28no—
102C2×C4×D53—48no—
103C2×Dic106—21no—
104C2×Dic106—22no—
105C2×Dic106C22×C414yes—
106C2×Dic106C22×C412yes—
107C2×C8×C53—216no—
108C2×C8×C53—24no—
109C2×C8×C53—24no—
110C2×C8×C53C22×C418yes—
111C42×C53C22⋊C414yes—
112C42×C53C22⋊C414yes—
113C42×C53C22×C414yes—
114C42×C53C2×D414yes—
115C22⋊Dic53—24no—
116C22⋊Dic53—28no—
117#11719921—28no—
118C4⋊Dic53—22no—
119C4⋊Dic53—24no—
120C4⋊Dic53C22×C418yes—
121C4⋊Dic53C22×C414yes—
122#11719919—28no—
123C4×Dic53—24no—
124C4×Dic53—28no—
125C4×Dic53—28no—
126C4×Dic53—48no—
127C23×C53—24no—
128C23×C53C2×C22⋊C412yes—
129C23×C53C4×D414yes—
130C23×C53C22×D411yescharacteristic
131C22×D53—22no—
132C22×D53—24no—
133C22×D53—44no—
134C2×C4×C53—24no—
135C2×C4×C53—24no—
136C2×C4×C53—48no—
137C2×C4×C53C2×C4214yes—
138C2×C4×C53C2×C22⋊C414yes—
139C2×C4×C53C2×C22⋊C414yes—
140C2×C4×C53C2×C22⋊C412yes—
141C2×C4×C53#11714yes—
142C2×C4×C53C4×D418yes—
143C2×C4×C53C4×D414yes—
144C2×C4×C53C23×C411yescharacteristic
145C2×C4×C53C23×C412yes—
146C2×C4×C53C22×D411yescharacteristic
147C2×C4×C53C2×#4112yes—
148C22⋊D53—216no—
149C2×Dic53—22no—
150C2×Dic53—24no—
151C2×Dic53—24no—
152C2×Dic53—28no—
153C2×Dic53—44no—
154C2×Dic53—48no—
155D212—22no—
156D212—24no—
157D212C2×C4214yes—
158C4×D53—216no—
159C4×D53—416no—
160Dic106—22no—
161Dic106—24no—
162Dic106C2×C4214yes—
163C8×C53—28no—
164#20411—531nomaximal, Sylow
165C22×C53—24no—
166C22×C53—28no—
167C22×C53C4×C22⋊C414yes—
168C22×C53C22×C22⋊C411yescharacteristic
169C22×C53C2×C4×D412yes—
170C22×C53#57514yes—
171C2×D53—24no—
172C2×D53—44no—
173C2×D53—48no—
174C4×C53—24no—
175C4×C53—24no—
176C4×C53C4×C22⋊C414yes—
177C4×C53C22×C4211yescharacteristic
178C4×C53C2×#11711yescharacteristic
179C4×C53C2×C4×D412yes—
180Dic53—24no—
181Dic53—44no—
182Dic53—48no—
183C2×C4×#41—531no—
184C2×#416—532no—
185C22×#97D5311yescharacteristic, complemented
186C4×#117—531no—
187#1763—534no—
188#1762—534no—
189#1735—532no—
190C2×C53—24no—
191C2×C53C2×C4×C22⋊C411yescharacteristic
192C2×C53#289612yes—
193C2×C53#294314yes—
194D53—44no—
195C22×#41—531no—
196C23×C8C2×D5312yes—
197C4×#41—538no—
198C4×#41—1064no—
199C2×C4×Q8—532no—
200C2×C4×D4—532no—
201C2×C4×D4—1062no—
202C2×#117—532no—
203C2×#117—531no—
204C2×#117—1061no—
205C22×C42—1061no—
206C22×C42C2×D5311yescharacteristic
207#416—5316no—
208C2×#103—534no—
209C2×#102—534no—
210C2×#97—24no—
211C2×#97C2×D5314yes—
212C4×C4⋊C4—534no—
213C4×C22⋊C4—1062no—
214#376—1061no—
215C43—1061no—
216#340—538no—
217C53#2041111yescharacteristic, complemented, Sylow
218C2×#41—534no—
219C2×#41—538no—
220C2×#41—1064no—
221C2×#41—1062no—
222C22×Q8—531no—
223C22×D4—531no—
224C22×D4—1061no—
225C23×C4—1061no—
226C23×C4—1062no—
227C23×C4D21212yes—
228C23×C4C22×D5311yescharacteristic
229C22×C8—24no—
230C22×C8—24no—
231C22×C8C4×D5318yescomplemented
232C22×C8C22×D5314yes—
233C4×Q8—538no—
234C4×Q8—1064no—
235C4×D4—538no—
236C4×D4—10616no—
237C4×D4—1064no—
238#117—534no—
239#117—538no—
240#117—1064no—
241#117—1062no—
242#117—1068no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4—534no—
245C2×C4⋊C4—1062no—
246C2×C22⋊C4—1061no—
247C2×C22⋊C4—1062no—
248C2×C22⋊C4—1062no—
249C2×C42—1062no—
250C2×C42—1061no—
251C2×C42—1062no—
252C2×C42—1068no—
253C2×C42—1064no—
254C2×C42C4×D5314yes—
255C2×C42D21211yescharacteristic
256C2×C42D21211yescharacteristic
257C2×C42C22⋊D5314yes—
258C2×C42C22×D5312yes—
259#103—5316no—
260#102—5316no—
261#97—28no—
262#95—1064no—
263C24—1061no—
264C24C2×D21211yescharacteristic
265#41—5316no—
266#41—1068no—
267#41—10616no—
268C2×Q8—534no—
269C2×Q8—532no—
270C2×Q8—1061no—
271C2×Q8—1062no—
272C2×D4—534no—
273C2×D4—532no—
274C2×D4—1064no—
275C2×D4—1068no—
276C2×D4—1061no—
277C2×D4—1062no—
278C22×C4—44no—
279C22×C4—1061no—
280C22×C4—1064no—
281C22×C4—1068no—
282C22×C4—1064no—
283C22×C4—1062no—
284C22×C4—1068no—
285C22×C4—2128no—
286C22×C4#1171992118yes—
287C22×C4C2×C4×D5312yes—
288C22×C4C2×C4×D5314yes—
289C22×C4C2×D21212yes—
290C22×C4C2×D21212yes—
291C22×C4C2×D21211yescharacteristic
292C22×C4#1171994514yes—
293C22×C4C2×C22⋊D5312yes—
294C22×C4C23×D5311yescharacteristic
295C2×C8—216no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8C2×C4×D5318yes—
299C4⋊C4—538no—
300C4⋊C4—534no—
301C4⋊C4—1068no—
302C4⋊C4—1062no—
303C4⋊C4—1064no—
304C22⋊C4—1064no—
305C22⋊C4—1068no—
306C22⋊C4—1068no—
307C42—1064no—
308C42—1068no—
309C42—1068no—
310C42—2128no—
311C42#1171992114yes—
312C42#1171992114yes—
313C42C2×C4×D5314yes—
314C42C2×C22⋊D5314yes—
315C23—24no—
316C23—1062no—
317C23—1064no—
318C23—2124no—
319C23C2×#1171992112yes—
320C23C4×D21214yes—
321C23C22×D21211yescharacteristic
322Q8—534no—
323Q8—1062no—
324Q8—1064no—
325D4—534no—
326D4—10616no—
327D4—1062no—
328D4—1064no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—48no—
332C2×C4—1062no—
333C2×C4—1064no—
334C2×C4—10616no—
335C2×C4—1064no—
336C2×C4—1068no—
337C2×C4—21216no—
338C2×C4—2128no—
339C2×C4—2124no—
340C2×C4C42×D5314yes—
341C2×C4C2×#1171992114yes—
342C2×C4C2×#1171992114yes—
343C2×C4C2×#1171992112yes—
344C2×C4#1313317414yes—
345C2×C4C4×C22⋊D5318yes—
346C2×C4C4×D21214yes—
347C2×C4C22×C4×D5311yescharacteristic
348C2×C4C22×C4×D5312yes—
349C2×C4C22×C22⋊D5311yescharacteristic
350C2×C4C2×#1171994512yes—
351C8—28no—
352C22—24no—
353C22—28no—
354C22—1064no—
355C22—2124no—
356C22—2128no—
357C22C4×#1171992114yes—
358C22C22×#1171992111yescharacteristic
359C22C2×C4×D21212yes—
360C22#1681142314yes—
361C4—24no—
362C4—24no—
363C4—1064no—
364C4—2124no—
365C4—2128no—
366C4C4×#1171992114yes—
367C4C2×C42×D5311yescharacteristic
368C4C2×#1313317411yescharacteristic
369C4C2×C4×C22⋊D5312yes—
370C2—24no—
371C2—2124no—
372C2C2×C4×#1171992111yescharacteristic
373C2#2028323512yes—
374C2#2028351714yes—
subgroupG/H
centerC22×C4C2×D212
commutatorC4×C53C22×C42
FrattiniC22×C4C23×D53
FittingC22×#97×C53C2
socleC23×C53C22×D4
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC22×#97D53
hypocenterC53#20411
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. C4×#13133176C2semidirect
4. #20274546C2semidirect
5. #20274541C2semidirect
6. #20274432C2semidirect
7. C22×#97×C53C2semidirect
10. C4×#11719945C4semidirect
18. #16810620C4semidirect
47. #13133176C2×C4semidirect
60. #13133142C2×C4semidirect
61. #13133139C2×C4semidirect
67. C22×C8×C53C2×C4semidirect
94. #11719945C42semidirect
185. C22×#97D53semidirect
217. C53#20411semidirect
231. C22×C8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#117199452
C2×#168106202
C4×#131331762
#202745462
#202745412
#202744322
C22×#97×C532
#2041153